Proof of stability conditions for token passing rings by Lyapunov functions

A token passing ring can be described as a system of M queues with one server that rotates around the queues sequentially. Georgiadis-Szpankowski (1992) considered rings where the token (server) performs x ∇ l_j services on queue j, where x is the size of queue j upon arrival of the token, and l_j is a fixed limit of service for queue j. The token then spends some random time switching to the next queue. For j=1, ..., M, arrivals to queue j are Poisson with rate λ_j, and service times have mean s _j and are independent of the arrival and switchover processes. The purpose of this paper is to give an alternate and simpler proof of the stability conditions given by Georgiadis-Szpankowski using Lyapunov functions. An additional assumption is made about the second moments of the service and switchover times being finite