On Poisson traffic processes in discrete-state Markovian systems by applications to queueing theory

This paper considers a regular Markov process with continuous parameter, countable state space, and stationary transition probabilities, and defines a class of traffic processes over it. The possibility that multiple traffic processes constitute mutually independent Poisson processes is investigated. A variety of independence conditions on a traffic process and the underlying Markov process are shown to lead to Poisson-related properties; these conditions include weak pointwise independence, and pointwise independence. Some examples of queueing-theoretic applications are given. For the class of traffic processes considered here in a queueing-theoretic context, Muntz's M M property, Gelenbe and Muntz's notion of completeness, and Kelly's notion of quasi-reversibility are shown to be essentially equivalent to pointwise independence of traffic and state. The relevance of the theory to queueing network decomposition is pointed out.