Polling systems with station breakdowns

Abstract

None of the works in the literature on Polling Systems deals with the important phenomenon of failing nodes. This paper addresses this issue. We assume that there are two modes of station breakdowns: node-dependent and general. Each node is subject to a random failure process and a failure of a node during a visit of the server causes the latter to move to the next station. We carry performance analyses of both the Gated and the Exhaustive service regimes and derive probability generating functions, Laplace transforms and means of variables such as queue sizes at polling and at arbitrary instants, durations of busy periods, number of jobs left behind at server's station-departure moments, and cycle times. The analysis of the Exhaustive case is achieved via an explicit solution of an infinite set of linear equations, where the unknowns are the state-dependent joint transforms of two key characteristics: the number of arrivals to a station during a busy period starting with a given number of jobs, and the number of jobs successfully served during that period. Stability conditions are indicated for each regime.