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Article Excerpt
We consider a call center model with multiple customer classes and multiple server pools. Calls arrive randomly over time, and the instantaneous arrival rates are allowed to vary both temporally and stochastically in an arbitrary manner. The objective is to minimize the sum of personnel costs arid expected abandonment penalties by selecting an appropriate staffing level for each server pool. We propose a simple and computationally tractable method for solving this problem that requires as input only a few system parameters and historical call arrival data for each customer class; in this sense the method is said to be data-driven. The efficacy of the proposed method is illustrated via numerical examples. An asymptotic-analysis establishes that the prescribed staffing levels achieve near-optimal performance and characterizes the magnitude of the optimally gap.
Subject classifications: stochastic model applications; stochastic networks; nonstationary queues; limit theorem; approximations.
Area of review: Stochastic Models
History: Received May 2006: revisions received February 2007 September 2007. February 2008; accepted March 2008.
Published online in Articles in Advance March 16, 2009.
1. Introduction
1.1. Introduction and Overview of the Main Contributions
This paper is concerned with the problem of staffing a telephone call center that serves multiple customer classes using agents of multiple skill sets In particular, we study a tractable solution method to this problem, whose salient feature is that it is data-driven. By that we mean that it essentially requires only historical call data to arrive at staffing decisions, and in doing so imposes minimal assumptions with regard to the nature of this data. To the best of our knowledge, this represents a departure from most studies of the starring problem in the operations research literature, which are typically model-based, insofar as solutions proposed there arc constructed using probabilistic structure that is assumed to characterize the data-generating process; see Gans et al. (2003) for a comprehensive survey on the topic of modeling and analysis of telephone call centers. (A review of literature relevant to this paper is deferred to the end of this section.)
Before we can explain in more detail the contributions of this paper, let us first describe in broad strokes the call center model that will be the locus of our attention. As indicated above, our model has multiple customer classes and multiple agent pools. Each of the pools consists of identical servers (agents) whose skills dictate the possible customer classes they can serve and the speed at which such service is delivered. Customers of various classes arrive randomly over time and upon arrival are either served immediately or wait in an infinite-capacity buffer.
Two important assumptions are made with regard to the call arrival process. First, arrival rates of incoming calls are not assumed to be constant or known. Rather, we allow these rates to be temporally varying and random; that is, there is inherent uncertainty with respect to their true value. Second, we assume that customers waiting for their service lo commence might abandon before they are assigned a server. Both of these modeling assumptions capture key characteristics present in actual telephone call center operating environments; see, e.g., Gans et al. (2003) and Steckley et al. (2004) for a discussion of the latter, and Avramidis et al. (2004) and Brown et al. (2005) for discussion of the former, as well as references therein.
To describe the staffing problem, consider a fixed and given time interval, hereafter referred lo as a "staffing segment," over which staffing decisions are held constant. That is, a segment represents the smallest time interval over which a staffing decision cannot be revised (typically this interval ranges from 30 minutes to two hours). We assume that there are two types of costs related to operating the call center: personnel costs and abandonment costs. The objective is then to find a staffing level for the various server pools that minimizes the sum of the two costs over a given staffing segment. (The solution of the staffing problem usually forms the basis for more detailed workforce management decisions that assign individual agents to specific work schedules, although this level of granularity is beyond the topic of this paper.)
A major obstacle in solving for the "optimal" staffing levels is that the performance of any proposed solution requires specification of a routing policy that describes how incoming calls will be assigned to agents at any point in time, so as to minimize abandonment-related costs. Unfortunately, this dynamic control problem can rarely be solved, and thus it has become common practice to consider only relatively simple call routing rules and then rely on simulation to evaluate the performance of a given staffing level; see Hans et al. (2003) and the literature review that follows. As a consequence of this limitation, it would be difficult to discern, for example, whether a given staffing level performs poorly because the associated routing logic is deficient, or whether this is due to the staffing level itself being strictly suboptimal. (In particular, it is not clear how one identifies, even in theory, the optimal solution of the staffing problem; the characterization of the latter is obviously useful for purposes of benchmarking any other proposed solution.) One of the contributions of this paper is that it provides an approach for studying the staffing problem essentially in "isolation."
