Call centres with balking and abandonment: from queueing to

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Call centres with balking and abandonment: from
queueing to queueing network models
A Thesis Submitted to the
College of Graduate Studies and Research
in Partial Ful(cid:12)llment of the Requirements
for the Degree of
Doctor of Philosophy
in the
Department of Mathematics and Statistics
University of Saskatchewan
Saskatoon, Saskatchewan
By
Zhidong Zhang
June 2010
c Zhidong Zhang, June 2010. All rights reserved.
(cid:13)
Permission to Use
In presenting this thesis in partial ful(cid:12)lment of the requirements for a Postgraduate
degree from the University of Saskatchewan, I agree that the Libraries of this University
may make it freely available for inspection. I further agree that permission for copying
of this thesis in any manner, in whole or in part, for scholarly purposes may be granted
by the professor or professors who supervised my thesis work or, in their absence, by the
Head of the Department or the Dean of the College in which my thesis work was done.
It is understood that any copying or publication or use of this thesis or parts thereof for
(cid:12)nancial gain shall not be allowed without my written permission. It is also understood
that due recognition shall be given to me and to the University of Saskatchewan in any
scholarly use which may be made of any material in my thesis.
Requests for permission to copy or to make other use of material in this thesis in whole
or part should be addressed to:
Head of the Department of Mathematics and Statistics
University of Saskatchewan
Saskatoon, Saskatchewan
Canada
S7N 5E6
i
Abstract
The research on call centres has attracted many researchers from di(cid:11)erent disciplines re-
cently. In this thesis, we focus on call centre modelling, analysis and design. In terms of
modelling, traditionally call centres have been modelled as single-node queueing systems.
Based on the Semiopen Queueing Network (SOQN) model proposed by Srinivasan et al.
[42], we propose and study SOQN models with balking and abandonment (both expo-
nential and general patience time distributions). In addition, we study the corresponding
single-node queueing systems and obtain new results. For each model, we study the queue
length distribution, waiting time distribution and the related performance measures. To
facilitate the computation, we express the performance measures in terms of special func-
tions. In terms of call centre design, we develop a design algorithm to determine the
minimal number of CSRs (S) and trunk lines (N) to satisfy a given set of service level
constraints.
The explicit expressions for performance measures obtained allow for theoretical analysis
of the performance measures. For example we prove monotonicity and convexity proper-
ties of performance measures for the M=M=S=N and M=M=S=N +M models. We also
studythecomparisonofdi(cid:11)erentpatiencetimedistributionsfortheM=M=S=N+Gmodel.
Weprovidenumericalexamplesforeachmodelanddiscussnumericalresultssuchasmono-
tonicity properties of performance measures. In particular, we illustrate the e(cid:14)cacy of our
design algorithm for various models including patient, balking and abandonment models.
The impact of model parameters on the design of call centres is also discussed based on
the numerical examples. The results are computed using Matlab, where special functions
are available.
ii
Acknowledgements
This thesis grew out of a research project provided by my supervisor Prof. Raj Srini-
vasan. I am sincerely grateful to Prof. Raj Srinivasan for his invaluable advice and patient
guidance. This thesis could not have been (cid:12)nished without his constant help and support.
I would like to thank the members of my committee, Prof. Mikelis G. Bickis, Prof. Chris
Soteros, Prof. William H. Laverty, Prof. Winfried K. Grassmann and my external exam-
iner Prof. Noah F. Gans for reading my thesis and valuable suggestions. Last but not
least, I want to thank my family and friends, for their support and encouragement.
