Telephone call centers are an integral part of many businesses, and their economic role is significant and growing. They are also fascinating socio-technical systems in which the behavior of customers and employees is closely intertwined with physical performance measures. In these environments traditional operational models are of great value – and at the same time fundamentally limited – in their ability to characterize system performance. We review the state of research on telephone call centers. We begin with a tutorial on how call centers function and proceed to survey academic research devoted to the management of their operations. We then outline important problems that have not been addressed and identify promising directions for future research. Acknowledgments The authors thank Lee Schwarz, Wallace Hopp and the editorial board of M&SOM for initiating this project, as well as the referees for their valuable comments. Thanks are also due to L. Brown, A. Sakov, H. Shen, S. Zeltyn and L. Zhao for their approval of importing pieces of [36, 112].

In this paper we present and compare two different approaches for the transient solution of Markov regenerative stochastic Petri Nets: the method based on Markov regenerative theory and the method of supplementary variables. In both cases the equations that govern the marking process of the non-Markovian stochastic Petri net are presented and then solved either in time-domain or using a Laplace-Stieltjes transformation. We develop expressions for asymptotic computational costs and storage requirements. We also perform experimental studies to compare accuracy, time, and space complexity of the methods. 1 Introduction Stochastic Petri nets (SPNs) are well suited for the model-based performance and dependability evaluation of complex systems. In the past few years, several papers were published dealing with the transient and stationary analysis of non-Markovian SPNs in which, under certain structural restrictions, the firing times may be generally distributed. Besides the approach of ap...

This paper presents new methods for the transient and stationary analysis of stochastic Petri nets with exponentially distributed and deterministic firing delays. The method of supplementary variables is used for the derivation of general state equations which describe the temporal behavior of the underlying stochastic process. Numerical algorithms are presented for the solution of the state equations and illustrated by numerical examples. The results of this paper allow an enhanced performance and reliability analysis based on stochastic Petri net models. 1 Introduction Deterministic and stochastic Petri nets (DSPNs) [1] represent a graphical method for the modeling of discrete event systems like computer architectures, communication systems, and manufacturing systems. The stochastic extensions to the pure Petri net formalism allow to model and evaluate the performance and dependability of these systems. In DSPNs transitions may fire either without consuming time (immediate transit...

Uniformization has been shown to be, in many cases, a good method to compute transient state probabilities of a continuous-time Markovchain. However, two issues limit its use: uniformization can be computationally very intensive, for instance, on sti# models, and uniformization cannot be used for all model classes, e.g., models with not uniformly bounded transition rates. In this paper weintroduce adaptive uniformization,avariation on standard uniformization, which can overcome these problems for some models. Adaptive uniformization di#ers from standard uniformization in that it uses a uniformization rate that adapts depending on the set of states that the process can be in after a particular number of jumps. Doing this can sometimes signi#cantly reduce the computational cost required to obtain a solution. A formal de#nition of adaptive uniformization is #rst given, along with a proof that adaptive uniformization yields correct results. Characteristics of models that can facilitate solution and alternative methods for computing the required #jump probabilities" are then discussed. Finally, the computational cost of adaptive uniformization #relative to standard uniformization# is illustrated, through its application to an extended machine-repairman model.

This paper presents an efficient and numerically reliable method for the transient analysis of deterministic and stochastic Petri nets. The transient behavior is described by state equations derived by the method of supplementary variables. Significant features of the proposed solution algorithm of fourth order are an automatic stepsize control and a two-stage relative error control. Furthermore, a formal way of dealing with discontinuities in the transient state equations is developed. This resolves the problems posed by initially enabled deterministic transitions and also improves the accuracy of numerical results. Experiments with a queueing system with failure and repair illustrate the efficiency (with respect to both CPU-time and memory space) and the numerical quality of the new algorithm.

We describe a method to find low cost employee shift schedules that guarantee that the fraction of customers who wait less than a specified time (the service level) is always at or above a specified minimum. Most previous approaches used a two-step procedure: (1) determine period-by-period staffing requirements, and (2) find a minimum cost schedule that provides the required number of employees in every period. Due to approximations used in the first step, the two-step approach sometimes results in infeasible or suboptimal solutions. Our method iterates between a schedule evaluator and a schedule generator. Each iteration begins with a schedule for which the schedule evaluator calculates transient service levels using the randomization method. The transient service levels are used to identify infeasible intervals, where the service level is lower than desired. The schedule generator solves a series of integer programs to produce improved schedules. One constraint is added to the integer program for every infeasible interval, in an attempt to eliminate infeasibility without eliminating the optimal solution. The procedure terminates when a feasible solution is found. We present results for a range of test problems and discuss factors that make our approach more likely to outperform previous approaches. 1.

Abstract. Quantitative analysis of quality-of-service metrics is an important tool in early evaluation of service provision. This analysis depends on being able to estimate the average duration of critical activities used by the service but at the earliest stages of service planning it may be impossible to obtain accurate estimates of the expected duration of these activities. We analyse the time-dependent behaviour of an automotive rescue service in the context of uncertainty about durations. We deploy a distributed computing platform to allow the efficient derivation of quantitative analysis results across the range of possible values for assignments of durations to the symbolic rates of our high-level formal model of the service expressed in a stochastic process algebra. 1

We compare the performance of six methods in computing or approximating service levels for nonstationary M/M/s queueing systems: an exact method (a Runge Kutta ordinary differential equation solver), the randomization method, a closure (or surrogate distribution) approximation, a direct infinite server approximation, a modified offered load infinite server approximation, and an effective arrival rate approximation. We used all of the methods to solve the same set of 128 test problems. The randomization method was almost as accurate as the exact method, and used less than half the computational time of the exact method. The closure approximation was less accurate, and in many cases slower, than the randomization method. The two infinite server based approximations and the effective arrival rate approximation had were less accurate but had computation times that were far shorter and less problem-dependent than for the other three methods.

Telephone call centers are an integral part of many businesses, and their economic role is significant and growing. They are also fascinating socio-technical systems in which the behavior of customers and employees is closely intertwined with physical performance measures. In these environments traditional operational models are of great value – and at the same time fundamentally limited – in their ability to characterize system performance. We characterize the state of research on telephone call centers. We begin with a tutorial on how call centers function and proceed to survey academic research devoted to the management of their operations. We then outline important problems that have not been addressed and identify promising directions for future research.

Krylov subspace techniques have been shown to yield robust methods for the numerical computation of large sparse matrix exponentials and especially the transient solutions of Markov Chains. The attractiveness of these methods results from the fact that they allow us to compute the action of a matrix exponential operator on an operand vector without having to compute, explicitly, the matrix exponential in isolation. In this paper we compare a Krylov-based method with some of the current approaches used for computing transient solutions of Markov chains. After a brief synthesis of the features of the methods used, numerical comparisons are performed on a POWER CHALLENGEarray supercomputer on three different models. Keywords. Matrix exponential, Markov chains, Krylov methods, Ordinary differential equations. AMS Subject Classification. 65F99, 65L05, 65U05. 1 Introduction There exists several numerical techniques for obtaining transient solutions of homogeneous, irreducible Markov chain...