Athanassios N. Avramidis

Modeling Daily Arrivals to a Telephone Call Center

We develop stochastic models of time-dependent arrivals, with focus on the application to call centers. Our models reproduce three essential features of call center arrivals observed in recent empirical studies: a variance larger than the mean for the number of arrivals in any given time interval, a time-varying arrival intensity over the course of a day, and nonzero correlation between the arrival counts in different periods within the same day. For each of the new models, we characterize the joint distribution of the vector of arrival counts, with particular focus on characterizing how the new models are more flexible than standard or previously proposed models. We report empirical results from a study on arrival data from a real-life call center, including the essential features of the arrival process, the goodness of fit of the estimated models, and the sensitivity of various simulated performance measures of the call center to the choice of arrival process model.

Optimizing daily agent scheduling in a multiskill call center

We examine and compare simulation-based algorithms for solving the agent scheduling problem in a multiskill call center. This problem consists in minimizing the total costs of agents under constraints on the expected service level per call type, per period, and aggregated. We propose a solution approach that combines simulation with integer or linear programming, with cut generation. In our numerical experiments with realistic problem instances, this approach performs better than all other methods proposed previously for this problem. We also show that the two-step approach, which is the standard method for solving this problem, sometimes yield solutions that are highly suboptimal and inferior to those obtained by our proposed method.

Markov chain models of a telephone call center with call blending

Motivated by a Bell Canada call center operating in blend mode, we consider a system with two types of traffic and two types of agents. Outbound calls are served only by blend agents, whereas inbound calls can be served by either inbound-only or blend agents. Inbound callers may balk or abandon. There are several performance measures of interest, including the rate of outbound calls and the proportion of inbound calls waiting more than some fixed number of seconds. We present a collection of continuous-time Markov chain (CTMC) models which capture many real-world characteristics while maintaining parsimony that results in fast computation. We discuss and explore the tradeoffs between model fidelity and efficacy and compare our different CTMC models with a realistic simulation model of a Bell Canada call center, used as a benchmark.

Efficient Monte Carlo and Quasi--Monte Carlo Option Pricing Under the Variance Gamma Model

We develop and study efficient Monte Carlo algorithms for pricing path-dependent options with the variance gamma model. The key ingredient is difference-of-gamma bridge sampling, based on the representation of a variance gamma process as the difference of two increasing gamma processes. For typical payoffs, we obtain a pair of estimators (named low and high) with expectations that (1) are monotone along any such bridge sampler, and (2) contain the continuous-time price. These estimators provide pathwise bounds on unbiased estimators that would be more expensive (infinitely expensive in some situations) to compute. By using these bounds with extrapolation techniques, we obtain significant efficiency improvements by work reduction. We then combine the gamma bridge sampling with randomized quasi--Monte Carlo to reduce the variance and thus further improve the efficiency. We illustrate the large efficiency improvements on numerical examples for Asian, lookback, and barrier options.

Correlation-Induction Techniques for Estimating Quantiles in Simulation Experiments

A simulation-based quantile estimator measures the level of system performance that can be delivered with a prespecified probability. To estimate selected quantiles of the response of a finite-horizon simulation, we develop procedures based on correlation induction techniques for variance reduction, with emphasis on antithetic variates and Latin hypercube sampling. These procedures achieve improved precision by controlling the simulations random-number inputs as an integral part of the experimental design. The proposed multiple-sample quantile estimator is the average of negatively correlated quantile estimators computed from disjoint samples of the simulation response, where negative correlation is induced between corresponding responses in different samples while mutual independence of responses is maintained within each sample. The proposed single-sample quantile estimator is computed from negatively correlated simulation responses within one all-inclusive sample. The single-sample estimator based on Latin hypercube sampling is shown to be asymptotically normal and unbiased with smaller variance than the comparable directsimulation estimator based on independent replications. Similar asymptotic comparisons of the multiple-sample and directsimulation estimators focus on bias and mean square error. Monte Carlo results suggest that the proposed procedures can yield significant reductions in bias, variance, and mean square error when estimating quantiles of the completion time of a stochastic activity network.