iii
To
My father
Dehua Zhang
My wife
Jiezhi Qi
My daughter
Erin Jiaqi Zhang
iv
Table of Contents
Permission to Use i
Abstract ii
Acknowledgements iii
Table of Contents v
List of Tables viii
List of Figures ix
List of Symbols and Abbreviations xi
1 Introduction 1
1.1 Introduction to call centres . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 What is a call centre? . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Operational process of an inbound call centre . . . . . . . . . . . . 2
1.2 Main research problems in call centres . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Call centre modelling and performance analysis problems . . . . . 4
1.2.2 Call centre design problem . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Outline and contributions of the thesis . . . . . . . . . . . . . . . . . . . . 12
2 Review of Single-node Markovian Queueing Models of Call Centres 14
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 M=M=S=S model and Erlang B formula . . . . . . . . . . . . . . . . . . . 16
2.3 M=M=S model and Erlang C formula . . . . . . . . . . . . . . . . . . . . 18
2.3.1 Queue length process. . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.2 Waiting time distribution . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 M=M=S=N model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.1 Queue length process. . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.2 Waiting time distribution . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.3 Monotonicity properties . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 SOQN Model of Call Centres 41
3.1 Modelling motivation and model description . . . . . . . . . . . . . . . . . 41
3.2 Product form solution of the queue length process . . . . . . . . . . . . . 42
v
3.2.1 Direct method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.2 Method of CQN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 The stationary distribution of the total number of calls in the system. . . 48
3.3.1 Direct method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.2 Throughput method . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.3 Calculation of the blocking probability . . . . . . . . . . . . . . . . 50
3.3.4 Other performance measures . . . . . . . . . . . . . . . . . . . . . 51
3.4 Waiting time distribution and mean waiting time . . . . . . . . . . . . . . 53
3.4.1 Using q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
j
3.4.2 Using (cid:31)(k;j) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4 Balking Models of Call Centres 60
4.1 Single-node state-dependent balking model M(n)=M=S=N . . . . . . . . . 60
4.1.1 Queue length process. . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.1.2 Waiting time distribution . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.3 Special cases of b . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
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4.2 Two-nodenetworkmodelwithstate-dependentbalkingandstate-dependent
service . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.2 Product form solution of the queue length process . . . . . . . . . 66
4.2.3 Semiopen case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3 SOQN model with state-dependent balking of call centres . . . . . . . . . 72
4.3.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3.2 Product form solution of the queue length process . . . . . . . . . 73
4.3.3 An alternative proof . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.4 Blocking probability . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3.5 Other performance measures . . . . . . . . . . . . . . . . . . . . . 79
4.3.6 Waiting time distribution and mean waiting time . . . . . . . . . . 79
4.3.7 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5 Exponential Abandonment Models of Call Centres 84
5.1 M=M=S +M(Erlang-A model) . . . . . . . . . . . . . . . . . . . . . . . . 85
5.1.1 Queue length process. . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.1.2 Waiting time distribution . . . . . . . . . . . . . . . . . . . . . . . 90
5.1.3 Mean waiting time . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1.4 Probability of abandonment . . . . . . . . . . . . . . . . . . . . . . 97
5.2 M=M=S=N +M model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2.1 Queue length process. . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2.2 Probability of abandonment . . . . . . . . . . . . . . . . . . . . . . 103
5.2.3 Waiting time distribution . . . . . . . . . . . . . . . . . . . . . . . 104
5.2.4 Mean waiting time . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.2.5 Response time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2.6 A numerical approximation method . . . . . . . . . . . . . . . . . 115
5.3 MonotonicityandconcavitypropertiesofperformancemeasuresforM=M=S=N+
M model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
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5.3.1 Monotonicity properties with respect to bu(cid:11)er size K . . . . . . . 118
5.3.2 Concavity property with respect to bu(cid:11)er size K . . . . . . . . . . 131
5.3.3 Monotonicity properties with respect to S . . . . . . . . . . . . . . 132
5.3.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.4 SOQN model with exponential abandonment of call centres (SOQN+M) . 135
5.4.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.4.2 Product form solution of the queue length process . . . . . . . . . 135
5.4.3 Blocking probability . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.4.4 Probability of abandonment and other performance measures . . . 139
5.4.5 Waiting time distribution . . . . . . . . . . . . . . . . . . . . . . . 142
5.4.6 Mean waiting time . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.4.7 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6 General Abandonment Models of Call Centres 150
