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AGGREGATION APPROACHES FOR INCORPORATING EMAIL PROCESSING HISTORY IN QUEUEING MODELS OF CUSTOMER CONTACT CENTERS By MOHAN RAJ CHINNASWAMY Bachelor of Engineering University of Madras Madras, India 2001 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE May, 2005 ii AGGREGATION APPROACHES FOR INCORPORATING EMAIL PROCESSING HISTORY IN QUEUEING MODELS OF CUSTOMER CONTACT CENTERS Thesis Approved: Dr. Manjunath Kamath Thesis Advisor Dr. David B. Pratt Dr. Ramesh Sharda Dr. A. Gordon Emslie Dean of the Graduate College iii ACKNOWLEDGEMENT No words can ever express my sincere gratitude and indebtness to my advisor Dr. Manjunath Kamath, Without his support and motivation the completion of this thesis would have not been possible. I also thank my committee members Dr. David Pratt and Dr. Ramesh Sharda for providing me with invaluable suggestions and guidance. I thank my friends and CCiMe members Sandeep, Parthi, Mahesh, Sarath, Ananth, Karthik, Vishal and Nandan for helping me in the various stages of my thesis completion. Fruitful conversations with Bob and Ashish are much appreciated. I sincerely thank my late father, mother Mrs. C. Muthulakshmi, sister Jamuna and younger brother Raju for their constant support and encouragement. Last but not least I appreciate the support I received from the faculty and staff at the Industrial Engineering and Management department during the course of my graduate study. iv Dedicated to my late father Sri. M. Chinnaswamy and to my mentor Dr. Manjunath Kamath v TABLE OF CONTENTS 1. INTRODUCTION.....................................................................................................................1 2. STATEMENT OF THE PROBLEM .......................................................................................5 3. LITERATURE REVIEW .........................................................................................................7 3.1 SKILL  BASED ROUTING ....................................................................................................... 7 3.1.1 Challenges in SkillBased Routing.................................................................................8 3.2 RESOURCE POOLING ............................................................................................................ 10 3.2.1 StochasticProcess Limits ............................................................................................11 3.3 WORKFORCE MANAGEMENT............................................................................................... 12 3.3.1 RealTime Staffing........................................................................................................13 3.3.2 ShortTerm Staffing......................................................................................................14 3.3.3 LongTerm Staffing ......................................................................................................14 3.4 ORGANIZATIONAL BEHAVIOR ............................................................................................. 15 3.5 PERFORMANCE EVALUATION .............................................................................................. 15 3.5.1 Queueing Models of Call Centers ................................................................................16 3.5.1.1 Arrival Process......................................................................................................16 3.5.1.2 Waiting Behavior of Customers............................................................................ 17 3.5.1.3 Service Time Distribution of Agents .................................................................... 18 3.5.2 Queueing Models of Inbound Email Contact Centers ................................................19 4. RESEARCH STATEMENT ...................................................................................................21 vi 4.1 RESEARCH OBJECTIVES ....................................................................................................... 21 4.2 RESEARCH SCOPE AND LIMITATIONS .................................................................................. 22 4.3 RESEARCH CONTRIBUTIONS ................................................................................................ 22 5. MODELING METHODOLOGY...........................................................................................24 5.1 THE PARAMETRIC DECOMPOSITION (PD) METHOD ............................................................ 24 5.2 AGGREGATION APPROACHES .............................................................................................. 26 5.3 NUMERICAL VALIDATION.................................................................................................... 27 6. A MULTICLASS OPEN QUEUEING NETWORK MODEL OF EMAIL CUSTOMER CONTACT CENTERS................................................................................................................29 6.1 CONTACT CENTER DESCRIPTION......................................................................................... 30 6.2 ASSUMPTIONS..................................................................................................................... 31 6.3 MODELING THE EMAIL CONTACT CENTER USING AN OPEN QUEUEING NETWORK ......... 31 6.3.1 Rate and SCV of the External Arrival Process at a Node ............................................34 6.3.2 Weights for Classbased Aggregation..........................................................................35 6.3.3 Markovian Routing Probabilities.................................................................................36 6.3.4 Mean Service Time at a Node ......................................................................................37 6.3.5 Squared Coefficient of Variation of the Service Time Distribution at a Node.............38 6.4 APPROACHES TO COMPUTE THE WEIGHTS FOR HISTORYBASED AGGREGATION.............. 39 6.4.1 Aggregation Method M2 ..............................................................................................40 6.5 DISCRETETIME MARKOV CHAIN MODEL OF HISTORYBASED EMAIL ROUTING ............. 43 6.5.1 Analysis of the Absorbing Markov Chain.....................................................................46 6.5.2 Performance Evaluation of the EMail Contact Center...............................................47 vii 6.6 NUMERICAL EXPERIMENTS.................................................................................................. 51 6.6.1 Email Contact Center Characteristics........................................................................51 6.6.2 Email Contact Center Example ..................................................................................52 6.6.3 Probabilities for HistoryBased Aggregation obtained from the DTMC.....................53 6.7 RESULTS AND DISCUSSIONS ................................................................................................ 58 6.7.1 Results ..........................................................................................................................58 6.7.2 Discussion ....................................................................................................................68 7. MODEL EXTENSIONS: SKILLBASED GROUPING OF AGENTS AND SERVICE INTERRUPTIONS ......................................................................................................................69 7.1 CONTACT CENTER MODEL WITH SKILLBASED GROUPING................................................ 70 7.2 NUMERICAL EXPERIMENTS.................................................................................................. 70 7.3 RESULTS AND DISCUSSION .................................................................................................. 71 7.3.1 Results ..........................................................................................................................72 7.4 MODELING SERVICE INTERRUPTIONS.................................................................................. 93 7.4.1 Contact Center Model with Service Interruptions .......................................................93 7.5 NUMERICAL EXPERIMENTS.................................................................................................. 94 7.6 RESULTS.............................................................................................................................. 94 8. CONCLUSIONS AND FUTURE RESEARCH..................................................................103 8.1 RESEARCH SUMMARY........................................................................................................ 103 8.2 RESEARCH CONTRIBUTIONS .............................................................................................. 104 8.3 FUTURE RESEARCH DIRECTIONS ....................................................................................... 105 8.3.1 Extensions to Model Agent Schedules ........................................................................105 viii 9. REFERENCES......................................................................................................................107 A1. FUNDAMENTAL MATRIX ENTRIES FOR SCENARIOS 1 AND 2..........................115 A1.1 MATLAB PROGRAM FOR FUNDAMENTAL MATRIX INVERSION ...................................... 132 A1.2 MATLAB CODE TO OBTAIN THE PARAMETERS FOR HYPEREXPONENTIAL DISTRIBUTION USED IN SIMULATION .............................................................................................................. 132 A2. SIMULATION MODEL ....................................................................................................133 A2.1 THE SIMULATION LOGIC................................................................................................. 133 A2.2 SERVICE TIME DISTRIBUTIONS AND THEIR PARAMETERS.............................................. 139 A2.2.1 Erlang Distribution..................................................................................................139 A2.2.2 Exponential Distribution..........................................................................................140 A2.2.3 Hyperexponential Distribution ................................................................................140 A3. DETERMINATION OF THE WARMUP PERIOD AND RUN LENGTH FOR SIMULATION EXPERIMENTS.............................................................................................142 A3.1 INTRODUCTION ............................................................................................................... 142 A3.2 WELCH’S TECHNIQUE TO DETERMINE WARMUP PERIOD ............................................. 143 A3.3 DETERMINATION OF RUN LENGTH ................................................................................. 144 ix LIST OF FIGURES FIGURE 1.1: OUTLINE OF THE THESIS .............................................................................................. 4 FIGURE 6.1: EMAIL HANDLING LOGIC ......................................................................................... 33 FIGURE 6.2: AGGREGATION METHOD M2 ..................................................................................... 41 FIGURE 6.3: ONESTEP TRANSITION PROBABILITY MATRIX......................................................... 46 FIGURE 6.4: CONTACT CENTER PERFORMANCE EVALUATION...................................................... 50 FIGURE 6.6: SCENARIO 1  AGENT UTILIZATION  ERLANG PROCESSING TIME ............................ 62 FIGURE 6.7: SCENARIO 1  AVERAGE RESPONSE TIME .................................................................. 62 FIGURE 6.8: SCENARIO 1  AVERAGE RESOLUTION TIME.............................................................. 63 FIGURE 6.9: SCENARIO 1  AVERAGE NUMBER OF EMAILS IN THE SYSTEM ................................ 63 FIGURE 6.10: SCENARIO 2  AGENT UTILIZATION  ERLANG PROCESSING TIMES......................... 66 FIGURE 6.11: SCENARIO 2  AVERAGE RESPONSE TIME ................................................................ 66 FIGURE 6.12: SCENARIO 2  AVERAGE RESOLUTION TIME............................................................ 67 FIGURE 6.13: SCENARIO 2  AVERAGE NUMBER OF EMAILS IN THE SYSTEM .............................. 67 FIGURE 7.1: SCENARIO 1, CONFIGURATION 1  AGENT UTILIZATION  ERLANG PROCESSING TIMES ................................................................................................................................... 73 FIGURE 7.2: SCENARIO 1, CONFIGURATION 1  AVERAGE RESPONSE TIME .................................. 74 FIGURE 7.3: SCENARIO 1, CONFIGURATION 1  AVERAGE RESOLUTION TIME.............................. 74 FIGURE 7.4: SCENARIO 1, CONFIGURATION 1  AVERAGE NUMBER OF EMAILS IN THE SYSTEM. 75 FIGURE 7.5: SCENARIO 1, CONFIGURATION 2  AGENT UTILIZATION  ERLANG PROCESSING TIMES ................................................................................................................................... 77 FIGURE 7.6: SCENARIO 1, CONFIGURATION 2  AVERAGE RESPONSE TIME .................................. 77 x FIGURE 7.7: SCENARIO 1, CONFIGURATION 2  AVERAGE RESOLUTION TIME.............................. 78 FIGURE 7.8: SCENARIO 1, CONFIGURATION 2  AVERAGE NUMBER OF EMAILS IN THE SYSTEM. 78 FIGURE 7.9: SCENARIO 1, CONFIGURATION 3  AGENT UTILIZATION AND ERLANG PROCESSING TIMES ................................................................................................................................... 80 FIGURE 7.10: SCENARIO 1, CONFIGURATION 3  AVERAGE RESPONSE TIME ................................ 81 FIGURE 7.11: SCENARIO 1, CONFIGURATION 3  AVERAGE RESOLUTION TIME ............................ 81 FIGURE 7.12: SCENARIO 1, CONFIGURATION 3  AVERAGE NUMBER OF EMAILS IN THE SYSTEM .............................................................................................................................................. 82 FIGURE 7.13: SCENARIO 2, CONFIGURATION 1  AGENT UTILIZATION  ERLANG PROCESSING TIMES ................................................................................................................................... 84 FIGURE 7.14: SCENARIO 2, CONFIGURATION 1  AVERAGE RESPONSE TIME ................................ 84 FIGURE 7.15: SCENARIO 2, CONFIGURATION 1  AVERAGE RESOLUTION TIME ............................ 85 FIGURE 7.16: SCENARIO 2, CONFIGURATION 1  AVERAGE NUMBER OF EMAILS IN THE SYSTEM .............................................................................................................................................. 85 FIGURE 7.17: SCENARIO 2, CONFIGURATION 2  AGENT UTILIZATION  ERLANG PROCESSING TIMES ................................................................................................................................... 87 FIGURE 7.18: SCENARIO 2, CONFIGURATION 2  AVERAGE RESPONSE TIME ................................ 88 FIGURE 7.19: SCENARIO 2, CONFIGURATION 2  AVERAGE RESOLUTION TIME ............................ 88 FIGURE 7.20: SCENARIO 2, CONFIGURATION 2  AVERAGE NUMBER OF EMAILS IN THE SYSTEM .............................................................................................................................................. 89 FIGURE 7.21: SCENARIO 2, CONFIGURATION 3  AGENT UTILIZATION  ERLANG PROCESSING TIMES ................................................................................................................................... 91 FIGURE 7.22: SCENARIO 2, CONFIGURATION 3  AVERAGE RESPONSE TIME ................................ 91 xi FIGURE 7.23: SCENARIO 2, CONFIGURATION 3  AVERAGE RESPONSE TIME ................................ 92 FIGURE 7.24: SCENARIO 2, CONFIGURATION 3  AVERAGE NUMBER OF EMAILS IN THE SYSTEM .............................................................................................................................................. 92 FIGURE 7.25: SCENARIO 1  AGENT UTILIZATION  ERLANG PROCESSING TIMES......................... 96 FIGURE 7.26: SCENARIO 1  AVERAGE RESPONSE TIME ................................................................ 97 FIGURE 7.27: SCENARIO 1  AVERAGE RESOLUTION TIME............................................................ 97 FIGURE 7.28: SCENARIO 1  AVERAGE NUMBER OF EMAILS IN THE SYSTEM .............................. 98 FIGURE 7.29: SCENARIO 2  AGENT UTILIZATION  ERLANG PROCESSING TIMES....................... 100 FIGURE 7.30: SCENARIO 2  AVERAGE RESPONSE TIME .............................................................. 100 FIGURE 7.31: SCENARIO 2  AVERAGE RESOLUTION TIME.......................................................... 101 FIGURE 7.32: SCENARIO 2  AVERAGE NUMBER OF EMAILS IN THE SYSTEM ............................ 101 FIGURE A2.1: SIMULATION MODEL............................................................................................. 135 FIGURE A2.2: ARRIVAL AND ROUTING SUB MODE ..................................................................... 136 FIGURE A2.3: REP I (AGENT I) SUB MODEL I = 1, 2 AND 3 .......................................................... 137 FIGURE A2.4: STATISTICS AND RESOLUTION SUB MODEL.......................................................... 138 FIGURE A2.5: ERLANG SERVICE TIME DISTRIBUTION................................................................ 139 FIGURE A2.6: HYPEREXPONENTIAL SERVICE TIME DISTRIBUTION............................................. 141 FIGURE A3.1: PLOT OF RESOLUTION TIME IN SYSTEM................................................................ 145 FIGURE A3.2: PLOT OF RESOLUTION TIME IN SYSTEM (CONTD) ................................................. 145 FIGURE A3.3: PLOT OF RESOLUTION TIME IN SYSTEM (CONTD) ................................................. 146 FIGURE A3.4: PLOT OF RESOLUTION TIME IN SYSTEM (CONTD) ................................................. 146 FIGURE A3.5: PLOT OF RESOLUTION TIME IN SYSTEM (CONTD) ................................................. 147 FIGURE A3.6: PLOT OF RESOLUTION TIME IN SYSTEM (CONTD) ................................................. 147 xii FIGURE A3.7: PLOT OF RESOLUTION TIME IN SYSTEM (CONTD) ................................................. 148 FIGURE A3.8: PLOT OF RESOLUTION TIME IN SYSTEM (CONTD) ................................................. 148 FIGURE A3.9: PLOT OF RESOLUTION TIME IN SYSTEM (CONTD) ................................................. 149 FIGURE A3.10: PLOT OF RESOLUTION TIME IN SYSTEM (CONTD) ............................................... 149 xiii LIST OF TABLES TABLE 5.1: SIMULATION PARAMETERS ......................................................................................... 28 TABLE 5.2: SCV LEVELS AND CORRESPONDING DISTRIBUTIONS ................................................. 28 TABLE 6.1: SCENARIO 1  MEAN PROCESSING TIMES.................................................................... 54 TABLE 6.2: SCENARIO 1  FORWARDING PROBABILITIES .............................................................. 54 TABLE 6.3: SCENARIO 1  RESOLUTION PROBABILITIES................................................................ 54 TABLE 6.4: SCENARIO 1, TYPE 1 EMAILS  AGGREGATION PROBABILITIES FOR METHOD M3.... 55 TABLE 6.5: SCENARIO 1, TYPE 2 EMAILS  AGGREGATION PROBABILITIES FOR METHOD M3.... 55 TABLE 6.6: SCENARIO 2  MEAN PROCESSING TIMES.................................................................... 56 TABLE 6.7: SCENARIO 2  FORWARDING PROBABILITIES .............................................................. 56 TABLE 6.8: SCENARIO 2  RESOLUTION PROBABILITIES................................................................ 56 TABLE 6.9: SCENARIO 2, TYPE 1 EMAILS  AGGREGATION PROBABILITIES FOR METHOD M3.... 57 TABLE 6.10: SCENARIO 2, TYPE 2 EMAILS  AGGREGATION PROBABILITIES FOR METHOD M3.. 57 TABLE 6.11: SCENARIO 1  NODELEVEL PERFORMANCE MEASURES........................................... 60 TABLE 6.12: SCENARIO 1  SYSTEMLEVEL PERFORMANCE MEASURES ....................................... 61 TABLE 6.13: SCENARIO 2  NODELEVEL PERFORMANCE MEASURES........................................... 64 TABLE 6.14: SCENARIO 2  SYSTEMLEVEL PERFORMANCE MEASURES ....................................... 65 TABLE 7.1: SCENARIO 1, CONFIGURATION 1  PERFORMANCE MEASURES FOR ERLANG PROCESSING TIMES ............................................................................................................... 72 TABLE 7.2: SCENARIO 1, CONFIGURATION 1  PERFORMANCE MEASURES FOR EXPONENTIAL PROCESSING TIMES ............................................................................................................... 72 xiv TABLE 7.3: SCENARIO 1, CONFIGURATION 1  PERFORMANCE MEASURES FOR HYPEREXPONENTIAL PROCESSING TIMES............................................................................. 73 TABLE 7.4: SCENARIO 1, CONFIGURATION 2  PERFORMANCE MEASURES FOR ERLANG PROCESSING TIMES ............................................................................................................... 75 TABLE 7.5: SCENARIO 1, CONFIGURATION 2  PERFORMANCE MEASURES FOR EXPONENTIAL PROCESSING TIMES ............................................................................................................... 76 TABLE 7.6: SCENARIO 1, CONFIGURATION 2  PERFORMANCE MEASURES FOR HYPEREXPONENTIAL PROCESSING TIMES............................................................................. 76 TABLE 7.7: SCENARIO 1, CONFIGURATION 3  PERFORMANCE MEASURES FOR ERLANG PROCESSING TIMES ............................................................................................................... 79 TABLE 7.8: SCENARIO 1, CONFIGURATION 3  PERFORMANCE MEASURES FOR EXPONENTIAL PROCESSING TIMES ............................................................................................................... 79 TABLE 7.9: SCENARIO 1, CONFIGURATION 3  PERFORMANCE MEASURES FOR HYPEREXPONENTIAL PROCESSING TIMES............................................................................. 80 TABLE 7.10: SCENARIO 2, CONFIGURATION 1  PERFORMANCE MEASURES FOR ERLANG PROCESSING TIMES ............................................................................................................... 82 TABLE 7.11: SCENARIO 2, CONFIGURATION 1  PERFORMANCE MEASURES FOR EXPONENTIAL PROCESSING TIMES ............................................................................................................... 83 TABLE 7.12: SCENARIO 2, CONFIGURATION 1  PERFORMANCE MEASURES FOR HYPEREXPONENTIAL PROCESSING TIMES............................................................................. 83 TABLE 7.13: SCENARIO 2, CONFIGURATION 2  PERFORMANCE MEASURES FOR ERLANG PROCESSING TIMES ............................................................................................................... 86 xv TABLE 7.14: SCENARIO 2, CONFIGURATION 2  PERFORMANCE MEASURES FOR EXPONENTIAL PROCESSING TIMES ............................................................................................................... 86 TABLE 7.15: SCENARIO 2, CONFIGURATION 2  PERFORMANCE MEASURES FOR HYPEREXPONENTIAL PROCESSING TIMES............................................................................. 87 TABLE 7.16: SCENARIO 2, CONFIGURATION 3  PERFORMANCE MEASURES FOR ERLANG PROCESSING TIMES ............................................................................................................... 89 TABLE 7.17: SCENARIO 2, CONFIGURATION 3  PERFORMANCE MEASURES FOR EXPONENTIAL PROCESSING TIMES ............................................................................................................... 90 TABLE 7.18: SCENARIO 2, CONFIGURATION 3  PERFORMANCE MEASURES FOR HYPEREXPONENTIAL PROCESSING TIMES............................................................................. 90 TABLE 7.19: SCENARIO 1  PERFORMANCE MEASURES FOR ERLANG PROCESSING TIMES .......... 95 TABLE 7.20: SCENARIO 1  PERFORMANCE MEASURES FOR EXPONENTIAL PROCESSING TIMES.. 95 TABLE 7.21: SCENARIO 1  PERFORMANCE MEASURES FOR HYPEREXPONENTIAL PROCESSING TIMES ................................................................................................................................... 96 TABLE 7.22: SCENARIO 2  PERFORMANCE MEASURES FOR ERLANG PROCESSING TIMES ........... 98 TABLE 7.23: SCENARIO 2  PERFORMANCE MEASURES FOR EXPONENTIAL PROCESSING TIMES.. 99 TABLE 7.19: SCENARIO 2  PERFORMANCE MEASURES FOR HYPEREXPONENTIAL PROCESSING TIMES ................................................................................................................................... 99 TABLE A1.1: SCENARIO 1, TYPE 1 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC................................................................................................................................ 116 TABLE A1.2: SCENARIO 1, TYPE 1 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 117 xvi TABLE A1.3: SCENARIO 1, TYPE 1 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 118 TABLE A1.4: SCENARIO 1, TYPE 1 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 119 TABLE A1.5: SCENARIO 1, TYPE 2 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC................................................................................................................................ 120 TABLE A1.6: SCENARIO 1, TYPE 2 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 121 TABLE A1.7: SCENARIO 1, TYPE 2 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 122 TABLE A1.8: SCENARIO 1, TYPE 2 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 123 TABLE A1.9: SCENARIO 2, TYPE 1 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC................................................................................................................................ 124 TABLE A1.10: SCENARIO 2, TYPE 1 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 125 TABLE A1.11: SCENARIO 2, TYPE 1 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 126 TABLE A1.12: SCENARIO 2, TYPE 1 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 127 TABLE A1.13: SCENARIO 2, TYPE 2 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC................................................................................................................................ 128 xvii TABLE A1.14: SCENARIO 2, TYPE 2 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 129 TABLE A1.15: SCENARIO 2, TYPE 2 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 130 TABLE A1.16: SCENARIO 2, TYPE 2 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 131 xviii NOTATION c type of email; c = 1, 2, ..., C (These types may represent different classes of email product inquiry, technical support, etc.) A number of customer service agents k,i, j index for previous, current and next agent processing an email; k = 0,1, ..., A; i, j =1, 2, ..., A; k = 0 represents a new email λ (c) external arrival rate of (new) emails belonging to type c (c) i λ external arrival rate of type c emails at agent i i λ external arrival rate of emails at agent i (c) i γ total ( external plus internal) arrival rate of type c emails at agent i r (c) i probability that an email received by agent i is of type c β (c) represents the total (new plus previously processed) arrival rate of type c emails for Scenario 1 (c) i α probability that a new type c email will be routed to agent i w (c, k) i proportion of type c emails received by agent i that were previously processed by agent k or new (k = 0); i = A +1 represents the delay node ( , ) 1 w c k A+ probability that a type c email at the delay node (A +1) came from agent k (c, k) i υ average number of visits to agent i made by a type c email that was previously processed by agent k (c, k) Di υ average number of times a type c email was processed by agent i and not resolved xix ( , ) , p c k i j probability that agent i forwards a type c email that was previously processed by agent k to agent j ( ) , p c i j classspecific routing probability from agent i to agent j for a type c email i j p , aggregate routing probability from agent i to agent j p(c) represents the aggregate forwarding probability for type c emails q (c, k) i probability that a type c email currently processed by agent i and previously processed by agent k ends in problem resolution q(c) represents the aggregate resolution probability for a type c email ( ) , 1 p c i A+ classspecific routing probability from agent i to delay node (A +1) for type c email i,A+1 p aggregate routing probability from agent i to delay node (A +1) ( ) 1, p c A+ i classspecific routing probability from the delay node (A +1) to agent i for a type c email A i p +1, aggregate routing probability from delay node (A +1) to agent i P (c) i random variable that represents the preprocessing time at agent i for a type c email Z (c, k) i indicator random variable that takes the value 1 if agent i processes a type c email that was previously processed by agent k and the value 0 otherwise S (c, k) i random variable that represents the processing time at agent i for a type c email that was previously processed by agent k xx T (c, k) i random variable that represents the overall service time at agent i for a type c email that was previously processed by agent k (c) i θ E[P (c)] i p (c, k) i E[Z (c, k)] i = − Σ ≠ = A i j j i j p c k 1 , 1 ( , ) s (c, k) i E[S (c, k)] i t (c, k) i E[T (c, k)] i c2 squared coefficient of variation (SCV) = variance/mean2 c2 (c) o SCV of the interarrival time for new, external type c emails c2 (c) oi SCV of the interarrival time for external type c emails at agent i 2 oi c SCV of the aggregate interarrival time for external emails at agent i 2 ( ) , c c pre i SCV of P (c) i 2 ( , ) , c c k ser i SCV of S (c, k) i c2 (c, k) i SCV of T (c, k) i c2 (c) i SCV of the overall classdependent service time at agent i 2 i c SCV of the overall service time at agent i The following additional notation is needed for the Markov chain analysis presented in Section 6.5. N represents the “new” state  the lifecycle of a new email starts in this state xxi R represents the “resolution” node  the lifecycle of an email ends in this state i D represents the delay state linked to agent i  emails processed by agent i that are unresolved pass through this state P onestep transition probability matrix U unit matrix of size (1×1) M column matrix of size (n×1) , where n = A3 + A2 + A+1 Q truncated matrix associated with P and of size (n × n) , where n = A3 + A2 + A+1 I identity matrix of size (n × n) , where n = A3 + A2 + A+1 F fundamental matrix of size (n× n) , where n = A3 + A2 + A+1 1 CHAPTER 1 INTRODUCTION The service sector has become a dominant part of the US economy, due in part to the ecommerce revolution of the late nineties. Better quality of service delivered virtually with very little or no waiting time is what customers frequently expect today. One notable facet of the service industry is the call center industry. A call center is any “group whose principal business is talking on the telephone to customers or prospects (Mehrotra 1997).” Customer call centers, which represent a multibillion dollar industry, are evolving into customer contact centers. “It is estimated about four million people in the United States 3% of the workforce  work in contact centers, with the number growing by about 20% per year. A contact center is a collection of resources providing an interface between the service provider and its remote customers (Whitt 2002a).” The interface can be through any one or combination of media  telephone, email, fax, paper, chat sessions and the Web. In the private sector, contact centers are used in various industries and are an important communication channel to acquire new customers as well as to support existing customers. In email contact centers, the traffic can be inbound or outbound. In an inbound email contact center, agents respond to emails from customers. Examples of inbound email contact centers are technical product support centers and travel 2 reservation centers. In outbound email contact centers, agents initiate email that is sent to customers. Examples of outbound email contact centers are companies conducting surveys and market research. In inbound email contact centers the response time is flexible, i.e., customers do not expect an email to be answered within minutes, whereas they could get frustrated waiting on the telephone even for a few minutes. This flexibility allows for the possibility of postponing a response. The average time an agent takes to respond to a customer’s email is known as the response time. This time needs to be as small as possible and is an important performance measures in both call and contact centers. The other measure of interest is the resolution time, i.e., the average time that is taken to resolve the problem represented by the email. Shorter response times and resolution times can help contact centers to better serve and retain their customers. These system performance measures can be improved if the email contact center employs more agents. But employing more agents leads to higher operating costs. In email contact centers, more than half of the operating costs are driven by the costs of employing agents. Therefore, agent utilization is often used to indicate the economic performance measure of an inbound email contact center. The number of agents is an important decision variable in designing email contact centers. Both queueing theory and simulation have been used to model the operations of call centers. These models describe the behavior of the system over time, which helps in designing call center operations. Bulk of the existing literature focuses on modeling traditional telephone call centers. As pointed out by Whitt (2002a), more research is needed in the stochastic modeling of customer contact centers, where the contact is 3 through media other than the telephone. The focus of this thesis was on modeling an important characteristic of email contact centers, namely the dependence of email processing times and routing on email history. A novel historybased aggregation approach was developed to handle the dependence on processing history within a multiclass open queueing network model. Through extensive numerical experimentation, this thesis shows the importance of modeling email history in accurately predicting the performance of email contact centers. The numerical investigations also demonstrate the accuracy and robustness of the historybased aggregation approach. The remainder of this thesis document is organized as follows (also see Figure 1.1). Chapter 2 describes the problem statement. Chapter 3 presents an extensive literature review of the work carried to date in modeling call and contact centers. Chapter 4 presents the research statement. Also included in this chapter are the research objectives, scope and limitations, and contributions of the research conducted. Chapter 5 describes the modeling approach that was followed in developing and solving the open queueing network models of email contact centers. Chapter 6 presents the nucleus of this thesis effort. It includes three analytical methods ranging from a simple averaging technique to a very detailed Markov chain based aggregation approach to model dependence on email history. These analytical methods are integrated into the solution method for a multiclass, open queueing network model of email contact centers. Chapter 7 extends the network model presented in Chapter 6 to include (i) multiserver nodes that represent skillbased pools of agents and (ii) random interruptions that affect agent’s availability for email processing. Chapter 8 presents research contributions, conclusions and directions for future research. 