Bayesian nonparametric inference for the M/G/1 queueing systems based on the marked departure process

TL;DR

This paper develops Bayesian nonparametric methods to infer the unobservable service time distribution in M/G/1 queueing systems using marked departure data, providing a new statistical framework with validated large-sample properties.

Contribution

It introduces a novel Bayesian nonparametric approach for inferring service times in M/G/1 queues based on marked departure data, including the development of sufficient statistics and posterior analysis.

Findings

Posterior consistency and normality established for large samples.

New statistical structure identified via the S-structure.

Method enables updating prior distributions to posterior based on observed data.

Abstract

In the present work we study Bayesian nonparametric inference for the continuous-time M/G/1 queueing system. In the focus of the study is the unobservable service time distribution. We assume that the only available data of the system are the marked departure process of customers with the marks being the queue lengths just after departure instants. These marks constitute an embedded Markov chain whose distribution may be parametrized by stochastic matrices of a special delta form. We develop the theory in order to obtain integral mixtures of Markov measures with respect to suitable prior distributions. We have found a sufficient statistic with a distribution of a so-called S-structure sheding some new light on the inner statistical structure of the M/G/1 queue. Moreover, it allows to update suitable prior distributions to the posterior. Our inference methods are validated by large sample results as posterior consistency and posterior normality.