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IBM Journal of Research and Development  
Volume 19, Number 3, Page 283 (1975)
System Performance Evaluation
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Queuing Networks with Multiple Closed Chains: Theory and Computational Algorithms

by M. Reiser, H. Kobayashi
In this paper a recent result of Baskett, Chandy, Muntz, and Palacios is generalized to the case in which customer transitions are characterized by more than one closed Markov chain. Generating functions are used to derive closed-form solutions to stability, normalization constant, and marginal distributions. For such a system with N servers and L chains the solutions are considerably more complicated than those for systems with one subchain only. It is shown how open and closed subchains interact with each other in such systems. Efficient algorithms are then derived from our generating function representation.
Related Subjects: Analytical models; Markov process analysis; Queuing theory and applications