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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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References

[1] Baccelli, F. and Brémaud, P. (1987) Palm Probabilities and Stationary Queues. Lecture Notes in Statistics 41, Springer-Verlag, Berlin.Google Scholar
[2] Barbour, A. D. (1982) Generalized semi-Markov schemes and open queueing networks. J. Appl. Prob. 19, 469–474.Google Scholar
[3] Barbour, A. D. and Schassberger, R. (1981) Insensitive average residence times in generalized semi-Markov processes. Adv. Appl. Prob. 13, 720–735.CrossRefGoogle Scholar
[4] Baskett, A. D., Chandy, K. M., Muntz, R. R. and Palacios, F. G. (1982) Open, closed and mixed networks of queues with different classes of customers. J. Assoc. Comput. Mach. 22, 248–260.CrossRefGoogle Scholar
[5] Foley, R. and Klutke, G.-A. (1989) Stationary increments in the accumulated work process in processor sharing queues. J. Appl. Prob. 26, 671–677.CrossRefGoogle Scholar
[6] Foley, R., Klutke, G.-A. and König, D. (1991) Stationary increments of accumulation processes in queues and generalized semi-Markov schemes. J. Appl. Prob. 28, 864–872.Google Scholar
[7] Franken, P., König, D., Arndt, U. and Schmidt, V. (1982) Queues and Point Processes. Wiley New York.Google Scholar
[8] Helm, W. E. and Schassberger, R. (1982) Insensitive generalized semi-Markov schemes with point process input. Math. Operat. Res. 7, 129–138.Google Scholar
[9] Jansen, U. (1984) Conditional expected sojourn times in insensitive queueing systems and networks. Adv. Appl. Prob. 16, 906–919.CrossRefGoogle Scholar
[10] Jansen, U., König, D. and Nawrotzki, K. (1979) A criterion of insensitivity for a class of queueing systems with random marked point processes. Math. Operationsforsch. Statist., Ser. Optimization 10, 379–403.Google Scholar
[11] Kelly, F. P. (1976) Networks of queues. Adv. Appl. Prob. 8, 416–432.Google Scholar
[12] Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
[13] Kleinrock, L. (1976) Queueing Systems, Vol. 2: Computer Applications. Wiley, New York.Google Scholar
[14] König, D. and Schmidt, V. (1992) Random Point Processes. Teubner, Stuttgart (in German).Google Scholar
[15] Matthes, K. (1962) On the theory of queueing processes. Trans. 3rd Prague Conf. Inform. Theory, Statist. Dec. Funct. Random Processes. Prague , 513–528 (in German).Google Scholar
[16] Miyazawa, M. (1983) The derivation of invariance relations in complex queueing systems with stationary inputs. Adv. Appl. Prob. 15, 874–885.Google Scholar
[17] Miyazawa, M. (1991) The characterization of the stationary distributions of the supplemented self-clocking jump process. Math. Operat. Res. 16, 547–565.Google Scholar
[18] Miyazawa, M. (1993) Insensitivity and product-form decomposability of reallocatable GSMP. Adv. Appl. Prob. 25, 415–437.Google Scholar
[19] Schassberger, R. (1978) Insensitivity of steady-state distributions of generalized semi-Markov processes with speeds. Adv. Appl. Prob. 10, 836–851.Google Scholar
[20] Schassberger, R. (1986) Two remarks on insensitive stochastic models. Adv. Appl. Prob. 18, 791–814.Google Scholar
[21] Sevcik, K. C. and Mitrani, I. (1981) The distribution of queueing network states at input and output instants. J. Assoc. Comput. Mach. 28, 358–371.Google Scholar