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Sojourn times in closed queueing networks

Published online by Cambridge University Press:  01 July 2016

F. P. Kelly  and
P. K. Pollett
F. P. Kelly*
Affiliation:
University of Cambridge
P. K. Pollett*
Affiliation:
University of Cambridge
*
∗Postal address: Statistical Laboratory, 16 Mill Lane, Cambridge CB2 1SB, U.K.
∗∗Present address: Department of Mathematical Statistics and Operational Research, University College, P.O. Box 78, Cardiff CF1 1XL, U.K.
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Abstract

This paper obtains the stationary joint distribution of a customer's sojourn times along an overtake-free path in a closed multiclass Jackson network. The distribution has a simple representation in terms of the product form distribution for the state of the network at an arrival instant.

Keywords

RESPONSE TIMESJACKSON NETWORKSOVERTAKE-FREE PATHS
Type
Research Article
Information
Advances in Applied Probability , Volume 15 , Issue 3 , September 1983 , pp. 638 - 656
Copyright
Copyright © Applied Probability Trust 1983 

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References

[1] Baskett, F., Chandy, K. M., Muntz, R. R. and Palacios, F. G. (1975) Open, closed, and mixed networks of queues with different classes of customers. J. Assoc. Comput. Mach. 22, 248–260.CrossRefGoogle Scholar
[2] Boxma, O. J., Kelly, F. P. and Konheim, A. G. (1983) The product form for sojourn time distributions in cyclic exponential queues. J. Assoc. Comput. Mach. Google Scholar
[3] Brown, T. C. and Pollett, P. K. (1982) Some distributional approximations in Markovian queueing networks. Adv. Appl. Prob. 14, 654–671.Google Scholar
[4] Burke, P. J. (1968) The output process of a stationary M/M/s queueing system. Ann. Math. Statist. 39, 1144–1152.Google Scholar
[5] Burke, P. J. (1972) Output processes and tandem queues. In Proc. Symp. Computer-Communication Networks and Teletraffic, ed. Fox, J., Polytechnic Press of the Polytechnic Institute of Brooklyn, distributed by Wiley, New York, 419–428.Google Scholar
[6] Chandy, M., Herzog, U. and Woo, L. (1975) Parametric analysis of queueing network models. IBM J. Res. Develop. 19, 36–42.Google Scholar
[7] Daduna, H. (1982) Passage times for overtake-free paths in Gordon-Newell networks. Adv. Appl. Prob. 14, 672–686.Google Scholar
[8] Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn. Wiley, New York.Google Scholar
[9] Franken, P., König, D., Arndt, U. and Schmidt, V. (1981) Queues and Point Processes. Akademie-Verlag, Berlin.Google Scholar
[10] Freedman, D. (1971) Markov Chains. Holden-Day, San Francisco.Google Scholar
[11] Iglehart, D. L. and Shedler, G. S. (1980) Regenerative Simulation of Response Times in Networks of Queues. Lecture Notes in Control and Information Sciences 26, Springer-Verlag, Berlin.CrossRefGoogle Scholar
[12] Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
[13] Kelly, F. P. (1982) Networks of quasi-reversible nodes. In Applied Probability-Computer Science, the Interface: Proceedings of the ORSA-TIMS Boca Raton Symposium, ed. Disney, R., Birkhäuser Boston, Cambridge, Mass.Google Scholar
[14] Kelly, F. P. (1983) Invariant measures and the q-matrix. In Probability, Statistics and Analysis, ed. Kingman, J. F. C. and Reuter, G. E. H., London Mathematical Society Lecture Notes Series 79, Cambridge University Press, 143–160.Google Scholar
[15] Khintchine, A. Y. (1960) Mathematical Methods in the Theory of Queueing. Griffin, London.Google Scholar
[16] 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
[17] Lemoine, A. J. (1977) Networks of queues–a survey of equilibrium analysis. Management Sci. 24, 464–481.Google Scholar
[18] Lemoine, A. J. (1979) On total sojourn time in networks of queues. Management Sci. 25, 1034–1035.Google Scholar
[19] Melamed, B. (1982) On Markov jump processes imbedded at jump epochs and their queueing-theoretic applications. Math. Operat. Res. 7, 111–128.CrossRefGoogle Scholar
[20] Melamed, B. (1982) Sojourn times in queueing networks. Math. Operat. Res. 7, 233–244.CrossRefGoogle Scholar
[21] Pollett, P. K. (1982) Distributional Approximations for Networks of Queues. , University of Cambridge.Google Scholar
[22] Reich, E. (1957) Waiting times when queues are in tandem. Ann. Math. Statist. 28, 768–773.Google Scholar
[23] Reich, E. (1963) Notes on queues in tandem. Ann. Math. Statist. 34, 338–341.Google Scholar
[24] Ryll-Nardzewski, C. (1961) Remarks on processes of calls. Proc. 4th Berkeley Symp. Math. Statist. Prob. 2, 455–465.Google Scholar
[25] Sevcik, K. C. and Mitrani, J. (1981) The distribution of queueing network states at input and output instants. J. Assoc. Comput. Mach. 28, 358–371.Google Scholar
[26] Walrand, J. and Varaiya, P. (1980) Sojourn times and the overtaking condition in Jacksonian networks. Adv. Appl. Prob. 12, 1000–1018.Google Scholar
[27] Walrand, J. and Varaiya, P. (1980) Interconnections of Markov chains and quasi-reversible queueing networks. Stoc. Proc. Appl. 10, 209–219.CrossRefGoogle Scholar
[28] Whittle, P. (1968) Equilibrium distributions for an open migration process. J. Appl. Prob. 5, 567–571.Google Scholar