Mean-value analysis and convolution method for queue-dependent servers in closed queueing networks

Abstract

In this paper, we formulate a recursive relation for the marginal probabilities of a closed network with K customers in terms of the same network with K − 1 customers. Mean-value analysis (MVA) is an application of this relation, together with Little's formula. It is shown that the convolution method, too, can be based on the same recursive result. This leads to a new convolution algorithm called normalized convolution algorithm (NCA), which like MVA works entirely with probabilities and throughputs rather than with quantities such as normalization contacts. NCA avoids a difficult problem, the occurrence of floating-point over-flows in the original convolution algorithm. We shall also solve a numerical stability problem found in MVA. Finally, we show how MVA and the convolution algorithm can be combined in the same problem to yield a hybrid method retaining the best properties of both methods.