4 Chapters 1 & 2 INTRODUCTION PROBLEM STATEMENT Chapter 3 LITERATURE REVIEW Skillbased Routing, Resource Pooling, Workforce Management, Organizational Behaviour, and Performance Evaluation Chapter 4 RESEARCH STATEMENT Objectives, Scope & Limitations, Contributions Chapter 5 MODELING APPROACH Parametric Decomposition (PD) Method Chapters 6 & 7 MULTICLASS, OPEN QUEUEING NETWORK MODELS  Historybased Aggregation  Classbased Aggregation  Single / MultiServer Nodes  Service Interruptions Chapter 8 CONCLUSIONS AND FUTURE RESEARCH Figure 1.1: Outline of the Thesis 5 CHAPTER 2 STATEMENT OF THE PROBLEM Modeling call centers using queueing theory is not new. Many researchers have modeled call centers using the standard M/M/c/∞ queues to obtain steadystate performance measures such as average number of calls in the system and average waiting time for calls. However, only very limited literature exists on queueing models of contact centers because of their recent emergence as an alternative to call centers. In addition to analytical approaches, both call and contact centers can be modeled using simulation. Simulation models require more detailed information when compared to queueing models. The model development and model execution activities in simulation could be time consuming. On the other hand, analytical models such as queueing networks provide more insight and understanding of the system. Analytical models may require simplifying assumptions of the system, and the results obtained are generally less accurate than those estimated via simulation. But analytical models yield results quickly and “are appropriate for rapid and rough cut analysis (Suri et al. 1993).” Hence, analytical models can be used for preliminary design and simulation for fine tuning. The evolution of email contact centers has thrown light on many new modeling issues that are not typically addressed in the study of traditional call centers. Customer or problem history can be more useful with asynchronous communication tools such as e 6 mail and can influence routing strategies and processing times. For example, it is possible for an agent who previously attended to a customer’s problem (email) to also deal with the followup email because of the asynchronous nature of email processing. With regard to system performance measures, the average resolution time, i.e., the average time needed to resolve a customer’s problem is not normally addressed in the call center literature. Problem resolution, while important in both call and contact centers, has much more visibility in an email contact center because of the history embedded in the email reply. In a call center, because of the nature of the customer contact, the response time for a call becomes more important. In a contact center, both response and resolution times become important as the latter is a function of the former. Also, because of the asynchronous nature of email, the customer service agent has more flexibility in terms of the time to generate a response to an email. Also, it is possible to direct an email to the appropriate agent if it contains information about previous response(s). These new modeling issues present unique challenges while developing queueing models of contact center operations. This thesis has addressed some of these issues. The problem statement can be summarized as follows: “To model the performance of an inbound email contact center in which the routing of an email and email processing times at the various agents can depend on the email history.” 7 CHAPTER 3 LITERATURE REVIEW The research on modeling call centers and contact centers can be broadly classified into the following categories  1) SkillBased Routing, 2) Resource Pooling, 3) Workforce Management, 4) Performance Evaluation and 5) Organizational Behavior. Within each category, a formal subcategory on email contact centers is created only when there is a need to distinguish the work from that for the call center classification. Otherwise, the description for call centers holds good for the email contact centers also. Though the scope of the literature review presented in this chapter is much broader than the performance evaluation theme of this research, it nevertheless illustrated the importance of analytical models in the analysis and design of customer call centers. This chapter also explores some new research directions and issues in the area of customer contact center modeling. 3.1 Skill  Based Routing Modern call centers have multiple call types and multiple types of agents. One way of classifying customer calls is by language. With the globalization of many businesses, call centers receive calls in different languages from their customers throughout the world. Another way of classifying customer calls is based on special promotions. The customer may be calling a tollfree number designated for a special promotion purpose. Training 8 agents for a special promotion purpose is a common practice. To match the caller’s need and agent’s skill set, modern call centers have automatic call distributors (ACDs) that have the capability of routing calls to agents with the appropriate skills. This capability of ACDs is known as skillbased routing (SBR). In email contact centers, the emails are routed to the appropriate agents with the help of information and communication technologies. An important issue to be addressed pertains to the optimality of email/call routing policies. 3.1.1 Challenges in SkillBased Routing There are many challenges and issues to be addressed in performing skillbased routing well. First, with a given collection of agents, it is difficult to route multiple calls/emails to an agent in an optimal manner. This is due to elementary skillbased routing algorithms that are used in call/contact centers. According to “(Whitt 2002a), there remains a great opportunity for devising better routing algorithms.” The routing of calls/emails depends on the number of agents in the center. Second, it is difficult to determine exactly the number of agents with appropriate skill sets. With multiple call/email types, not only the prediction of the overall arrival rate, but also the prediction of arrival rates for individual types becomes important. Koole et al. (2003) presented an approximation method for analyzing the performance of call centers with skillbased routing. They considered two types of call arrivals. Arriving calls abandon the system if the agents with the right skill are busy. But under these conditions, it was difficult to compute the optimal routing policies of calls because of statespace explosion. Therefore, the authors considered a different queueing system where the call gets queued if the agents are not available in any of the groups. This is equivalent to calls overflowing from the groups without available agents. This type of 9 routing is known as overflow routing. This system is easy to characterize and optimal policies can be practically implemented, but allows less flexibility in routing policies. Koole and Talim (2000) studied a multiskill call center as a network of queues. They approximated each queue as an M/M/r loss system, because an arriving call left if all agents in the network are busy. The interoverflow time distribution at each node in the network was approximated by an exponential distribution, and the efficiency of the approximation was illustrated using simulation. Bhulai and Koole (2003) developed a queueing model from a call blending perspective for schedule agents either to incoming or outgoing calls in order to increase productivity and reduce the call waiting times. In addition, scheduling policies and their implementation within call center software were also discussed. Garnett and Mandelbaum (2000) illustrated the operational complexities of skillbased routing using simulation studies. Perry and Nilsson (1992) considered simple strategies to overcome the dimensionality of call centers. They considered a twochannel agent system, where the waiting customer was assigned an agingfactor. This factor was directly proportional to the waiting time of the customer in the system. The customer with the largest aging factor (considering both queues) was chosen for service. They determined the expected waiting time for each call type, and the number of agents required for answering calls to maintain an acceptable grade of service. To conclude, there is scope for more research in developing simple and better routing algorithms, finding optimal routing policies using approximations and strategies, and implementing those in call/contact centers. 10 3.2 Resource Pooling Resource pooling is overlapping of agents between groups so as to meet customer needs and to increase the efficiency of call/contact centers. In a call center, it is customary to have a large group of agents dedicated to a particular skill set. This can be viewed as a big call center with large number of smaller independent call centers. Upon call arrival, the calls can be routed to the group of agents with the appropriate skill sets. When the arrival process to the big call center is Poisson and with independent Bernoulli routing to the smaller call centers, the arrival process to the smaller call center is Poisson and tends to act independently of other smaller units. Partitioning into subgroups tends to make call centers less efficient. While operating, some of the smaller centers tend to overloaded, while others may be underutilized. The efficiency of larger service groups is explained by Smith and Whitt (1981) and Whitt (1992). Smith and Whitt (1981) argued that if two systems were combined together into a single system, the efficiency of the combined system is higher than the efficiency of the individual systems. This seems intuitive and trivial, but becomes difficult to prove. The authors used stochasticorder relations to prove the result and concluded that the results apply to general arrival processes and general servicetime distributions. The combination of systems assumes common servicetime distribution, since it may be disadvantageous to combine systems with different servicetime distributions. When the service time distributions are different it is advantageous to partition the system as in supermarket checkouts and reservation centers. Whitt (1992) explained the economy of scale, and gave a quantitative characterization of a multiserver queueing system with unlimited waiting space. He showed that the increased variability in the arrival and service processes degrade server utilization with a 11 given grade of service. He also presented a proof for finding the number of servers as a function of the arrival rate for a given service time distribution and grade of service. The proof is based on heavytraffic limit theorems and uses infiniteserver (IS) approximation to heuristically derive the result. The author developed simple approximations for the expected waiting time and the probability of delay using the infiniteserver approximation, and conditional waiting time distribution using a singleserver approximation. The next section throws light on the application of StochasticProcess limits in resource pooling. 3.2.1 StochasticProcess Limits It is natural to study a system by allowing the number of customers and servers to grow large. This is done to gain more knowledge and insight about the behavior of the system under consideration. The mathematical results on resource pooling are based on the asymptotic regime in which the number of servers is allowed to approach infinity. This is the principle behind stochasticprocess limits. Significant work on resource pooling has been done till now. The paper by Vvedenskaya et al. (1996) serves as a good example for its significance, and it considered a single stream of customers with a Poisson arrival process. Upon arrival, each customer joins one of the many identical singleserver queues. The service time distribution in all queues follows an exponential distribution and all queues have an unlimited waiting space. The standard approach is joining the queue with very few customers. The number of customers is determined based on the states of all queues. Indeed, joining the shortest queue is optimal (Winston 1977). However, optimality is ceased if each queue has different servicetime distribution (Whitt 1986). A difficulty in joining the shortest queue is that it may require a large amount of state information. Therefore techniques which require less amount of state information 12 have to be adopted. For example, each arrival can be randomly assigned to one of the queues with equal probability of selecting a server. This can be modeled as an M/M/1 queue and all required performance measures can be calculated. Resource pooling has to be done very carefully. Sometimes it helps, but sometimes it hurts. The effect (good or bad) can be unbounded. The paper by Mandelbaum and Reiman (1998) clearly assesses the value of resource pooling. Halfin and Whitt (1981) obtained limiting regimes for the Erlang C delay and GI/M/s models. This was carried out by allowing the number of servers to approach infinity, while letting the probability of delay approach a number strictly between 0 and 1. For more about StochasticProcess limits, the reader is referred to Whitt (2002b). Heavy traffic limit regimes for the Erlang B (loss) model can be found in Srikant and Whitt (1996). Related results for queueing networks, statedependent queues can also be found in Mandelbaum and Pats (1995). Papers about heavy traffic limit regimes on contact centers are very few. Whitt (2002a) stated that there remains a great opportunity for research in establishing some new interesting regimes. 3.3 Workforce Management Workforce management plays a very significant role in design and maintenance of call/contact centers. Cleveland and Mayben (1997) addressed some mathematical issues in workforce management. Staffing is a challenging issue in call/contact centers. Whitt (2002a) described three types of staffing according to time scale. They are 1) realtime 2) shortterm and 3) longterm. 13 3.3.1 RealTime Staffing Realtime staffing has sufficient flexibility to add agents when needed, and alternatively, pull them off to do some other work. Whitt (1999b) addressed the modeling of dynamic staffing of a call center with the aim of immediately answering the calls. The system state is exploited to obtain estimates for mean and variance of demand in near future. The staffing needs can be predicted from the information about recent demand and current calls in progress, as well as historical data. The information and telecommunication equipment makes it possible to obtain the required information. It is possible to classify the call by identifying the calling or called customer, purpose of the call and the agent who will be serving or served the call. By calculating the time length that the call has been in service before service completion, it is possible to predict the conditional probability distribution of the remaining call holding time. By combining the information over many calls and agents, it is possible to predict the staff demands in the near future, providing a basis for realtime staffing. The paper by Whitt (1999b) “shows how stochastic models can be exploited to facilitate the process” and the author stated the idea needs be explored more thoroughly. Jennings et al. (1996) determined the number of servers as a function of time for a multiserver, timevarying demand service system based on an infiniteserver (IS) approximation. The IS approximation averages the timevarying demand into an effective arrival rate which remain the same at all times. An approximate busy period server distribution is obtained by allowing the number of servers to grow large and by approximating the delay probability to a specific target value. The busy period server distribution is approximated by a timedependent normal distribution where the mean and variance are determined by 14 IS approximations. Wallace and Whitt (2004) discussed a staffing algorithm for the skillbased routing call center. Realtime staffing poses great challenges in applying queueing theory, because it requires the analysis of timedependent behavior of queueing systems. Therefore, people seek numerical algorithms and approximations to describe the timedependent behavior of queues. Papers in this regard include Whitt (1999a), and Abate and Whitt (1998, 1999) where they apply decomposition approximations and numerical transform inversion techniques respectively to study the timedependent behavior of queues. 3.3.2 ShortTerm Staffing In shortterm staffing, the daily staffing is carried out in response to the forecasted demand of calls and the availability agents. As the call arrivals vary significantly from day to day, one can use the steadystate behavior of queueing systems instead of the timedependent one. This is because the call holding times are much shorter, and the time dependence can be safely ignored. In some cases of shortterm staffing, it is important to analyze the system with a timevarying arrival rate. A significant amount of effort in this area has been done by Abate and Whitt (1998), Mandelbaum and Pats (1995) and Whitt (1999b). A significant challenge in shortterm staffing is scheduling agents for small breaks for example, lunch and coffee. Mathematical programming tools have been widely used in modeling this; see for example Segal (1974). 3.3.3 LongTerm Staffing There are various challenges in longterm staffing. Training new agents is an important decision variable, which can be handled using a dynamic programming technique. On a longterm basis it is important to consider the agent’s career paths and attrition. The ideal 15 situation is to have both satisfied customers and agents. Gans and Zhou (1999) developed a Markov decision model for callcenter staffing where they address learning and turnover issues and optimal policies for longterm staffing. 3.4 Organizational Behavior The role of human element is very important in call/contact centers. Customers are people and the service reps (agents) are people. “We can easily relate to contact centers because we ourselves often are customers of contact centers (Whitt 2002a).” The human behavior is really very important in studying the psychology of queueing systems (Gail and Scott 1997, Larson 1987). Enough time should be devoted to analyzing why customers abandon or revisit. Whitt (2002a) expressed an opinion that few papers have been published in this area. 3.5 Performance Evaluation Considerable research has been done in performance modeling of call centers. Call center performance modeling studies have focused on (i) analyzing customer waiting times and customer impatience because of agent unavailability, (ii) finding optimal staffing to meet customer demands, and (iii) determining routing policies to serve customers at the earliest possible time. Traditional analysis techniques are typically based on the standard Erlang formula (Koole 2001). For example, the Erlang formula can be used to determine the upper and lower bounds on the number of employees needed, which are useful in employee scheduling (Koole 2001). = Σ= s k k s k l s l P s l 0 ! ( , ) ! (1) 16 where s number of servers in the queueing system l offered load in Erlangs, given by (l = λτ ), where λ is the arrival rate of customers and τ is the average service time. The Erlang formula is insensitive to the service time distribution. When the servers and the offered load become large, it becomes difficult to calculate the blocking probability. Therefore, recursive techniques are used to numerically calculate the blocking probability. 3.5.1 Queueing Models of Call Centers According to Stolletz (2003), queueing models of an inbound call center can be described using customer profile, agent characteristics, routing policies, and limitation of waiting room. Customer profile describes the customer arrival process to the call center and the patience/impatience characteristics of customers of a particular class. The agent characteristics describe the agent skill set and the service time distribution of the agent. The routing policy defines which agent needs to serve which customer. These policies may depend on the number of busy agents and the number of waiting customers of different classes. The size of the waiting rooms defines the maximum number of customers in the system and may depend on the customer class. 3.5.1.1 Arrival Process Customers of a call center cannot see others being served or waiting for service in the queue. Therefore, customers call independently of others and for this simple reason the arrival process in inbound call centers can be modeled as a timeinhomogeneous Poisson process (Koole and Mandelbaum 2001). As the call arrivals vary from time to time, day to day, the common approach is to approximate the timevarying arrival process by a 17 stationary, independent period by period (SIPP) approximation (Green et al. 2001). They also discussed a situation where server staffing requirements are done based on a random cyclic demand and concluded that the SIPP approximation was not accurate for many real situations. Other than SIPP approximations, point wise approximations (PSA), simple stationary approximation (SSA) and infiniteserver approximation (IS) can be found in the literature. The usage of these approximations depends on how the arrival rate varies from time to time. But all the approximations average the timevarying arrival rate into an effective arrival rate which remains constant at all times. In each time interval, arrivals occur according to a homogeneous Poisson process and it is assumed that the steadystate arrival rate does not change in each time interval. The standard steadystate approaches can then be effectively used to calculate the various performance measures of a particular queueing model (Koole and Mandelbaum 2001). 3.5.1.2 Waiting Behavior of Customers In call centers customers can be patient or impatient. Impatient customers are of two types. Balking occurs when the arriving customer finds the server busy and leaves the system immediately without being served. Reneging is when the customer finds the server busy, waits in the queue for certain random time and leaves the system without being served. The process of reneging is described by the random waiting time distribution in the queue. Both balking and reneging may be state dependent or constant. MontazerHaghighi et al. (1986) considered a multiserver queueing system with balking and reneging and obtained the average number of customers in the system under steadystate. They also presented expressions for the average loss of customers during a fixed interval of time. AbouElAta and Hariri (1992) expressed the steadystate distribution 18 for the number in the system in terms of hypergeometric function for an M/M/c/N queue with balking and reneging. Brandt and Brandt (1999) studied customer impatience in a finiteserver queueing system where the arrival and service rates could depend on the number of callers in the system. Movaghar (1998) explained customer impatience for a queueing system with statedependent Poisson arrivals, exponential service times, multiple servers and FCFS service discipline. Brandt and Brandt (2002) extended the system considered in Movaghar (1998) by assuming statedependent exponential servers. They presented asymptotic results for the number of calls leaving the system and presented a Markovian approximation for the system. Boots and Tijms (1999) presented a simple approximation for the blocking probability in an M/G/c queue with customer impatience. Bae et al. (2001) presented limiting virtual waiting time distribution for an M/G/1 queue with impatient customers. 3.5.1.3 Service Time Distribution of Agents Traditionally almost all papers in call center literature model service times of agents using the exponential distribution. In a majority of the models published in the call center literature, service times of agents have been typically modeled by the exponential distribution for model tractability. While some studies have shown that the exponential service time distribution is a good fit (e.g., see Koole and Mandelbaum 2001) arrivals are, other studies (e.g., Mandelbaum et al. 2001) have concluded that service time distribution cannot be approximated by the exponential distribution based on empirical data, and that the usefulness of exponential service time distribution may vary from one inbound call center to the other. Harris et al. (1987) analyzed data from a telephone taxpayer information system in which both the talk time and aftercall work time (time an agent takes to fill a form or mail order) followed a Weibull distribution. They compared the 19 performance measures obtained from the Weibull distribution to the performance measures obtained from the exponential distribution and observed a high level of insensitivity. 3.5.2 Queueing Models of Inbound Email Contact Centers Very few papers have been published to date in analyzing email contact centers using queueing models. This is due to the recent emergence of contact centers as an extension of traditional call centers. Most of the call center description holds good for the email contact center except for the behavior of customers. This is because it is difficult to characterize customer impatience (balking and reneging). Once the customer sends an email, two things can happen. The email can reach the agents inbox and the agent responds to the email or it can bounce back due to lack of space in agent’s inbox. The second condition is of course rare, but there are some chances of its occurrence. This in a way can be thought of as balking. Once the customer’s problem is not resolved after many email exchanges with the agent, he/she may renege, i.e., leave the system permanently. Whitt (2002a) explained the many challenging research issues in the area of customer contact center modeling and suggested research directions related to skillbased routing, resource pooling and agent staffing. Armony and Maglaras (2004a, 2004b) focused on a customer contact (call) center that offers two modes of service: realtime telephone service and callback service. In Armony and Maglaras (2004b) arriving customers are informed about the delay, and the contact center is modeled as a twoclass M/M/r queueing system with statedependent arrival rates. Armony and Maglaras (2004a) proposed an estimation scheme for the anticipated delay time based on the heavy traffic regime, approximated the system performance, and presented a staffing rule that picks the minimum number of agents. 