Table Of ContentLecture Notes in Economics
and Mathematical Systems 528
Founding Editors:
M.Beckmann
H. P. Kiinzi
Managing Editors:
Prof. Dr. G. Fandel
Fachbereich Wirtschaftswissenschaften
Fernuniversitat Hagen
Feithstr. 140lAVZ II, 58084 Hagen, Germany
Prof. Dr. W. Trockel
Institut fiir Mathematische Wirtschaftsforschung (IMW)
Universitat Bielefeld
Universitatsstr. 25,33615 Bielefeld, Germany
Co-Editors:
C. D. Aliprantis
Editorial Board:
A. Basile, A. Drexl, G. Feichtinger, W. Giith, K. Inderfurth, P. Korhonen,
W. Kiirsten, U. Schittko, P. SchOnfeld, R. Seiten, R. Steuer, F. Vega-Redondo
Springer-Verlag Berlin Heidelberg GmbH
Raik Stolletz
Performance Analysis
and Optimization
of Inbound Call Centers
Springer
Author
Raik Stolletz
Technical University of Clausthal
Institute of Business Administration and Economics
Julius-Albert-StraBe 2
38678 Clausthal-Zellerfeld
Cataloging-in-Publication Data applied for
A catalog record for this book is available from the Library of Congress.
Bibliographic information published by Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the Internet at http://dnb.ddb.de
ISSN 0075-8450
ISBN 978-3-540-00812-5 ISBN 978-3-642-55506-0 (eBook)
DOI 10.1007/978-3-642-55506-0
This work is subject to copyright. AII rights are reserved, whether the whole or part
of the material is concemed, specificalIy the rights of translation, reprinting, re-use
of illustrations, recitation, broadcasting, reproduction on microfilms or in any other
way, and storage in data banks. Duplication of this publication or parts thereof is
permitted only under the provisions of the German Copyright Law of September 9,
1965, in its current version, and permission for use must always be obtained from
Springer-Verlag. Violations are liable for prosecution under the German Copyright
Law.
http://www.springer.de
© Springer-Verlag Berlin Heidelberg 2003
Originally published by Springer-Verlag Berlin Heidelberg New York in 2003
The use of general descriptive names, registered names, trademarks, etc. in this
publication does not imply, even in the absence of a specific statement, that such
names are exempt from the relevant protective laws and regulations and therefore
free for general use.
Typesetting: Camera ready by author
Cover design: Erich Kirchner, Heidelberg
Printed on acid-free paper 41/3143/du 5432 1 O
Preface
The material presented in this book is a result of my work in the field of call
center management during the period 1999-2002. The focus is on the perfor
mance analysis and optimization of inbound call centers. Since call arrivals
and call-handling times are often random in inbound call centers, this thesis
concentrates on the performance analysis and optimization using queueing
models. This book describes mathematical methods and algorithms to relate
the number of agents and telephone trunks of a given call center configuration
to technical as well as economic performance measures.
This book has been accepted as a PhD thesis in Business Administration at
the Technical University of Clausthal, Germany. I am indebted to many people
for their support during the process of writing this thesis. First of all, I would
like to thank my advisor, Prof. Dr. Stefan Helber, for motivating my research
to call center related problems. He gently pushed me in fruitful directions
and encouraged me to strike a balance between mathematical results and
economic implications. Many other helpful suggestions came from him, and
his constructive comments on draft versions of this book are invaluable. I
am thankful to him and to Prof. Dr. Rolf Schwinn for refereeing this thesis.
I am also much indebted to my brother Lars for reviewing several drafts of
this book and his outstanding criticism. I would like to thank my colleagues
at the Department of Business Administration in Clausthal for many helpful
discussions, especially Kirsten Block for her support while writing the final
version-her patience and comments are sincerely appreciated. Furthermore,
the following reviewers have provided very useful comments on the drafts of
the manuscript: Nicole Bippig, Georg N. Krieg, and Michael Manitz. I am
grateful that Joanne Baste, Ian M. Langella, and J. Philipp Siemer made
several suggestions for improving my English.
Finally, I thank Silvia for her steady encouragement during the time I have
worked on this thesis.
