Queues

Abstract

Queueing systems are simplified mathematical models to explain congestion. Broadly speaking, a queueing system occurs any time ‘customers’ demand ‘service’ from some facility; usually both the arrival of the customers and the service times are assumed to be random. If all of the ‘servers’ are busy when new customers arrive, these will generally wait in line for the next available server. Simple queueing systems are defined by specifying the following (a) the arrival pattern, (b) the service mechanism, and (c) queue discipline. From the probabilistic point of view, properties of queues are usually derived from the properties of stochastic processes associated with them. However, in all but the simplest queues, determination of the state probabilities is extremely difficult. Often, however, it is possible to determine their large time limit, the so-called equilibrium or steady-state distribution. This distribution does not depend on the initial conditions of the system and it is stationary. The ergodic conditions give the restrictions on the parameters under which the system will eventually reach the equilibrium. For the most part, queueing theory deals with computations of the steady-state probabilities and their use in computing other (steady-state) measures of performance of the queue. When only the expected values are required, an extremely useful formula for systems in equilibrium is Little's law. Most of the vast effort in queueing theory has been devoted to the probabilistic development of queueing models and to the study of its mathematical properties; that is, the parameters governing the models are, for the most part, assumed given. Statistical analyses, in which uncertainty is introduced, are comparatively very scarce. Inference in queueing systems is not easy: development of the necessary sampling distributions can be very involved and often the analysis is restricted to asymptotic results. The statistical analysis is simpler if approached from the Bayesian perspective. Since Bayesian analyses are insensitive to (noninformative) stopping rules, all that it is required from the data is a likelihood function, which combined with the prior distribution on the parameters produces the posterior distribution from which inferences are derived. This is an important simplification in the analysis of queues where there are a variety of possible ways of observing the system, many providing proportional likelihood functions but very different sampling distributions. The prior distribution quantifies whatever is known about the system before the data is collected. Usually, there is plenty of information a priori about the queue, especially if it is assumed to be in equilibrium. However, it is also possible to keep the parallelism with a likelihood analysis and to avoid the incorporation of further subjective inputs, by carrying out a Bayesian analysis usually called ‘objective’ because the prior distribution used is of the ‘non-informative’ or ‘objective’ type. From the posterior distribution, computation of estimates and standard errors is immediate. Also, probabilities of direct interest (like the probability that the ergodic condition holds) can be computed. Most importantly, restrictions in the parameter space imposed by the assumption of equilibrium are readily incorporated into the analysis. Prediction of measures of congestion of the system (number of customers waiting, time spent queuing, number of busy servers, and to on) is carried out from the corresponding predictive distributions which are also very useful for design and intervention in the system.