Optimal Capacities of Tokens at Tandem-Queue Models of General Service Times

Abstract

We consider a queueing system with two stations in series. Assume the service time distributions are general at one station and a finite mixture of Erlang distributions at the other. Exogenous customers should snatch tokens at a token buffer of finite capacity in order to enter the system. Customers are lost if there are no tokens available in the token buffer while they arrive. To obtain the stationary probability distribution of number of customers in the system, we construct an embedded Markov chain at the departure times. The solution is solved analytically and its analysis is extended to semi-Markovian representation of performance measures in queueing networks. A formula of the loss probability is derived to describe the probability of an arriving customer who finds no token in the token buffer, by which the throughput and the optimal number of tokens are also studied.