Paper Info
Title
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 approxi- mations. For example, insensitivity properties established for special classes of generalized semi-Markov processes by Schassberger (1977), (1978), Konig and Jansen (1976) and Burman (1981) extend to a larger class of generalized semi-Markov processes, 1. Introduction and summary. Among the most promising stochastic processes for modeling complex phenomena in operations research are the generalized semi-Markov processes introduced by Matthes (19) and investigated further by Konig, Matthes and Nawrotzki (15), (16), Konig and Jansen (17), Schassberger (23)-(25), Burman (6) and Fossett (8). A GSMP moves from state to state with the destination and duration of each transition depending on which of several possible events associated with the occupied state occurs first. Several different events compete for causing the next jump and imposing their own particular jump distribution for determining the next state. An ordinary SMP (semi-Markov process) is the special case in which there is only one event associated with each state. At each transition of a GSMP, new events may be scheduled. For each of these new events, a clock indicating the time until the event is scheduled to occur is set by an independent chance mechanism. An event which is scheduled but does not initiate a transition is either abandoned or it is associated with the next state and its clock just continues running. We think of a GSMP as a model of discrete-even t simulation. A good example of a GSMP is provided by the general multiple-heterogeneous-channel queue studied in Iglehart and Whitt (11). A state could be the number of customers in the system and an indication of which servers are busy. Possible events associated with such a state would be an arrival in one of the arrival channels or a service completion by one of the occupied servers. With the usual independence assumptions and without any Markov assumptions, as in (11), this representation yields a GSMP which is not a SMP. Furthermore, this GSMP is not regenerative; there does not exist an embedded renewal process. (This statement may be confusing, however, because after appending appropriate supplementary variables to the GSMP we obtain an associated Markov process, and recent results of Athreya, McDonald and Ney (1), (2), and Nummelin (21) show that there will often exist a regenerative structure for this Markov process. At
Year
DOI
Venue
1980
10.1287/moor.5.4.494
MATHEMATICS OF OPERATIONS RESEARCH
DocType
Volume
Issue
Journal
5
4
ISSN
Citations 
PageRank 
0364-765X
63
24.09
References 
Authors
1
1
Name
Order
Citations
PageRank
Ward Whitt11509658.94