Periodicity in the Transient Regime of Exhaustive Polling Systems

We consider an exhaustive polling system with three nodes in its transient regime under a switching rule of generalized greedy type. We show that, for the system with Poisson arrivals and service times with finite second moment, the sequence of nodes visited by the server is eventually periodic almost surely. To do this, we construct a dy-namical system, the triangle process, which we show has eventually periodic trajectories for almost all sets of parameters and in this case we show that the stochastic trajectories follow the deterministic ones a.s. We also show there are infinitely many sets of parameters where the triangle process has aperiodic trajectories and in such cases trajectories of the stochastic model are aperiodic with positive probability. 1. Introduction. A polling system has N nodes where jobs arrive and queue and a single server which switches between the nodes to process the jobs. In exhaustive polling systems the server processes all jobs at its current node i, say, including any that arrive while jobs there are being processed, before switching to another node j, chosen by some rule. Conditions for transience/recurrence of polling systems using a greedy switching rule were given in [5, 6]. There is a critical case which has been investigated in [8] and [9]. In this paper we show that, for an exhaustive polling system with N = 3 nodes, with arrival streams and service times putting the system in its transient regime and switching according to a threshold rule (a type of generalized greedy rule), the sequence in which the server visits the nodes is eventually periodic for almost all choices of threshold parameters.