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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.
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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 

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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