20 When modeling email contact centers, we assume that emails that represent spam have been filtered and only useful emails arrive at the contact center. For more information about nonspam, workrelated to filtering of emails, the reader is referred to Sharda et al. (1999). Many companies use an email filtering language that can be supported in an email client and client software. Greve et al. (2004) focus on email response management problems in customer contact centers, where they specifically address the problem of processing emails in a timely manner. They use simulation to evaluate different routing policy and email processing strategies that can be employed by a contact center. In summary, work on analytical performance modeling of customer contact centers is very limited. As explained in Chapters 1 and 2, the asynchronous nature of email introduces new possibilities in operating customer contact centers and consequently, new modeling challenges. This thesis effort focused on one such challenge related on modeling the dependence of routing and processing on email history. 21 CHAPTER 4 RESEARCH STATEMENT This chapter presents the specific objectives, scope, limitations, and contributions of the research conducted as part of this thesis effort. The overall goals of this research were (i) to develop queueing network models of inbound email customer contact operations that are capable of capturing the dependence of routing and processing on email history, and (ii) to support the development of rapid whatif analysis tools that can assist the decisionmaker in designing and improving customer contact center operations. 4.1 Research Objectives The specific objectives of this research were as follows. Objective 1: To perform a thorough investigation of the literature related to the modeling of customer call and contact centers. Objective 2: To develop queueing network models of inbound email contact centers with the following characteristics. • Multiple types of email inquiries. • Heterogeneous agents with random service interruptions: The processing of emails can be interrupted when agents have to handle other knowledge work or decide to take a break. • Grouping of agents: Agents with similar skills are grouped to form an agent pool. 22 • Routing and service times are dependent on history of email (new or previously processed). Objective 3: To suggest extensions to the queueing network model to approximately handle daily and/or weekly schedules of agents. 4.2 Research Scope and Limitations The scope of this thesis was limited by the following assumptions. 1. Emails arrive continuously and the contact center agents are available 24/7. The reference to a particular agent is only with respect to a rule that requires a specificskill set with memory augmented by customized CRM tools. 2. Emails are selected from an inbox by an agent according to the FIFO (First in First out) service discipline. 3. Priorities of emails are not modeled. 4. Modeling of email history is limited to capturing the identity of the previous agent in the case of a previously processed email. 5. Modeling of agent schedules is limited to suggestions of potential extensions to the network models developed. 4.3 Research Contributions The purpose of this thesis was to contribute towards the development of queueing network models of inbound email customer contact center operations. The following contributions have been made by this thesis effort. 1. Development of a novel modeling and solution approach to handle routing and processing schemes that are dependent on email history. The approach developed is very general and extends the power of existing queueing network models. 23 2. Development of queueing models that can be incorporated into rapid analysis tools that can support the analysis and design of customer contact center operations. 24 CHAPTER 5 MODELING METHODOLOGY This chapter explains the methodology that was used to develop and solve the queueing network models in this thesis. It also includes a list of important contact performance measures that were addressed in this thesis effort. The inbound email customer contact center was modeled as a multiclass, open queueing network. The parametric decomposition (PD) method and its extensions presented in Whitt (1983, 1994) were used to solve the multiclass, open queueing network model. While the extensions presented in Whitt (1983, 1994) can handle multiple customer classes, the dependence of the processing times and routing probabilities on email history is not addressed by the PD method and its extensions. Modeling this dependence was a key contribution of this thesis, and details are discussed in the next chapter. The PD method is briefly explained next. 5.1 The Parametric Decomposition (PD) Method From the late 1950s to the mid 1980s, the analysis of queueing networks was dominated by the wellknown productform method (Baskett et al. 1975; Jackson 1957). The main problem with the productform analysis method and its extensions was the assumption of Poisson arrivals and exponential service times. There was no convenient mechanism for modeling the variability present in realworld processes. A fundamental change occurred 25 in the mid eighties. There was a paradigm shift from "exact analysis of an approximate model" to "approximate analysis of a more exact model (Whitt 1983).” The basic reason behind the distributional assumptions of the productform analysis was the tractability of the mathematical model. In the product form analysis, the focus was more on developing a model that can be solved exactly. The method made popular by the Queueing Network Analyzer (QNA) software developed at Bell Laboratories focused on the development of a more realistic model at the cost of our ability to solve the model exactly (Whitt 1983). An analysis method known as the parametric decomposition (PD) method based on twomoment queueing approximations (using mean and SCV  Squared Coefficient of Variation = Variance/mean2) became popular. The PD method was first proposed by Reiser and Kobayashi (1974) and subsequently extended by Kuehn (1979), Whitt (1983, 1994) and many others (see for example references in Suri et al. 1993). The PD method is the basis of many of the recent tools and techniques developed for queueing network analysis (Suri et al. 1993). The main reason for the success of the PD method is that it does not make any distributional assumption and uses only the mean and variance information of processing times and interarrival times. Through extensions to the PD method, several features relevant to real world systems have been incorporated including multiclass networks with deterministic routing, equipment breakdown and repair, changing lot sizes, inspection and testing, batch service, and overtime (Kamath et al. 1995, Suri et al. 1993). The PD method for a singleclass, open queueing network is based on 1) analysis of interactions between the nodes to obtain the mean and SCV of the interarrival time at each node, and 2) decomposition of the network into individual nodes and calculation of 26 node measures and network performance measures using GI/G/1 (general arrival, general service time distribution with single server) or (Whitt 1983) GI/G/m (general arrival, general service time distribution with multiple servers) approximations (Whitt 1993). The rate and the variability parameters of the combined (external plus internal) arrival processes approximately capture interactions among the nodes. The total arrival rate at each node can be obtained by solving the traffic flow rate equations, which represent the conservation of flow. Utilizations are calculated at each node to check system stability. The system is stable if the utilization at each node is strictly less than one. Up to this point, the analysis is similar to the one carried out in the product form analysis method (Jackson 1957) for solving open networks and involves no approximations. The SCVs of interarrival times at each node are calculated by solving the traffic variability equations which are linear. The traffic variability equations involve approximations for the basic network operations like a) flow through a node, b) merging of flow and c) splitting of flow. These approximations can be found in Whitt (1983, 1994).In calculating the performance measures, all nodes are assumed to be stochastically independent. The performance measures at each node are calculated from the GI/G/1 or GI/G/m results given in (Whitt 1983, 1993). 5.2 Aggregation Approaches The PD method described in the previous section essentially solves a singleclass network with Markovian routing probabilities. As explained in Whitt (1983, 1994), the general approach to solving multiclass networks is to aggregate the multiclass information (service and arrival) to define an aggregate singleclass network; solve the singleclass network using the PD method; and disaggregate to calculate classspecific 27 performance measures. Whitt (1983, 1994) presented extensions to the PD method to handle several classes of customers. Each customer class is described by a deterministic route. Whitt’s extension (1983, 1994) not only allows different service time parameters for different classes at a node, but also different service time parameters for different visits to the same node by the same class. Whit (1983) also mentioned that the routing could be probabilistic in the multiclass case. If so, then a routing probability matrix and parameters for the external arrival processes and node service times must be specified for each customer class. Whitt’s (1983) classbased aggregation method was modified to handle probabilistic routing in the case of the contact center model. This thesis effort has added a new extension to the PD method to treat an open queueing network in which routing probabilities and processing times depend on email history. This extension involves a new historybased aggregation step within each customer class before the classbased aggregation step. Details of this extension are presented in Chapter 6. 5.3 Numerical Validation The performance measures computed using the queueing network models were compared with steadystate simulation results obtained using an Arena 7.0 simulation model to evaluate the accuracy of the analytical results. The analytical results for different scenarios and different levels of service time SCVs were compare with the corresponding simulation estimates. Relative percentage error was used as an indication of the accuracy of the analytical model. Relative percentage error = 100% simulation estimate (analytical result simulation estimate) ⋅ − 28 The performance measures that were of interest include the average response time, average resolution time, agent utilizations, and the average number of emails in the system. The average resolution time is the average time an email spends in the system (customer and contact center) before eventual “resolution”. The parameters used for all the simulation experiments are presented in Table 5.1. The warmup period was determined by the application of Welch’s procedure (Welch 1983). Further details regarding the application of Welch’s procedure are contained in Appendix A3. Table 5.1: Simulation Parameters Number of Replications Warmup period (hours) Replication length (hours) 10 1,680 18,480 Table 5.2 presents the different levels of service time variability that were tested in all the scenarios. The details about the specific distributions used, and the procedure to calculate their parameters for the simulation model are given in Appendix A2. Table 5.2: SCV Levels and Corresponding Distributions SCV Distribution 0.25 4stage Erlang 1 Exponential 2.00 Hyperexponential 29 CHAPTER 6 A MULTICLASS OPEN QUEUEING NETWORK MODEL OF EMAIL CUSTOMER CONTACT CENTERS A multiclass open queueing network model was developed to model email contact centers. The nodes of the network represent customer service agents with the exception of one special delay node that models the elapsed time at the customer end. The routing probabilities as well as the processing time parameters could depend on email history. A novel historybased aggregation approach to model the dependence on email history was developed. The aggregation approach extends the popular parametric decomposition (PD) method for solving multiclass open queueing networks to more general situations. A discretetime Markov chain (DTMC) with an expanded state space was developed to model the nonMarkovian routing of a new email through the contact center until its eventual resolution. The analysis of this absorbing Markov chain allows the computation of the proportion of emails in an agent’s inbox that are new, previously processed by the same agent, or previously processed by another agent. Using these proportions, a new “historybased” aggregation step for each customer class was introduced. This step precedes the existing classbased aggregation step that extends the original PD method. The resulting queueing network model was solved using the RAQS software package that implements the PD method and its extensions. The accuracy and robustness of the 30 analytical model was demonstrated by comparing the analytical results with simulation estimates of performance measures for a variety of scenarios. The contact center considered was similar to the example that was the subject of an extensive simulation study in Greve et al. (2004). The model developed in this chapter does not consider the pooling of agents or service interruptions. These issues are addressed in Chapter 7. The remainder of this chapter is organized as follows. Section 6.1 gives a description of the contact center that was modeled. Section 6.2 describes some additional assumptions. Section 6.3 describes the modeling of the email contact center using a multiclass open queueing network. Section 6.4 presents approximate approaches to compute the weights for historybased aggregation. Section 6.5 presents the discretetime Markov chain model of the historybased email routing. The numerical experiments are presented in Section 6.6 and Section 6.7 presents a summary of the results and discussions. 6.1 Contact Center Description A contact center with multiple types of arriving emails is considered. Within each type, the emails received are identified, by software, as new or previously processed. If emails are new, they are routed with equal probability to one of the agents. If an email has been previously processed, then the agent who previously processed the email can be identified, and the email is routed to that agent. Alternatively, the email could also be routed to one of the agents randomly, just like a new email. Once the email is routed to an agent, the agent preprocesses it. Preprocessing involves reviewing the email type and its history. The history of the email can be any one of the following; the email can be brand new, processed by the same agent, or processed by a different agent. From this 31 information the agent determines whether to process the email or to forward the email to another agent. The time required to process an email is random and is influenced by both the email type and history. When the email response provided by an agent is sufficient to resolve the customer’s problem the email leaves the system permanently. If the email response is not enough to address the customer’s concerns, email returns to the system (as another email, e.g., a reply) after a random delay. This random delay represents the time for the customer to receive the agent’s response and send a reply. Without any loss of generality, we could assume for modeling purposes that “resolution” includes both the actual resolution of the customer’s problem and the customer’s decision to not pursue the problem resolution any further. The flowchart depicting the email handling logic is shown in Figure 6.1. 6.2 Assumptions • An individual agent processes emails in his/her inbox according to a first in first out (FIFO) queueing discipline. The FIFO discipline holds only for the emails in an agent’s inbox and not for the system. • For an unresolved problem that will be pursued by the customer, the email (response) enters the system after a random delay independent of the email processing history. • The preprocessing and processing times are independent random variables. 6.3 Modeling the EMail Contact Center Using an Open Queueing Network The situation explained in Section 6.1 was modeled as an open queueing network where the nodes represent the agents and customers represent emails. The open network has 32 (A +1) nodes where nodes 1 through A represent the customer service agents and node (A +1) represents a delay node. The delay node was used to model the time for the customer to receive a response and send a followup email. The different types of emails were modeled using different customer classes, each having their own arrival, routing, and service characteristics. For a given email type, the routing probabilities and the service time distributions depend on the history of the email. This dependence on email history is a feature that cannot be handled with the current extensions to the PD method. Hence, the basic idea behind the solution approach is as follows. If the history and classbased parameters were somehow aggregated into only classbased parameters, then Whitt’s (1983, 1994) method could be used to solve the multiclass open queueing network. The input for the PD method includes the number of nodes, the number of servers (one in this chapter) at each node, the SCVs of the external interarrival and service time distributions at each node, and the Markovian routing probability matrix. To perform the historybased aggregation within a customerclass, the following probability at each node or agent was needed. It is the probability that an email of type c currently at agent i was previously processed by agent k ( k = 0 represents a new email). These probabilities can also be thought of as “weights” to be used within the aggregation approach. A new technique was developed for computing these probabilities and it is presented in Section 6.5. In the remainder of this section the aggregation of the detailed parameters of the contact center model to yield the singleclass arrival, service, and routing parameters for solution using the PD method is discussed. The overall aggregation process is a twolevel technique, where the first level takes care of the 33 dependence on history within a class and the second level deals with the aggregation of classspecific information. IDENTIFY EMAIL TYPE USING SOFTWARE START RECEIVE EMAIL DIRECT EMAIL TO AN AGENT PREPROCESS EMAIL FORWARD EMAIL ? DELAY RESOLVED ? END PROCESS EMAIL YES NO YES NO Figure 6.1: EMail Handling Logic 34 A simple aggregation scheme was first developed to convert the class and nodespecific detailed routing probabilities and service time information into an approximately equivalent singleclass, nodespecific service time parameters and Markovian routing probabilities. The PD approach was then used to analyze the model. 6.3.1 Rate and SCV of the External Arrival Process at a Node First, we need to compute the rate and the SCV for the external arrival process at each node that represents an agent in the singleclass open queueing network model. The external arrivals correspond to the arrival of new emails. The delay node or node (A + 1) has no external arrivals because all previously processed emails (or customer replies) are considered to be internal arrivals as far as the open queueing network model is concerned. If new external emails of type c are routed to agent i with a fixed probability distribution, say, { (c), i 1,2, ..., A}, i α = then the classspecific and total external arrival rates at node i are given by (c) (c) (c) i i λ α λ ⋅ = ; Σ= = C c i i c 1 λ λ ( ) i =1, 2, ..., A The SCV calculations for the interarrival time for new, external emails at node i follow two steps. When a new email of type c enters the system, it is split according to the fixed probability distribution (c) i α . First, we obtain the SCV of the split arrival process of new type c emails at node i by assuming that the external arrival process is a renewal process. c2 (c) (c) c2 (c) 1 (c) oi i o i = α ⋅ + −α i =1, 2, ..., A At node i , the split arrival processes of new external emails of different types get merged. For the second step, we use results from Whitt (1983) to obtain the SCV of the (2) (3) 35 external arrivals at node i oi i hi i c2 =ψ ⋅ c2 + 1−ψ where Σ Σ = = ⋅ = C c i C c oi i hi c c c c c 1 1 2 2 ( ) ( ) ( ) λ λ i =1, 2, ..., A; = [1 + 4(1− )2 ⋅ ( −1)]−1; i i i ψ ρ χ 2 1 1 ( ) ( ) − = = ΣC c i i i c c λ λ χ ; and i ρ is the utilization of node i. It should be noted that the calculation of the SCV, 2 oi c , can be performed after i ρ is available from the solution of the traffic equations for the aggregate singleclass network. 6.3.2 Weights for Classbased Aggregation Whitt (1983) presented an approach to derive the parameters for an aggregate singleclass network in the case of a multiclass network with deterministic routes. We extend Whitt’s (1983) approach to a multiclass network with probabilistic routing. To compute the probabilities or weights for classbased aggregation, we need, (c), j γ the total (internal plus external) arrival rate of type c emails at node j. This can be obtained by solving traffic equations for a particular class. (c) j γ can be obtained by solving the following system of linear equations. ( ) ( ) ( ) ( ) , 1 1 c c c p c i j A j i i j j i ⋅ + = Σ+ ≠ = γ λ γ j = 1, 2, ..., A +1 For i, j =1, 2, ..., A; i ≠ j, the routing probability ( ) , p c i j from agent i to agent j for a type c email is given by ( ) ( , ) ( , ). , 0 , p c w c k p c k i j A k i j i Σ= = ⋅ For each class, the routing (4) (3) 36 probability from node i to the delay node (A +1) is obtained by aggregating the product of the probability that agent i processes the email and the probability that the problem is not resolved. Σ= + = ⋅ ⋅ − A k i A i i i p c w c k p c k q c k 0 , 1 ( ) ( , ) ( , ) (1 ( , )) i = 1, 2, ..., A The routing probability ( ) 1, p c A+ i depends on the routing strategy for previously processed emails. If previously processed emails (i.e., customer responses to agents’ emails) are routed to agent i with a fixed probability distribution {p i A} i , = 1, 2, ..., , then we have ( ) . A 1, i i p c = p + A special case of this strategy is to set all p s i ' to be the same. In this case, we have p 1/ A. i = If previously processed emails are routed to the agent that processed them, then the routing probability for a type c email from the delay node to agent i is equal to the probability that a type c email at the delay node came from agent i , i.e., ( ) ( , ). 1, 1 p c w c i A+ i A+ = Finally, we can compute the probabilities needed for the classbased aggregation as follows Σ= = C c i i i c c r c 1 ( ) ( ) ( ) γ γ i = 1, 2, ..., A +1 6.3.3 Markovian Routing Probabilities We calculate the Markovian routing probabilities for the singleclass open network by aggregating the classspecific probabilities computed in the previous section. The routing probabilities among the agent nodes are given by Σ= = ⋅ C c i j i i j p r c p c 1 , , ( ) ( ) i, j = 1, 2, ..., A; i ≠ j (5) (6) (7) 37 Similarly, the classindependent routing probability from agent node i to the delay node (A +1) is Σ= + + = ⋅ C c i A i i A p r c p c 1 , 1 , 1 ( ) ( ) i =1, 2, ..., A And the classindependent routing probability from the delay node (A +1) to agent node i is Σ= + + = ⋅ C c A i i A i p r c p c 1 1, 1, ( ) ( ) i = 1, 2, ..., A 6.3.4 Mean Service Time at a Node The overall service time is equal to the preprocessing time plus the actual processing time needed by the agent if the agent decides to process the email himself/herself. The overall service time at agent i for a new or previously processed type c email is given by T (c, k) P (c) Z (c, k) S (c, k) i i i i = + ⋅ i = 1, 2, ..., A; k =0,1,..., A The random variables P (c), i Z (c, k) i and S (c, k) i are assumed to be mutually independent. That is, the preprocessing time, the agent’s decision to process or forward the email, and the email processing time are all independent random variables. The mean service time at agent i for a new or previously processed type c email is given by E[T (c, k)] E[P (c)] E[Z (c, k)] E[S (c, k)] i i i i = + ⋅ i = 1, 2, ..., A; k =0,1,..., A Using the notation defined earlier we get t (c, k) (c) p (c, k) s (c, k) i i i i =θ + ⋅ i = 1, 2, ..., A; k =0,1,..., A (10) (11) (12) (8) (9) 38 Once again, the mean service time is calculated by using the twolevel aggregation method. The mean service time for a type c email at node/agent i is given by ( ) ( , ) ( , ) 0 t c w c k t c k i A k i i ⋅ =Σ= i = 1, 2, ..., A Next, the classspecific mean service times are aggregated to obtain the mean (overall) service time at node/agent i. Σ= = ⋅ C c i i i t r c t c 1 ( ) ( ) i = 1, 2, ..., A 6.3.5 Squared Coefficient of Variation of the Service Time Distribution at a Node For each class, the variance of the overall service time is first computed by using the fact that the effective processing time distribution is a mixture of processing time distributions for new and previously processed emails. The SCV of the overall service time is obtained by once again using the fact that it is a mixture of classspecific distributions. By using the property that the “second moment of a mixture of distributions is the mixture of the second moments (Whitt 1983)” the first step is to obtain the service time SCV at node/agent i for a type c email. ( ) ( ( ) 1) ( , ) 2 ( , ) ( 2 ( , ) 1) 0 2 2 + ⋅ ⋅ = + ⋅ Σ= t c c c w c k t c k c c k i i A k i i i i = 1, 2, ..., A where, c2 (c, k) i is obtained as follows T (c, k) P (c) Z (c, k) S (c, k) i i i i = + ⋅ i = 1, 2, ..., A Var(T (c, k)) Var(P (c) Z (c, k) S (c, k)) i i i i = + ⋅ Var(T (c, k)) Var(P (c)) Var(Z (c, k)) Var(S (c, k)) i i i i = + ⋅ Using the independence assumptions stated earlier, we have (13) (14) (15) 39 [ ( , )] ( ( , )) [ ( , )] ( ( , )) ( ( , )) ( ( )) ( ( , )) ( ( , )) E2 Z c k Var S c k E2 S c k Var Z c k Var T c k Var P c Var Z c k Var S c k i i i i i i i i ⋅ + ⋅ = + ⋅ + ( , ) ( , ) ( , ) ( , ) ( , ) [1 ( , )] ( ) ( ) ( , ) [1 ( , )] ( , ) ( , ) 2 2 2 , 2 2 2 , 2 2 , p c k c c k s c k s c k p c k p c k c c c p c k p c k c c k s c k i ser i i i i i prei i i i ser i i ⋅ ⋅ + ⋅ ⋅ − = ⋅θ + ⋅ − ⋅ ⋅ + ( , ) ( , ) [1 ( , )]} ( ) ( ) ( , ) ( , ) {[1 ( , )] ( , ) 2 , 2 , 2 2 2 , p c k c c k p c k c c c p c k s c k p c k c c k i ser i i pre i i i i i ser i ⋅ + − = ⋅θ + ⋅ ⋅ − ⋅ + ( ) ( ) ( , ) ( , ) { 2 ( , ) [1 ( , )]} , 2 2 2 , c c c p c k s c k c c k p c k pre i i i i ser i i = ⋅θ + ⋅ ⋅ + − Using the relation, Var(T (c, k)) c2 (c, k) t 2 (c, k) i i i = ⋅ have ( , ) ( ) ( ) ( , ) ( , ) { ( , ) [1 ( , )]} ( , ) 2 2 , 2 2 2 2 , t c k c c c p c k s c k c c k p c k c c k i pre i i i i ser i i i ⋅ + ⋅ ⋅ + − = θ By substituting c2 (c, k) i in equation [15] [1 ( , )] ) ( , )} ( ) ( ( ) 1) ( , ) { ( ) ( ) ( , ) ( , ).( ( , ) 2 2 , 2 2 0 2 , 2 2 p c k t c k t c c c w c k c c c p c k s c k c c k i i i i i ser i A k i i i pre i + − + ⋅ + ⋅ ⋅ = + ⋅ Σ= θ The second step is to aggregate the classspecific service time SCV, c2 (c) i to obtain the overall service time SCV, 2 i c at node/agent i is given by Σ Σ = = ⋅ ⋅ + ⋅ + = C c c C c c i i i i t c c c t c 1 1 2 2 2 2 ( ) ( ( ) 1) ( 1) λ λ i = 1, 2, ..., A 6.4 Approaches to Compute the Weights for HistoryBased Aggregation To demonstrate the importance of accurately modeling the dependence on the email history, two additional approximate methods to compute the weights w (c, k) i are presented here and included in the numerical experimentation. M1 is a naïve method, (17) (18) (16) 40 which involves nothing but a simple average. In method M1, w (c, k) i is equal to +1 1 A for all i, k and c , i.e., the probability that a type c email is new or previously processed by agent k remains the same irrespective of the email processing history. Similarly at the delay node (A+1), the probability that a type c email was previously processed by agent k , i.e., ( , ) 1 w c k A+ is equal to A 1 . This simple method was used in Chinnaswamy et al. (2004). M2 is a method with medium complexity, where w (c, k) i are calculated using aggregate resolution and forwarding probabilities. In method M2, the processing history of emails is taken into account in an approximate way for calculatingw (c, k). i As in method M1, the weights w (c, k) i are obtained for a type c new email and previously processed emails. Because of aggregation, the probability that email was previously processed by agent k is the same for k =1,2,..., A. Again because of aggregation, the probability that a type c email was previously processed by agent k, i.e., ( , ) 1 w c k A+ is equal to 1 . A M3 is the DTMCbased approach and it is presented in Section 6.5. In method M3, the complete history of an email is indirectly captured from its entry till its resolution. This information is used in calculating the weights w (c, k). i 6.4.1 Aggregation Method M2 In this section, the M2 method is explained for the two scenarios that will be considered later in numerical experimentation. Scenario 1 is the case when a previously processed email is equally likely to be routed to one of the agents. Scenario 2 is the case when a previously processed email is routed to the agent that processed it. Let p(c) represent 41 the aggregate forwarding probability and q(c) the aggregate resolution probability for class c. p(c) and q(c) are computed as follows. − − = ΣΣ ΣΣΣ = ≠ = = = ≠ = for Scenario 2 ( ,0) for Scenario 1 ( , ) ( ) 2 1 1 , 3 0 1 1 , A A p c A A p c k p c A i A i j j i j A k A i A i j j i j + = Σ ΣΣ = = = for Scenario 2 ( , ) for Scenario 1 ( , ) ( ) 1 2 0 1 A q c i A A q c k q c A i i A k A i i Figure 6.2 illustrates the flow of new and previouslyprocessed emails through the contact center from an aggregate point of view. Figure 6.2: Aggregation Method M2 (23) (24) (25) (26) Contact Center Resolved Previously processed, New, forwarded New, External λ(c) p(c) (1  p(c)).