A compensation procedure for multiprogramming queues

In this paper we study a multiprogramming system consisting of an input-output unit (10 unit) and a central processor (CP). This system can be represented by a continuous time Marlmv process with states (m,n). where m and n denote the number of jobs at the CP and the 10 unit respectively. The computation of the equilibrium distribution {p_{m,n}} of this Markov process is the purpose of the analysis in this paper. The analysis consists of two parts. In the first part, we use a compensation procedure to show that the equilibrium distribution {p_{m,n}} in those states (m,n) for which m+n is not too small, can be expressed as an infinite linear combination of product forms. Explicit formulae are given for the product forms and the coefficients of this infinite linear combination. In the second part of the analysis, we pay attention to some numerical aspects of the computation of the equilibrium distribution. For the computation of the equilibrium probabilities that can be expressed as infinite linear combinations of product forms, we derive bounds for the errors caused by cutting off these infinite linear combinations, and after that we present numerically stable formulae to compute one by one the remaining equilibrium probabilities. Keywords: multiprogramming queues problem, Markov process, equilibrium distribution, product form, convergence of series, embedded Markov process, bounds