Polling systems with parameter regeneration, the general case

Iain Macphee (1) , Mikhail Menshikov (1) , Dimitri Petritis (2) , Serguei Popov (3)

1 Department of Mathematics
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 Instituto de Matematica

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~\cite{MMPP}. 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.

Mots clés: Polling system; parameter regeneration; stability; time-inhomogeneous Markov chains; recurrence; Lyapunov functions; random matrices

Domaines: Probabilités [math.PR]

Iain Macphee, Mikhail Menshikov, Dimitri Petritis, Serguei Popov. Polling systems with parameter regeneration, the general case. The Annals of Applied Probability, 2008, 18 (6), pp.2131-2155. ⟨10.1214/08-AAP519⟩. ⟨hal-00260810⟩