On Poisson Traffic Processes in Discrete State Markovian Systems with Applications to Queueing Theory.
MICHIGAN UNIV ANN ARBOR COMPUTER INFORMATION AND CONTROL ENGINEERING
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We consider a regular Markov process with continuous parameter, countable state space, and stationary transition probabilities, over which we define a class of traffic processes. The feasibility that multiple traffic processes constitute mutually independent Poisson processes is investigated in some detail. We show that a variety of independence conditions on a traffic process and the underlying Markov process are equivalent or sufficient to ensure Poisson related properties these conditions include independent increments, renewal, weak pointwise independence, and pointwise independence. Two computational criteria for Poisson traffic are developed a necessary condition in terms of weak pointwise independence, and a sufficient condition in terms of pointwise independence. The utility of these criteria is demonstrated by sample applications of queueing-theoretic models. It follows that, for the class of traffic processes as per this paper in queueing-theoretic contexts, Muntzs M yields M property, Gelenbe and Muntzs notion of completeness, and Kellys notion of quasi-reversibility and essentially equivalent to pointwise independence of traffic and state. The relevance of the theory to queueing network decomposition is also noted. Author
- Statistics and Probability