On Poisson Traffic Processes in Discrete State Markovian Systems with Applications to Queueing Theory.

摘要

A regular Markov process with continuous parameter, countable state space, and stationary transition probabilities is considered, 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. It is shown 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, Muntz's M yields M property, Gelenbe and Muntz's notion of completeness, and Kelly's 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)