Modeling and simulation of call centers

In this review, we introduce key notions and describe the decision problems commonly encountered in call center management. Main themes are the central role of uncertainty throughout the decision hierarchy and the many operational complexities and relationships between decisions. We make connections to analytical models in the literature, emphasizing insights gained and model limitations. The high operational complexity and the prevalent uncertainty suggest that simulation modeling and simulation-based decision making could have a central role in the management of call centers. We formulate some common decision problems and point to recently developed simulation-based solution techniques. We review recent work that supports modeling the primitive inputs to a call center and highlight call center modeling difficulties.

Integrated Variance Reduction Strategies for Simulation

We develop strategies for integrated use of certain well-known variance reduction techniques to estimate a mean response in a finite-horizon simulation experiment. The building blocks for these integrated variance reduction strategies are the techniques of conditional expectation, correlation induction including antithetic variates and Latin hypercube sampling, and control variates; all pairings of these techniques are examined. For each integrated strategy, we establish sufficient conditions under which that strategy will yield a smaller response variance than its constituent variance reduction techniques will yield individually. We also provide asymptotic variance comparisons between many of the methods discussed, with emphasis on integrated strategies that incorporate Latin hypercube sampling. An experimental performance evaluation reveals that in the simulation of stochastic activity networks, substantial variance reductions can be achieved with these integrated strategies. Both the theoretical and experimental results indicate that superior performance is obtained via joint application of the techniques of conditional expectation and Latin hypercube sampling.

Modelling and simulation of a telephone call center

We consider a system with two types of traffic and two types of agents. Outbound calls are served only by blend agents, whereas inbound calls can be served by either inbound-only or blend agents. Our objective is to allocate a number of agents such that some service requirement is satisfied. We have taken two approaches in analyzing this staffing problem: we developed a simulation model of the call center, which allows us to do a what-if analysis, as well as continuous-time Markov chain (CTMC) queueing models, which provide approximations of system performance measures. We describe the simulation model here.

CONTROL OF A BATCH-PROCESSING MACHINE : A COMPUTATIONAL APPROACH

Batch processing machines, where a number of jobs are processed simultaneously as a batch, occur frequently in semiconductor manufacturing environments, particularly at diffusion in wafer fabrication and at burn-in in final test. In this paper we consider a batch-processing machine subject to uncertain (Poisson) job arrivals. Two different cases are studied: (1) the processing times of batches are independent and identically distributed (IID), corresponding to a diffusion tube; and (2) the processing time of each batch is the maximum of the processing times of its constituent jobs, where the processing times of jobs are IID, modelling a burn-in oven. We develop computational procedures to minimize the expected long-run-average number of jobs in the system under a particular family of control policies. The control policies considered are threshold policies, where processing of a batch is initiated once a certain number of jobs have accumulated in the system. We present numerical examples of our methods and...

Efficient Correlation Matching for Fitting Discrete Multivariate Distributions with Arbitrary Marginals and Normal-Copula Dependence

A popular approach for modeling dependence in a finite-dimensional random vector X with given univariate marginals is via a normal copula that fits the rank or linear correlations for the bivariate marginals of X. In this approach, known as the NORTA method, the normal distribution function is applied to each coordinate of a vector Z of correlated standard normals to produce a vector U of correlated uniform random variables over (0,1); then X is obtained by applying the inverse of the target marginal distribution function for each coordinate of U. The fitting requires finding the appropriate correlation ρ between any two given coordinates of Z that would yield the target rank or linear correlation r between the corresponding coordinates of X. This root-finding problem is easy to solve when the marginals are continuous but not when they are discrete. In this paper, we provide a detailed analysis of this root-finding problem for the case of discrete marginals. We prove key properties of r and of its derivative as a function of ρ. It turns out that the derivative is easier to evaluate than the function itself. Based on that, we propose and compare alternative methods for finding or approximating the appropriate ρ. The case of discrete distributions with unbounded support is covered as well. In our numerical experiments, a derivative-supported method is faster and more accurate than a state-of-the-art, nonderivative-based method. We also characterize the asymptotic convergence rate of the function r (as a function of ρ) to the continuous-marginals limiting function, when the discrete marginals converge to continuous distributions.