Clausthal-Zellerfeld, February 2003 Raik Stolletz
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Characterization of Inbound Call Centers. . . . . . . . . . . . . . . . . . 3
2.1 What is an Inbound Call Center? . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Performance Measures and Objective Functions. . . . . . . . . . . .. 5
2.2.1 Technical Performance Measures. . . . . . . . . . . . . . . . . . . . 5
2.2.2 Economic Performance Measures. . . . . . . . . . . . . . . . . . . . 8
2.2.3 Objective Functions for Operational Planning. . . . . . . .. 11
2.3 Operational Personnel Planning in Inbound Call Centers ..... 12
2.3.1 Main Tasks of Operational Personnel Planning. . . . . . .. 12
2.3.2 Call Forecasting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13
2.3.3 Weekly Personnel Scheduling ....................... 15
2.3.3.1 Current Planning Process in Practice. . . . . . . .. 15
2.3.3.2 Agent Requirements Planning. . . . . . . . . . . . . .. 16
2.3.3.3 Shift Scheduling and Rostering . . . . . . . . . . . . .. 18
2.3.3.4 A Simultaneous Planning Approach. . . . . . . . .. 19
3 Classification of Queueing Models of Inbound Call Centers 21
3.1 Characteristics of Queueing Models of Call Centers. . . . . . . . .. 21
3.2 Classification by Customers and Agents . . . . . . . . . . . . . . . . . . .. 22
3.2.1 Arrival Process .................................... 22
3.2.2 Waiting Behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23
3.2.3 Distribution of Service Time. . . . . . . . . . . . . . . . . . . . . . .. 24
3.2.4 Homogeneity of Customers and Agents ............... 26
3.3 Routing Decisions in Call Centers ......................... 28
3.3.1 Overview .......................................... 28
3.3.2 Customer Selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 29
3.3.3 Agent Selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31
3.4 Limitation of the Waiting Room. . . . . . . . . . . . . . . . . . . . . . . . . .. 32
3.5 Review of the Literature ................................. 37
3.5.1 Overview......................................... 37
VIII Contents
3.5.2 Homogeneous Customers and Homogeneous Agents .... 37
3.5.3 Heterogeneous Customers and Heterogeneous Agents .. 40
3.5.3.1 Introduction of the M-Design ............... 40
3.5.3.2 The M-Design with Priority-Based Routing
Policies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40
3.5.3.3 Special Cases of the M-Design with
Priority-Based Routing Policies. . . . . . . . . . . . .. 43
4 Queueing Models of Call Centers with Homogeneous
Customers and Homogeneous Agents . . . . . . . . . . . . . . . . . . . . .. 47
4.1 Common Features ....................................... 47
4.2 The M / M / c and M / M / c/ K Queueing Models with Patient
Customers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49
4.2.1 Description and Derivation of Performance Measures
for the M / M / c/ K Queueing Model. . . . . . . . . . . . . . . . .. 49
4.2.2 Description and Derivation of Performance Measures
for the M / M / c/ 00 Queueing Model. . . . . . . . . . . . . . . . .. 53
4.2.3 Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 55
4.2.3.1 Impact of the Number of Trunks. . . . . . . . . . . .. 55
4.2.3.2 Economies of Scale. . . . . . . . . . . . . . . . . . . . . . . .. 61
4.2.3.3 Impact of Talk Time. . . . . . . . . . . . . . . . . . . . . .. 68
4.2.3.4 Optimal Number of Agents and Trunks in
Large Call Centers . . . . . . . . . . . . . . . . . . . . . . . .. 70
4.3 An M / M / c/ K Queueing Model with Impatient Customers ... 74
4.3.1 Model Description and Derivation of Performance
Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 74
4.3.2 Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 82
4.3.2.1 Impact of Customer Impatience: Reneging .... 82
4.3.2.2 Impact of Customer Impatience: Balking. . . . .. 87
4.3.2.3 Impact of Customer Impatience:
Dependencies between Reneging and Balking.. 88
4.3.2.4 Optimal Number of Agents and the Impact
of the Number of Trunks ................... 90
4.4 Management Implications of the Numerical Results. . . . . . . . .. 96
5 Queueing Model of a Call Center with two Classes of
Customers and Skill-Based Routing. . . . . . . . . . . . . . . . . . . . . . .. 97
5.1 Description of the Queueing System ....................... 97
5.2 Description of the State Space ............................ 101
5.2.1 Representation of the States ........................ 101
5.2.2 Division and Size of the State Space ................. 103
5.3 Steady-State Equations .................................. 106
5.3.1 Steady-State Equations for States with Waiting A-
and B-Customers .................................. 108
Contents IX
5.3.2 Steady-State Equations for States with Waiting A-
and without Waiting B-Customers ................... 111
5.3.3 Steady-State Equations for States with Waiting B-
and without Waiting A-Customers ................... 113
5.3.4 Steady-State Equations for States without Waiting
Customers ........................................ 114
5.4 Determination of Performance Measures .................... 116
