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.

Many primary care offices and other medical practices regularly experience long backlogs for appointments. These backlogs are exacerbated by a significant level of last-minute cancellations or "no-shows," which have the effect of wasting capacity. In this paper, we conceptualize such an appointment system as a single-server queueing system in which customers who are about to enter service have a state-dependent probability of not being served and may rejoin the queue. We derive stationary distributions of the queue size, assuming both deterministic as well as exponential service times, and compare the performance metrics to the results of a simulation of the appointment system. Our results demonstrate the usefulness of the queueing models in providing guidance on identifying patient panel sizes for medical practices that are trying to implement a policy of "advanced access."Subject classifications: health care; queues: applications.

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.

Hospital diagnostic facilities, such as magnetic resonance imaging centers, typically provide service to several diverse patient groups: outpatients, who are scheduled in advance; inpatients, whose demands are generated randomly during the day; and emergency patients, who must be served as soon as possible. Our analysis focuses on two interrelated tasks: designing the outpatient appointment schedule, and establishing dynamic priority rules for admitting patients into service.We formulate the problem of managing patient demand for diagnostic service as a finite-horizon dynamic program and identify properties of the optimal policies. Using empirical data from a major urban hospital, we conduct numerical studies to develop insights into the sensitivity of the optimal policies to the various cost and probability parameters and to evaluate the performance of several heuristic rules for appointment acceptance and patient scheduling. AbstractHospital diagnostic facilities, such as magnetic resonance imaging centers, typically provide service to several diverse patient groups: outpatients, who are scheduled in advance; inpatients, whose demands are generated randomly during the day; and emergency patients, who must be served as soon as possible. Our analysis focuses on two inter-related tasks: designing the outpatient appointment schedule, and establishing dynamic priority rules for admitting patients into service.We formulate the problem of managing patient demand for diagnostic service as a finite horizon dynamic program and identify properties of the optimal policies.Using empirical data from a major urban hospital, we conduct numerical studies to develop insights on the sensitivity of the optimal policies to the various cost and probability parameters and to evaluate the performance of several heuristic rules for appointment acceptance and patient scheduling.

Timely access to a provider is a critical dimension of ED quality performance. In an environment in which EDs are often understaffed, analyses of arrival patterns and the use of queueing models can be extremely useful in identifying the most effective allocation of staff.