Fast and Scalable Nonparametric Bayesian Prediction for the M/G/1 Queue

In this article we develop a nonparametric Bayesian approach to prediction for the M/G/1 queue, focusing on the imbedded semiMarkov process of the queue at the departure times. Our approach is motivated by queues with a large number of data points and highfrequency systems, where times consuming MCMC/ABC algorithms might be infeasible and a nonparametric approach is desirable to avoid parametric assumptions. We define a reinforced stochastic model for the analysis of the M/G/1 queue through a system of predictive distributions. Using the theory of partial exchangeable processes, we prove that the reinforced stochastic process is the predictive probability model of a Bayesian semi-Markov mixture model. This result enabled fast and scalable Bayesian nonparametric prediction based on the reinforced stochastic process.