Simulation Methods for Optimization and Learning, Globex 2021
Service Center Design Assignment
1 Introduction
Deadline 19-7-2021, 23h hrs
Bernd Heidergott
The call center industry is a large and rapidly growing sector that provides a variety of services to organizations and customers, e.g., handling of orders, complaints, and questions, providing technical support, etc. This is an integral part of the customer relationship management of many organizations, which is an easy concept but a hard reality. As more and more organizations diversify and their products and services become more complex, the efficient operational control of call centers has grown in complexity as well. In particular, efficient workforce management has been the center of attention, since this forms a substantial part of the operational costs. Generally speaking, the objective of the call center is to constrain the expected waiting time in the queue of an arbitrary customer. On the other hand, modern call centers are also faced with other tasks with a less strict requirement, such as emails, web messages, and outbound calls. This work typically has a less tight constraint, and therefore the objective of the call center is to serve as many jobs as possible per time unit (called throughput).
For this project, we study a call center where we can distinguish the two types of work by type 1 (incoming calls) and type 2 (emails, outbound calls, etc.) jobs. Both job types have different service requirements, and there is a common pool of agents to serve both of them in a non-preemptive regime. The system is depicted in Figure 1. More specifically, the two types of jobs have independent exponentially distributed service requirements with rates μ1 and μ2, and we let μ1 ̸= μ2. Type 1 jobs arrive according to a Poisson process with rate λ, and there is an infinite waiting capacity for jobs that cannot be served yet. There is an infinite supply of type 2 jobs. There are a total of C identical agents (servers). The question is how to efficiently use the workforce to maximize the throughput of type 2 jobs while guaranteeing that the long-term average waiting time of type 1 jobs is below a predefined constant α. While this type of system has been intensively studied in the literature, no exact optimal policy has been identified yet in the case of unequal service rates (i.e., μ1 ̸= μ2), which is due to the complexity of the solution and the model.
The objective for type 2 jobs is to maximize its throughput, i.e., to serve on average per unit of time as many type 2 jobs as possible, of course at the same time obeying the constraint on type 1 waiting time. Due to the fact that we are considering long-term average performance it is only optimal to schedule jobs at completion or arrival instants. Indeed, if it is optimal to keep a server idle at a certain instant, then this remains optimal until the next event in the system. Therefore it suffices to consider the system only at completion or arrival instants. Note that the policy for assigning jobs to vacant servers is not specified at the moment and we will discuss in the subsequent section two standard policies.
Trunk Reservation Policy: In this study we will explore the performance of a trunk reservation for the case of unequal service requirements, see [2]. For trunk reservation, there will be always K agents reserved
1
Server Pool
Arrival rate
Service rate 1
Service rate 2
Incoming calls
Outgoing calls
Job Queue 1
Job Queue 2
Serve which queue?
Figure 1: A call center system
for type 1 jobs. Only when there are more than K agents idle, those extra idle agents will serve type 2 jobs.
Flow Rate Policy: If there is a type 1 customer in the queue, idle agents will always serve them; only when the queue is empty, idle agents have probability θ to serve type 2 customers. More specifically: at each arrival and departure instant, assume that there are N idle agents, and there are M type 1 jobs in the queue. If N ≤ M, all idle agents will serve type 1 jobs. If N > M, M idle agents will serve type 1, and at the same time, each of the extra N − M idle agents will simultaneously and independently flip a coin, where with probability θ this idle agent will serve a type 2 job and with probability 1 − θ this agent will remain idle in order to wait for the next possible arriving type 1 job. We call this the θ-flow rate policy, see [2].
2 Assignment
Use for your experiments
λ=1,μ1=4,μ2=3,and C=5. 2 10 10
(1) To keep the analysis simple, we will approximate the long-run average cost by a transient simulation experiment over a fixed time horizon T . Determine T large enough so that effect to the initial phase is no longer present in your performance measurements.
(2) Consider the Trunk Reservation Policy and determine for α = 0.05, α = 0.1 and α = 0.2 the best value for K. Make sure that you base you advice on K on a proper statistical analysis.
(3) Study the problem again for the same values of α but now for the flow rate policy. Use stochastic approximation for finding the optimal value for θ.
(4) Discuss your findings and interpret the relation between the solutions found in (2) and (3).
3 Further Reading
The call center model under discussion has been initially studied for the case of equal service rates (i.e., μ1 = μ2) by [2], [5], [9], and by [6] who study multiple classes of calls. [10], [11], and [12] consider fixed, static priority policies. A similar approach is adopted by [8] and [3], who provide an approximate
2
Agent 1
Agent 2
Agent C
analysis of the overflow behavior from one pool of agents to another. For a literature survey on asymptotic heavy-traffic regimes we refer to [7] and [4]. This assignment is based on [1].
References
[1] S. Bhulai, Y. Farenhorst-Yuan, B. Heidergott, and D. van der Laan. Optimal balanced control for call centers. Annals of Operations Research, 2012.
[2] S. Bhulai and G. Koole. A queueing model for call blending in call centers. IEEE Transactions on Automatic Control, 48:1434 –1438, 2003.
[3] G. Franx, G. Koole, and S. Pot. Approximating multi-skill blocking systems by hyperexponential decomposition. Performance Evaluation, 63(8):799–824, 2006.
[4] N. Gans, G. Koole, and A. Mandelbaum. Telephone call centers: tutorial, review, and research prospects. Manufacturing and Service Operations Management, 5:79–141, 2003.
[5] N. Gans and Y. Zhou. A call-routing problem with service-level constraints. Operations Research, 51:255–271, 2003.
[6] I. Gurvich, M. Armony, and A. Mandelbaum. Service level differentation in call centers with fully flexible servers. Management Science, 54(2):279–294, 2008.
[7] G. Koole and A. Mandelbaum. Queueing models of call centers: an introduction. Annals of Opera- tions Research, 113:41–59, 2002.
[8] G. Koole and J. Talim. Exponential approximation of multi-skill call centers architecture. In QNETs 2000: Fourth International Workshop on Queueing Networks with Finite Capacity, pages 23/1–10, Craiglands Hotel, Ilkley, West Yorkshire, UK, 2000.
[9] M. Perry and A. Nilsson. Performance modeling of automatic call distributors: Assignable grade of service staffing. In 14th International Switching Symposium, pages 294–298, Yokohama, 1992.
[10] R. Shumsky. Approximation and analysis of a queueing system with flexible and specialized servers. OR Spektrum, 26:307–330, 2004.
[11] D. Stanford and W. Grassmann. Bilingual server call centers. In D. McDonald and S. Turner, editors, Analysis of Communication Networks: Call Centers, Traffic and Performance, volume 208, pages 31–47. Fields Institute Communications, 2000.
[12] R. Wallace and W. Whitt. A staffing algorithm for call centers with skill-based routing. Manufac- turing and Service Operations Management, 7(4):276–294, 2005.
3