Stability of Polling Networks with State-Dependent Server Routing

Abstract

The polling network considered here consists of a finite collection of stations visited successively by a single server according to a state-dependent routing scheme. At every visit of a station, a possibly random number of customers are served and depart from the network thereafter. The server takes a possibly random time to walk from one station to the next. The network receives groups of customers at Poisson instants. These customers are distributed randomly over the stations and wait until served. A necessary condition for stability is derived, which proves to be also sufficient in certain special cases. Such cases include a starlike network and a ringlike one with a greedy service policy. Mathematically, positive recurrence of a Markov chain is studied.

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