(1  q(c)) (1  p(c)). q(c) 42 Let β (c) represent the total (new plus previously processed) email arrival rate. From Figure 6.2 we can see that the following flow conversation relation needs to hold. β (c) = λ (c)+ β (c) ⋅{p(c)+(1− p(c)) ⋅ (1− q(c))} This yields (1 ( )) ( ) ( ) ( ) p c q c c c − ⋅ = λ β By focusing on the solid lines in Figure 6.2, we can see that Total arrival rate of new emails = (1 ( )) ( ) p c c − λ Total arrival rate of previously processed emails = (1 ( )) ( ) ( ) (1 ( )) p c q c c q c − ⋅ λ ⋅ − Hence, from (27) and (28), the proportion of new emails = q(c) And the proportion of previously processed emails = (1− q(c) ) The weights w (c, k) i are given by w c q c i A i ( ,0) = ( ) = 1,2,..., For Scenario 1, we have ( , ) (1 ( )) i , k 1,2,..., A; A w c k q c i = − = k A A w c k A ( , ) 1 1,2,..., 1 = = + For Scenario 2, we have = ≠ − = = = i k A i k q c i k A w c k i 0 , 1,2,..., ; 1 ( ) 1,2,..., ( , ) (29) (30) (27) (28) (32) (34) (33) (36) (31) (37) (35) 43 k A A w c k A ( , ) 1 1,2,..., 1 = = + 6.5 DiscreteTime Markov Chain Model of Historybased Email Routing One of the critical steps in the successful application of the solution approach is the historybased aggregation of routing and servicetime parameters within a customer class or email type. As explained in Section 6.3, the key element in this aggregation step is w (c, k), i the probability that a type c email currently at agent i was previously processed by agent k (k =0 represents a new email). This section presents the method M3, where the probability distribution {w (c, k), k 0,1, ..., A} i = is computed by analyzing a discretetime Markov chain (DTMC) model of the routing of an email during its entire lifecycle, i.e., from a new email to a series of agent responses and customer replies to eventual “resolution.” It is important to note that the dependence on history is limited to only the identity of the previous agent visited in the case of a previously processed email. The remainder of this section describes the definition and solution of this DTMC model. The statespace of the DTMC can be divided into internal and boundary states. The internal states can be described by the 3tuple (k, i, j) where k describes the email history; k = 0 represents a new email and 1 ≤ k ≤ A represents an email previously processed by agent k . i denotes the current location of the email, i.e., with agent i ; i = 1, 2, ..., A. 44 j denotes the current agent’s decision regarding the processing of the email; if agent i decides to process the email then j = i else agent i forwards the email to agent j ( j ≠ i) ; j = 1, 2, ..., A. The boundary states are represented by the set { , , , ..., , }, 1 2 N D D D R A where N represents the state in which all new emails begin their journey. R represents the state in which all emails end their journey. A D , D , ...,D 1 2 represents the state in which customer receives an agent’s response and sends a reply. The entire statespace of the DTMC is {N} {(k, i, j); k 0,1, 2, ..., A, i 1, 2, ..., A, j 1, 2, ..., A,} {D ; i 1, 2, ..., A} {R}. i ∪ = = = ∪ = ∪ The size of the state space is equal to (A + 1) ⋅ A⋅ A + A + 2 or A3 + A2 + A + 2. A DTMC with (A3 + A2 + A + 2) states for each customer class or email type was constructed and the possible state transitions were defined in order to construct the onestep transition probability matrix for the DTMC. First, transitions within internal states are considered. Such transitions occur only as a result of one agent forwarding an email to another. The recipient can forward it again or decide to process it. ∀(k, i, j) where j ≠ i ( ) ( , , ) ,( , * , * ) P c k i j k i j = = = = = ≠ 0 otherwise ( , ) ; ( , ) ; 1,2,..., ; * * * * * , * p c k i j j j p c k i j j A j j j j j (38) 45 Next, transitions from the internal states to the boundary states are considered. These occur when an agent decides to process an email. If the email is resolved, then the transition is to the resolved state R ; otherwise it is to the delay node i D . ∀(k, i, j) where j = i ( ) ( , ) ( , , ) , P c q c k k i i R i = ( ) 1 ( , ) ( , , ) , P c q c k k i i Di i = − P c j A k i i k i j ( ) 0 ; * 1, 2, ..., ( , , ) , ( , , * ) = = Next, the transitions from the boundary states to the internal states are considered. (a) From state N to an internal state ( ) , ( , , ) P c N k i j = ≠ = ⋅ = = = ⋅ = ≠ = k i j A c p c k i j A j i k c p c k i j A j i k i i i i j 0 0; , 1,2,..., ( ) ( , ) , 1,2,..., ; ; 0 ( ) ( , ) , 1,2,..., ; ; 0 , α α (b) From state D (i 1,2,..., A) i = to an internal state. (i) Previously processed emails are routed to agent i* with a fixed probability distribution { , * 1, 2,..., } * p i A i = independent of processing history. ( ) , ( , * , ) P c Di k i j ∗ = ⋅ = = = ⋅ = = ≠ 0 otherwise ( , ) ; , 1, 2,..., ; ( , ) ; , 1, 2,..., ; * * * * * * * * , * * * * * p p c k k i i j A j i p p c k k i i j A j i i i i i j (ii) Previously processed emails are routed to the agent that processed them. ( ) , ( , * , * ) P c Di k i j = = = = = = = ≠ 0 otherwise ( , ) ; ; ( , ) ; ; 1, 2,..., ; * * * * * * , * * * p c k k i i i j i p c k k i i i j A j i i i j (40) (39) (41) (42) (44) (43) 46 Finally the transitions within the boundary states are considered. The only possible transition is R,R = 1 P . All other transitions probabilities from one boundary state to the same state or any other boundary state is zero. Using the transition probabilities, a (n' × n' ) onestep transition probability matrix P, where n' = A3 + A2 + A + 2 , was defined. Because state R is an absorbing state, the DTMC represented by P is a reducible chain with all states except state R being transient. Next, the focus is on the analysis of this absorbing Markov chain. 6.5.1 Analysis of the Absorbing Markov Chain The DTMC defined in the previous section consists of an absorbing state and (A3 + A2 + A +1) transient states. The onestep transition probability matrix, P, can be rearranged in such a way that the absorbing state is the first state, without any loss of generality. States are arranged in the following way  resolution node R , followed by A delay nodes numbered from A D ,D , ..., D 1 2 , followed by the new node N and finally, the internal nodes (k,i, j). U 0 1 row M Q A3 + A2 + A +1 rows 1 column A3 + A2 + A +1 columns Figure 6.3: OneStep Transition Probability Matrix P = 47 By analyzing this absorbing DTMC, the average number of visits an email makes to each node before getting resolved or absorbed can be computed. The steps outlined in Ramakumar (1993) were followed to obtain the fundamental matrix F associated with the absorbing Markov chain. The fundamental matrix F is given by F = [I  Q]1 The row of the Fundamental matrix, F that corresponds to the state N gives the average number of visits an email makes to the all other states before absorption. The average number of visits an email that was last processed by agent k (k =0 is a new email) makes to agent i is given by Σ= = A j i N k i j v c k F c 1 , ( , , ) ( , ) ( ) The probability that a type c email received by agent i was previously processed by agent k (k =0 is a new email) is given by Σ= = A k i i i v c k v c k w c k 0 ( , ) ( , ) ( , ) i = 1, 2, ..., A; k =0,1, ..., A; The above results can be easily derived by observing that the arrival rate of type c emails with a processing history k at agent i is proportional to v (c, k). i Similarly, the probability that a type c email received at the delay node (A +1)was previously processed by agent k is given by k A c k c k w c k A j D D A j k 1, 2,..., ( , ) ( , ) ( , ) 1 1 = = Σ= + ν ν where ( ) ( ) , , c F c Dj j N Dj ν = 6.5.2 Performance Evaluation of the EMail Contact Center Figure 6.4 explain the stepbystep procedure followed to obtain the performance measures of an inbound email contact center. It was assumed that the forwarding (45) (46) (47) 48 probabilities, resolution probabilities and service time parameters for a particular type email could be obtained from the available email contact center data. It was assumed that these parameters could be dependent on email history, i.e., the last agent, k who previously processed the email, (k =0 for new emails). The external arrival rates and SCVs for different email types, the forwarding and resolution probabilities, number of agents along with the mean service time and servicetime SCV for each agent were the inputs for the analytical model. These inputs known as parameters were used for the entire aggregation procedure. For a particular email type the onestep, historyembedded transition probability matrix was constructed based on possible transitions made by an email between the agents’ inboxes and the delay nodes representing customers. The construction of the above matrix was based on the number of agents in the contact center and historyembedded resolution and forwarding probabilities of an email. A truncated portion of the onestep transition probability matrix, i.e., Q then was inverted using MATLAB 7.0.1 (Lipsman 2001). The truncated matrix was based on the number of absorbing nodes (one in this case). The inverted matrix gives the average number of visits an email makes to a node before absorption or problem “resolution.” In particular, the row corresponding to the new email node gives the average number of visits a new email makes to agents’ inboxes and to the customer inbox (modeled as delay node) before getting resolved. These visits were then converted into probabilities or weights of new and previously processed emails as seen by an agent for a known email type. These weights were used in the historybased aggregation procedure to obtain parameters that are based on only the class or email type information. For the classbased aggregation, the external arrival rates and parameters obtained from the historybased 49 aggregation were used. The classbased aggregation was done according to the method outlined in Whitt (1983) in order to obtain the routing matrix, means and SCVs of service times, and interarrival time SCVs at each agent. These were then fed into RAQS software package to obtain the performance measures for the email contact center. Many steps in the overall procedure, including calculation of the onestep transition probability matrix, calculation of weights after solving the DTMC, historybased aggregation, and classbased aggregation were coded in EXCEL spread sheets. 50 EMPIRICAL CONTACT CENTER DATA pij(c, k), qi(c, k), mean and SCVs of preprocessing and processing times, number of agents, number of email types, external arrival rates and SCVs pij(c, k), qi(c, k), number of agents wik(c) pij(c), qi(c), ti(c), ci 2(c), coi 2(c) pij, ti, ci 2, coi 2 INPUT TO THE PD METHOD Number of nodes, mean and SCV of service time at each node, external arrival rates and SCVs, probability routing matrix CONTACT CENTER PERFORMANCE MEASURES HISTORYBASED AGGREGATION OBTAINED BY SOLVING THE DTMC CLASSBASED AGGREGATION SOLVED USING RAQS PARAMETRIC SOFTWARE DECOMPOSITION METHOD Historybased aggregation for each email type Classbased aggregation for each email type Figure 6.4: Contact Center Performance Evaluation 51 6.6 Numerical Experiments One of the important tasks in the numerical evaluation of the accuracy and robustness of the approach is to construct a contact center example that is representative of realworld contact center operations. The parameter selection approach was based on email contact center characteristics that are available in the public domain. Details are described in the next section. 6.6.1 Email Contact Center Characteristics Survey data available at “BenchmarkPortal.com” was used to gain an initial understanding of a typical email response center. Each month, “BenchmarkPortal.com” sends 1,000 surveys to randomly selected customer response centers, and typically receives a 15 to 25 % response rate. In one such survey, 53% of surveyed contact centers indicated that 80100% of email contacts were resolved with the first response (Benchmark Portal 2004b). From this statistic an estimate for the resolution probability for a typical email agent was obtained. Also, 34.62% of surveyed contact centers indicated that their average email response time was less than 6 hours. Another 15.38% indicated an average response time of 6 to 12 hours, and 35.38% indicated an average response time of 12 to 24 hours (Benchmark Portal 2004c). Also, the capacity of a dedicated email associate is comparable to that of a telephone agent (Benchmark Portal 2004a). Given an eighthour day, this translates to between 2.4 and 9.6 minutes of an email agent’s time being spent on each email response. Based on the above information and the contact center example analyzed in Greve et al. (2004) and Chinnaswamy et al. (2004), the parameter values for a small contact center were defined. 52 6.6.2 Email Contact Center Example A contact center with three agents that handles two types of emails was considered. We have A = 3 agents and C = 2 . Although the analytical approach can handle general arrival processes, Poisson arrivals were assumed for this example. The arrival rates for the two types of emails were 6.0 / hr 1 λ = and 4.5 / hr. 2 λ = A new email of either type was assumed to be equally likely to be routed to one of the agents. Two scenarios were defined to model two different strategies with regard to the handling of previously processed emails. In Scenario 1, a previouslyprocessed email was routed to any one of the agents with equal probability. In Scenario 2, a previouslyprocessed email was routed to the agent that processed it. Hence, we have ( ) ( ) 0.33; ( ) 0.34 1 2 3 α c =α c = α c = for c = 1,2. The preprocessing time distribution for both email types at all three agents was assumed to be the uniform distribution. Hence, P (c) ~ i Uniform (0.01, 0.50) for i = 1,2,3 ; c = 1,2.With regard to forwarding probabilities, sample values were generated using uniform (0, 0.25) for the two email types. While assigning the sampled values, it was assumed that an agent is more likely to forward an email previously processed by another agent. The complete set of forwarding probabilities is presented in Table 6.2 for Scenario 1 and Table 6.7 for Scenario 2. In the case of resolution probabilities, samples from the following uniform distributions were used. • Type 1, New  Uniform (0.60, 0.90) • Type 1, Previously processed  Uniform (0.75, 0.95) • Type 2, New  Uniform (0.50, 0.80) • Type 2, Previously processed  Uniform (0.70, 0.90) 53 While assigning the sampled probabilities, it was assumed that an agent was more likely to resolve an email previously processed by them. The full set of resolution probabilities are summarized in Table 6.3 for Scenario 1 and Table 6.8 for Scenario 2. To test the robustness of the approach developed, experiments with three processingtime distribution were chosen; Erlang (SCV =0.25) to represent low variability situations, exponential (SCV =1.00) to represent medium variability situations, and Hyperexponential (SCV = 2.00) to represent high variability situations. To capture the dependence on the history and class, samples for the mean processing times were drawn from the following uniform distributions. • Type 1, New  Uniform (0.10, 0.20) • Type 1, Previously processed  Uniform (0.05, 0.15) • Type 2, New  Uniform (0.15, 0.25) • Type 2, Previously processed  Uniform (0.10, 0.20) While assigning the sampled mean processing times, it was assumed that the mean would be lower for emails previously processed by the same agent. The complete set of mean processing times can be viewed in Table 6.1 for Scenario1 and Table 6.6 for Scenario 2. Finally, the delay time was assumed to be exponentially distributed with a mean of 4 hours, i.e., Di ~ Exponential (4) for i = 1,2,3. With regard to the routing of previously processed emails, two scenarios were considered as described earlier. 6.6.3 Probabilities for HistoryBased Aggregation obtained from the DTMC In this section, the probabilities or weights used for historybased aggregation obtained from the fundamental matrix are presented for both Scenarios 1 and 2. Tables 6.4 and 6.5 give the average number of visits and probabilities for Scenario 1 obtained using the 54 analytical method M3 for Type 1 and 2 emails, respectively. Tables 6.9 and 6.10 give the average number of visits and probabilities for Scenario 2 obtained using the analytical method M3 for Type 1 and 2 emails, respectively. The reader is referred to Appendix A1; Tables A1.1 through A1.18 for complete details related to the fundamental matrix F. Table 6.1: Scenario 1  Mean Processing Times ( ) , s c i k (c,k) c = 1 c = 2 i k=0 k=1 k=2 k=3 k=0 k=1 k=2 k=3 1 0.15 0.11 0.16 0.15 0.18 0.06 0.13 0.10 2 0.19 0.13 0.15 0.14 0.20 0.11 0.07 0.14 3 0.14 0.20 0.18 0.13 0.17 0.08 0.13 0.07 Table 6.2: Scenario 1  Forwarding Probabilities ( , ) , p c k i j (c,k) c = 1 c = 2 (i,j) k=0 k=1 k=2 k=3 k=0 k=1 k=2 k=3 (1,2) 0.01 0.02 0.08 0.19 0.06 0.03 0.03 0.12 (1,3) 0.14 0.02 0.24 0.12 0.05 0.06 0.14 0.13 (2,1) 0.05 0.24 0.01 0.14 0.13 0.13 0.00 0.15 (2,3) 0.03 0.10 0.06 0.25 0.10 0.09 0.03 0.13 (3,1) 0.08 0.24 0.08 0.01 0.11 0.15 0.14 0.07 (3,2) 0.10 0.09 0.10 0.07 0.02 0.13 0.11 0.01 Table 6.3: Scenario 1  Resolution Probabilities q (c, k) i (c,k) c = 1 c = 2 i k=0 k=1 k=2 k=3 k=0 k=1 k=2 k=3 1 0.71 0.88 0.79 0.74 0.75 0.94 0.76 0.78 2 0.74 0.85 0.85 0.85 0.78 0.89 0.95 0.81 3 0.72 0.87 0.77 0.88 0.79 0.78 0.83 0.86 55 Table 6.4: Scenario 1, Type 1 Emails  Aggregation Probabilities for Method M3 011 0.32392000 021 0.01871300 031 0.03236600 012 0.00381080 022 0.34433000 032 0.04045800 013 0.05335100 023 0.01122800 033 0.33175000 v1,0(1) 0.38108180 v2,0(1) 0.37427100 v3,0(1) 0.40457400 111 0.05622500 121 0.01027700 131 0.01062300 112 0.00117140 122 0.02826300 132 0.00398360 113 0.00117140 123 0.00428230 133 0.02965600 v1,1(1) 0.05856780 v2,1(1) 0.04282230 v3,1(1) 0.04426260 211 0.02622600 221 0.00042175 231 0.00377580 212 0.00308540 222 0.03922300 232 0.00471980 213 0.00925610 223 0.00253050 233 0.03870200 v1,2(1) 0.03856750 v2,2(1) 0.04217525 v3,2(1) 0.04719760 311 0.03059200 321 0.00688920 331 0.00055631 312 0.00842390 322 0.03001700 332 0.00389420 313 0.00532030 323 0.01230200 333 0.05118100 v1,3(1) 0.04433620 v2,3(1) 0.04920820 v3,3(1) 0.05563151 overall sum 0.52255330 overall sum 0.50847675 overall sum 0.55166571 w1,0(1) 0.72926877 w2,0(1) 0.73606315 w3,0(1) 0.73336804 w1,1(1) 0.11208005 w2,1(1) 0.08421683 w3,1(1) 0.08023446 w1,2(1) 0.07380587 w2,2(1) 0.08294430 w3,2(1) 0.08555471 w1,3(1) 0.08484532 w2,3(1) 0.09677571 w3,3(1) 0.10084279 Email Type 1  Weights from the Fundamental Matrix Table 6.5: Scenario 1, Type 2 Emails  Aggregation Probabilities for Method M3 011 0.37462000 021 0.04721600 031 0.04371000 012 0.02525600 022 0.27967000 032 0.00794730 013 0.02104600 023 0.03632000 033 0.34571000 v1,0(2) 0.42092200 v2,0(2) 0.36320600 v3,0(2) 0.39736730 111 0.04334700 121 0.00554470 131 0.00650410 112 0.00142900 122 0.03326800 132 0.00563690 113 0.00285810 123 0.00383860 133 0.03122000 v1,1(2) 0.04763410 v2,1(2) 0.04265130 v3,1(2) 0.04336100 211 0.02284600 221 0.00000000 231 0.00404220 212 0.00082576 222 0.02666000 232 0.00317600 213 0.00385350 223 0.00082454 233 0.02165500 v1,2(2) 0.02752526 v2,2(2) 0.02748454 v3,2(2) 0.02887320 311 0.02770400 321 0.00508950 331 0.00274400 312 0.00443260 322 0.02442900 332 0.00039200 313 0.00480200 323 0.00441090 333 0.03606400 v1,3(2) 0.03693860 v2,3(2) 0.03392940 v3,3(2) 0.03920000 overall sum 0.53301996 overall sum 0.46727124 overall sum 0.50880150 w1,0(2) 0.78969275 w2,0(2) 0.77729158 w3,0(2) 0.78098689 w1,1(2) 0.08936645 w2,1(2) 0.09127739 w3,1(2) 0.08522184 w1,2(2) 0.05164020 w2,2(2) 0.05881924 w3,2(2) 0.05674747 w1,3(2) 0.06930059 w2,3(2) 0.07261179 w3,3(2) 0.07704380 Email Type 2  Weights from the Fundamental Matrix 56 Table 6.6: Scenario 2  Mean Processing Times ( ) , s c i k (c,k) c = 1 c = 2 i k=0 k=1 k=2 k=3 k=0 k=1 k=2 k=3 1 0.15 0.11 0.00 0.00 0.18 0.06 0.00 0.00 2 0.19 0.00 0.15 0.00 0.20 0.00 0.07 0.00 3 0.14 0.00 0.00 0.13 0.17 0.00 0.00 0.07 Table 6.7: Scenario 2  Forwarding Probabilities ( , ) , p c k i j (c,k) c = 1 c = 2 (i,j) k=0 k=1 k=2 k=3 k=0 k=1 k=2 k=3 (1,2) 0.01 0.00 0.00 0.00 0.06 0.00 0.00 0.00 (1,3) 0.14 0.00 0.00 0.00 0.05 0.00 0.00 0.00 (2,1) 0.05 0.00 0.00 0.00 0.13 0.00 0.00 0.00 (2,3) 0.03 0.00 0.00 0.00 0.10 0.00 0.00 0.00 (3,1) 0.08 0.00 0.00 0.00 0.11 0.00 0.00 0.00 (3,2) 0.10 0.00 0.00 0.00 0.02 0.00 0.00 0.00 Table 6.8: Scenario 2  Resolution Probabilities q (c, k) i (c,k) c = 1 c = 2 i k=0 k=1 k=2 k=3 k=0 k=1 k=2 k=3 1 0.71 0.88 0.00 0.00 0.75 0.94 0.00 0.00 2 0.74 0.00 0.85 0.00 0.78 0.00 0.95 0.00 3 0.72 0.00 0.00 0.88 0.79 0.00 0.00 0.86 57 Table 6.9: Scenario 2, Type 1 Emails  Aggregation Probabilities for Method M3 011 0.32392000 021 0.01871300 031 0.03236600 012 0.00381080 022 0.34433000 032 0.04045800 013 0.05335100 023 0.01122800 033 0.33175000 v1,0(1) 0.38108180 v2,0(1) 0.37427100 v3,0(1) 0.40457400 111 0.10675000 121 0.00000000 131 0.00000000 112 0.00000000 122 0.00000000 132 0.00000000 113 0.00000000 123 0.00000000 133 0.00000000 v1,1(1) 0.10675000 v2,1(1) 0.00000000 v3,1(1) 0.00000000 211 0.00000000 221 0.00000000 231 0.00000000 212 0.00000000 222 0.10532000 232 0.00000000 213 0.00000000 223 0.00000000 233 0.00000000 v1,2(1) 0.00000000 v2,2(1) 0.10532000 v3,2(1) 0.00000000 311 0.00000000 321 0.00000000 331 0.00000000 312 0.00000000 322 0.00000000 332 0.00000000 313 0.00000000 323 0.00000000 333 0.10556000 sum 0.00000000 0.00000000 0.10556000 overall sum 0.48783180 overall sum 0.47959100 overall sum 0.51013400 w1,0(1) 0.78117458 w2,0(1) 0.78039621 w3,0(1) 0.79307398 w1,1(1) 0.21882542 w2,1(1) 0.00000000 w3,1(1) 0.00000000 w1,2(1) 0.00000000 w2,2(1) 0.21960379 w3,2(1) 0.00000000 w1,3(1) 0.00000000 w2,3(1) 0.00000000 w3,3(1) 0.20692602 Email Type 1  Weights from solving the DiscreteTime Markov Chain Table 6.10: Scenario 2, Type 2 Emails  Aggregation Probabilities for Method M3 011 0.37462000 021 0.04721600 031 0.04371000 012 0.02525600 022 0.27967000 032 0.00794730 013 0.02104600 023 0.03632000 033 0.34571000 v1,0(2) 0.42092200 v2,0(2) 0.36320600 v3,0(2) 0.39736730 111 0.09963400 121 0.00000000 131 0.00000000 112 0.00000000 122 0.00000000 132 0.00000000 113 0.00000000 123 0.00000000 133 0.00000000 v1,1(2) 0.09963400 v2,1(2) 0.00000000 v3,1(2) 0.00000000 211 0.00000000 221 0.00000000 231 0.00000000 212 0.00000000 222 0.06476500 232 0.00000000 213 0.00000000 223 0.00000000 233 0.00000000 v1,2(2) 0.00000000 v2,2(2) 0.06476500 v3,2(2) 0.00000000 311 0.00000000 321 0.00000000 331 0.00000000 312 0.00000000 322 0.00000000 332 0.00000000 313 0.00000000 323 0.00000000 333 0.08441700 v1,3(2) 0.00000000 v2,3(2) 0.00000000 v3,3(2) 0.08441700 overall sum 0.52055600 overall sum 0.42797100 overall sum 0.48178430 w1,0(2) 0.80860080 w2,0(2) 0.84866965 w3,0(2) 0.82478258 w1,1(2) 0.19139920 w2,1(2) 0.00000000 w3,1(2) 0.00000000 w1,2(2) 0.00000000 w2,2(2) 0.15133035 w3,2(2) 0.00000000 w1,3(2) 0.00000000 w2,3(2) 0.00000000 w3,3(2) 0.17521742 Email Type 2  Weights from solving the DiscreteTime Markov Chain 58 The next section presents the performance evaluation of the contact center example using both analytical and simulation models and examines the accuracy of the analytical results. 6.7 Results and Discussions The aggregated singleclass open queueing network model was solved using the RAQS software package (Kamath et al. 1995). RAQS is a software package for analyzing general queueing network models using the PD method (http://www.okstate.edu/cocim/raqs/). Simulation estimates represent averages over ten independent replications. Each replication was simulated for 18,480 hours of operation with a warm up of 1,680 hours. The analytical and simulation results are shown for three processing time distributions  Erlang (SCV =0.25), exponential (SCV =1.00), and Hyperexponential (SCV = 2.00). 6.7.1 Results The analytical and simulation results are presented in Tables 6.11 and 6.12 and in Figures 6.6 through 6.10 for Scenario 1, where a previously processed email is routed to any one of the agents with equal probability. Tables 6.13 and 6.14 and Figures 6.11 through 6.15 shows the results for Scenario 2, where a previously processed email is routed to the agent that processed it. The quantity in parentheses below a simulation estimate is the halfwidth of the 95% confidence interval. The quantity in parentheses below an analytical value is the relative percentage error. The columns labeled M1 contain analytical results obtained using the analytical method M1, i.e., the naïve method based on a simple average. The columns labeled M2 contain analytical results obtained using the analytical method M2, i.e., the approximate method based on aggregate forwarding 59 and resolution probabilities. Finally, the column labeled M3 contain analytical results obtained using the analytical method M3, i.e., the DTMCbased method for computing the probabilities /weights for historybased aggregation. Table 6.11: Scenario 1  Nodelevel Performance Measures M1 M2 M3 Sim M1 M2 M3 Sim M1 M2 M3 Sim 0.735 0.910 0.890 0.887 0.264 1.091 0.866 0.897 1.423 6.080 4.800 4.962 (17.14%) (+2.59%) (+0.34%) (±0.003) (70.57%) (+21.63%) (3.46%) (±0.045) (71.32%) (+22.53%) (3.26%) (±0.258) Erlang 0.728 0.951 0.923 0.921 0.276 2.438 1.476 1.479 1.400 12.659 7.621 7.618 (20.96%) (+3.26%) (0.22%) (±0.003) (81.34%) (+64.84%) (0.20%) (±0.073) (81.62%) (+66.17%) (+0.04%) (±0.391) 0.793 0.876 0.865 0.863 0.393 0.740 0.669 0.689 2.169 4.176 3.752 3.861 (8.11%) (+1.51%) (+0.23%) (±0.002) (42.96%) (+7.40%) (2.90%) (±0.018) (43.82%) (+8.16%) (2.82%) (±0.108) 0.735 0.910 0.890 0.888 0.378 1.591 1.261 1.237 2.043 8.870 6.992 6.850 (17.23%) (+2.48%) (+0.22%) (±0.004) (69.44%) (+28.62%) (+1.94%) (±0.034) (70.18%) (+29.49%) (+2.07%) (±0.196) Exponential 0.728 0.951 0.923 0.921 0.399 3.588 2.170 2.147 2.023 18.631 11.201 11.082 (20.96%) (+3.26%) (+0.22%) (±0.004) (81.42%) (+67.12%) (+1.07%) (±0.162) (81.75%) (+68.12%) (+1.07%) (±0.872) 0.793 0.876 0.865 0.864 0.569 1.074 0.971 0.982 3.139 6.065 5.447 5.507 (8.22%) (+1.39%) (+0.12%) (±0.003) (42.06%) (+9.37%) (1.12%) (±0.039) (43.00%) (+10.13%) (1.09%) (±0.224) 0.735 0.910 0.890 0.892 0.531 2.258 1.788 1.831 2.868 12.587 9.913 10.161 (17.60%) (+2.02%) (0.22%) (±0.004) (71.00%) (+23.32%) (2.35%) (±0.104) (71.77%) (+23.88%) (2.44%) (±0.586) Hyperexponential 0.728 0.951 0.923 0.924 0.563 5.122 3.094 3.061 2.854 26.593 15.973 15.809 (21.21%) (+2.92%) (0.11%) (±0.003) (81.61%) (+67.33%) (+1.08%) (±0.223) (81.95%) (+68.21%) (+1.04%) (±1.155) 0.793 0.876 0.865 0.862 0.803 1.520 1.374 1.317 4.430 8.583 7.706 7.389 (8.00%) (+1.62%) (+0.35%) (±0.004) (39.03%) (+15.41%) (+4.33%) (±0.059) (40.05%) (+16.16%) (+4.29%) (±0.332) Processing Time Distribution Node (Agent) Utilization Average Queueing Delay (hours) Average Number in Queue 1 2 3 1 2 3 1 2 3 60 Table 6.12: Scenario 1  Systemlevel Performance Measures M1 M2 M3 Sim M1 M2 M3 Sim M1 M2 M3 Sim 0.562 1.854 1.375 1.398 1.605 3.713 3.017 3.030 16.858 38.990 31.681 31.803 (59.80%) (+32.62%) (1.64%) (±0.029) (47.03%) (+22.54%) (0.43%) (±0.040) (46.99%) (+22.60%) (0.38%) (±0.029) 0.733 2.624 1.920 1.907 1.816 4.728 3.728 3.698 19.070 49.641 39.147 38.861 (61.56%) (+37.60%) (+0.68%) (±0.077) (50.89%) (+27.85%) (+0.81%) (±0.098) (50.93%) (+27.74%) (+0.74%) (±1.097) 0.962 3.650 2.646 2.630 2.097 6.080 4.676 4.646 22.017 63.838 49.099 48.826 (63.42%) (+38.78%) (+0.61%) (±0.106) (54.86%) (+30.87%) (+0.65%) (±0.139) (54.91%) (+30.75%) (+0.56%) (±0.106) Exponential Hyperexponential Processing Time Distribution Average Response Time (hours) Average Resolution Time (hours) Erlang Average Number of Emails in the System 61 62 Agent Utilization (Erlang Processing Times) 0.000 0.200 0.400 0.600 0.800 1.000 1 2 3 Agent Number Utilization (%) M1 M2 M3 Sim Figure 6.6: Scenario 1  Agent Utilization  Erlang Processing Time Average Response Time 0.000 0.500 1.000 1.500 2.000 2.500 3.000 3.500 4.000 Erlang Exponential Hyperexponential Processing Time Distribution Average Response Time M1 M2 M3 Sim Figure 6.7: Scenario 1  Average Response Time 63 Average Resolution Time 0.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 Erlang Exponential Hyperexponential Processing Time Distribution Average Resolution Time M1 M2 M3 Sim Figure 6.8: Scenario 1  Average Resolution Time Average Number of Emails in the System 0 10 20 30 40 50 60 70 Erlang Exponential Hyperexponential Processing Time Distribution Average Number of Emails in the System M1 M2 M3 Sim Figure 6.9: Scenario 1  Average Number of Emails in the System Table 6.13: Scenario 2  Nodelevel Performance Measures M1 M2 M3 Sim M1 M2 M3 Sim M1 M2 M3 Sim 1 0.670 0.864 0.853 0.853 0.195 0.687 0.617 0.654 0.905 3.610 3.261 3.460 (21.45%) (+1.29%) (0.00%) (±0.003) (70.18%) (+5.05%) (5.66%) (±0.024) (73.84%) (+4.33%) (5.75%) (±0.133) Erlang 2 0.747 0.910 0.906 0.905 0.335 1.269 1.202 1.260 1.491 6.216 5.786 6.065 (17.46%) (+0.55%) (+0.11%) (±0.002) (73.41%) (+0.71%) (4.60%) (±0.053) (75.42%) (+2.49%) (4.60%) (±0.263) 3 0.684 0.822 0.811 0.808 0.206 0.476 0.437 0.440 0.964 2.495 2.292 2.230 (15.35%) (+1.73%) (+0.37%) (±0.003) (53.18%) (+8.18%) (0.68%) (±0.014) (56.77%) (+11.88%) (+2.78%) (±0.079) 1 0.670 0.864 0.853 0.853 0.279 0.999 0.895 0.894 1.295 5.248 4.731 4.720 (21.45%) (+1.29%) (0.00%) (±0.003) (68.79%) (+11.74%) (+0.11%) (±0.037) (72.56%) (+11.19%) (+0.23%) (±0.210) Exponential 2 0.747 0.910 0.906 0.903 0.488 1.865 1.765 1.760 2.168 9.133 8.499 8.463 (17.28%) (+0.77%) (+0.33%) (±0.004) (72.27%) (+5.97%) (+0.28%) (±0.162) (74.38%) (+7.92%) (+0.42%) (±0.794) 3 0.684 0.822 0.811 0.808 0.294 0.687 0.631 0.631 1.377 3.606 3.308 3.300 (15.35%) (+1.73%) (+0.37%) (±0.003) (53.41%) (+8.87%) (0.00%) (±0.018) (58.27%) (+9.27%) (+0.24%) (±0.100) 1 0.670 0.864 0.853 0.850 0.391 1.415 1.266 1.202 1.815 7.432 6.689 6.338 (21.18%) (+1.65%) (+0.35%) (±0.005) (67.47%) (+17.72%) (+5.32%) (±0.076) (71.36%) (+17.26%) (+5.54%) (±0.421) Hyperexponential 2 0.747 0.910 0.906 0.904 0.691 2.659 2.516 2.415 3.071 13.021 12.115 11.601 (17.37%) (+0.66%) (+0.22%) (±0
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Title  Aggregation Approaches for Incorporating Email Processing History in Queueing Models of Customer Contact Centers 
Date  20050501 
Author  Chinnaswamy, Mohan Raj 
Department  Industrial Engineering & Management 
Document Type  
Full Text Type  Open Access 
Abstract  The operation of a customer contact center where the interface between the agents' and the customer happens through email was modeled using a multiclass open queueing network model. A novel approach to model the dependence of processing times and routing on email history called the historybased aggregation method was developed. This aggregation method extends the popular parametric decomposition (PD) method for solving multiclass open queueing networks to more general situations. A discretetime Markov chain (DTMC) with an expanded state space was developed to model the nonMarkovian routing of a new email through the contact center until its eventual resolution. The analysis of this absorbing Markov chain led to the computation of the proportion of emails in an agent's inbox that are new, previously processed by the same agent, or previously processed by another agent. Using these proportions, a new "historybased" aggregation step for each customer class in the PD method was introduced. This step precedes the existing classbased aggregation step in the PD method. The resulting queueing network model was solved using the RAQS software package. The accuracy and robustness of the analytical approach was demonstrated by comparing the analytical results with simulation estimates of performance measures for a variety of scenarios. Typical contact center situations like grouping of agents and server interruptions were also modeled within the above approach. The DTMCbased analytical method showed excellent prediction capability across the various cased examined. The relative percentage error in utilization was less than 1% in all cases and less than 17% for other performance measures. This work has laid the foundation for the development of rapid performance analysis tools for customer contact centers which can be effectively used for improving customer contact center operations. The historybased aggregation method represents a significant extension to the PD method for solving queueing network models. 
