Semi-Markov Processes

A Primer

by Bennett L. Fox

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An introduction to semi-Markov processes (SMPs) for an audience primarily interested in applications. SMPs, useful in problems of inventory control and queueing, result from a union of renewal processes and Markov chains; they consider only the states corresponding to instantaneous regeneration points and do so in continuous, rather than discrete, time. The presentation is deliberately unconventional to permit a more heuristic discussion of the subject. The approach used eliminates all nonzero holding times and is thus not the same as the state definition used in more fully rigorous treatments of probability theory in general and SMPs in particular. The underlying mathematics remains the same, however, and the approach permits the focus to be on aspects of problem-solving interest. The states are defined so that they can be represented graphically by nodes of a network with stochastic arc lengths; for example, traversing an arc might correspond to a customer completing service. In specifying the states, a detailed description is often required to make all decision points regeneration points, and linear programs with hundreds or thousands of constraints may result. Generally, analysis shows that the constraint matrix is sparse and structured so as to be amenable to decomposition.

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