Performance analysis of the parallel cyclic two-stage queueing model

Abstract

The 2-stage cyclic queueing network model is an analytical tool that has been widely used for the performance analysis of sequential processing systems in computer, manufacturing, logistic and communication modeling contexts. No similar tool exists for the parallel processing case, because of the structural complexity of the underlying stochastic process. Consequently, only numerical approaches can be used. On the other hand, numerical solutions are difficult to deal with for at least three reasons: (1) because of the growth of the process' rate matrix with the number of jobs N; (2) because of the fact that the N + 1, N + 2,…, N + k job solutions cannot be derived from the N-job solution, and (3) because of the recalculations that are needed each time new parameters are introduced. This paper deals with the parallel version of the 2-stage cyclic queueing model. It shows that the parallel cyclic server belongs to the class of Quasi-Birth-and-Death finite-state processes for which an explicit expression can be found for the invariant probability vector, without recourse to the more expensive Matrix-Geometric method. It then provides an analytical solution, in the time domain, based on an efficient recurrence which solves the model without use of the N-dependent rate matrix. It is in symbolic form. The effect of parameter change is therefore easily evaluated, and the N + 1, N + 2, etc., job analysis is easily derived from the N-job solution.