Note  Thesis 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  AGGREGATION APPROACHES FOR INCORPORATING EMAIL PROCESSING HISTORY IN QUEUEING MODELS OF CUSTOMER CONTACT CENTERS By MOHAN RAJ CHINNASWAMY Bachelor of Engineering University of Madras Madras, India 2001 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE May, 2005 ii AGGREGATION APPROACHES FOR INCORPORATING EMAIL PROCESSING HISTORY IN QUEUEING MODELS OF CUSTOMER CONTACT CENTERS Thesis Approved: Dr. Manjunath Kamath Thesis Advisor Dr. David B. Pratt Dr. Ramesh Sharda Dr. A. Gordon Emslie Dean of the Graduate College iii ACKNOWLEDGEMENT No words can ever express my sincere gratitude and indebtness to my advisor Dr. Manjunath Kamath, Without his support and motivation the completion of this thesis would have not been possible. I also thank my committee members Dr. David Pratt and Dr. Ramesh Sharda for providing me with invaluable suggestions and guidance. I thank my friends and CCiMe members Sandeep, Parthi, Mahesh, Sarath, Ananth, Karthik, Vishal and Nandan for helping me in the various stages of my thesis completion. Fruitful conversations with Bob and Ashish are much appreciated. I sincerely thank my late father, mother Mrs. C. Muthulakshmi, sister Jamuna and younger brother Raju for their constant support and encouragement. Last but not least I appreciate the support I received from the faculty and staff at the Industrial Engineering and Management department during the course of my graduate study. iv Dedicated to my late father Sri. M. Chinnaswamy and to my mentor Dr. Manjunath Kamath v TABLE OF CONTENTS 1. INTRODUCTION.....................................................................................................................1 2. STATEMENT OF THE PROBLEM .......................................................................................5 3. LITERATURE REVIEW .........................................................................................................7 3.1 SKILL  BASED ROUTING ....................................................................................................... 7 3.1.1 Challenges in SkillBased Routing.................................................................................8 3.2 RESOURCE POOLING ............................................................................................................ 10 3.2.1 StochasticProcess Limits ............................................................................................11 3.3 WORKFORCE MANAGEMENT............................................................................................... 12 3.3.1 RealTime Staffing........................................................................................................13 3.3.2 ShortTerm Staffing......................................................................................................14 3.3.3 LongTerm Staffing ......................................................................................................14 3.4 ORGANIZATIONAL BEHAVIOR ............................................................................................. 15 3.5 PERFORMANCE EVALUATION .............................................................................................. 15 3.5.1 Queueing Models of Call Centers ................................................................................16 3.5.1.1 Arrival Process......................................................................................................16 3.5.1.2 Waiting Behavior of Customers............................................................................ 17 3.5.1.3 Service Time Distribution of Agents .................................................................... 18 3.5.2 Queueing Models of Inbound Email Contact Centers ................................................19 4. RESEARCH STATEMENT ...................................................................................................21 vi 4.1 RESEARCH OBJECTIVES ....................................................................................................... 21 4.2 RESEARCH SCOPE AND LIMITATIONS .................................................................................. 22 4.3 RESEARCH CONTRIBUTIONS ................................................................................................ 22 5. MODELING METHODOLOGY...........................................................................................24 5.1 THE PARAMETRIC DECOMPOSITION (PD) METHOD ............................................................ 24 5.2 AGGREGATION APPROACHES .............................................................................................. 26 5.3 NUMERICAL VALIDATION.................................................................................................... 27 6. A MULTICLASS OPEN QUEUEING NETWORK MODEL OF EMAIL CUSTOMER CONTACT CENTERS................................................................................................................29 6.1 CONTACT CENTER DESCRIPTION......................................................................................... 30 6.2 ASSUMPTIONS..................................................................................................................... 31 6.3 MODELING THE EMAIL CONTACT CENTER USING AN OPEN QUEUEING NETWORK ......... 31 6.3.1 Rate and SCV of the External Arrival Process at a Node ............................................34 6.3.2 Weights for Classbased Aggregation..........................................................................35 6.3.3 Markovian Routing Probabilities.................................................................................36 6.3.4 Mean Service Time at a Node ......................................................................................37 6.3.5 Squared Coefficient of Variation of the Service Time Distribution at a Node.............38 6.4 APPROACHES TO COMPUTE THE WEIGHTS FOR HISTORYBASED AGGREGATION.............. 39 6.4.1 Aggregation Method M2 ..............................................................................................40 6.5 DISCRETETIME MARKOV CHAIN MODEL OF HISTORYBASED EMAIL ROUTING ............. 43 6.5.1 Analysis of the Absorbing Markov Chain.....................................................................46 6.5.2 Performance Evaluation of the EMail Contact Center...............................................47 vii 6.6 NUMERICAL EXPERIMENTS.................................................................................................. 51 6.6.1 Email Contact Center Characteristics........................................................................51 6.6.2 Email Contact Center Example ..................................................................................52 6.6.3 Probabilities for HistoryBased Aggregation obtained from the DTMC.....................53 6.7 RESULTS AND DISCUSSIONS ................................................................................................ 58 6.7.1 Results ..........................................................................................................................58 6.7.2 Discussion ....................................................................................................................68 7. MODEL EXTENSIONS: SKILLBASED GROUPING OF AGENTS AND SERVICE INTERRUPTIONS ......................................................................................................................69 7.1 CONTACT CENTER MODEL WITH SKILLBASED GROUPING................................................ 70 7.2 NUMERICAL EXPERIMENTS.................................................................................................. 70 7.3 RESULTS AND DISCUSSION .................................................................................................. 71 7.3.1 Results ..........................................................................................................................72 7.4 MODELING SERVICE INTERRUPTIONS.................................................................................. 93 7.4.1 Contact Center Model with Service Interruptions .......................................................93 7.5 NUMERICAL EXPERIMENTS.................................................................................................. 94 7.6 RESULTS.............................................................................................................................. 94 8. CONCLUSIONS AND FUTURE RESEARCH..................................................................103 8.1 RESEARCH SUMMARY........................................................................................................ 103 8.2 RESEARCH CONTRIBUTIONS .............................................................................................. 104 8.3 FUTURE RESEARCH DIRECTIONS ....................................................................................... 105 8.3.1 Extensions to Model Agent Schedules ........................................................................105 viii 9. REFERENCES......................................................................................................................107 A1. FUNDAMENTAL MATRIX ENTRIES FOR SCENARIOS 1 AND 2..........................115 A1.1 MATLAB PROGRAM FOR FUNDAMENTAL MATRIX INVERSION ...................................... 132 A1.2 MATLAB CODE TO OBTAIN THE PARAMETERS FOR HYPEREXPONENTIAL DISTRIBUTION USED IN SIMULATION .............................................................................................................. 132 A2. SIMULATION MODEL ....................................................................................................133 A2.1 THE SIMULATION LOGIC................................................................................................. 133 A2.2 SERVICE TIME DISTRIBUTIONS AND THEIR PARAMETERS.............................................. 139 A2.2.1 Erlang Distribution..................................................................................................139 A2.2.2 Exponential Distribution..........................................................................................140 A2.2.3 Hyperexponential Distribution ................................................................................140 A3. DETERMINATION OF THE WARMUP PERIOD AND RUN LENGTH FOR SIMULATION EXPERIMENTS.............................................................................................142 A3.1 INTRODUCTION ............................................................................................................... 142 A3.2 WELCH’S TECHNIQUE TO DETERMINE WARMUP PERIOD ............................................. 143 A3.3 DETERMINATION OF RUN LENGTH ................................................................................. 144 ix LIST OF FIGURES FIGURE 1.1: OUTLINE OF THE THESIS .............................................................................................. 4 FIGURE 6.1: EMAIL HANDLING LOGIC ......................................................................................... 33 FIGURE 6.2: AGGREGATION METHOD M2 ..................................................................................... 41 FIGURE 6.3: ONESTEP TRANSITION PROBABILITY MATRIX......................................................... 46 FIGURE 6.4: CONTACT CENTER PERFORMANCE EVALUATION...................................................... 50 FIGURE 6.6: SCENARIO 1  AGENT UTILIZATION  ERLANG PROCESSING TIME ............................ 62 FIGURE 6.7: SCENARIO 1  AVERAGE RESPONSE TIME .................................................................. 62 FIGURE 6.8: SCENARIO 1  AVERAGE RESOLUTION TIME.............................................................. 63 FIGURE 6.9: SCENARIO 1  AVERAGE NUMBER OF EMAILS IN THE SYSTEM ................................ 63 FIGURE 6.10: SCENARIO 2  AGENT UTILIZATION  ERLANG PROCESSING TIMES......................... 66 FIGURE 6.11: SCENARIO 2  AVERAGE RESPONSE TIME ................................................................ 66 FIGURE 6.12: SCENARIO 2  AVERAGE RESOLUTION TIME............................................................ 67 FIGURE 6.13: SCENARIO 2  AVERAGE NUMBER OF EMAILS IN THE SYSTEM .............................. 67 FIGURE 7.1: SCENARIO 1, CONFIGURATION 1  AGENT UTILIZATION  ERLANG PROCESSING TIMES ................................................................................................................................... 73 FIGURE 7.2: SCENARIO 1, CONFIGURATION 1  AVERAGE RESPONSE TIME .................................. 74 FIGURE 7.3: SCENARIO 1, CONFIGURATION 1  AVERAGE RESOLUTION TIME.............................. 74 FIGURE 7.4: SCENARIO 1, CONFIGURATION 1  AVERAGE NUMBER OF EMAILS IN THE SYSTEM. 75 FIGURE 7.5: SCENARIO 1, CONFIGURATION 2  AGENT UTILIZATION  ERLANG PROCESSING TIMES ................................................................................................................................... 77 FIGURE 7.6: SCENARIO 1, CONFIGURATION 2  AVERAGE RESPONSE TIME .................................. 77 x FIGURE 7.7: SCENARIO 1, CONFIGURATION 2  AVERAGE RESOLUTION TIME.............................. 78 FIGURE 7.8: SCENARIO 1, CONFIGURATION 2  AVERAGE NUMBER OF EMAILS IN THE SYSTEM. 78 FIGURE 7.9: SCENARIO 1, CONFIGURATION 3  AGENT UTILIZATION AND ERLANG PROCESSING TIMES ................................................................................................................................... 80 FIGURE 7.10: SCENARIO 1, CONFIGURATION 3  AVERAGE RESPONSE TIME ................................ 81 FIGURE 7.11: SCENARIO 1, CONFIGURATION 3  AVERAGE RESOLUTION TIME ............................ 81 FIGURE 7.12: SCENARIO 1, CONFIGURATION 3  AVERAGE NUMBER OF EMAILS IN THE SYSTEM .............................................................................................................................................. 82 FIGURE 7.13: SCENARIO 2, CONFIGURATION 1  AGENT UTILIZATION  ERLANG PROCESSING TIMES ................................................................................................................................... 84 FIGURE 7.14: SCENARIO 2, CONFIGURATION 1  AVERAGE RESPONSE TIME ................................ 84 FIGURE 7.15: SCENARIO 2, CONFIGURATION 1  AVERAGE RESOLUTION TIME ............................ 85 FIGURE 7.16: SCENARIO 2, CONFIGURATION 1  AVERAGE NUMBER OF EMAILS IN THE SYSTEM .............................................................................................................................................. 85 FIGURE 7.17: SCENARIO 2, CONFIGURATION 2  AGENT UTILIZATION  ERLANG PROCESSING TIMES ................................................................................................................................... 87 FIGURE 7.18: SCENARIO 2, CONFIGURATION 2  AVERAGE RESPONSE TIME ................................ 88 FIGURE 7.19: SCENARIO 2, CONFIGURATION 2  AVERAGE RESOLUTION TIME ............................ 88 FIGURE 7.20: SCENARIO 2, CONFIGURATION 2  AVERAGE NUMBER OF EMAILS IN THE SYSTEM .............................................................................................................................................. 89 FIGURE 7.21: SCENARIO 2, CONFIGURATION 3  AGENT UTILIZATION  ERLANG PROCESSING TIMES ................................................................................................................................... 91 FIGURE 7.22: SCENARIO 2, CONFIGURATION 3  AVERAGE RESPONSE TIME ................................ 91 xi FIGURE 7.23: SCENARIO 2, CONFIGURATION 3  AVERAGE RESPONSE TIME ................................ 92 FIGURE 7.24: SCENARIO 2, CONFIGURATION 3  AVERAGE NUMBER OF EMAILS IN THE SYSTEM .............................................................................................................................................. 92 FIGURE 7.25: SCENARIO 1  AGENT UTILIZATION  ERLANG PROCESSING TIMES......................... 96 FIGURE 7.26: SCENARIO 1  AVERAGE RESPONSE TIME ................................................................ 97 FIGURE 7.27: SCENARIO 1  AVERAGE RESOLUTION TIME............................................................ 97 FIGURE 7.28: SCENARIO 1  AVERAGE NUMBER OF EMAILS IN THE SYSTEM .............................. 98 FIGURE 7.29: SCENARIO 2  AGENT UTILIZATION  ERLANG PROCESSING TIMES....................... 100 FIGURE 7.30: SCENARIO 2  AVERAGE RESPONSE TIME .............................................................. 100 FIGURE 7.31: SCENARIO 2  AVERAGE RESOLUTION TIME.......................................................... 101 FIGURE 7.32: SCENARIO 2  AVERAGE NUMBER OF EMAILS IN THE SYSTEM ............................ 101 FIGURE A2.1: SIMULATION MODEL............................................................................................. 135 FIGURE A2.2: ARRIVAL AND ROUTING SUB MODE ..................................................................... 136 FIGURE A2.3: REP I (AGENT I) SUB MODEL I = 1, 2 AND 3 .......................................................... 137 FIGURE A2.4: STATISTICS AND RESOLUTION SUB MODEL.......................................................... 138 FIGURE A2.5: ERLANG SERVICE TIME DISTRIBUTION................................................................ 139 FIGURE A2.6: HYPEREXPONENTIAL SERVICE TIME DISTRIBUTION............................................. 141 FIGURE A3.1: PLOT OF RESOLUTION TIME IN SYSTEM................................................................ 145 FIGURE A3.2: PLOT OF RESOLUTION TIME IN SYSTEM (CONTD) ................................................. 145 FIGURE A3.3: PLOT OF RESOLUTION TIME IN SYSTEM (CONTD) ................................................. 146 FIGURE A3.4: PLOT OF RESOLUTION TIME IN SYSTEM (CONTD) ................................................. 146 FIGURE A3.5: PLOT OF RESOLUTION TIME IN SYSTEM (CONTD) ................................................. 147 FIGURE A3.6: PLOT OF RESOLUTION TIME IN SYSTEM (CONTD) ................................................. 147 xii FIGURE A3.7: PLOT OF RESOLUTION TIME IN SYSTEM (CONTD) ................................................. 148 FIGURE A3.8: PLOT OF RESOLUTION TIME IN SYSTEM (CONTD) ................................................. 148 FIGURE A3.9: PLOT OF RESOLUTION TIME IN SYSTEM (CONTD) ................................................. 149 FIGURE A3.10: PLOT OF RESOLUTION TIME IN SYSTEM (CONTD) ............................................... 149 xiii LIST OF TABLES TABLE 5.1: SIMULATION PARAMETERS ......................................................................................... 28 TABLE 5.2: SCV LEVELS AND CORRESPONDING DISTRIBUTIONS ................................................. 28 TABLE 6.1: SCENARIO 1  MEAN PROCESSING TIMES.................................................................... 54 TABLE 6.2: SCENARIO 1  FORWARDING PROBABILITIES .............................................................. 54 TABLE 6.3: SCENARIO 1  RESOLUTION PROBABILITIES................................................................ 54 TABLE 6.4: SCENARIO 1, TYPE 1 EMAILS  AGGREGATION PROBABILITIES FOR METHOD M3.... 55 TABLE 6.5: SCENARIO 1, TYPE 2 EMAILS  AGGREGATION PROBABILITIES FOR METHOD M3.... 55 TABLE 6.6: SCENARIO 2  MEAN PROCESSING TIMES.................................................................... 56 TABLE 6.7: SCENARIO 2  FORWARDING PROBABILITIES .............................................................. 56 TABLE 6.8: SCENARIO 2  RESOLUTION PROBABILITIES................................................................ 56 TABLE 6.9: SCENARIO 2, TYPE 1 EMAILS  AGGREGATION PROBABILITIES FOR METHOD M3.... 57 TABLE 6.10: SCENARIO 2, TYPE 2 EMAILS  AGGREGATION PROBABILITIES FOR METHOD M3.. 57 TABLE 6.11: SCENARIO 1  NODELEVEL PERFORMANCE MEASURES........................................... 60 TABLE 6.12: SCENARIO 1  SYSTEMLEVEL PERFORMANCE MEASURES ....................................... 61 TABLE 6.13: SCENARIO 2  NODELEVEL PERFORMANCE MEASURES........................................... 64 TABLE 6.14: SCENARIO 2  SYSTEMLEVEL PERFORMANCE MEASURES ....................................... 65 TABLE 7.1: SCENARIO 1, CONFIGURATION 1  PERFORMANCE MEASURES FOR ERLANG PROCESSING TIMES ............................................................................................................... 72 TABLE 7.2: SCENARIO 1, CONFIGURATION 1  PERFORMANCE MEASURES FOR EXPONENTIAL PROCESSING TIMES ............................................................................................................... 72 xiv TABLE 7.3: SCENARIO 1, CONFIGURATION 1  PERFORMANCE MEASURES FOR HYPEREXPONENTIAL PROCESSING TIMES............................................................................. 73 TABLE 7.4: SCENARIO 1, CONFIGURATION 2  PERFORMANCE MEASURES FOR ERLANG PROCESSING TIMES ............................................................................................................... 75 TABLE 7.5: SCENARIO 1, CONFIGURATION 2  PERFORMANCE MEASURES FOR EXPONENTIAL PROCESSING TIMES ............................................................................................................... 76 TABLE 7.6: SCENARIO 1, CONFIGURATION 2  PERFORMANCE MEASURES FOR HYPEREXPONENTIAL PROCESSING TIMES............................................................................. 76 TABLE 7.7: SCENARIO 1, CONFIGURATION 3  PERFORMANCE MEASURES FOR ERLANG PROCESSING TIMES ............................................................................................................... 79 TABLE 7.8: SCENARIO 1, CONFIGURATION 3  PERFORMANCE MEASURES FOR EXPONENTIAL PROCESSING TIMES ............................................................................................................... 79 TABLE 7.9: SCENARIO 1, CONFIGURATION 3  PERFORMANCE MEASURES FOR HYPEREXPONENTIAL PROCESSING TIMES............................................................................. 80 TABLE 7.10: SCENARIO 2, CONFIGURATION 1  PERFORMANCE MEASURES FOR ERLANG PROCESSING TIMES ............................................................................................................... 82 TABLE 7.11: SCENARIO 2, CONFIGURATION 1  PERFORMANCE MEASURES FOR EXPONENTIAL PROCESSING TIMES ............................................................................................................... 83 TABLE 7.12: SCENARIO 2, CONFIGURATION 1  PERFORMANCE MEASURES FOR HYPEREXPONENTIAL PROCESSING TIMES............................................................................. 83 TABLE 7.13: SCENARIO 2, CONFIGURATION 2  PERFORMANCE MEASURES FOR ERLANG PROCESSING TIMES ............................................................................................................... 86 xv TABLE 7.14: SCENARIO 2, CONFIGURATION 2  PERFORMANCE MEASURES FOR EXPONENTIAL PROCESSING TIMES ............................................................................................................... 86 TABLE 7.15: SCENARIO 2, CONFIGURATION 2  PERFORMANCE MEASURES FOR HYPEREXPONENTIAL PROCESSING TIMES............................................................................. 87 TABLE 7.16: SCENARIO 2, CONFIGURATION 3  PERFORMANCE MEASURES FOR ERLANG PROCESSING TIMES ............................................................................................................... 89 TABLE 7.17: SCENARIO 2, CONFIGURATION 3  PERFORMANCE MEASURES FOR EXPONENTIAL PROCESSING TIMES ............................................................................................................... 90 TABLE 7.18: SCENARIO 2, CONFIGURATION 3  PERFORMANCE MEASURES FOR HYPEREXPONENTIAL PROCESSING TIMES............................................................................. 90 TABLE 7.19: SCENARIO 1  PERFORMANCE MEASURES FOR ERLANG PROCESSING TIMES .......... 95 TABLE 7.20: SCENARIO 1  PERFORMANCE MEASURES FOR EXPONENTIAL PROCESSING TIMES.. 95 TABLE 7.21: SCENARIO 1  PERFORMANCE MEASURES FOR HYPEREXPONENTIAL PROCESSING TIMES ................................................................................................................................... 96 TABLE 7.22: SCENARIO 2  PERFORMANCE MEASURES FOR ERLANG PROCESSING TIMES ........... 98 TABLE 7.23: SCENARIO 2  PERFORMANCE MEASURES FOR EXPONENTIAL PROCESSING TIMES.. 99 TABLE 7.19: SCENARIO 2  PERFORMANCE MEASURES FOR HYPEREXPONENTIAL PROCESSING TIMES ................................................................................................................................... 99 TABLE A1.1: SCENARIO 1, TYPE 1 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC................................................................................................................................ 116 TABLE A1.2: SCENARIO 1, TYPE 1 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 117 xvi TABLE A1.3: SCENARIO 1, TYPE 1 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 118 TABLE A1.4: SCENARIO 1, TYPE 1 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 119 TABLE A1.5: SCENARIO 1, TYPE 2 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC................................................................................................................................ 120 TABLE A1.6: SCENARIO 1, TYPE 2 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 121 TABLE A1.7: SCENARIO 1, TYPE 2 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 122 TABLE A1.8: SCENARIO 1, TYPE 2 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 123 TABLE A1.9: SCENARIO 2, TYPE 1 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC................................................................................................................................ 124 TABLE A1.10: SCENARIO 2, TYPE 1 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 125 TABLE A1.11: SCENARIO 2, TYPE 1 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 126 TABLE A1.12: SCENARIO 2, TYPE 1 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 127 TABLE A1.13: SCENARIO 2, TYPE 2 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC................................................................................................................................ 128 xvii TABLE A1.14: SCENARIO 2, TYPE 2 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 129 TABLE A1.15: SCENARIO 2, TYPE 2 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 130 TABLE A1.16: SCENARIO 2, TYPE 2 EMAILS  AVERAGE NUMBER OF VISITS BY SOLVING THE DTMC (CONTD) .................................................................................................................. 