Quality Technology & Quantitative Management Vol. 2, No. 1, pp. 109-121, 2005 QTQM © ICAQM 2005 Optimal Capacities of Tokens at Tandem-Queue Models of General Service Times Hsing Luh and Chun-Lian Huang Department of Mathematical Sciences, National ChengChi University, Taiwan (Received May 2003, accepted August 2004) ____________________________________________________________________ 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. Keywords: Embedded Markov chains, probability distributions, queueing networks. ____________________________________________________________________ 1. Introduction n this paper we consider an open queueing model of finite capacity. The system consists of two stations in series and a token buffer whose size is limited. The simple structure of the model is depicted in Figure 1 and explained as follows . There are only one server and one queue at each station. Customers arrive from the exterior of the system following a Poisson distribution, entering the system at station 1 and departing at station 2. We assume the service time follows a finite mixture of Erlang distributions at serve 1 but a general distribution at the serve 2. The service discipline is First-Come-First-Served. Queue 1 Token Buffer Server 1 Server 2 Queue 2 Figure 1. Suppose the size of the token buffer is fixed and denoted by N. An arrival may enter station 1 if there is at least one token available at the token buffer, or it will be rejected by the system, thus resulting in a lost customer. Customers attend stations in sequence to obtain two services provided by server 1 and server 2. They leave the system and return tokens to the token buffer immediately when they complete their works. When a customer takes a token he will join server 1 if the server is idle, but wait for service in queue 1 if server 1 is busy. After I