Main Contributions. The main algorithmic contribution of this paper is in proposing a computationally tractable method for obtaining prescriptive solutions to die staffing problem described above. (See [section]4.2.) The method build on recent work of Harrison and Zeevi (2005), but whereas that paper develops a model-based approach (namely, it assumes knowledge of the probabilistic structure characterizing the mean call arrival patterns), this paper relies only on past call data as an input.
The main theoretical and methodological contribution of this paper is in establishing that the proposed data-driven staffing method yields prescriptions that are provably near-optimal in a suitable asymptotic sense. This analysis blends two different types of asymptotics that form the basis of our main results. The first asymptotic considers a sequence of systems in which call arrival volume, as well as abandonment and processing rates, increase without bound. This type of asymptotic is of the variety used in operations research studies of high volume large-scale systems. In particular, we use multiscale fluid limit machinery developed in Bassamboo et al. (2005) (see also Bassamboo et at. 2006) to characterize the "approximation error" that results from applying our method to finite sized systems; see Theorems 1 and 2. The second type of asymptotic is one that is frequently used to study properties of statistical estimators, and it involves the size of the data growing large. In particular, we rely on machinery from empirical process theory (see, e.g., van der Vaart and Wellner 1996) to characterize the "estimation error" that stems from having access only to historical call arrival data; .see Theorems 3 and 4.
To the best of our knowledge, the combination of the two types of asymptotics discussed above is new in the operation research literature, and this paper illustrates the benefits of this synergistic approach; see in particular Theorem 4. Using these results the performance of the proposed solution is seen to be eventually "close" to the best achievable performance, and in that sense it is asymptotically optimal.
The above claim of asymptotic optimality appears to be somewhat peculiar in light of the earlier discussion on the difficulties of characterizing the optimal solution in the staffing problem. In particular, recall that the latter requires knowledge of the optimal routing logic that should be paired with it. To this end, imagine that one has access to an oracle that provides the optimal routing policy associated with any given .staffing vector and hence can compute the optimal solution to the staffing problem. With this aid of the oracle, it is possible 10 assess the loss in performance that stems from using our proposed solution as opposed to the optimal one. The surprising observation is that we can characterize this optimality gap without ever having to compute the (oracle based) optimal staffing solution. The techniques used to establish this might be of independent interest and could prove useful in other related problems of design and control of stochastic systems.
The Remainder of the Paper (and a Reading Guide).
This section concludes with a review of related literature. Section 2 provides a mathematical description of our call center model, and [section]3 describes the staffing problem. Heading both sections is in some sense necessary to follow [section]4, which explains our data-driven solution method. Section 5 provides the intuition behind the proposed method. Those who are not in need of such intuition can skip this section and move on to [section]6, which presents numerical examples illustrating the performance of the method. Those seeking theory that supports the results observed in the numerical examples can find that in [section]7. Some qualitative insights and other points pertaining to these theoretical results are summarized in $8. Finally, all proofs are collected in two appendices, which are pan m the online companion for the paper (available at http://or.journal.informs.org/): the main results are proved in Appendix A. and auxiliary results are proved in Appendix B.
1.2. Literature Review
Most of the literature on call center starting focuses on a single pool of identical servers. In that realm, the ease where there is only a single class of customers leads to trivial control decisions, and if the system is Markovian then the Erlang-C formula provides the main mathematical tool for solving the staffing problem. An important rule-of-thumb that arises from the Erlang-C formula is the so-called square.-root staffing rule see Gans et al (2003, [section]4.1.1) for further discussion. Borst et al. (2004) refine the square-root rule to balance queuing and staffing costs; this type of objective function is...
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