An analysis of closed queueing networks with product form solution

It is a surprising discovery that tractable queueing models can accurately predict the behavior of computer systems despite unrealistic assumptions used in the construction of stochastic models for such systems. The models of most important computer systems are, however, closed queueing networks which can be usually analyzed only by using computational algorithms. A unified (applicable to all known algorithms) and comprehensive (using simple rules) treatment of computational algorithms is highly desirable. This paper provides such a treatment. This treatment is unified because not only the convolution algorithm but also the mean value algorithm and the coalesce algorithm can also be derived and extended to higher moments, using solely the generating function and the Z-transformation techniques. The treatment is comprehensive since it is based on the differentiation of elementary functions and the application of simple rules to the Z-inverse transformation. The generality of the methods permits the discovery of some new computational algorithms. The work constitutes a simple synthesis and extension of computational algorithms for closed queueing networks.