5.4.1 Solution of the Steady-State Equations ............... 116
5.4.2 Derivation of Technical Performance Measures ........ 119
5.4.3 Validation of the Derivation for Special Cases ......... 124
5.5 Numerical Results ....................................... 128
5.5.1 Impact of Flexible Agents: Effects of Priority-Based
Customer Selection and Pooling ..................... 129
5.5.1.1 Impact of Priority-Based Customer Selection .. 129
5.5.1.2 Comparison of Call Centers with Completely
Crosstrained or Completely Specialized Agents 134
5.5.1.3 Advantages of Adding Generalists Instead of
Specialists ................................ 141
5.5.2 Impact of the Processing Times of Generalists ........ 145
5.5.3 Impact of the Allocation of Trunks .................. 149
5.5.4 Impact of the Allocation of a Fixed Number of Agents. 154
5.5.4.1 Identically Distributed Processing Times for
Specialists and Generalists .................. 154
5.5.4.2 Different Processing Times for Generalists
and Specialists ............................ 159
5.6 Management Implications of the Numerical Results .......... 163
6 Conclusions and Suggestions for Further Research ......... 165
A Algorithms for Call Center Models with Homogeneous
Customers and Agents ..................................... 167
A.1 Computations for the M/M/c/K Model with Patient
Customers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
A.2 Computations for the M / M / c/ 00 Model with Patient
Customers .............................................. 170
A.3 Computations for the M / M / c/K Model with Impatient
Customers .............................................. 171
A.3.1 Computation of Steady-State Probabilities ............ 171
A.3.2 Computation of the Waiting Time Distributions ....... 173
B Appendix for the Queueing Model of a Call Center with
two Classes of Customers and Skill-Based Routing ......... 179
B.1 Derivation of the Number of States ....................... 179
B.2 Derivation of the Remaining Steady-State Equations ......... 182
X Contents
B.2.1 Steady-State Equations for States with Waiting A-
and without Waiting B-Customers ................... 182
B.2.2 Steady-State Equations for States with Waiting B-
and without Waiting A-Customers ................... 185
B.2.3 Steady-State Equations for States without Waiting
Customers ........................................ 188
B.3 Algorithms Used for Performance Analysis .................. 193
B.3.1 Description of Data Structures and Algorithms ........ 193
B.3.2 Behavior of the Algorithm .......................... 200
Glossary of Notation .......................................... 203
List of Figures ................................................. 205
List of Tables .................................................. 211
References ..................................................... 213
1
Introduction
Call centers provide phone-based services to customers or clients in the private
or public sector. In the private sector, call centers are used in various indus
tries to manage relations between customers and the company. A call center
is an important communication channel to acquire new customers as well as
to support existing ones. This customer support may include all after-sales
activities aiming at customer loyalty, for example, help desk services and the
management of customer complaints. During the past few years the number
of call centers and the number of agents has been growing rapidly, demon
strating the increasing importance of call centers.
By the direction in which the contact between the customer and the agent
is initiated, we distinguish inbound and outbound call centers. In inbound
call centers the employees or agents receive calls from outside customers, and
therefore these call centers are driven by random customer call arrivals. For
this reason (on average) we have both customers waiting for an agent and
agents waiting for a customer.
Short expected waiting times are important to achieve customer satisfac
tion. Performance can be measured by waiting times, availability of service,
or customer abandonment. These technical performance measures improve if
the call center management employs more agents. However, the operating ex
penses of call centers are mainly driven by the costs of these agents. About
60 to 70 percent of operating costs are personnel-related. Call center manage
ment has to adjust imbalances of acceptable technical performance measures
and the economic performance of the call center.
Based on the forecasts of the number of calls for different time intervals,
the main operational planning task is to find an agent schedule subject to
a particular objective function and certain restrictions. Both the number of
agents and the number of offered phone lines are important decision vari
ables in this context. Performance analysis relates these decision variables to
economic as well as technical performance measures, for example, related to
waiting times. Based on technical performance measures, this thesis devel-
