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On the structure of an insensitive generalized semi-Markov process with reallocation and point-process input

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  01 July 2016

Masakiyo Miyazawa ,
Rolf Schassberger  and
Volker Schmidt
Masakiyo Miyazawa*
Affiliation:
Science University of Tokyo
Rolf Schassberger*
Affiliation:
Technical University of Braunschweig
Volker Schmidt*
Affiliation:
University of Ulm
*
*Postal address: Dept. of Information Sci., Science University of Tokyo, Noda City, Chiba 278, Japan.
**Postal address: Inst, of Math. Stochastics, Technical University of Braunschweig, Pockelsstr. 14, 38106 Braunschweig, Germany.
***Postal address: Dept. of Stochastics, University of Ulm, 89069 Ulm, Germany.
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Abstract

A generalized semi-Markov process with reallocation (RGSMP) was introduced to accommodate a large class of stochastic processes which cannot be analyzed by the well-known model of an ordinary generalized semi-Markov process (GSMP). For stationary RGSMP whose initial distribution has a product form, we show that, for a randomly chosen clock of a fixed insensitive type, if the lifetime of this clock is changed to infinity, then the background process is stationary under a certain time change. This implies that the expected time required for the tagged clock to consume a given amount x of resource, called the attained sojourn time, is a linear function of x. Such stationarity and linearity results are known for two special RGSMPs: ordinary GSMP and Kelly's symmetric queue. Our results not only extend them to a general RGSMP but also give more detailed formulas, which allow us to calculate for instance the expected attained sojourn time while the background process is in a given state. Furthermore, we remark that analogous results hold for GSMP with point-process input, in which the lifetimes of clocks of a fixed type form an arbitrary stationary sequence (of not necessarily independent random variables).

Keywords

PALM PROBABILITIESCAMPBELL'S THEOREMPRODUCT-FORM DISTRIBUTIONINSENSITIVITYATTAINED SOJOURN TIMEQUEUEING NETWORK

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60G55: Point processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 27 , Issue 1 , March 1995 , pp. 203 - 225
Copyright
Copyright © Applied Probability Trust 1995 

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