Ergodicity of a polling network
A queueing system with a single server and with \(N\) service stations is considered which are successively visited by the server who is polling them according to a Markovian routing scheme. For every \(i\in \{1,\dots, N\}\), the server can serve no more than a random maximum number \(D_ i\) of customers during each visit at station \(i\). The \(D_ i\) and the service times at this station are drawn independently from general distributions which may depend on \(i\). For switching from station \(i\) to station \(j\), the server needs a random switching time whose distribution may depend on the pair \((i,j)\). At each station, groups of customers arrive according to a homogeneous Poisson process whose intensity and group-size distribution may be station-dependent, too. Arrival processes, routing, switching times, and service times are mutually independent. The temporal evolution of the system is described by the Markov chain \((S, {\mathcal X})= \{(S(n), X_ 1(n),\) \(\dots, X_ i (n))\); \(n\geq 0\}\) where \(S(n)\) denotes the \(n\)th station polled by the server, and \(X_ i(n)\) the queue length at station \(i\) at the moment when the \(n\)th switching time is terminated. Conditions are derived under which \((S,{\mathcal X})\) is positive recurrent.
scientific article; zbMATH DE number 590988