Derivation of equilibrium and time - dependent solutions to MIMI 001 IN and MIMI 00 queueing systems using entropy . maximization

Queueing theory has provided the basis for remarkable successes in the performance modeling and analysis of computer systems.6,19,21 Because it is clear that computer systems do not satisfy assumptions made by the stochastic process models that are used, this success has been somewhat puzzling; it appears that queueing theory equations have wider applicability than is suggested by their classical derivations. Buzen has offered one possible explanation in terms of "operational analysis."4 In this paper we point out that certain results in elementary queueing theory can be derived simply and with relatively few assumptions by means of entropy maximization. This entropy maximization viewpoint provides the basis of another possible explanation for the widespread applicability of queueing theory formulas. Analysis by means of entropy maximization has been of interest since Shannon24 showed, for discrete noiseless systems, that the best encoding of an information source, in the sense of enabling the highest information rate over a fixed capacity channel, is the one that maximizes the source entropy. In addition to continuing applications in communication theory, there has been a growing interest in the use of entropy maximization techniques for probabilistic analysis and problem solving in other fields. Much of this work has been stimulated by that of E. T. Jaynes. Detailed discussions concerning the motivation, justification, and validity of various entropy maximization techniques are available elsewhere.3',9,12-15,17,26,27 There have been applications in statistical mechanics,lO,11,16 traffic networks,3,8 reliability estimation,27 production line decision making,13,29 system simulation,5 statistics ,9,20,22 spectral analysis ,1 image reconstruction,30 and general probabilistic problem solving. 12-15,17,28 After summarizing the entropy maximization technique in the second section, we obtain in the third section the maximum entropy solution for the state probabilities of an abstract, general system. All of our results are obtained by