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Conditional job-observer property for multitype closed queueing networks

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  14 July 2016

Hans Daduna  and
Ryszard Szekli
Hans Daduna*
Affiliation:
Universität Hamburg
Ryszard Szekli*
Affiliation:
Wrocław University
*
∗ Postal address: Fachbereich Mathematik Stochastik, Universität Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany.
∗∗ Postal address: Mathematical Institute, Wrocław University, 2/4 PL Grunwaldzki, 50-384 Wrocław, Poland. Email address: szekli@math.uni.wroc.pl
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Abstract

For functionals of multitype closed queueing networks, a conditional job-observer property is shown which provides more insight into the classical job-observer property. Applications and examples are given, including the classical job-observer property for the number of customers in a network, a representation of cycle time distributions and a basic formula for sojourn times.

Keywords

Gordon-Newell networkcyclic queuesproduct formjob-observer propertypoint processstochastic intensityPapangelou formulaPalm measure

MSC classification

Primary: 60K25: Queueing theory 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 4 , December 2002 , pp. 865 - 881
Copyright
Copyright © Applied Probability Trust 2002 

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Footnotes

Work supported by an Alexander von Humboldt Fellowship and by KBN grant 2P03A04915.

References

Baccelli, F. and Brémaud, P. (1994). Elements of Queueing Theory. Springer, Berlin.Google Scholar
Boxma, O. J. (1984). Analysis of successive cycles in a cyclic queue. In Performance of Computer Communication Systems, eds Rudin, H. and Bux, W., North Holland, Amsterdam, pp. 293–306.Google Scholar
Boxma, O. J., and Donk, P. (1982). On the response time and cycle time distributions in a two stage cyclic queue. Performance Evaluation 2, 181–194.CrossRefGoogle Scholar
Boxma, O. J., Kelly, F. P., and Konheim, A. G. (1984). The product form for sojourn time distributions in cyclic exponential queues. J. Assoc. Comput. Mach. 31, 128–133.CrossRefGoogle Scholar
Brémaud, P., Kannurpatti, R., and Mazumdar, R. (1992). Event and time averages: a review. Adv. Appl. Prob. 24, 377–411.CrossRefGoogle Scholar
Daduna, H. (1997). Sojourn time distributions in non-product-form queueing networks. In Frontiers in Queueing, ed. Dshalalow, J. H., CRC, Boca Raton, FL, pp. 197–224.Google Scholar
Daduna, H., and Szekli, R. (2000a). Conditional job-observer properties in multitype closed queueing networks. Preprint, Mathematical Institute, University of Wrocław.Google Scholar
Daduna, H., and Szekli, R. (2000b). Monotonicity and dependence properties of sojourn and cycle times in closed networks. Preprint, Schwerpunkt Mathematische Statistik und Stochastische Prozesse, Fachbereich Mathematik, Universität Hamburg.Google Scholar
Jacod, J. (1975). Multivariate point processes: predictable projection, Radon–Nikodym derivatives, representation of martingales. Z. Wahrscheinlichkeitsth. 31, 235–253.CrossRefGoogle Scholar
Kelly, F. P. (1979). Reversibility and Stochastic Networks. John Wiley, New York.Google Scholar
Kelly, F. P., and Pollet, P. K. (1983). Sojourn times in closed queueing networks. Adv. Appl. Prob. 15, 638–653.CrossRefGoogle Scholar
Kook, K. H., and Serfozo, R. F. (1993). Travel and sojourn times in stochastic networks. Ann. Appl. Prob. 3, 228–252.CrossRefGoogle Scholar
Lavenberg, S. S., and Reiser, M. (1980). Stationary state probabilities at arrival instants for closed queueing networks with multiple types of customers. J. Appl. Prob. 17, 1048–1061.CrossRefGoogle Scholar
Mitrani, I. (1987). Modelling of Computer and Communications Systems. Cambridge University Press.Google Scholar
Schassberger, R., and Daduna, H. (1987). Sojourn times in queueing networks with multiserver nodes. J. Appl. Prob. 24, 511–521.CrossRefGoogle Scholar
Serfozo, R. F. (1999). Introduction to Stochastic Networks. Springer, New York.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Walrand, J. (1988). An Introduction to Queueing Networks. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
Zazanis, M. (1997). Formulas and representations for cyclic Markovian networks via Palm calculus. Queueing Systems 26, 151–167.CrossRefGoogle Scholar