On Poisson Traffic Processes in Discrete State Markovian Systems with Applications to Queueing Theory.
MICHIGAN UNIV ANN ARBOR DEPT OF INDUSTRIAL AND OPERATIONS ENGINEERING
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A regular Markov process is considered with continuous parameter, countable state space, and stationary transition probabilities, over which a class of traffic processes is defined. The feasibility that multiple traffic processes constitute mutually independent Poisson processes is investigated in some detail. A variety of independence conditions on the traffic process and the underlying Markov process are shown to be 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 to queueing-theoretic models. It follows that, for the class of traffic processes as per this paper in a queueing-theoretic context, Kellys notion of quasi-reversibility and Gelenbe and Muntzs notion of completeness are essentially equivalent to pointwise independence of traffic and state. The latter concept, however, is the most general one. The relevance of the theory developed to queueing network decomposition is also pointed out.
- Statistics and Probability
- Operations Research