Invariance Properties Of Queueing Networks And Their Application To Computer/Communications Systems

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Abstract
During the past decade there has been a quantitative leap in the development of queueing theory as it relates to the design, control, and operation of computer communications systems. Much of the research centres on the so-called “product form solution” of the network state probabilities. This, in turn, has led to the discovery, in many specific cases, that the state probabilities are invariant with respect to the form of the service time distribution. No systematic investigation of this invariance has appeared in tJie computer/communications literature as of this date. Our main objectives in this paper, therefore, are to describe and define the class of queueing models which has this invariance property and to demonstrate that almost all such models have a common basis. The theoretical foundation for the classification of invariance (developed by Kovalenko (19, 20) and, independently, by Konig (17)) has not received much recognition. In this paper, Kovalenko’s main theorem is stated, the required conditions for invariance are explained, and examples of solutions which can be derived quite easily from the theorem are given. The connection between the invariance property and the product form of the state probabilities is demonstrated. Finally, two extensions of Kovalenko’s theorem are presented; one by Guseinov (10) and one by the authors.