Polling systems with parameter regeneration, the general case

TL;DR

This paper analyzes a polling system where service parameters are randomly regenerated at each station, providing criteria for recurrence and transience, and revealing complex behaviors like null recurrence in such stochastic models.

Contribution

It introduces a generalized polling model with random parameter regeneration and establishes recurrence criteria using Lyapunov exponents, extending previous two-station results.

Findings

Criteria for recurrence, transience, and moments of return times.

Model can exhibit null recurrence over a broad parameter region.

Generalizes earlier two-station models with new stochastic dynamics.

Abstract

We consider a polling model with multiple stations, each with Poisson arrivals and a queue of infinite capacity. The service regime is exhaustive and there is Jacksonian feedback of served customers. What is new here is that when the server comes to a station it chooses the service rate and the feedback parameters at random; these remain valid during the whole stay of the server at that station. We give criteria for recurrence, transience and existence of the $s$th moment of the return time to the empty state for this model. This paper generalizes the model, when only two stations accept arriving jobs, which was considered in [Ann. Appl. Probab. 17 (2007) 1447--1473]. Our results are stated in terms of Lyapunov exponents for random matrices. From the recurrence criteria it can be seen that the polling model with parameter regeneration can exhibit the unusual phenomenon of null recurrence over a thick region of parameter space.