A compensation procedure for multiprogramming queues (Q684386)

We study a multiprogramming system consisting of an input-output unit (IO unit) and a central processor (CP). This system can be represented by a continuous time Markov process with states \((m,n)\), where \(m\) and \(n\) denote the number of jobs at the CP and the IO unit respectively. The computation of the equilibrium distribution \(\{p_{m,n}\}\) of this Markov process is the purpose of the analysis. 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. Some numerical aspects of the computation of the equilibrium distribution are considered.