6.1 M=M=S +G model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.2 M=M=S=N +G model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.2.1 Queue length process. . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.2.2 Probability of abandonment . . . . . . . . . . . . . . . . . . . . . . 162
6.2.3 Waiting time distribution . . . . . . . . . . . . . . . . . . . . . . . 164
6.2.4 Mean waiting time . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.2.5 M=M=S=N +M model . . . . . . . . . . . . . . . . . . . . . . . . 171
6.2.6 M=M=S=N +D model . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.2.7 An alternative method . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.2.8 Types of patience time distributions . . . . . . . . . . . . . . . . . 185
6.3 SOQN model with general abandonment of call centres (SOQN+G) . . . 193
6.3.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.3.2 Product form solution of the queue length process . . . . . . . . . 194
6.3.3 Blocking probability . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.3.4 Probability of abandonment and other performance measures . . . 197
6.3.5 Waiting time distribution . . . . . . . . . . . . . . . . . . . . . . . 199
6.3.6 Mean waiting time . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
6.3.7 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . 204
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7 An Algorithm for Call Centre Design Problem 208
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
7.2 Design algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
7.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
7.3.1 M=M=S=N and SOQN models . . . . . . . . . . . . . . . . . . . . 213
7.3.2 SOQN model with balking . . . . . . . . . . . . . . . . . . . . . . . 217
7.3.3 Exponential abandonment models . . . . . . . . . . . . . . . . . . 217
7.3.4 SOQN model with general abandonment . . . . . . . . . . . . . . . 220
7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8 Summary and Future Work 222
8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
8.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
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List of Tables
1.1 Common service level constraints . . . . . . . . . . . . . . . . . . . . . . . 10
2.1 Some Markovian queueing models . . . . . . . . . . . . . . . . . . . . . . 14
5.1 Four-dimensional waiting time performance measure . . . . . . . . . . . . 93
7.1 P(blocking) and P(W > 20) corresponding to the optimal design parame-
q
ters (S;N) for all models . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
7.2 The optimal design parameters (S;N) for models with exponential aban-
donment when (cid:18) = 0:01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
7.3 The optimal design parameters (S;N) for models with exponential aban-
donment when (cid:18) = 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
7.4 Comparisonoftheoptimaldesignparameters(S;N)forSOQNmodelswith
general abandonment when (cid:11) = 0:01 . . . . . . . . . . . . . . . . . . . . . 220
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List of Figures
1.1 Operational process of an inbound call centre . . . . . . . . . . . . . . . . 2
1.2 A natural queueing model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Semiopen queueing network model description and parameters . . . . . . 7
2.1 M=M=S=S model description and parameters . . . . . . . . . . . . . . . . 16
2.2 M=M=S=S model stationary state transition diagram. . . . . . . . . . . . 16
2.3 M=M=S model description and parameters . . . . . . . . . . . . . . . . . 18
2.4 M=M=S model stationary state transition diagram . . . . . . . . . . . . . 19
2.5 M=M=S=N model description and parameters . . . . . . . . . . . . . . . . 22
2.6 M=M=S=N model stationary state transition diagram . . . . . . . . . . . 22
2.7 From SOQN to CQN for M=M=S=N model . . . . . . . . . . . . . . . . . 27
2.8 P(blocking);P(nodelay) and P(delay) of Example 2.1 for M=M=S=N model 36
2.9 P(blocking);P(nodelay) and P(delay) of Example 2.2 for M=M=S=N model 37
2.10 P(blocking);P(nodelay) and P(delay) of Example 2.3 for M=M=S=N model 37
2.11 P(nodelay non-blocking);P(delay non-blocking)andP(W > 0:5)ofExam-
j j q
ple 2.1 for M=M=S=N model . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.12 P(nodelay non-blocking);P(delay non-blocking)andP(W > 0:5)ofExam-
j j q
ple 2.2 for M=M=S=N model . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.13 P(nodelay non-blocking);P(delay non-blocking)andP(W > 0:5)ofExam-
j j q
ple 2.3 for M=M=S=N model . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1 Semiopen queueing network model description and parameters . . . . . . 42
3.2 The equivalent CQN for the SOQN model . . . . . . . . . . . . . . . . . . 45
3.3 P(blocking) and P(W > 20) for Example 3.1 . . . . . . . . . . . . . . . . 57
q
3.4 P(blocking) and P(W > 20) for Example 3.2 . . . . . . . . . . . . . . . . 57
q
3.5 P(blocking) and P(W > 20) for Example 3.3 . . . . . . . . . . . . . . . . 58
q
3.6 P(blocking) and P(W > 20) for Example 3.4 . . . . . . . . . . . . . . . . 59
q
4.1 M(n)=M=S=N model description and parameters . . . . . . . . . . . . . . 61
4.2 M(n)=M=S=N model stationary state transition diagram . . . . . . . . . 61
4.3 Model description and parameters for the two-node network model . . . . 66
4.4 Stationary state transition diagram for the two-node network model . . . 67
4.5 SOQN model with state-dependent balking . . . . . . . . . . . . . . . . . 73
4.6 P(blocking) and P(W > 20) for Example 4.1 . . . . . . . . . . . . . . . . 82
q
4.7 P(blocking) and P(W > 20) for Example 4.2 . . . . . . . . . . . . . . . . 83
q
5.1 M=M=S +M model description and parameters . . . . . . . . . . . . . . 86
5.2 M=M=S +M model stationary state transition diagram . . . . . . . . . . 87
5.3 M=M=S=N +M model description and parameters . . . . . . . . . . . . . 99
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