Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-24T07:02:29.688Z Has data issue: false hasContentIssue false
  • English
  • Français

Burke's theorem on passage times in Gordon–Newell networks

Published online by Cambridge University Press:  01 July 2016

Hans Daduna
Hans Daduna*
Affiliation:
Technische Universität Berlin
*
∗Present address: Institut für Mathematische Stochastik der Universität Hamburg, Bundes-strasse 55, 2000 Hamburg 13, West Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

In a closed cycle of exponential queues where the first and the last nodes are multiserver queues while the other nodes are single-server queues, the cycle-time distribution has a simple product form. The same result holds for passage-time distributions on overtake-free paths in Gordon–Newell networks. In brief, we prove Burke's theorem on passage times in closed networks.

Keywords

CYCLIC QUEUESMULTISERVER QUEUESCYCLE TIMES
Type
Research Article
Information
Advances in Applied Probability , Volume 16 , Issue 4 , December 1984 , pp. 867 - 886
Copyright
Copyright © Applied Probability Trust 1984 

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

Barbour, A. (1982) Generalized semi-Markov schemes and open queueing networks. J. Appl. Prob. 19, 469–474.CrossRefGoogle Scholar
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.Google Scholar
Boxma, O. J. (1983) Response time distributions in cyclic queues. Paper presented at the 44th Session of the International Statistical Institute, Madrid.Google Scholar
Boxma, O. J. and Donk, P. (1982) On response-time and cycle-time distributions in cyclic queues. Performance Evaluation 2, 181–194.Google 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.Google Scholar
Burke, P. J. (1964) The dependence of delays in tandem queues. Ann. Math. Statist. 35, 874–875.Google Scholar
Burke, P. J. (1968) The output process of a stationary M/M/s queueing system. Ann. Math. Statist. 39, 1144–1152.CrossRefGoogle Scholar
Burke, P. J. (1972) Output processes and tandem queues. In Proc. Symp. Computer Communications Networks and Teletraffic, Brooklyn 1972, ed. Fox, J., 419–428.Google Scholar
Chow, W. M. (1981) The cycle time distribution of exponential cyclic queues. J. Assoc. Comput. Mach. 27, 281–286.CrossRefGoogle Scholar
Daduna, H. (1982) Passage times for overtake-free paths in Gordon-Newell networks. Adv. Appl. Prob. 14, 672–686.Google Scholar
Daduna, H. (1983) On passage times in Jackson networks: Two stations network and overtake-free paths. Z. Operat. Res. 27, 239–256.Google Scholar
Gordon, W. J. and Newell, G. F. (1967) Closed queueing systems with exponential servers. Operat. Res. 15, 254–265.Google Scholar
Kawashima, T. and Torigoe, N. (1983) Cycle time distribution in a central server queuing system with multi-server stations.Google Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Kelly, F. P. and Pollett, P. K. (1983) Sojourn times in closed queueing networks. Adv. Appl. Prob. 15, 638–656.Google Scholar
Melamed, B. (1979) Characterisation of Poisson traffic streams in Jackson queueing networks. Adv. Appl. Prob. 11, 422–438.Google Scholar
Melamed, B. (1982a) Sojourn times in queueing networks. Math. Operat. Res. 7, 223–244.Google Scholar
Melamed, B. (1982b) On Markov jump processes imbedded at jump epochs and their queueing-theoretic applications. Math. Operat. Res. 7, 111–128.Google Scholar
Reich, E. (1957) Waiting times when queues are in tandem. Ann. Math. Statist. 28, 768–773.Google Scholar
Reich, E. (1963) Note on queues in tandem. Ann. Math. Statist. 34, 338–341.CrossRefGoogle Scholar
Schassberger, R. and Daduna, H. (1983) The time for a round trip in a cycle of exponential queues. J. Assoc. Comput. Mach. 30, 146–150.Google Scholar
Sekino, A. (1972) Response time distribution of multiprogrammed time-shared computer systems. Proc. 6th Annual Princeton Conf. Information Science and Systems, 613–619.Google 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
Simon, B. and Foley, R. D. (1979) Some results on sojourn times in acyclic Jackson networks. Management Sci. 25, 1027–1034.Google Scholar
Walrand, J. and Varaiya, P. (1980) Sojourn times and the overtaking condition in Jacksonian networks. Adv. Appl. Prob. 12, 1000–1018.Google Scholar
Wong, J. (1979) Response time distribution of the M/M/m/N queueing model. Operat. Res. 27, 1196–1202.CrossRefGoogle Scholar