131 xviii NOTATION c type of email; c = 1, 2, ..., C (These types may represent different classes of email product inquiry, technical support, etc.) A number of customer service agents k,i, j index for previous, current and next agent processing an email; k = 0,1, ..., A; i, j =1, 2, ..., A; k = 0 represents a new email λ (c) external arrival rate of (new) emails belonging to type c (c) i λ external arrival rate of type c emails at agent i i λ external arrival rate of emails at agent i (c) i γ total ( external plus internal) arrival rate of type c emails at agent i r (c) i probability that an email received by agent i is of type c β (c) represents the total (new plus previously processed) arrival rate of type c emails for Scenario 1 (c) i α probability that a new type c email will be routed to agent i w (c, k) i proportion of type c emails received by agent i that were previously processed by agent k or new (k = 0); i = A +1 represents the delay node ( , ) 1 w c k A+ probability that a type c email at the delay node (A +1) came from agent k (c, k) i υ average number of visits to agent i made by a type c email that was previously processed by agent k (c, k) Di υ average number of times a type c email was processed by agent i and not resolved xix ( , ) , p c k i j probability that agent i forwards a type c email that was previously processed by agent k to agent j ( ) , p c i j classspecific routing probability from agent i to agent j for a type c email i j p , aggregate routing probability from agent i to agent j p(c) represents the aggregate forwarding probability for type c emails q (c, k) i probability that a type c email currently processed by agent i and previously processed by agent k ends in problem resolution q(c) represents the aggregate resolution probability for a type c email ( ) , 1 p c i A+ classspecific routing probability from agent i to delay node (A +1) for type c email i,A+1 p aggregate routing probability from agent i to delay node (A +1) ( ) 1, p c A+ i classspecific routing probability from the delay node (A +1) to agent i for a type c email A i p +1, aggregate routing probability from delay node (A +1) to agent i P (c) i random variable that represents the preprocessing time at agent i for a type c email Z (c, k) i indicator random variable that takes the value 1 if agent i processes a type c email that was previously processed by agent k and the value 0 otherwise S (c, k) i random variable that represents the processing time at agent i for a type c email that was previously processed by agent k xx T (c, k) i random variable that represents the overall service time at agent i for a type c email that was previously processed by agent k (c) i θ E[P (c)] i p (c, k) i E[Z (c, k)] i = − Σ ≠ = A i j j i j p c k 1 , 1 ( , ) s (c, k) i E[S (c, k)] i t (c, k) i E[T (c, k)] i c2 squared coefficient of variation (SCV) = variance/mean2 c2 (c) o SCV of the interarrival time for new, external type c emails c2 (c) oi SCV of the interarrival time for external type c emails at agent i 2 oi c SCV of the aggregate interarrival time for external emails at agent i 2 ( ) , c c pre i SCV of P (c) i 2 ( , ) , c c k ser i SCV of S (c, k) i c2 (c, k) i SCV of T (c, k) i c2 (c) i SCV of the overall classdependent service time at agent i 2 i c SCV of the overall service time at agent i The following additional notation is needed for the Markov chain analysis presented in Section 6.5. N represents the “new” state  the lifecycle of a new email starts in this state xxi R represents the “resolution” node  the lifecycle of an email ends in this state i D represents the delay state linked to agent i  emails processed by agent i that are unresolved pass through this state P onestep transition probability matrix U unit matrix of size (1×1) M column matrix of size (n×1) , where n = A3 + A2 + A+1 Q truncated matrix associated with P and of size (n × n) , where n = A3 + A2 + A+1 I identity matrix of size (n × n) , where n = A3 + A2 + A+1 F fundamental matrix of size (n× n) , where n = A3 + A2 + A+1 1 CHAPTER 1 INTRODUCTION The service sector has become a dominant part of the US economy, due in part to the ecommerce revolution of the late nineties. Better quality of service delivered virtually with very little or no waiting time is what customers frequently expect today. One notable facet of the service industry is the call center industry. A call center is any “group whose principal business is talking on the telephone to customers or prospects (Mehrotra 1997).” Customer call centers, which represent a multibillion dollar industry, are evolving into customer contact centers. “It is estimated about four million people in the United States 3% of the workforce  work in contact centers, with the number growing by about 20% per year. A contact center is a collection of resources providing an interface between the service provider and its remote customers (Whitt 2002a).” The interface can be through any one or combination of media  telephone, email, fax, paper, chat sessions and the Web. In the private sector, contact centers are used in various industries and are an important communication channel to acquire new customers as well as to support existing customers. In email contact centers, the traffic can be inbound or outbound. In an inbound email contact center, agents respond to emails from customers. Examples of inbound email contact centers are technical product support centers and travel 2 reservation centers. In outbound email contact centers, agents initiate email that is sent to customers. Examples of outbound email contact centers are companies conducting surveys and market research. In inbound email contact centers the response time is flexible, i.e., customers do not expect an email to be answered within minutes, whereas they could get frustrated waiting on the telephone even for a few minutes. This flexibility allows for the possibility of postponing a response. The average time an agent takes to respond to a customer’s email is known as the response time. This time needs to be as small as possible and is an important performance measures in both call and contact centers. The other measure of interest is the resolution time, i.e., the average time that is taken to resolve the problem represented by the email. Shorter response times and resolution times can help contact centers to better serve and retain their customers. These system performance measures can be improved if the email contact center employs more agents. But employing more agents leads to higher operating costs. In email contact centers, more than half of the operating costs are driven by the costs of employing agents. Therefore, agent utilization is often used to indicate the economic performance measure of an inbound email contact center. The number of agents is an important decision variable in designing email contact centers. Both queueing theory and simulation have been used to model the operations of call centers. These models describe the behavior of the system over time, which helps in designing call center operations. Bulk of the existing literature focuses on modeling traditional telephone call centers. As pointed out by Whitt (2002a), more research is needed in the stochastic modeling of customer contact centers, where the contact is 3 through media other than the telephone. The focus of this thesis was on modeling an important characteristic of email contact centers, namely the dependence of email processing times and routing on email history. A novel historybased aggregation approach was developed to handle the dependence on processing history within a multiclass open queueing network model. Through extensive numerical experimentation, this thesis shows the importance of modeling email history in accurately predicting the performance of email contact centers. The numerical investigations also demonstrate the accuracy and robustness of the historybased aggregation approach. The remainder of this thesis document is organized as follows (also see Figure 1.1). Chapter 2 describes the problem statement. Chapter 3 presents an extensive literature review of the work carried to date in modeling call and contact centers. Chapter 4 presents the research statement. Also included in this chapter are the research objectives, scope and limitations, and contributions of the research conducted. Chapter 5 describes the modeling approach that was followed in developing and solving the open queueing network models of email contact centers. Chapter 6 presents the nucleus of this thesis effort. It includes three analytical methods ranging from a simple averaging technique to a very detailed Markov chain based aggregation approach to model dependence on email history. These analytical methods are integrated into the solution method for a multiclass, open queueing network model of email contact centers. Chapter 7 extends the network model presented in Chapter 6 to include (i) multiserver nodes that represent skillbased pools of agents and (ii) random interruptions that affect agent’s availability for email processing. Chapter 8 presents research contributions, conclusions and directions for future research. 4 Chapters 1 & 2 INTRODUCTION PROBLEM STATEMENT Chapter 3 LITERATURE REVIEW Skillbased Routing, Resource Pooling, Workforce Management, Organizational Behaviour, and Performance Evaluation Chapter 4 RESEARCH STATEMENT Objectives, Scope & Limitations, Contributions Chapter 5 MODELING APPROACH Parametric Decomposition (PD) Method Chapters 6 & 7 MULTICLASS, OPEN QUEUEING NETWORK MODELS  Historybased Aggregation  Classbased Aggregation  Single / MultiServer Nodes  Service Interruptions Chapter 8 CONCLUSIONS AND FUTURE RESEARCH Figure 1.1: Outline of the Thesis 5 CHAPTER 2 STATEMENT OF THE PROBLEM Modeling call centers using queueing theory is not new. Many researchers have modeled call centers using the standard M/M/c/∞ queues to obtain steadystate performance measures such as average number of calls in the system and average waiting time for calls. However, only very limited literature exists on queueing models of contact centers because of their recent emergence as an alternative to call centers. In addition to analytical approaches, both call and contact centers can be modeled using simulation. Simulation models require more detailed information when compared to queueing models. The model development and model execution activities in simulation could be time consuming. On the other hand, analytical models such as queueing networks provide more insight and understanding of the system. Analytical models may require simplifying assumptions of the system, and the results obtained are generally less accurate than those estimated via simulation. But analytical models yield results quickly and “are appropriate for rapid and rough cut analysis (Suri et al. 1993).” Hence, analytical models can be used for preliminary design and simulation for fine tuning. The evolution of email contact centers has thrown light on many new modeling issues that are not typically addressed in the study of traditional call centers. Customer or problem history can be more useful with asynchronous communication tools such as e 6 mail and can influence routing strategies and processing times. For example, it is possible for an agent who previously attended to a customer’s problem (email) to also deal with the followup email because of the asynchronous nature of email processing. With regard to system performance measures, the average resolution time, i.e., the average time needed to resolve a customer’s problem is not normally addressed in the call center literature. Problem resolution, while important in both call and contact centers, has much more visibility in an email contact center because of the history embedded in the email reply. In a call center, because of the nature of the customer contact, the response time for a call becomes more important. In a contact center, both response and resolution times become important as the latter is a function of the former. Also, because of the asynchronous nature of email, the customer service agent has more flexibility in terms of the time to generate a response to an email. Also, it is possible to direct an email to the appropriate agent if it contains information about previous response(s). These new modeling issues present unique challenges while developing queueing models of contact center operations. This thesis has addressed some of these issues. The problem statement can be summarized as follows: “To model the performance of an inbound email contact center in which the routing of an email and email processing times at the various agents can depend on the email history.” 7 CHAPTER 3 LITERATURE REVIEW The research on modeling call centers and contact centers can be broadly classified into the following categories  1) SkillBased Routing, 2) Resource Pooling, 3) Workforce Management, 4) Performance Evaluation and 5) Organizational Behavior. Within each category, a formal subcategory on email contact centers is created only when there is a need to distinguish the work from that for the call center classification. Otherwise, the description for call centers holds good for the email contact centers also. Though the scope of the literature review presented in this chapter is much broader than the performance evaluation theme of this research, it nevertheless illustrated the importance of analytical models in the analysis and design of customer call centers. This chapter also explores some new research directions and issues in the area of customer contact center modeling. 3.1 Skill  Based Routing Modern call centers have multiple call types and multiple types of agents. One way of classifying customer calls is by language. With the globalization of many businesses, call centers receive calls in different languages from their customers throughout the world. Another way of classifying customer calls is based on special promotions. The customer may be calling a tollfree number designated for a special promotion purpose. Training 8 agents for a special promotion purpose is a common practice. To match the caller’s need and agent’s skill set, modern call centers have automatic call distributors (ACDs) that have the capability of routing calls to agents with the appropriate skills. This capability of ACDs is known as skillbased routing (SBR). In email contact centers, the emails are routed to the appropriate agents with the help of information and communication technologies. An important issue to be addressed pertains to the optimality of email/call routing policies. 3.1.1 Challenges in SkillBased Routing There are many challenges and issues to be addressed in performing skillbased routing well. First, with a given collection of agents, it is difficult to route multiple calls/emails to an agent in an optimal manner. This is due to elementary skillbased routing algorithms that are used in call/contact centers. According to “(Whitt 2002a), there remains a great opportunity for devising better routing algorithms.” The routing of calls/emails depends on the number of agents in the center. Second, it is difficult to determine exactly the number of agents with appropriate skill sets. With multiple call/email types, not only the prediction of the overall arrival rate, but also the prediction of arrival rates for individual types becomes important. Koole et al. (2003) presented an approximation method for analyzing the performance of call centers with skillbased routing. They considered two types of call arrivals. Arriving calls abandon the system if the agents with the right skill are busy. But under these conditions, it was difficult to compute the optimal routing policies of calls because of statespace explosion. Therefore, the authors considered a different queueing system where the call gets queued if the agents are not available in any of the groups. This is equivalent to calls overflowing from the groups without available agents. This type of 9 routing is known as overflow routing. This system is easy to characterize and optimal policies can be practically implemented, but allows less flexibility in routing policies. Koole and Talim (2000) studied a multiskill call center as a network of queues. They approximated each queue as an M/M/r loss system, because an arriving call left if all agents in the network are busy. The interoverflow time distribution at each node in the network was approximated by an exponential distribution, and the efficiency of the approximation was illustrated using simulation. Bhulai and Koole (2003) developed a queueing model from a call blending perspective for schedule agents either to incoming or outgoing calls in order to increase productivity and reduce the call waiting times. In addition, scheduling policies and their implementation within call center software were also discussed. Garnett and Mandelbaum (2000) illustrated the operational complexities of skillbased routing using simulation studies. Perry and Nilsson (1992) considered simple strategies to overcome the dimensionality of call centers. They considered a twochannel agent system, where the waiting customer was assigned an agingfactor. This factor was directly proportional to the waiting time of the customer in the system. The customer with the largest aging factor (considering both queues) was chosen for service. They determined the expected waiting time for each call type, and the number of agents required for answering calls to maintain an acceptable grade of service. To conclude, there is scope for more research in developing simple and better routing algorithms, finding optimal routing policies using approximations and strategies, and implementing those in call/contact centers. 10 3.2 Resource Pooling Resource pooling is overlapping of agents between groups so as to meet customer needs and to increase the efficiency of call/contact centers. In a call center, it is customary to have a large group of agents dedicated to a particular skill set. This can be viewed as a big call center with large number of smaller independent call centers. Upon call arrival, the calls can be routed to the group of agents with the appropriate skill sets. When the arrival process to the big call center is Poisson and with independent Bernoulli routing to the smaller call centers, the arrival process to the smaller call center is Poisson and tends to act independently of other smaller units. Partitioning into subgroups tends to make call centers less efficient. While operating, some of the smaller centers tend to overloaded, while others may be underutilized. The efficiency of larger service groups is explained by Smith and Whitt (1981) and Whitt (1992). Smith and Whitt (1981) argued that if two systems were combined together into a single system, the efficiency of the combined system is higher than the efficiency of the individual systems. This seems intuitive and trivial, but becomes difficult to prove. The authors used stochasticorder relations to prove the result and concluded that the results apply to general arrival processes and general servicetime distributions. The combination of systems assumes common servicetime distribution, since it may be disadvantageous to combine systems with different servicetime distributions. When the service time distributions are different it is advantageous to partition the system as in supermarket checkouts and reservation centers. Whitt (1992) explained the economy of scale, and gave a quantitative characterization of a multiserver queueing system with unlimited waiting space. He showed that the increased variability in the arrival and service processes degrade server utilization with a 11 given grade of service. He also presented a proof for finding the number of servers as a function of the arrival rate for a given service time distribution and grade of service. The proof is based on heavytraffic limit theorems and uses infiniteserver (IS) approximation to heuristically derive the result. The author developed simple approximations for the expected waiting time and the probability of delay using the infiniteserver approximation, and conditional waiting time distribution using a singleserver approximation. The next section throws light on the application of StochasticProcess limits in resource pooling. 3.2.1 StochasticProcess Limits It is natural to study a system by allowing the number of customers and servers to grow large. This is done to gain more knowledge and insight about the behavior of the system under consideration. The mathematical results on resource pooling are based on the asymptotic regime in which the number of servers is allowed to approach infinity. This is the principle behind stochasticprocess limits. Significant work on resource pooling has been done till now. The paper by Vvedenskaya et al. (1996) serves as a good example for its significance, and it considered a single stream of customers with a Poisson arrival process. Upon arrival, each customer joins one of the many identical singleserver queues. The service time distribution in all queues follows an exponential distribution and all queues have an unlimited waiting space. The standard approach is joining the queue with very few customers. The number of customers is determined based on the states of all queues. Indeed, joining the shortest queue is optimal (Winston 1977). However, optimality is ceased if each queue has different servicetime distribution (Whitt 1986). A difficulty in joining the shortest queue is that it may require a large amount of state information. Therefore techniques which require less amount of state information 12 have to be adopted. For example, each arrival can be randomly assigned to one of the queues with equal probability of selecting a server. This can be modeled as an M/M/1 queue and all required performance measures can be calculated. Resource pooling has to be done very carefully. Sometimes it helps, but sometimes it hurts. The effect (good or bad) can be unbounded. The paper by Mandelbaum and Reiman (1998) clearly assesses the value of resource pooling. Halfin and Whitt (1981) obtained limiting regimes for the Erlang C delay and GI/M/s models. This was carried out by allowing the number of servers to approach infinity, while letting the probability of delay approach a number strictly between 0 and 1. For more about StochasticProcess limits, the reader is referred to Whitt (2002b). Heavy traffic limit regimes for the Erlang B (loss) model can be found in Srikant and Whitt (1996). Related results for queueing networks, statedependent queues can also be found in Mandelbaum and Pats (1995). Papers about heavy traffic limit regimes on contact centers are very few. Whitt (2002a) stated that there remains a great opportunity for research in establishing some new interesting regimes. 3.3 Workforce Management Workforce management plays a very significant role in design and maintenance of call/contact centers. Cleveland and Mayben (1997) addressed some mathematical issues in workforce management. Staffing is a challenging issue in call/contact centers. Whitt (2002a) described three types of staffing according to time scale. They are 1) realtime 2) shortterm and 3) longterm. 13 3.3.1 RealTime Staffing Realtime staffing has sufficient flexibility to add agents when needed, and alternatively, pull them off to do some other work. Whitt (1999b) addressed the modeling of dynamic staffing of a call center with the aim of immediately answering the calls. The system state is exploited to obtain estimates for mean and variance of demand in near future. The staffing needs can be predicted from the information about recent demand and current calls in progress, as well as historical data. The information and telecommunication equipment makes it possible to obtain the required information. It is possible to classify the call by identifying the calling or called customer, purpose of the call and the agent who will be serving or served the call. By calculating the time length that the call has been in service before service completion, it is possible to predict the conditional probability distribution of the remaining call holding time. By combining the information over many calls and agents, it is possible to predict the staff demands in the near future, providing a basis for realtime staffing. The paper by Whitt (1999b) “shows how stochastic models can be exploited to facilitate the process” and the author stated the idea needs be explored more thoroughly. Jennings et al. (1996) determined the number of servers as a function of time for a multiserver, timevarying demand service system based on an infiniteserver (IS) approximation. The IS approximation averages the timevarying demand into an effective arrival rate which remain the same at all times. An approximate busy period server distribution is obtained by allowing the number of servers to grow large and by approximating the delay probability to a specific target value. The busy period server distribution is approximated by a timedependent normal distribution where the mean and variance are determined by 14 IS approximations. Wallace and Whitt (2004) discussed a staffing algorithm for the skillbased routing call center. Realtime staffing poses great challenges in applying queueing theory, because it requires the analysis of timedependent behavior of queueing systems. Therefore, people seek numerical algorithms and approximations to describe the timedependent behavior of queues. Papers in this regard include Whitt (1999a), and Abate and Whitt (1998, 1999) where they apply decomposition approximations and numerical transform inversion techniques respectively to study the timedependent behavior of queues. 3.3.2 ShortTerm Staffing In shortterm staffing, the daily staffing is carried out in response to the forecasted demand of calls and the availability agents. As the call arrivals vary significantly from day to day, one can use the steadystate behavior of queueing systems instead of the timedependent one. This is because the call holding times are much shorter, and the time dependence can be safely ignored. In some cases of shortterm staffing, it is important to analyze the system with a timevarying arrival rate. A significant amount of effort in this area has been done by Abate and Whitt (1998), Mandelbaum and Pats (1995) and Whitt (1999b). A significant challenge in shortterm staffing is scheduling agents for small breaks for example, lunch and coffee. Mathematical programming tools have been widely used in modeling this; see for example Segal (1974). 3.3.3 LongTerm Staffing There are various challenges in longterm staffing. Training new agents is an important decision variable, which can be handled using a dynamic programming technique. On a longterm basis it is important to consider the agent’s career paths and attrition. The ideal 15 situation is to have both satisfied customers and agents. Gans and Zhou (1999) developed a Markov decision model for callcenter staffing where they address learning and turnover issues and optimal policies for longterm staffing. 3.4 Organizational Behavior The role of human element is very important in call/contact centers. Customers are people and the service reps (agents) are people. “We can easily relate to contact centers because we ourselves often are customers of contact centers (Whitt 2002a).” The human behavior is really very important in studying the psychology of queueing systems (Gail and Scott 1997, Larson 1987). Enough time should be devoted to analyzing why customers abandon or revisit. Whitt (2002a) expressed an opinion that few papers have been published in this area. 3.5 Performance Evaluation Considerable research has been done in performance modeling of call centers. Call center performance modeling studies have focused on (i) analyzing customer waiting times and customer impatience because of agent unavailability, (ii) finding optimal staffing to meet customer demands, and (iii) determining routing policies to serve customers at the earliest possible time. Traditional analysis techniques are typically based on the standard Erlang formula (Koole 2001). For example, the Erlang formula can be used to determine the upper and lower bounds on the number of employees needed, which are useful in employee scheduling (Koole 2001). = Σ= s k k s k l s l P s l 0 ! ( , ) ! (1) 16 where s number of servers in the queueing system l offered load in Erlangs, given by (l = λτ ), where λ is the arrival rate of customers and τ is the average service time. The Erlang formula is insensitive to the service time distribution. When the servers and the offered load become large, it becomes difficult to calculate the blocking probability. Therefore, recursive techniques are used to numerically calculate the blocking probability. 3.5.1 Queueing Models of Call Centers According to Stolletz (2003), queueing models of an inbound call center can be described using customer profile, agent characteristics, routing policies, and limitation of waiting room. Customer profile describes the customer arrival process to the call center and the patience/impatience characteristics of customers of a particular class. The agent characteristics describe the agent skill set and the service time distribution of the agent. The routing policy defines which agent needs to serve which customer. These policies may depend on the number of busy agents and the number of waiting customers of different classes. The size of the waiting rooms defines the maximum number of customers in the system and may depend on the customer class. 3.5.1.1 Arrival Process Customers of a call center cannot see others being served or waiting for service in the queue. Therefore, customers call independently of others and for this simple reason the arrival process in inbound call centers can be modeled as a timeinhomogeneous Poisson process (Koole and Mandelbaum 2001). As the call arrivals vary from time to time, day to day, the common approach is to approximate the timevarying arrival process by a 17 stationary, independent period by period (SIPP) approximation (Green et al. 2001). They also discussed a situation where server staffing requirements are done based on a random cyclic demand and concluded that the SIPP approximation was not accurate for many real situations. Other than SIPP approximations, point wise approximations (PSA), simple stationary approximation (SSA) and infiniteserver approximation (IS) can be found in the literature. The usage of these approximations depends on how the arrival rate varies from time to time. But all the approximations average the timevarying arrival rate into an effective arrival rate which remains constant at all times. In each time interval, arrivals occur according to a homogeneous Poisson process and it is assumed that the steadystate arrival rate does not change in each time interval. The standard steadystate approaches can then be effectively used to calculate the various performance measures of a particular queueing model (Koole and Mandelbaum 2001). 3.5.1.2 Waiting Behavior of Customers In call centers customers can be patient or impatient. Impatient customers are of two types. Balking occurs when the arriving customer finds the server busy and leaves the system immediately without being served. Reneging is when the customer finds the server busy, waits in the queue for certain random time and leaves the system without being served. The process of reneging is described by the random waiting time distribution in the queue. Both balking and reneging may be state dependent or constant. MontazerHaghighi et al. (1986) considered a multiserver queueing system with balking and reneging and obtained the average number of customers in the system under steadystate. They also presented expressions for the average loss of customers during a fixed interval of time. AbouElAta and Hariri (1992) expressed the steadystate distribution 18 for the number in the system in terms of hypergeometric function for an M/M/c/N queue with balking and reneging. Brandt and Brandt (1999) studied customer impatience in a finiteserver queueing system where the arrival and service rates could depend on the number of callers in the system. Movaghar (1998) explained customer impatience for a queueing system with statedependent Poisson arrivals, exponential service times, multiple servers and FCFS service discipline. Brandt and Brandt (2002) extended the system considered in Movaghar (1998) by assuming statedependent exponential servers. They presented asymptotic results for the number of calls leaving the system and presented a Markovian approximation for the system. Boots and Tijms (1999) presented a simple approximation for the blocking probability in an M/G/c queue with customer impatience. Bae et al. (2001) presented limiting virtual waiting time distribution for an M/G/1 queue with impatient customers. 3.5.1.3 Service Time Distribution of Agents Traditionally almost all papers in call center literature model service times of agents using the exponential distribution. In a majority of the models published in the call center literature, service times of agents have been typically modeled by the exponential distribution for model tractability. While some studies have shown that the exponential service time distribution is a good fit (e.g., see Koole and Mandelbaum 2001) arrivals are, other studies (e.g., Mandelbaum et al. 2001) have concluded that service time distribution cannot be approximated by the exponential distribution based on empirical data, and that the usefulness of exponential service time distribution may vary from one inbound call center to the other. Harris et al. (1987) analyzed data from a telephone taxpayer information system in which both the talk time and aftercall work time (time an agent takes to fill a form or mail order) followed a Weibull distribution. They compared the 19 performance measures obtained from the Weibull distribution to the performance measures obtained from the exponential distribution and observed a high level of insensitivity. 