W e review queueing-theory methods for setting staffing requirements in service systems where customer demand varies in a predictable pattern over the day. Analyzing these systems is not straightforward, because standard queueing theory focuses on the long-run steady-state behavior of stationary models. We show how to adapt stationary queueing models for use in nonstationary environments so that time-dependent performance is captured and staffing requirements can be set. Relatively little modification of straightforward stationary analysis applies in systems where service times are short and the targeted quality of service is high. When service times are moderate and the targeted quality of service is still high, time-lag refinements can improve traditional stationary independent period-by-period and peak-hour approximations. Time-varying infinite-server models help develop refinements, because closed-form expressions exist for their time-dependent behavior. More difficult cases with very long service times and other complicated features, such as end-of-day effects, can often be treated by a modified-offered-load approximation, which is based on an associated infinite-server model. Numerical algorithms and deterministic fluid models are useful when the system is overloaded for an extensive period of time. Our discussion focuses on telephone call centers, but applications to police patrol, banking, and hospital emergency rooms are also mentioned.

This paper evaluates the practice of determining staffing requirements in service systems with random cyclic demands by using a series of stationary queueing models. We consider Markovian models with sinusoidal arrival rates and use numerical methods to show that the commonly used "stationary independent period by period" (SIPP) approach to setting staffing requirements is inaccurate for parameter values corresponding to many real situations. Specifically, using the SIPP approach can result in staffing levels that do not meet specified period by period probability of delay targets during a significant fraction of the cycle. We determine the manner in which the various system parameters affect SIPP reliability and identify domains for which SIPP will be accurate. After exploring several alternatives, we propose two simple modifications of SIPP that will produce reliable staffing levels in models whose parameters span a broad range of practical situations. Our conclusions from the sinusoidal model are tested against some empirical data.

We empirically explore the accuracy of an easily computed approximation for long run, average performance measures such as expected delay and probability of delay in multiserver queueing systems with exponential service times and periodic (sinusoidal) Poisson arrival processes. The pointwise stationary approximation is computed by integrating over time (that is taking the expectation of) the formula for the stationary performance measure with the arrival rate that applies at each point in time. This approximation, which has been empirically confirmed as a tight upper bound of the true value, is shown to be very accurate for a range of parameter values corresponding to a reasonably broad spectrum of real systems.queues, nonstationarity, approximation

While the goal of OR/MS is to aid decision makers, implementation of published models occurs less frequently than one might hope. However, one area that has been significantly impacted by management science is emergency response systems. Dozens of papers on emergency service management appeared in the OR/MS literature in the 1970s alone, many of which were published in Management Science. Three of these papers won major prizes. More importantly, many of these papers led to the implementation of substantially new policies and practices, particularly in policing and firefighting. Much of this work originated in New York City, though many other cities subsequently adopted the resulting models and strategies. In this paper, we look at the context, content, and nature of the research and the factors that led to these early implementation successes. We then track the extent to which these original models are still affecting decision making in emergency response systems. We also examine the pace of development of new OR/MS models and applications in the area. Finally, we look at issues in emergency responsiveness that have emerged recently as a result of the national focus on terrorism and discuss the potential for future OR/MS modeling and application.applications, emergency services, fire, police, public sector, urban

When all the fire companies in a region are engaged in fighting fires, protection against a future fire is considerably reduced. It is standard practice in many urban fire departments to protect the exposed region by relocating outside fire companies temporarily to some of the vacant houses. In New York City, situations requiring such relocations arise ten times a day on the average. The Fire Department of the City of New York (FDNY) currently makes its relocations according to a system of preplanned moves. This system was designed at a time when alarm rates were low and is based on the assumption that only one fire is in progress at a time. Because of the high alarm rates currently being experienced in parts of New York City, this assumption is no longer valid, and the preplanned relocation system breaks down at the times when it is needed most. This paper describes a computer-based method for determining relocations that overcomes the deficiencies of the existing method by utilizing the computer’s ability to (1) store up-to-date information about the status of all fires in progress and the location and activity of all fire companies, (2) generate and compare many alternative relocation plans quickly. The method, which will become part of the FDNY’s real-time Management Information and Control System (MICS), is designed to be fast and to require little computer memory. After giving some background of the problem and the objectives of relocation, we give the problem a mathematical programming formulation and then describe the heuristic algorithm to be used for generating relocations in the MICS. The remainder of the paper is devoted to a discussion of an example illustrating how the algorithm works, a rigorous test of the algorithm using a computer simulation model of Fire-Department operations, and a description of the current use of the computer algorithm by dispatchers in an interactive time-shared environment. The results of the testing indicate that the proposed algorithm is a significant improvement over existing methods, particularly in crisis situations. Although designed to solve a problem for the New York City Fire Department, the algorithm should be applicable to other cities.