Bayesian inference and prediction in an M/D/1 queueing system

In many real-life situations, certain infinite queueing systems arise in which the clients arrive according to a Poisson process and are served by a single server under an approximately constant service time. An example of such system includes the cycle of a washing machine that takes a fixed length of time to finish one service. This example represents the kind of queueing system that this article attempts to describe, by estimating its traffic intensity ρ, which represents the fraction of time that is busy and can be used to determine other quality measures such as the average queue Lq and the expected number of customers in the system Ls. Bayes estimators for ρ are obtained under the squared error loss function assuming three forms of prior information about ρ, i.e., incomplete gamma prior, left-truncated beta prior, and the improper Jeffreys prior. Furthermore, the Bayes factor as a model comparison criterion is proposed to select a suitable model for Bayesian analysis. A comprehensive set of Monte Carlo simulations comprises our experimental set designed to attest to the efficacy and efficiency of the proposed algorithms. A real case situation is analyzed to illustrate the applicability of the methods developed.