3.5.2 Queueing Models of Inbound Email Contact Centers Very few papers have been published to date in analyzing email contact centers using queueing models. This is due to the recent emergence of contact centers as an extension of traditional call centers. Most of the call center description holds good for the email contact center except for the behavior of customers. This is because it is difficult to characterize customer impatience (balking and reneging). Once the customer sends an email, two things can happen. The email can reach the agents inbox and the agent responds to the email or it can bounce back due to lack of space in agent’s inbox. The second condition is of course rare, but there are some chances of its occurrence. This in a way can be thought of as balking. Once the customer’s problem is not resolved after many email exchanges with the agent, he/she may renege, i.e., leave the system permanently. Whitt (2002a) explained the many challenging research issues in the area of customer contact center modeling and suggested research directions related to skillbased routing, resource pooling and agent staffing. Armony and Maglaras (2004a, 2004b) focused on a customer contact (call) center that offers two modes of service: realtime telephone service and callback service. In Armony and Maglaras (2004b) arriving customers are informed about the delay, and the contact center is modeled as a twoclass M/M/r queueing system with statedependent arrival rates. Armony and Maglaras (2004a) proposed an estimation scheme for the anticipated delay time based on the heavy traffic regime, approximated the system performance, and presented a staffing rule that picks the minimum number of agents. 20 When modeling email contact centers, we assume that emails that represent spam have been filtered and only useful emails arrive at the contact center. For more information about nonspam, workrelated to filtering of emails, the reader is referred to Sharda et al. (1999). Many companies use an email filtering language that can be supported in an email client and client software. Greve et al. (2004) focus on email response management problems in customer contact centers, where they specifically address the problem of processing emails in a timely manner. They use simulation to evaluate different routing policy and email processing strategies that can be employed by a contact center. In summary, work on analytical performance modeling of customer contact centers is very limited. As explained in Chapters 1 and 2, the asynchronous nature of email introduces new possibilities in operating customer contact centers and consequently, new modeling challenges. This thesis effort focused on one such challenge related on modeling the dependence of routing and processing on email history. 21 CHAPTER 4 RESEARCH STATEMENT This chapter presents the specific objectives, scope, limitations, and contributions of the research conducted as part of this thesis effort. The overall goals of this research were (i) to develop queueing network models of inbound email customer contact operations that are capable of capturing the dependence of routing and processing on email history, and (ii) to support the development of rapid whatif analysis tools that can assist the decisionmaker in designing and improving customer contact center operations. 4.1 Research Objectives The specific objectives of this research were as follows. Objective 1: To perform a thorough investigation of the literature related to the modeling of customer call and contact centers. Objective 2: To develop queueing network models of inbound email contact centers with the following characteristics. • Multiple types of email inquiries. • Heterogeneous agents with random service interruptions: The processing of emails can be interrupted when agents have to handle other knowledge work or decide to take a break. • Grouping of agents: Agents with similar skills are grouped to form an agent pool. 22 • Routing and service times are dependent on history of email (new or previously processed). Objective 3: To suggest extensions to the queueing network model to approximately handle daily and/or weekly schedules of agents. 4.2 Research Scope and Limitations The scope of this thesis was limited by the following assumptions. 1. Emails arrive continuously and the contact center agents are available 24/7. The reference to a particular agent is only with respect to a rule that requires a specificskill set with memory augmented by customized CRM tools. 2. Emails are selected from an inbox by an agent according to the FIFO (First in First out) service discipline. 3. Priorities of emails are not modeled. 4. Modeling of email history is limited to capturing the identity of the previous agent in the case of a previously processed email. 5. Modeling of agent schedules is limited to suggestions of potential extensions to the network models developed. 4.3 Research Contributions The purpose of this thesis was to contribute towards the development of queueing network models of inbound email customer contact center operations. The following contributions have been made by this thesis effort. 1. Development of a novel modeling and solution approach to handle routing and processing schemes that are dependent on email history. The approach developed is very general and extends the power of existing queueing network models. 23 2. Development of queueing models that can be incorporated into rapid analysis tools that can support the analysis and design of customer contact center operations. 24 CHAPTER 5 MODELING METHODOLOGY This chapter explains the methodology that was used to develop and solve the queueing network models in this thesis. It also includes a list of important contact performance measures that were addressed in this thesis effort. The inbound email customer contact center was modeled as a multiclass, open queueing network. The parametric decomposition (PD) method and its extensions presented in Whitt (1983, 1994) were used to solve the multiclass, open queueing network model. While the extensions presented in Whitt (1983, 1994) can handle multiple customer classes, the dependence of the processing times and routing probabilities on email history is not addressed by the PD method and its extensions. Modeling this dependence was a key contribution of this thesis, and details are discussed in the next chapter. The PD method is briefly explained next. 5.1 The Parametric Decomposition (PD) Method From the late 1950s to the mid 1980s, the analysis of queueing networks was dominated by the wellknown productform method (Baskett et al. 1975; Jackson 1957). The main problem with the productform analysis method and its extensions was the assumption of Poisson arrivals and exponential service times. There was no convenient mechanism for modeling the variability present in realworld processes. A fundamental change occurred 25 in the mid eighties. There was a paradigm shift from "exact analysis of an approximate model" to "approximate analysis of a more exact model (Whitt 1983).” The basic reason behind the distributional assumptions of the productform analysis was the tractability of the mathematical model. In the product form analysis, the focus was more on developing a model that can be solved exactly. The method made popular by the Queueing Network Analyzer (QNA) software developed at Bell Laboratories focused on the development of a more realistic model at the cost of our ability to solve the model exactly (Whitt 1983). An analysis method known as the parametric decomposition (PD) method based on twomoment queueing approximations (using mean and SCV  Squared Coefficient of Variation = Variance/mean2) became popular. The PD method was first proposed by Reiser and Kobayashi (1974) and subsequently extended by Kuehn (1979), Whitt (1983, 1994) and many others (see for example references in Suri et al. 1993). The PD method is the basis of many of the recent tools and techniques developed for queueing network analysis (Suri et al. 1993). The main reason for the success of the PD method is that it does not make any distributional assumption and uses only the mean and variance information of processing times and interarrival times. Through extensions to the PD method, several features relevant to real world systems have been incorporated including multiclass networks with deterministic routing, equipment breakdown and repair, changing lot sizes, inspection and testing, batch service, and overtime (Kamath et al. 1995, Suri et al. 1993). The PD method for a singleclass, open queueing network is based on 1) analysis of interactions between the nodes to obtain the mean and SCV of the interarrival time at each node, and 2) decomposition of the network into individual nodes and calculation of 26 node measures and network performance measures using GI/G/1 (general arrival, general service time distribution with single server) or (Whitt 1983) GI/G/m (general arrival, general service time distribution with multiple servers) approximations (Whitt 1993). The rate and the variability parameters of the combined (external plus internal) arrival processes approximately capture interactions among the nodes. The total arrival rate at each node can be obtained by solving the traffic flow rate equations, which represent the conservation of flow. Utilizations are calculated at each node to check system stability. The system is stable if the utilization at each node is strictly less than one. Up to this point, the analysis is similar to the one carried out in the product form analysis method (Jackson 1957) for solving open networks and involves no approximations. The SCVs of interarrival times at each node are calculated by solving the traffic variability equations which are linear. The traffic variability equations involve approximations for the basic network operations like a) flow through a node, b) merging of flow and c) splitting of flow. These approximations can be found in Whitt (1983, 1994).In calculating the performance measures, all nodes are assumed to be stochastically independent. The performance measures at each node are calculated from the GI/G/1 or GI/G/m results given in (Whitt 1983, 1993). 5.2 Aggregation Approaches The PD method described in the previous section essentially solves a singleclass network with Markovian routing probabilities. As explained in Whitt (1983, 1994), the general approach to solving multiclass networks is to aggregate the multiclass information (service and arrival) to define an aggregate singleclass network; solve the singleclass network using the PD method; and disaggregate to calculate classspecific 27 performance measures. Whitt (1983, 1994) presented extensions to the PD method to handle several classes of customers. Each customer class is described by a deterministic route. Whitt’s extension (1983, 1994) not only allows different service time parameters for different classes at a node, but also different service time parameters for different visits to the same node by the same class. Whit (1983) also mentioned that the routing could be probabilistic in the multiclass case. If so, then a routing probability matrix and parameters for the external arrival processes and node service times must be specified for each customer class. Whitt’s (1983) classbased aggregation method was modified to handle probabilistic routing in the case of the contact center model. This thesis effort has added a new extension to the PD method to treat an open queueing network in which routing probabilities and processing times depend on email history. This extension involves a new historybased aggregation step within each customer class before the classbased aggregation step. Details of this extension are presented in Chapter 6. 5.3 Numerical Validation The performance measures computed using the queueing network models were compared with steadystate simulation results obtained using an Arena 7.0 simulation model to evaluate the accuracy of the analytical results. The analytical results for different scenarios and different levels of service time SCVs were compare with the corresponding simulation estimates. Relative percentage error was used as an indication of the accuracy of the analytical model. Relative percentage error = 100% simulation estimate (analytical result simulation estimate) ⋅ − 28 The performance measures that were of interest include the average response time, average resolution time, agent utilizations, and the average number of emails in the system. The average resolution time is the average time an email spends in the system (customer and contact center) before eventual “resolution”. The parameters used for all the simulation experiments are presented in Table 5.1. The warmup period was determined by the application of Welch’s procedure (Welch 1983). Further details regarding the application of Welch’s procedure are contained in Appendix A3. Table 5.1: Simulation Parameters Number of Replications Warmup period (hours) Replication length (hours) 10 1,680 18,480 Table 5.2 presents the different levels of service time variability that were tested in all the scenarios. The details about the specific distributions used, and the procedure to calculate their parameters for the simulation model are given in Appendix A2. Table 5.2: SCV Levels and Corresponding Distributions SCV Distribution 0.25 4stage Erlang 1 Exponential 2.00 Hyperexponential 29 CHAPTER 6 A MULTICLASS OPEN QUEUEING NETWORK MODEL OF EMAIL CUSTOMER CONTACT CENTERS A multiclass open queueing network model was developed to model email contact centers. The nodes of the network represent customer service agents with the exception of one special delay node that models the elapsed time at the customer end. The routing probabilities as well as the processing time parameters could depend on email history. A novel historybased aggregation approach to model the dependence on email history was developed. The aggregation approach extends the popular parametric decomposition (PD) method for solving multiclass open queueing networks to more general situations. A discretetime Markov chain (DTMC) with an expanded state space was developed to model the nonMarkovian routing of a new email through the contact center until its eventual resolution. The analysis of this absorbing Markov chain allows the computation of the proportion of emails in an agent’s inbox that are new, previously processed by the same agent, or previously processed by another agent. Using these proportions, a new “historybased” aggregation step for each customer class was introduced. This step precedes the existing classbased aggregation step that extends the original PD method. The resulting queueing network model was solved using the RAQS software package that implements the PD method and its extensions. The accuracy and robustness of the 30 analytical model was demonstrated by comparing the analytical results with simulation estimates of performance measures for a variety of scenarios. The contact center considered was similar to the example that was the subject of an extensive simulation study in Greve et al. (2004). The model developed in this chapter does not consider the pooling of agents or service interruptions. These issues are addressed in Chapter 7. The remainder of this chapter is organized as follows. Section 6.1 gives a description of the contact center that was modeled. Section 6.2 describes some additional assumptions. Section 6.3 describes the modeling of the email contact center using a multiclass open queueing network. Section 6.4 presents approximate approaches to compute the weights for historybased aggregation. Section 6.5 presents the discretetime Markov chain model of the historybased email routing. The numerical experiments are presented in Section 6.6 and Section 6.7 presents a summary of the results and discussions. 6.1 Contact Center Description A contact center with multiple types of arriving emails is considered. Within each type, the emails received are identified, by software, as new or previously processed. If emails are new, they are routed with equal probability to one of the agents. If an email has been previously processed, then the agent who previously processed the email can be identified, and the email is routed to that agent. Alternatively, the email could also be routed to one of the agents randomly, just like a new email. Once the email is routed to an agent, the agent preprocesses it. Preprocessing involves reviewing the email type and its history. The history of the email can be any one of the following; the email can be brand new, processed by the same agent, or processed by a different agent. From this 31 information the agent determines whether to process the email or to forward the email to another agent. The time required to process an email is random and is influenced by both the email type and history. When the email response provided by an agent is sufficient to resolve the customer’s problem the email leaves the system permanently. If the email response is not enough to address the customer’s concerns, email returns to the system (as another email, e.g., a reply) after a random delay. This random delay represents the time for the customer to receive the agent’s response and send a reply. Without any loss of generality, we could assume for modeling purposes that “resolution” includes both the actual resolution of the customer’s problem and the customer’s decision to not pursue the problem resolution any further. The flowchart depicting the email handling logic is shown in Figure 6.1. 6.2 Assumptions • An individual agent processes emails in his/her inbox according to a first in first out (FIFO) queueing discipline. The FIFO discipline holds only for the emails in an agent’s inbox and not for the system. • For an unresolved problem that will be pursued by the customer, the email (response) enters the system after a random delay independent of the email processing history. • The preprocessing and processing times are independent random variables. 6.3 Modeling the EMail Contact Center Using an Open Queueing Network The situation explained in Section 6.1 was modeled as an open queueing network where the nodes represent the agents and customers represent emails. The open network has 32 (A +1) nodes where nodes 1 through A represent the customer service agents and node (A +1) represents a delay node. The delay node was used to model the time for the customer to receive a response and send a followup email. The different types of emails were modeled using different customer classes, each having their own arrival, routing, and service characteristics. For a given email type, the routing probabilities and the service time distributions depend on the history of the email. This dependence on email history is a feature that cannot be handled with the current extensions to the PD method. Hence, the basic idea behind the solution approach is as follows. If the history and classbased parameters were somehow aggregated into only classbased parameters, then Whitt’s (1983, 1994) method could be used to solve the multiclass open queueing network. The input for the PD method includes the number of nodes, the number of servers (one in this chapter) at each node, the SCVs of the external interarrival and service time distributions at each node, and the Markovian routing probability matrix. To perform the historybased aggregation within a customerclass, the following probability at each node or agent was needed. It is the probability that an email of type c currently at agent i was previously processed by agent k ( k = 0 represents a new email). These probabilities can also be thought of as “weights” to be used within the aggregation approach. A new technique was developed for computing these probabilities and it is presented in Section 6.5. In the remainder of this section the aggregation of the detailed parameters of the contact center model to yield the singleclass arrival, service, and routing parameters for solution using the PD method is discussed. The overall aggregation process is a twolevel technique, where the first level takes care of the 33 dependence on history within a class and the second level deals with the aggregation of classspecific information. IDENTIFY EMAIL TYPE USING SOFTWARE START RECEIVE EMAIL DIRECT EMAIL TO AN AGENT PREPROCESS EMAIL FORWARD EMAIL ? DELAY RESOLVED ? END PROCESS EMAIL YES NO YES NO Figure 6.1: EMail Handling Logic 34 A simple aggregation scheme was first developed to convert the class and nodespecific detailed routing probabilities and service time information into an approximately equivalent singleclass, nodespecific service time parameters and Markovian routing probabilities. The PD approach was then used to analyze the model. 6.3.1 Rate and SCV of the External Arrival Process at a Node First, we need to compute the rate and the SCV for the external arrival process at each node that represents an agent in the singleclass open queueing network model. The external arrivals correspond to the arrival of new emails. The delay node or node (A + 1) has no external arrivals because all previously processed emails (or customer replies) are considered to be internal arrivals as far as the open queueing network model is concerned. If new external emails of type c are routed to agent i with a fixed probability distribution, say, { (c), i 1,2, ..., A}, i α = then the classspecific and total external arrival rates at node i are given by (c) (c) (c) i i λ α λ ⋅ = ; Σ= = C c i i c 1 λ λ ( ) i =1, 2, ..., A The SCV calculations for the interarrival time for new, external emails at node i follow two steps. When a new email of type c enters the system, it is split according to the fixed probability distribution (c) i α . First, we obtain the SCV of the split arrival process of new type c emails at node i by assuming that the external arrival process is a renewal process. c2 (c) (c) c2 (c) 1 (c) oi i o i = α ⋅ + −α i =1, 2, ..., A At node i , the split arrival processes of new external emails of different types get merged. For the second step, we use results from Whitt (1983) to obtain the SCV of the (2) (3) 35 external arrivals at node i oi i hi i c2 =ψ ⋅ c2 + 1−ψ where Σ Σ = = ⋅ = C c i C c oi i hi c c c c c 1 1 2 2 ( ) ( ) ( ) λ λ i =1, 2, ..., A; = [1 + 4(1− )2 ⋅ ( −1)]−1; i i i ψ ρ χ 2 1 1 ( ) ( ) − = = ΣC c i i i c c λ λ χ ; and i ρ is the utilization of node i. It should be noted that the calculation of the SCV, 2 oi c , can be performed after i ρ is available from the solution of the traffic equations for the aggregate singleclass network. 6.3.2 Weights for Classbased Aggregation Whitt (1983) presented an approach to derive the parameters for an aggregate singleclass network in the case of a multiclass network with deterministic routes. We extend Whitt’s (1983) approach to a multiclass network with probabilistic routing. To compute the probabilities or weights for classbased aggregation, we need, (c), j γ the total (internal plus external) arrival rate of type c emails at node j. This can be obtained by solving traffic equations for a particular class. (c) j γ can be obtained by solving the following system of linear equations. ( ) ( ) ( ) ( ) , 1 1 c c c p c i j A j i i j j i ⋅ + = Σ+ ≠ = γ λ γ j = 1, 2, ..., A +1 For i, j =1, 2, ..., A; i ≠ j, the routing probability ( ) , p c i j from agent i to agent j for a type c email is given by ( ) ( , ) ( , ). , 0 , p c w c k p c k i j A k i j i Σ= = ⋅ For each class, the routing (4) (3) 36 probability from node i to the delay node (A +1) is obtained by aggregating the product of the probability that agent i processes the email and the probability that the problem is not resolved. Σ= + = ⋅ ⋅ − A k i A i i i p c w c k p c k q c k 0 , 1 ( ) ( , ) ( , ) (1 ( , )) i = 1, 2, ..., A The routing probability ( ) 1, p c A+ i depends on the routing strategy for previously processed emails. If previously processed emails (i.e., customer responses to agents’ emails) are routed to agent i with a fixed probability distribution {p i A} i , = 1, 2, ..., , then we have ( ) . A 1, i i p c = p + A special case of this strategy is to set all p s i ' to be the same. In this case, we have p 1/ A. i = If previously processed emails are routed to the agent that processed them, then the routing probability for a type c email from the delay node to agent i is equal to the probability that a type c email at the delay node came from agent i , i.e., ( ) ( , ). 1, 1 p c w c i A+ i A+ = Finally, we can compute the probabilities needed for the classbased aggregation as follows Σ= = C c i i i c c r c 1 ( ) ( ) ( ) γ γ i = 1, 2, ..., A +1 6.3.3 Markovian Routing Probabilities We calculate the Markovian routing probabilities for the singleclass open network by aggregating the classspecific probabilities computed in the previous section. The routing probabilities among the agent nodes are given by Σ= = ⋅ C c i j i i j p r c p c 1 , , ( ) ( ) i, j = 1, 2, ..., A; i ≠ j (5) (6) (7) 37 Similarly, the classindependent routing probability from agent node i to the delay node (A +1) is Σ= + + = ⋅ C c i A i i A p r c p c 1 , 1 , 1 ( ) ( ) i =1, 2, ..., A And the classindependent routing probability from the delay node (A +1) to agent node i is Σ= + + = ⋅ C c A i i A i p r c p c 1 1, 1, ( ) ( ) i = 1, 2, ..., A 6.3.4 Mean Service Time at a Node The overall service time is equal to the preprocessing time plus the actual processing time needed by the agent if the agent decides to process the email himself/herself. The overall service time at agent i for a new or previously processed type c email is given by T (c, k) P (c) Z (c, k) S (c, k) i i i i = + ⋅ i = 1, 2, ..., A; k =0,1,..., A The random variables P (c), i Z (c, k) i and S (c, k) i are assumed to be mutually independent. That is, the preprocessing time, the agent’s decision to process or forward the email, and the email processing time are all independent random variables. The mean service time at agent i for a new or previously processed type c email is given by E[T (c, k)] E[P (c)] E[Z (c, k)] E[S (c, k)] i i i i = + ⋅ i = 1, 2, ..., A; k =0,1,..., A Using the notation defined earlier we get t (c, k) (c) p (c, k) s (c, k) i i i i =θ + ⋅ i = 1, 2, ..., A; k =0,1,..., A (10) (11) (12) (8) (9) 38 Once again, the mean service time is calculated by using the twolevel aggregation method. The mean service time for a type c email at node/agent i is given by ( ) ( , ) ( , ) 0 t c w c k t c k i A k i i ⋅ =Σ= i = 1, 2, ..., A Next, the classspecific mean service times are aggregated to obtain the mean (overall) service time at node/agent i. Σ= = ⋅ C c i i i t r c t c 1 ( ) ( ) i = 1, 2, ..., A 6.3.5 Squared Coefficient of Variation of the Service Time Distribution at a Node For each class, the variance of the overall service time is first computed by using the fact that the effective processing time distribution is a mixture of processing time distributions for new and previously processed emails. The SCV of the overall service time is obtained by once again using the fact that it is a mixture of classspecific distributions. By using the property that the “second moment of a mixture of distributions is the mixture of the second moments (Whitt 1983)” the first step is to obtain the service time SCV at node/agent i for a type c email. ( ) ( ( ) 1) ( , ) 2 ( , ) ( 2 ( , ) 1) 0 2 2 + ⋅ ⋅ = + ⋅ Σ= t c c c w c k t c k c c k i i A k i i i i = 1, 2, ..., A where, c2 (c, k) i is obtained as follows T (c, k) P (c) Z (c, k) S (c, k) i i i i = + ⋅ i = 1, 2, ..., A Var(T (c, k)) Var(P (c) Z (c, k) S (c, k)) i i i i = + ⋅ Var(T (c, k)) Var(P (c)) Var(Z (c, k)) Var(S (c, k)) i i i i = + ⋅ Using the independence assumptions stated earlier, we have (13) (14) (15) 39 [ ( , )] ( ( , )) [ ( , )] ( ( , )) ( ( , )) ( ( )) ( ( , )) ( ( , )) E2 Z c k Var S c k E2 S c k Var Z c k Var T c k Var P c Var Z c k Var S c k i i i i i i i i ⋅ + ⋅ = + ⋅ + ( , ) ( , ) ( , ) ( , ) ( , ) [1 ( , )] ( ) ( ) ( , ) [1 ( , )] ( , ) ( , ) 2 2 2 , 2 2 2 , 2 2 , p c k c c k s c k s c k p c k p c k c c c p c k p c k c c k s c k i ser i i i i i prei i i i ser i i ⋅ ⋅ + ⋅ ⋅ − = ⋅θ + ⋅ − ⋅ ⋅ + ( , ) ( , ) [1 ( , )]} ( ) ( ) ( , ) ( , ) {[1 ( , )] ( , ) 2 , 2 , 2 2 2 , p c k c c k p c k c c c p c k s c k p c k c c k i ser i i pre i i i i i ser i ⋅ + − = ⋅θ + ⋅ ⋅ − ⋅ + ( ) ( ) ( , ) ( , ) { 2 ( , ) [1 ( , )]} , 2 2 2 , c c c p c k s c k c c k p c k pre i i i i ser i i = ⋅θ + ⋅ ⋅ + − Using the relation, Var(T (c, k)) c2 (c, k) t 2 (c, k) i i i = ⋅ have ( , ) ( ) ( ) ( , ) ( , ) { ( , ) [1 ( , )]} ( , ) 2 2 , 2 2 2 2 , t c k c c c p c k s c k c c k p c k c c k i pre i i i i ser i i i ⋅ + ⋅ ⋅ + − = θ By substituting c2 (c, k) i in equation [15] [1 ( , )] ) ( , )} ( ) ( ( ) 1) ( , ) { ( ) ( ) ( , ) ( , ).( ( , ) 2 2 , 2 2 0 2 , 2 2 p c k t c k t c c c w c k c c c p c k s c k c c k i i i i i ser i A k i i i pre i + − + ⋅ + ⋅ ⋅ = + ⋅ Σ= θ The second step is to aggregate the classspecific service time SCV, c2 (c) i to obtain the overall service time SCV, 2 i c at node/agent i is given by Σ Σ = = ⋅ ⋅ + ⋅ + = C c c C c c i i i i t c c c t c 1 1 2 2 2 2 ( ) ( ( ) 1) ( 1) λ λ i = 1, 2, ..., A 6.4 Approaches to Compute the Weights for HistoryBased Aggregation To demonstrate the importance of accurately modeling the dependence on the email history, two additional approximate methods to compute the weights w (c, k) i are presented here and included in the numerical experimentation. M1 is a naïve method, (17) (18) (16) 40 which involves nothing but a simple average. In method M1, w (c, k) i is equal to +1 1 A for all i, k and c , i.e., the probability that a type c email is new or previously processed by agent k remains the same irrespective of the email processing history. Similarly at the delay node (A+1), the probability that a type c email was previously processed by agent k , i.e., ( , ) 1 w c k A+ is equal to A 1 . This simple method was used in Chinnaswamy et al. (2004). M2 is a method with medium complexity, where w (c, k) i are calculated using aggregate resolution and forwarding probabilities. In method M2, the processing history of emails is taken into account in an approximate way for calculatingw (c, k). i As in method M1, the weights w (c, k) i are obtained for a type c new email and previously processed emails. Because of aggregation, the probability that email was previously processed by agent k is the same for k =1,2,..., A. Again because of aggregation, the probability that a type c email was previously processed by agent k, i.e., ( , ) 1 w c k A+ is equal to 1 . A M3 is the DTMCbased approach and it is presented in Section 6.5. In method M3, the complete history of an email is indirectly captured from its entry till its resolution. This information is used in calculating the weights w (c, k). i 6.4.1 Aggregation Method M2 In this section, the M2 method is explained for the two scenarios that will be considered later in numerical experimentation. Scenario 1 is the case when a previously processed email is equally likely to be routed to one of the agents. Scenario 2 is the case when a previously processed email is routed to the agent that processed it. Let p(c) represent 41 the aggregate forwarding probability and q(c) the aggregate resolution probability for class c. p(c) and q(c) are computed as follows. − − = ΣΣ ΣΣΣ = ≠ = = = ≠ = for Scenario 2 ( ,0) for Scenario 1 ( , ) ( ) 2 1 1 , 3 0 1 1 , A A p c A A p c k p c A i A i j j i j A k A i A i j j i j + = Σ ΣΣ = = = for Scenario 2 ( , ) for Scenario 1 ( , ) ( ) 1 2 0 1 A q c i A A q c k q c A i i A k A i i Figure 6.2 illustrates the flow of new and previouslyprocessed emails through the contact center from an aggregate point of view. Figure 6.2: Aggregation Method M2 (23) (24) (25) (26) Contact Center Resolved Previously processed, New, forwarded New, External λ(c) p(c) (1  p(c)).