Continuity of Generalized Semi-Markov Processes

Abstract

It is shown that sequences of generalized semi-Markov processes converge in the sense of weak convergence of random functions if associated sequences of defining elements (initial distributions, transition functions and clock time distributions) converge. This continuity or stability is used to obtain information about invariant probability measures. It is shown that there exists an invariant probability measure for any finite-state generalized semi-Markov process in which each clock time distribution has a continuous c.d.f. and a finite mean. For generalized semi-Markov processes with unique invariant probability measures, sequences of invariant probability measures converge when associated sequences of defining elements converge. Hence, properties of invariant measures can be deduced from convenient approximations. For example, insensitivity properties established for special classes of generalized semi-Markov processes by Schassberger (Schassberger, R. 1977. Insensitivity of steady-state distributions of generalized semi-Markov processes, I. Ann. Probab. 5 87–99.), (Schassberger, R. 1978. Insensitivity of steady-state distributions of generalized semi-Markov processes, II. Ann. Probab. 6 85–93.), Konig and Jansen (König, D., U. Jansen. 1976. Eine Invarianzeigenschaft Zufälliger Bedienungs-Prozesse mit Positiven Geschwindigkeiten. Math Nachr. 70 321–364.), and Burman (Burman, D. Y. 1981. Insensitivity in queueing systems. Adv. Appl. Probab. To appear.) extend to a larger class of generalized semi-Markov processes.