A call center is a service network in which agents provide telephone-based services. Customers who seek these services are delayed in tele-queues. This article summarizes an analysis of a unique record of call center operations. The data comprise a complete operational history of a small banking call center, call by call, over a full year. Taking the perspective of queueing theory, we decompose the service process into three fundamental components: arrivals, customer patience, and service durations. Each component involves different basic mathematical structures and requires a different style of statistical analysis. Some of the key empirical results are sketched, along with descriptions of the varied techniques required. Several statistical techniques are developed for analysis of the basic components. One of these techniques is a test that a point process is a Poisson process. Another involves estimation of the mean function in a nonparametric regression with lognormal errors. A new graphical technique is introduced for nonparametric hazard rate estimation with censored data. Models are developed and implemented for forecasting of Poisson arrival rates. Finally, the article surveys how the characteristics deduced from the statistical analyses form the building blocks for theoretically interesting and practically useful mathematical models for call center operations.

We consider a multiserver service system with general nonstationary arrival and service-time processes in which s(t), the number of servers as a function of time, needs to be selected to meet projected loads. We try to choose s(t) so that the probability of a delay (before beginning service) hits or falls just below a target probability at all times. We develop an approximate procedure based on a time-dependent normal distribution, where the mean and variance are determined by infinite-server approximations. We demonstrate that this approximation is effective by making comparisons with the exact numerical solution of the Markovian M t / M / s t model.

This paper studies alternative ways to manage a multi-server system such as a telephone call center. Three alternatives can be described succinctly by: (i) blocking, (ii) reneging and (iii) balk-ing. The first alternative – blocking – is to have no provision for waiting. The second alternative is to allow waiting, but neither inform customers about anticipated delays nor provide state infor-mation to allow arriving customers to predict delays. The second alternative tends to yield higher server utilizations. The first alternative tends to reduce to the second, without the first-come first-served service discipline, when customers can easily retry, as with automatic redialers in telephone access. The third alternative is to both allow waiting and inform customers about anticipated delays. The third alternative tends to cause balking when all servers are busy (abandonment upon arrival) instead of reneging (abandonment after waiting). Birth-and-death process models are pro-posed to describe the performance with each alternative. Algorithms are developed to compute the conditional distributions of the time to receive service and the time to renege given each outcome. Algorithms are also developed to help the service provider predict customer waiting times before beginning service, given estimated service-time distributions and the elapsed service times of the customers in service. Better predictions may be obtained by classifying customers and thereby obtaining better estimates of their service-time distributions.

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Motivated by telecommunication applications, we investigate ways to estimate the parameters of a nonhomogeneous Poisson process with linear rate over a finite interval, based on the number of counts in measurement subintervals. Such a linear arrival-rate function can serve as a component of a piecewise-linear approximation to a general arrival-rate function. We consider ordinary least squares (OLS), iterative weighted least squares (IWLS) and maximum likelihood (ML), all constrained to yield a nonnegative rate function. We prove that ML coincides with IWLS. As a reference point, we also consider the theoretically optimal weighted least squares (TWLS), which is least squares with weights inversely proportional to the variances (which would not be known with data). Overall, ML performs almost as well as TWLS. We describe computer simulations conducted to evaluate these estimation procedures. None of the procedures differ greatly when the rate function is not near 0 at either end, but when the rate function is near 0 at one end, TWLS and ML are significantly more effective than OLS. The number of measurement subintervals (with fixed total interval) makes surprisingly little difference when the rate function is not near 0 at either end. The variances are higher with only two or three

In this paper we discuss fairness in queues, view it in the perspective of social justice at large and survey the recently published research work and publications dealing with the issue of measuring fairness of queues. The emphasis is placed on the underlying principles of the different measuring approaches, on reviewing their methodology and on examining their applicability and intuitive appeal. Some quantitative results are also presented. The paper has three major parts (sections) and a short concluding discussion. In the first part, fairness in queues and its importance are discussed in the broader context of the prevailing conception of social justice at large. A special effort, including illustrative examples, is made to differentiate between fairness of the queue and fairness at large, which derives from favoring the more needy. The second part is dedicated to explaining and discussing the three main properties expected of a fairness measure: conformity to the general concept of social justice, granularity, and intuitive appeal and rationality. The third part reviews the fairness of the queue evaluation and

We study uniform acceleration (UA) expansions of finite-state continuous-time Markov chains with time-varying transition rates. The UA expansions can be used to justify, evaluate, and refine the pointwise stationary approximation, which is the steady-state distribution associated with the time-dependent generator at the time of interest. We obtain UA approximations from these UA asymptotic expansions. We derive a time-varying analog to the uniformization representation of transition probabilities for chains with constant transition rates, and apply it to establish asymptotic results related to the UA asymptotic expansion. These asymptotic results can serve as appropriate time-varying analogs to the notions of stationary distributions and limiting distributions. We illustrate the UA approximations by doing a numerical example for the time-varying Erlang loss model. 1 Accepted for publication in the Annals of Applied Probability. AMS 1991 subject classifications. 60J27, 60K30. Keywords...

Abstract: We consider queueing systems with multiple classes of customers and heterogeneous servers where customers have the flexibility of being processed by more than one server and servers possess the capability of processing more than one customer class. We provide a unified framework for the modeling and analysis of these systems under arbitrary customer and server flexibility and for a rich set of control policies that includes customer/server-specific priority schemes for server and customer selection. We use our models to generate several insights into the effect of system configuration and control policies. In particular, we examine the relationship between flexibility, control policies and throughput under varying assumptions for

Many service systems have demand that varies significantly by time of day, making it costly to provide sufficient capacity to be able to respond very quickly to each service re-quest. Fortunately, however, different service requests often have very different response-time requirements. Some service requests may need immediate response, while others can tolerate substantial delays. Thus it is often possible to smooth demand by partitioning the service requests into separate priority classes according to their response-time requirements. Classes with more stringent performance requirements are given higher priority for service. Lower capacity may be required if lower-priority-class demand can be met during off-peak periods. We show how the priority classes can be defined and the resulting required fixed capacity can be determined, directly accounting for the time-dependent behavior. For this purpose, we ex-ploit relatively simple analytical models, in particular, Mt/G/ ∞ and deterministic offered-load models. The analysis also provides an estimate of the capacity savings that can be obtained from partitioning time-varying demand into priority classes.

Physical concepts developed to describe instabilities in traffic flows can be generalized in a way that allows one to understand the wellknown instability of supply chains (the so-called “bullwhip effect”). That is, small variations in the consumption rate can cause large variations in the production rate of companies generating the requested product. Interestingly, the resulting oscillations have characteristic frequencies which are considerably lower than the variations in the consumption rate. This suggests that instabilities of supply chains may be the reason for the existence of business cycles. At the same time, we establish some link to queuing theory and between micro- and macroeconomics. 1