(1  q(c)) (1  p(c)). q(c) 42 Let β (c) represent the total (new plus previously processed) email arrival rate. From Figure 6.2 we can see that the following flow conversation relation needs to hold. β (c) = λ (c)+ β (c) ⋅{p(c)+(1− p(c)) ⋅ (1− q(c))} This yields (1 ( )) ( ) ( ) ( ) p c q c c c − ⋅ = λ β By focusing on the solid lines in Figure 6.2, we can see that Total arrival rate of new emails = (1 ( )) ( ) p c c − λ Total arrival rate of previously processed emails = (1 ( )) ( ) ( ) (1 ( )) p c q c c q c − ⋅ λ ⋅ − Hence, from (27) and (28), the proportion of new emails = q(c) And the proportion of previously processed emails = (1− q(c) ) The weights w (c, k) i are given by w c q c i A i ( ,0) = ( ) = 1,2,..., For Scenario 1, we have ( , ) (1 ( )) i , k 1,2,..., A; A w c k q c i = − = k A A w c k A ( , ) 1 1,2,..., 1 = = + For Scenario 2, we have = ≠ − = = = i k A i k q c i k A w c k i 0 , 1,2,..., ; 1 ( ) 1,2,..., ( , ) (29) (30) (27) (28) (32) (34) (33) (36) (31) (37) (35) 43 k A A w c k A ( , ) 1 1,2,..., 1 = = + 6.5 DiscreteTime Markov Chain Model of Historybased Email Routing One of the critical steps in the successful application of the solution approach is the historybased aggregation of routing and servicetime parameters within a customer class or email type. As explained in Section 6.3, the key element in this aggregation step is w (c, k), i the probability that a type c email currently at agent i was previously processed by agent k (k =0 represents a new email). This section presents the method M3, where the probability distribution {w (c, k), k 0,1, ..., A} i = is computed by analyzing a discretetime Markov chain (DTMC) model of the routing of an email during its entire lifecycle, i.e., from a new email to a series of agent responses and customer replies to eventual “resolution.” It is important to note that the dependence on history is limited to only the identity of the previous agent visited in the case of a previously processed email. The remainder of this section describes the definition and solution of this DTMC model. The statespace of the DTMC can be divided into internal and boundary states. The internal states can be described by the 3tuple (k, i, j) where k describes the email history; k = 0 represents a new email and 1 ≤ k ≤ A represents an email previously processed by agent k . i denotes the current location of the email, i.e., with agent i ; i = 1, 2, ..., A. 44 j denotes the current agent’s decision regarding the processing of the email; if agent i decides to process the email then j = i else agent i forwards the email to agent j ( j ≠ i) ; j = 1, 2, ..., A. The boundary states are represented by the set { , , , ..., , }, 1 2 N D D D R A where N represents the state in which all new emails begin their journey. R represents the state in which all emails end their journey. A D , D , ...,D 1 2 represents the state in which customer receives an agent’s response and sends a reply. The entire statespace of the DTMC is {N} {(k, i, j); k 0,1, 2, ..., A, i 1, 2, ..., A, j 1, 2, ..., A,} {D ; i 1, 2, ..., A} {R}. i ∪ = = = ∪ = ∪ The size of the state space is equal to (A + 1) ⋅ A⋅ A + A + 2 or A3 + A2 + A + 2. A DTMC with (A3 + A2 + A + 2) states for each customer class or email type was constructed and the possible state transitions were defined in order to construct the onestep transition probability matrix for the DTMC. First, transitions within internal states are considered. Such transitions occur only as a result of one agent forwarding an email to another. The recipient can forward it again or decide to process it. ∀(k, i, j) where j ≠ i ( ) ( , , ) ,( , * , * ) P c k i j k i j = = = = = ≠ 0 otherwise ( , ) ; ( , ) ; 1,2,..., ; * * * * * , * p c k i j j j p c k i j j A j j j j j (38) 45 Next, transitions from the internal states to the boundary states are considered. These occur when an agent decides to process an email. If the email is resolved, then the transition is to the resolved state R ; otherwise it is to the delay node i D . ∀(k, i, j) where j = i ( ) ( , ) ( , , ) , P c q c k k i i R i = ( ) 1 ( , ) ( , , ) , P c q c k k i i Di i = − P c j A k i i k i j ( ) 0 ; * 1, 2, ..., ( , , ) , ( , , * ) = = Next, the transitions from the boundary states to the internal states are considered. (a) From state N to an internal state ( ) , ( , , ) P c N k i j = ≠ = ⋅ = = = ⋅ = ≠ = k i j A c p c k i j A j i k c p c k i j A j i k i i i i j 0 0; , 1,2,..., ( ) ( , ) , 1,2,..., ; ; 0 ( ) ( , ) , 1,2,..., ; ; 0 , α α (b) From state D (i 1,2,..., A) i = to an internal state. (i) Previously processed emails are routed to agent i* with a fixed probability distribution { , * 1, 2,..., } * p i A i = independent of processing history. ( ) , ( , * , ) P c Di k i j ∗ = ⋅ = = = ⋅ = = ≠ 0 otherwise ( , ) ; , 1, 2,..., ; ( , ) ; , 1, 2,..., ; * * * * * * * * , * * * * * p p c k k i i j A j i p p c k k i i j A j i i i i i j (ii) Previously processed emails are routed to the agent that processed them. ( ) , ( , * , * ) P c Di k i j = = = = = = = ≠ 0 otherwise ( , ) ; ; ( , ) ; ; 1, 2,..., ; * * * * * * , * * * p c k k i i i j i p c k k i i i j A j i i i j (40) (39) (41) (42) (44) (43) 46 Finally the transitions within the boundary states are considered. The only possible transition is R,R = 1 P . All other transitions probabilities from one boundary state to the same state or any other boundary state is zero. Using the transition probabilities, a (n' × n' ) onestep transition probability matrix P, where n' = A3 + A2 + A + 2 , was defined. Because state R is an absorbing state, the DTMC represented by P is a reducible chain with all states except state R being transient. Next, the focus is on the analysis of this absorbing Markov chain. 6.5.1 Analysis of the Absorbing Markov Chain The DTMC defined in the previous section consists of an absorbing state and (A3 + A2 + A +1) transient states. The onestep transition probability matrix, P, can be rearranged in such a way that the absorbing state is the first state, without any loss of generality. States are arranged in the following way  resolution node R , followed by A delay nodes numbered from A D ,D , ..., D 1 2 , followed by the new node N and finally, the internal nodes (k,i, j). U 0 1 row M Q A3 + A2 + A +1 rows 1 column A3 + A2 + A +1 columns Figure 6.3: OneStep Transition Probability Matrix P = 47 By analyzing this absorbing DTMC, the average number of visits an email makes to each node before getting resolved or absorbed can be computed. The steps outlined in Ramakumar (1993) were followed to obtain the fundamental matrix F associated with the absorbing Markov chain. The fundamental matrix F is given by F = [I  Q]1 The row of the Fundamental matrix, F that corresponds to the state N gives the average number of visits an email makes to the all other states before absorption. The average number of visits an email that was last processed by agent k (k =0 is a new email) makes to agent i is given by Σ= = A j i N k i j v c k F c 1 , ( , , ) ( , ) ( ) The probability that a type c email received by agent i was previously processed by agent k (k =0 is a new email) is given by Σ= = A k i i i v c k v c k w c k 0 ( , ) ( , ) ( , ) i = 1, 2, ..., A; k =0,1, ..., A; The above results can be easily derived by observing that the arrival rate of type c emails with a processing history k at agent i is proportional to v (c, k). i Similarly, the probability that a type c email received at the delay node (A +1)was previously processed by agent k is given by k A c k c k w c k A j D D A j k 1, 2,..., ( , ) ( , ) ( , ) 1 1 = = Σ= + ν ν where ( ) ( ) , , c F c Dj j N Dj ν = 6.5.2 Performance Evaluation of the EMail Contact Center Figure 6.4 explain the stepbystep procedure followed to obtain the performance measures of an inbound email contact center. It was assumed that the forwarding (45) (46) (47) 48 probabilities, resolution probabilities and service time parameters for a particular type email could be obtained from the available email contact center data. It was assumed that these parameters could be dependent on email history, i.e., the last agent, k who previously processed the email, (k =0 for new emails). The external arrival rates and SCVs for different email types, the forwarding and resolution probabilities, number of agents along with the mean service time and servicetime SCV for each agent were the inputs for the analytical model. These inputs known as parameters were used for the entire aggregation procedure. For a particular email type the onestep, historyembedded transition probability matrix was constructed based on possible transitions made by an email between the agents’ inboxes and the delay nodes representing customers. The construction of the above matrix was based on the number of agents in the contact center and historyembedded resolution and forwarding probabilities of an email. A truncated portion of the onestep transition probability matrix, i.e., Q then was inverted using MATLAB 7.0.1 (Lipsman 2001). The truncated matrix was based on the number of absorbing nodes (one in this case). The inverted matrix gives the average number of visits an email makes to a node before absorption or problem “resolution.” In particular, the row corresponding to the new email node gives the average number of visits a new email makes to agents’ inboxes and to the customer inbox (modeled as delay node) before getting resolved. These visits were then converted into probabilities or weights of new and previously processed emails as seen by an agent for a known email type. These weights were used in the historybased aggregation procedure to obtain parameters that are based on only the class or email type information. For the classbased aggregation, the external arrival rates and parameters obtained from the historybased 49 aggregation were used. The classbased aggregation was done according to the method outlined in Whitt (1983) in order to obtain the routing matrix, means and SCVs of service times, and interarrival time SCVs at each agent. These were then fed into RAQS software package to obtain the performance measures for the email contact center. Many steps in the overall procedure, including calculation of the onestep transition probability matrix, calculation of weights after solving the DTMC, historybased aggregation, and classbased aggregation were coded in EXCEL spread sheets. 50 EMPIRICAL CONTACT CENTER DATA pij(c, k), qi(c, k), mean and SCVs of preprocessing and processing times, number of agents, number of email types, external arrival rates and SCVs pij(c, k), qi(c, k), number of agents wik(c) pij(c), qi(c), ti(c), ci 2(c), coi 2(c) pij, ti, ci 2, coi 2 INPUT TO THE PD METHOD Number of nodes, mean and SCV of service time at each node, external arrival rates and SCVs, probability routing matrix CONTACT CENTER PERFORMANCE MEASURES HISTORYBASED AGGREGATION OBTAINED BY SOLVING THE DTMC CLASSBASED AGGREGATION SOLVED USING RAQS PARAMETRIC SOFTWARE DECOMPOSITION METHOD Historybased aggregation for each email type Classbased aggregation for each email type Figure 6.4: Contact Center Performance Evaluation 51 6.6 Numerical Experiments One of the important tasks in the numerical evaluation of the accuracy and robustness of the approach is to construct a contact center example that is representative of realworld contact center operations. The parameter selection approach was based on email contact center characteristics that are available in the public domain. Details are described in the next section. 6.6.1 Email Contact Center Characteristics Survey data available at “BenchmarkPortal.com” was used to gain an initial understanding of a typical email response center. Each month, “BenchmarkPortal.com” sends 1,000 surveys to randomly selected customer response centers, and typically receives a 15 to 25 % response rate. In one such survey, 53% of surveyed contact centers indicated that 80100% of email contacts were resolved with the first response (Benchmark Portal 2004b). From this statistic an estimate for the resolution probability for a typical email agent was obtained. Also, 34.62% of surveyed contact centers indicated that their average email response time was less than 6 hours. Another 15.38% indicated an average response time of 6 to 12 hours, and 35.38% indicated an average response time of 12 to 24 hours (Benchmark Portal 2004c). Also, the capacity of a dedicated email associate is comparable to that of a telephone agent (Benchmark Portal 2004a). Given an eighthour day, this translates to between 2.4 and 9.6 minutes of an email agent’s time being spent on each email response. Based on the above information and the contact center example analyzed in Greve et al. (2004) and Chinnaswamy et al. (2004), the parameter values for a small contact center were defined. 52 6.6.2 Email Contact Center Example A contact center with three agents that handles two types of emails was considered. We have A = 3 agents and C = 2 . Although the analytical approach can handle general arrival processes, Poisson arrivals were assumed for this example. The arrival rates for the two types of emails were 6.0 / hr 1 λ = and 4.5 / hr. 2 λ = A new email of either type was assumed to be equally likely to be routed to one of the agents. Two scenarios were defined to model two different strategies with regard to the handling of previously processed emails. In Scenario 1, a previouslyprocessed email was routed to any one of the agents with equal probability. In Scenario 2, a previouslyprocessed email was routed to the agent that processed it. Hence, we have ( ) ( ) 0.33; ( ) 0.34 1 2 3 α c =α c = α c = for c = 1,2. The preprocessing time distribution for both email types at all three agents was assumed to be the uniform distribution. Hence, P (c) ~ i Uniform (0.01, 0.50) for i = 1,2,3 ; c = 1,2.With regard to forwarding probabilities, sample values were generated using uniform (0, 0.25) for the two email types. While assigning the sampled values, it was assumed that an agent is more likely to forward an email previously processed by another agent. The complete set of forwarding probabilities is presented in Table 6.2 for Scenario 1 and Table 6.7 for Scenario 2. In the case of resolution probabilities, samples from the following uniform distributions were used. • Type 1, New  Uniform (0.60, 0.90) • Type 1, Previously processed  Uniform (0.75, 0.95) • Type 2, New  Uniform (0.50, 0.80) • Type 2, Previously processed  Uniform (0.70, 0.90) 53 While assigning the sampled probabilities, it was assumed that an agent was more likely to resolve an email previously processed by them. The full set of resolution probabilities are summarized in Table 6.3 for Scenario 1 and Table 6.8 for Scenario 2. To test the robustness of the approach developed, experiments with three processingtime distribution were chosen; Erlang (SCV =0.25) to represent low variability situations, exponential (SCV =1.00) to represent medium variability situations, and Hyperexponential (SCV = 2.00) to represent high variability situations. To capture the dependence on the history and class, samples for the mean processing times were drawn from the following uniform distributions. • Type 1, New  Uniform (0.10, 0.20) • Type 1, Previously processed  Uniform (0.05, 0.15) • Type 2, New  Uniform (0.15, 0.25) • Type 2, Previously processed  Uniform (0.10, 0.20) While assigning the sampled mean processing times, it was assumed that the mean would be lower for emails previously processed by the same agent. The complete set of mean processing times can be viewed in Table 6.1 for Scenario1 and Table 6.6 for Scenario 2. Finally, the delay time was assumed to be exponentially distributed with a mean of 4 hours, i.e., Di ~ Exponential (4) for i = 1,2,3. With regard to the routing of previously processed emails, two scenarios were considered as described earlier. 6.6.3 Probabilities for HistoryBased Aggregation obtained from the DTMC In this section, the probabilities or weights used for historybased aggregation obtained from the fundamental matrix are presented for both Scenarios 1 and 2. Tables 6.4 and 6.5 give the average number of visits and probabilities for Scenario 1 obtained using the 54 analytical method M3 for Type 1 and 2 emails, respectively. Tables 6.9 and 6.10 give the average number of visits and probabilities for Scenario 2 obtained using the analytical method M3 for Type 1 and 2 emails, respectively. The reader is referred to Appendix A1; Tables A1.1 through A1.18 for complete details related to the fundamental matrix F. Table 6.1: Scenario 1  Mean Processing Times ( ) , s c i k (c,k) c = 1 c = 2 i k=0 k=1 k=2 k=3 k=0 k=1 k=2 k=3 1 0.15 0.11 0.16 0.15 0.18 0.06 0.13 0.10 2 0.19 0.13 0.15 0.14 0.20 0.11 0.07 0.14 3 0.14 0.20 0.18 0.13 0.17 0.08 0.13 0.07 Table 6.2: Scenario 1  Forwarding Probabilities ( , ) , p c k i j (c,k) c = 1 c = 2 (i,j) k=0 k=1 k=2 k=3 k=0 k=1 k=2 k=3 (1,2) 0.01 0.02 0.08 0.19 0.06 0.03 0.03 0.12 (1,3) 0.14 0.02 0.24 0.12 0.05 0.06 0.14 0.13 (2,1) 0.05 0.24 0.01 0.14 0.13 0.13 0.00 0.15 (2,3) 0.03 0.10 0.06 0.25 0.10 0.09 0.03 0.13 (3,1) 0.08 0.24 0.08 0.01 0.11 0.15 0.14 0.07 (3,2) 0.10 0.09 0.10 0.07 0.02 0.13 0.11 0.01 Table 6.3: Scenario 1  Resolution Probabilities q (c, k) i (c,k) c = 1 c = 2 i k=0 k=1 k=2 k=3 k=0 k=1 k=2 k=3 1 0.71 0.88 0.79 0.74 0.75 0.94 0.76 0.78 2 0.74 0.85 0.85 0.85 0.78 0.89 0.95 0.81 3 0.72 0.87 0.77 0.88 0.79 0.78 0.83 0.86 55 Table 6.4: Scenario 1, Type 1 Emails  Aggregation Probabilities for Method M3 011 0.32392000 021 0.01871300 031 0.03236600 012 0.00381080 022 0.34433000 032 0.04045800 013 0.05335100 023 0.01122800 033 0.33175000 v1,0(1) 0.38108180 v2,0(1) 0.37427100 v3,0(1) 0.40457400 111 0.05622500 121 0.01027700 131 0.01062300 112 0.00117140 122 0.02826300 132 0.00398360 113 0.00117140 123 0.00428230 133 0.02965600 v1,1(1) 0.05856780 v2,1(1) 0.04282230 v3,1(1) 0.04426260 211 0.02622600 221 0.00042175 231 0.00377580 212 0.00308540 222 0.03922300 232 0.00471980 213 0.00925610 223 0.00253050 233 0.03870200 v1,2(1) 0.03856750 v2,2(1) 0.04217525 v3,2(1) 0.04719760 311 0.03059200 321 0.00688920 331 0.00055631 312 0.00842390 322 0.03001700 332 0.00389420 313 0.00532030 323 0.01230200 333 0.05118100 v1,3(1) 0.04433620 v2,3(1) 0.04920820 v3,3(1) 0.05563151 overall sum 0.52255330 overall sum 0.50847675 overall sum 0.55166571 w1,0(1) 0.72926877 w2,0(1) 0.73606315 w3,0(1) 0.73336804 w1,1(1) 0.11208005 w2,1(1) 0.08421683 w3,1(1) 0.08023446 w1,2(1) 0.07380587 w2,2(1) 0.08294430 w3,2(1) 0.08555471 w1,3(1) 0.08484532 w2,3(1) 0.09677571 w3,3(1) 0.10084279 Email Type 1  Weights from the Fundamental Matrix Table 6.5: Scenario 1, Type 2 Emails  Aggregation Probabilities for Method M3 011 0.37462000 021 0.04721600 031 0.04371000 012 0.02525600 022 0.27967000 032 0.00794730 013 0.02104600 023 0.03632000 033 0.34571000 v1,0(2) 0.42092200 v2,0(2) 0.36320600 v3,0(2) 0.39736730 111 0.04334700 121 0.00554470 131 0.00650410 112 0.00142900 122 0.03326800 132 0.00563690 113 0.00285810 123 0.00383860 133 0.03122000 v1,1(2) 0.04763410 v2,1(2) 0.04265130 v3,1(2) 0.04336100 211 0.02284600 221 0.00000000 231 0.00404220 212 0.00082576 222 0.02666000 232 0.00317600 213 0.00385350 223 0.00082454 233 0.02165500 v1,2(2) 0.02752526 v2,2(2) 0.02748454 v3,2(2) 0.02887320 311 0.02770400 321 0.00508950 331 0.00274400 312 0.00443260 322 0.02442900 332 0.00039200 313 0.00480200 323 0.00441090 333 0.03606400 v1,3(2) 0.03693860 v2,3(2) 0.03392940 v3,3(2) 0.03920000 overall sum 0.53301996 overall sum 0.46727124 overall sum 0.50880150 w1,0(2) 0.78969275 w2,0(2) 0.77729158 w3,0(2) 0.78098689 w1,1(2) 0.08936645 w2,1(2) 0.09127739 w3,1(2) 0.08522184 w1,2(2) 0.05164020 w2,2(2) 0.05881924 w3,2(2) 0.05674747 w1,3(2) 0.06930059 w2,3(2) 0.07261179 w3,3(2) 0.07704380 Email Type 2  Weights from the Fundamental Matrix 56 Table 6.6: Scenario 2  Mean Processing Times ( ) , s c i k (c,k) c = 1 c = 2 i k=0 k=1 k=2 k=3 k=0 k=1 k=2 k=3 1 0.15 0.11 0.00 0.00 0.18 0.06 0.00 0.00 2 0.19 0.00 0.15 0.00 0.20 0.00 0.07 0.00 3 0.14 0.00 0.00 0.13 0.17 0.00 0.00 0.07 Table 6.7: Scenario 2  Forwarding Probabilities ( , ) , p c k i j (c,k) c = 1 c = 2 (i,j) k=0 k=1 k=2 k=3 k=0 k=1 k=2 k=3 (1,2) 0.01 0.00 0.00 0.00 0.06 0.00 0.00 0.00 (1,3) 0.14 0.00 0.00 0.00 0.05 0.00 0.00 0.00 (2,1) 0.05 0.00 0.00 0.00 0.13 0.00 0.00 0.00 (2,3) 0.03 0.00 0.00 0.00 0.10 0.00 0.00 0.00 (3,1) 0.08 0.00 0.00 0.00 0.11 0.00 0.00 0.00 (3,2) 0.10 0.00 0.00 0.00 0.02 0.00 0.00 0.00 Table 6.8: Scenario 2  Resolution Probabilities q (c, k) i (c,k) c = 1 c = 2 i k=0 k=1 k=2 k=3 k=0 k=1 k=2 k=3 1 0.71 0.88 0.00 0.00 0.75 0.94 0.00 0.00 2 0.74 0.00 0.85 0.00 0.78 0.00 0.95 0.00 3 0.72 0.00 0.00 0.88 0.79 0.00 0.00 0.86 57 Table 6.9: Scenario 2, Type 1 Emails  Aggregation Probabilities for Method M3 011 0.32392000 021 0.01871300 031 0.03236600 012 0.00381080 022 0.34433000 032 0.04045800 013 0.05335100 023 0.01122800 033 0.33175000 v1,0(1) 0.38108180 v2,0(1) 0.37427100 v3,0(1) 0.40457400 111 0.10675000 121 0.00000000 131 0.00000000 112 0.00000000 122 0.00000000 132 0.00000000 113 0.00000000 123 0.00000000 133 0.00000000 v1,1(1) 0.10675000 v2,1(1) 0.00000000 v3,1(1) 0.00000000 211 0.00000000 221 0.00000000 231 0.00000000 212 0.00000000 222 0.10532000 232 0.00000000 213 0.00000000 223 0.00000000 233 0.00000000 v1,2(1) 0.00000000 v2,2(1) 0.10532000 v3,2(1) 0.00000000 311 0.00000000 321 0.00000000 331 0.00000000 312 0.00000000 322 0.00000000 332 0.00000000 313 0.00000000 323 0.00000000 333 0.10556000 sum 0.00000000 0.00000000 0.10556000 overall sum 0.48783180 overall sum 0.47959100 overall sum 0.51013400 w1,0(1) 0.78117458 w2,0(1) 0.78039621 w3,0(1) 0.79307398 w1,1(1) 0.21882542 w2,1(1) 0.00000000 w3,1(1) 0.00000000 w1,2(1) 0.00000000 w2,2(1) 0.21960379 w3,2(1) 0.00000000 w1,3(1) 0.00000000 w2,3(1) 0.00000000 w3,3(1) 0.20692602 Email Type 1  Weights from solving the DiscreteTime Markov Chain Table 6.10: Scenario 2, Type 2 Emails  Aggregation Probabilities for Method M3 011 0.37462000 021 0.04721600 031 0.04371000 012 0.02525600 022 0.27967000 032 0.00794730 013 0.02104600 023 0.03632000 033 0.34571000 v1,0(2) 0.42092200 v2,0(2) 0.36320600 v3,0(2) 0.39736730 111 0.09963400 121 0.00000000 131 0.00000000 112 0.00000000 122 0.00000000 132 0.00000000 113 0.00000000 123 0.00000000 133 0.00000000 v1,1(2) 0.09963400 v2,1(2) 0.00000000 v3,1(2) 0.00000000 211 0.00000000 221 0.00000000 231 0.00000000 212 0.00000000 222 0.06476500 232 0.00000000 213 0.00000000 223 0.00000000 233 0.00000000 v1,2(2) 0.00000000 v2,2(2) 0.06476500 v3,2(2) 0.00000000 311 0.00000000 321 0.00000000 331 0.00000000 312 0.00000000 322 0.00000000 332 0.00000000 313 0.00000000 323 0.00000000 333 0.08441700 v1,3(2) 0.00000000 v2,3(2) 0.00000000 v3,3(2) 0.08441700 overall sum 0.52055600 overall sum 0.42797100 overall sum 0.48178430 w1,0(2) 0.80860080 w2,0(2) 0.84866965 w3,0(2) 0.82478258 w1,1(2) 0.19139920 w2,1(2) 0.00000000 w3,1(2) 0.00000000 w1,2(2) 0.00000000 w2,2(2) 0.15133035 w3,2(2) 0.00000000 w1,3(2) 0.00000000 w2,3(2) 0.00000000 w3,3(2) 0.17521742 Email Type 2  Weights from solving the DiscreteTime Markov Chain 58 The next section presents the performance evaluation of the contact center example using both analytical and simulation models and examines the accuracy of the analytical results. 6.7 Results and Discussions The aggregated singleclass open queueing network model was solved using the RAQS software package (Kamath et al. 1995). RAQS is a software package for analyzing general queueing network models using the PD method (http://www.okstate.edu/cocim/raqs/). Simulation estimates represent averages over ten independent replications. Each replication was simulated for 18,480 hours of operation with a warm up of 1,680 hours. The analytical and simulation results are shown for three processing time distributions  Erlang (SCV =0.25), exponential (SCV =1.00), and Hyperexponential (SCV = 2.00). 6.7.1 Results The analytical and simulation results are presented in Tables 6.11 and 6.12 and in Figures 6.6 through 6.10 for Scenario 1, where a previously processed email is routed to any one of the agents with equal probability. Tables 6.13 and 6.14 and Figures 6.11 through 6.15 shows the results for Scenario 2, where a previously processed email is routed to the agent that processed it. The quantity in parentheses below a simulation estimate is the halfwidth of the 95% confidence interval. The quantity in parentheses below an analytical value is the relative percentage error. The columns labeled M1 contain analytical results obtained using the analytical method M1, i.e., the naïve method based on a simple average. The columns labeled M2 contain analytical results obtained using the analytical method M2, i.e., the approximate method based on aggregate forwarding 59 and resolution probabilities. Finally, the column labeled M3 contain analytical results obtained using the analytical method M3, i.e., the DTMCbased method for computing the probabilities /weights for historybased aggregation. Table 6.11: Scenario 1  Nodelevel Performance Measures M1 M2 M3 Sim M1 M2 M3 Sim M1 M2 M3 Sim 0.735 0.910 0.890 0.887 0.264 1.091 0.866 0.897 1.423 6.080 4.800 4.962 (17.14%) (+2.59%) (+0.34%) (±0.003) (70.57%) (+21.63%) (3.46%) (±0.045) (71.32%) (+22.53%) (3.26%) (±0.258) Erlang 0.728 0.951 0.923 0.921 0.276 2.438 1.476 1.479 1.400 12.659 7.621 7.618 (20.96%) (+3.26%) (0.22%) (±0.003) (81.34%) (+64.84%) (0.20%) (±0.073) (81.62%) (+66.17%) (+0.04%) (±0.391) 0.793 0.876 0.865 0.863 0.393 0.740 0.669 0.689 2.169 4.176 3.752 3.861 (8.11%) (+1.51%) (+0.23%) (±0.002) (42.96%) (+7.40%) (2.90%) (±0.018) (43.82%) (+8.16%) (2.82%) (±0.108) 0.735 0.910 0.890 0.888 0.378 1.591 1.261 1.237 2.043 8.870 6.992 6.850 (17.23%) (+2.48%) (+0.22%) (±0.004) (69.44%) (+28.62%) (+1.94%) (±0.034) (70.18%) (+29.49%) (+2.07%) (±0.196) Exponential 0.728 0.951 0.923 0.921 0.399 3.588 2.170 2.147 2.023 18.631 11.201 11.082 (20.96%) (+3.26%) (+0.22%) (±0.004) (81.42%) (+67.12%) (+1.07%) (±0.162) (81.75%) (+68.12%) (+1.07%) (±0.872) 0.793 0.876 0.865 0.864 0.569 1.074 0.971 0.982 3.139 6.065 5.447 5.507 (8.22%) (+1.39%) (+0.12%) (±0.003) (42.06%) (+9.37%) (1.12%) (±0.039) (43.00%) (+10.13%) (1.09%) (±0.224) 0.735 0.910 0.890 0.892 0.531 2.258 1.788 1.831 2.868 12.587 9.913 10.161 (17.60%) (+2.02%) (0.22%) (±0.004) (71.00%) (+23.32%) (2.35%) (±0.104) (71.77%) (+23.88%) (2.44%) (±0.586) Hyperexponential 0.728 0.951 0.923 0.924 0.563 5.122 3.094 3.061 2.854 26.593 15.973 15.809 (21.21%) (+2.92%) (0.11%) (±0.003) (81.61%) (+67.33%) (+1.08%) (±0.223) (81.95%) (+68.21%) (+1.04%) (±1.155) 0.793 0.876 0.865 0.862 0.803 1.520 1.374 1.317 4.430 8.583 7.706 7.389 (8.00%) (+1.62%) (+0.35%) (±0.004) (39.03%) (+15.41%) (+4.33%) (±0.059) (40.05%) (+16.16%) (+4.29%) (±0.332) Processing Time Distribution Node (Agent) Utilization Average Queueing Delay (hours) Average Number in Queue 1 2 3 1 2 3 1 2 3 60 Table 6.12: Scenario 1  Systemlevel Performance Measures M1 M2 M3 Sim M1 M2 M3 Sim M1 M2 M3 Sim 0.562 1.854 1.375 1.398 1.605 3.713 3.017 3.030 16.858 38.990 31.681 31.803 (59.80%) (+32.62%) (1.64%) (±0.029) (47.03%) (+22.54%) (0.43%) (±0.040) (46.99%) (+22.60%) (0.38%) (±0.029) 0.733 2.624 1.920 1.907 1.816 4.728 3.728 3.698 19.070 49.641 39.147 38.861 (61.56%) (+37.60%) (+0.68%) (±0.077) (50.89%) (+27.85%) (+0.81%) (±0.098) (50.93%) (+27.74%) (+0.74%) (±1.097) 0.962 3.650 2.646 2.630 2.097 6.080 4.676 4.646 22.017 63.838 49.099 48.826 (63.42%) (+38.78%) (+0.61%) (±0.106) (54.86%) (+30.87%) (+0.65%) (±0.139) (54.91%) (+30.75%) (+0.56%) (±0.106) Exponential Hyperexponential Processing Time Distribution Average Response Time (hours) Average Resolution Time (hours) Erlang Average Number of Emails in the System 61 62 Agent Utilization (Erlang Processing Times) 0.000 0.200 0.400 0.600 0.800 1.000 1 2 3 Agent Number Utilization (%) M1 M2 M3 Sim Figure 6.6: Scenario 1  Agent Utilization  Erlang Processing Time Average Response Time 0.000 0.500 1.000 1.500 2.000 2.500 3.000 3.500 4.000 Erlang Exponential Hyperexponential Processing Time Distribution Average Response Time M1 M2 M3 Sim Figure 6.7: Scenario 1  Average Response Time 63 Average Resolution Time 0.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 Erlang Exponential Hyperexponential Processing Time Distribution Average Resolution Time M1 M2 M3 Sim Figure 6.8: Scenario 1  Average Resolution Time Average Number of Emails in the System 0 10 20 30 40 50 60 70 Erlang Exponential Hyperexponential Processing Time Distribution Average Number of Emails in the System M1 M2 M3 Sim Figure 6.9: Scenario 1  Average Number of Emails in the System Table 6.13: Scenario 2  Nodelevel Performance Measures M1 M2 M3 Sim M1 M2 M3 Sim M1 M2 M3 Sim 1 0.670 0.864 0.853 0.853 0.195 0.687 0.617 0.654 0.905 3.610 3.261 3.460 (21.45%) (+1.29%) (0.00%) (±0.003) (70.18%) (+5.05%) (5.66%) (±0.024) (73.84%) (+4.33%) (5.75%) (±0.133) Erlang 2 0.747 0.910 0.906 0.905 0.335 1.269 1.202 1.260 1.491 6.216 5.786 6.065 (17.46%) (+0.55%) (+0.11%) (±0.002) (73.41%) (+0.71%) (4.60%) (±0.053) (75.42%) (+2.49%) (4.60%) (±0.263) 3 0.684 0.822 0.811 0.808 0.206 0.476 0.437 0.440 0.964 2.495 2.292 2.230 (15.35%) (+1.73%) (+0.37%) (±0.003) (53.18%) (+8.18%) (0.68%) (±0.014) (56.77%) (+11.88%) (+2.78%) (±0.079) 1 0.670 0.864 0.853 0.853 0.279 0.999 0.895 0.894 1.295 5.248 4.731 4.720 (21.45%) (+1.29%) (0.00%) (±0.003) (68.79%) (+11.74%) (+0.11%) (±0.037) (72.56%) (+11.19%) (+0.23%) (±0.210) Exponential 2 0.747 0.910 0.906 0.903 0.488 1.865 1.765 1.760 2.168 9.133 8.499 8.463 (17.28%) (+0.77%) (+0.33%) (±0.004) (72.27%) (+5.97%) (+0.28%) (±0.162) (74.38%) (+7.92%) (+0.42%) (±0.794) 3 0.684 0.822 0.811 0.808 0.294 0.687 0.631 0.631 1.377 3.606 3.308 3.300 (15.35%) (+1.73%) (+0.37%) (±0.003) (53.41%) (+8.87%) (0.00%) (±0.018) (58.27%) (+9.27%) (+0.24%) (±0.100) 1 0.670 0.864 0.853 0.850 0.391 1.415 1.266 1.202 1.815 7.432 6.689 6.338 (21.18%) (+1.65%) (+0.35%) (±0.005) (67.47%) (+17.72%) (+5.32%) (±0.076) (71.36%) (+17.26%) (+5.54%) (±0.421) Hyperexponential 2 0.747 0.910 0.906 0.904 0.691 2.659 2.516 2.415 3.071 13.021 12.115 11.601 (17.37%) (+0.66%) (+0.22%) (±0 



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