Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-23T05:34:58.577Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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