Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T23:39:57.706Z Has data issue: false hasContentIssue false
  • English
  • Français

On sojourn time in Jackson networks of queues

Published online by Cambridge University Press:  14 July 2016

Austin J. Lemoine
Austin J. Lemoine*
Affiliation:
Ford Aerospace and Communications Corporation
*
∗Postal address: Ford Aerospace and Communications Corporation, Western Development Laboratories Division, 3939 Fabian Way, Palo Alto, CA 94303, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is about representations for equilibrium sojourn time distributions in Jackson networks of queues. For a network with N single-server nodes let hi be the Laplace transform of the residual system sojourn time for a customer ‘arriving' to node i, ‘arrival' meaning external input or internal transfer. The transforms {hi : i = 1, ···, N} are shown to satisfy a system of equations we call the network flow equations. These equations lead to a general recursive representation for the higher moments of the sojourn time variables {Ti : i = 1, ···, N}. This recursion is discussed and then, by way of illustration, applied to the single-server Markovian queue with feedback.

Keywords

EQUILIBRIUM DISTRIBUTIONSNETWORK FLOW EQUATIONSRECURSIVE REPRESENTATIONHIGHER MOMENTSMARTINGALE
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 2 , June 1987 , pp. 495 - 510
Copyright
Copyright © Applied Probability Trust 1987 

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.)

Footnotes

The partial support of Army Research Office Contract DAAG29-82-K-0151 and Air Force Office of Scientific Research Contract F49620-86-C-0022 is gratefully acknowledged.

References

[1] Breiman, L. (1968) Probability. Addison-Wesley, Reading, MA.Google Scholar
[2] Jackson, J. R. (1957) Networks of waiting lines. Operat. Res. 5, 518–521.Google Scholar
[3] Lemoine, A. J. (1977) Networks of queues — A survey of equilibrium analysis. Management Sci. 24, 464–481.Google Scholar
[4] Lemoine, A. J. (1979) Total Sojourn Time in Networks of Queues. TR 79-020-1. Systems Control, Inc., Palo Alto, CA.CrossRefGoogle Scholar
[5] Lemoine, A. J. (1986) A stochastic network model for complex sequential processes. Naval. Res. Log. Quart. 33, 431–443.Google Scholar
[6] Melamed, B. (1979) Characterizations of Poisson traffic streams in Jackson queueing networks. Adv. Appl. Prob. 11, 422–438.Google Scholar
[7] Melamed, B. (1982) On Markov jump processes imbedded at jump epochs and their queueing-theoretic applications. Math. Operat. Res. 7, 111–128.Google Scholar
[8] Melamed, B. (1982) Sojourn times in queueing networks. Math. Operat. Res. 7, 223–244.CrossRefGoogle Scholar
[9] Melamed, B. and Yadin, M. (1984) Randomization procedures in the computation of cumulative-time distributions over discrete state Markov processes. Operat. Res. 32, 926–944.Google Scholar
[10] Melamed, B. and Yadin, M. (1984) Numerical computation of sojourn-time distributions in queueing networks. J. Assoc. Comput. Mach. 31, 839–854.CrossRefGoogle Scholar
[11] Picard, P. (1980) Applications of martingales to some epidemic models. J. Appl. Prob. 17, 583–599.CrossRefGoogle Scholar
[12] Simon, B. and Foley, R. D. (1979) Some results on sojourn times in acyclic Jackson networks. Management Sci. 25, 1027–1034.CrossRefGoogle Scholar
[13] Takács, L. (1963) A single server queue with feedback. Bell System Tech. J. 42, 505–519.CrossRefGoogle Scholar
[14] Walrand, J. and Varaiya, P. (1980) Sojourn times and the overtaking condition in Jacksonian networks. Adv. Appl. Prob. 12, 1000–1018.CrossRefGoogle Scholar