ABSTRACT
In this paper we analyze an advanced multi-server queueing system with finite capacity and non-preemptive priorities . Recent studies of multi-server priority models (e.g. Gail, HantIer and Taylor, 1988 , Kao and Narayanan, 1990, Ngo and Lee, 1991) assume infinite buffer capacity and use exponentially distributed service times as well as Poisson arrival streams. Such models do not reflect systems with more general characteristics regarding the arrival process and the service time distributions . Even if the service time distributions are different for each priority class they are exponentially distributed which restricts the applicability. Often the service times are equal but generally distributed. Furthermore
it is important that the arrival process of each type of customer is general and different for the different types of customers . In Wagner ( 1994) we analyzed such a model where the arrival processes of the different priority classes are non-renewal and the service times are generally distributed. But these types of models with infinite buffer capacity are not suitable for computing performance measures of highly loaded systems . Therefore it is necessary to analyze a multi-server model with these general characteristics and a finite buffer. For the model presented in this paper the arrival process will be described by a generalized Markovian arrival process with p absorbing states to indicate the arrival events of the different priority classes . The service time distribution is of phase-type and identical for each of the different types of customers . Our main issue is a new approach to calculate the moments of the actual waiting times of the different priority classes . To our best knowledge the waiting times for models with finite capacity where a non-preemptive head of the line priority is assumed (cf. Kleinrock, 1976 , pp. 1 19) , have not been analyzed yet . The methods we present are derived from applying first passage time analysis based on matrix analytic methods. The paper is organized as follows. In section 1 we give the description of the multi-server priority system. It follows that the model can be described by a homogeneous continuous time Markov chain (CTMC ) . In section 2 we derive the methods for the computation of the performance characteristics , such as stationary distributions of the queue lengths of the different types of customers at an arbitrary time instant and immediately after arrival instants. Furthermore the numbers of customers in the system at an arbitrary time instant and immediately after arrival instants are calculated . We derive recursive formulas for the computation of the moments of the actual waiting times for each type of customers . It will be shown that the matrix analytic methods for the computation of the" Laplace-Stieltjes Transform of the waiting times for the infinite capacity model used by Wagner ( 1994) can be derived as a special case of the results presented in this paper. We illustrate some numerical examples . Finally we summarize the findings of the paper in the conclusion.
