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Stationary increments of accumulation processes in queues and generalized semi-Markov schemes

Published online by Cambridge University Press:  14 July 2016

R. D. Foley ,
Georgia-Ann Klutke  and
Dieter König
R. D. Foley*
Affiliation:
Georgia Institute of Technology
Georgia-Ann Klutke*
Affiliation:
University of Texas at Austin
Dieter König*
Affiliation:
Mining Academy Freiberg
*
∗ Postal address: Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332–0205, USA.
∗∗ Postal address: Operations Research Group, Department of Mechanical Engineering, University of Texas at Austin, Austin, TX 78712, USA.
∗∗∗ Postal address: Department of Mathematics, Mining Academy Freiberg, D-O 9200 Freiberg, Germany.
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Abstract

Let Tx be the length of time to accumulate x units of a resource. In queueing, the resource could be service. We derive a sufficient condition for the process to have stationary increments where Tx is an additive functional of a Markov process. This condition is satisfied in symmetric queues and generalized semi-Markov schemes with insensitive components. As a corollary, we show that the conditional expected response time in a symmetric queue is linear in the service requirement. A similar result holds for the conditional average residence time of an insensitive component in a GSMS.

Keywords

RANDOM TIME CHANGESYMMETRIC QUEUES
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 4 , December 1991 , pp. 864 - 872
Copyright
Copyright © Applied Probability Trust 1991 

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References

Baccelli, F. and Bremaud, P. (1987) Palm Probabilities and Stationary Queues. Springer-Verlag, Berlin.Google Scholar
Barbour, A. D. and Schassberger, R. (1981) Insensitive average residence times in generalized semi-Markov processes. Adv. Appl. Prob. 13, 720–735.Google Scholar
Foley, R. D. and Klutke, G.-A. (1989) Stationary increments in the accumulated work process in processor sharing queues. J. Appl. Prob. 26, 671–677.Google Scholar
Franken, P., König, D., Arndt, U. and Schmidt, V. (1981) Queues and Point Processes. Wiley, New York; Akademie-Verlag, Berlin.Google Scholar
Glynn, P. (1989) A GSMP formalism for discrete event systems. Proc. IEEE 70, 14–23.Google Scholar
Helm, W. E. and Schassberger, R. (1982) Insensitive generalized semi-Markov schemes with point process input, Math. Operat. Res. 7, 129–138.Google Scholar
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
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Kleinrock, L. (1976) Queueing Systems, Vol. 2: Computer Applications. Wiley, New York.Google Scholar
König, D. and Jansen, U. (1976) Eine Invarianzeigenschaft zufälliger Bedienungsprozesse mit positiven Geschwindigkeiten. Math. Nachr. 70, 321–364.Google Scholar
König, D., Matthes, K. and Nawrotzki, K. (1967) Verallgemeinerungen der Erlangschen und Engsetschen Formeln (Eine Methode in der Bedienungstheorie). Akademie-Verlag, Berlin.Google Scholar
Matthes, K. (1962) Zur Theorie der Bedienungsprozesse. Trans. 3rd Prague Conf. Information Theory.Google Scholar
Ross, S. (1981) Stochastic Processes. Wiley, New York.Google Scholar
Schassberger, R. (1978) Insensitivity of steady-state distribution of generalized semi-Markov processes with speeds. Adv. Appl. Prob. 10, 836–851.Google Scholar