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A perpetuity and the M/M/∞ ranked server system

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

J. Preater
J. Preater*
Affiliation:
University of Keele
*
∗Postal address: Department of Mathematics, University of Keele, Keele, Staffordshire, ST5 5BG, UK.
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Abstract

We relate the equilibrium size of an M/M/8 type queue having an intermittent arrival stream to a perpetuity, the solution of a random difference equation. One consequence is a classical result for ranked server systems, previously obtained by generating function methods.

Keywords

M/M/8 QUEUERANKED SERVERSRANDOM DIFFERENCE EQUATIONS

MSC classification

Primary: 60K25: Queueing theory
Type
Research Article
Information
Journal of Applied Probability , Volume 34 , Issue 2 , June 1997 , pp. 508 - 513
Copyright
Copyright © Applied Probability Trust 1997 

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References

Aldous, D. (1986) Some interesting processes arising as heavy traffic limits in an M/M/8 storage process. Stoch. Proc. Appl. 22, 291–313.CrossRefGoogle Scholar
ÇInlar, E. (1992) Sunset over Brownistan. Stoch. Proc. Appl. 40, 45–53.CrossRefGoogle Scholar
Coffman, E. G., Kadota, T. T. and Shepp, L. A. (1985) A stochastic model of fragmentation in dynamic storage allocation. SIAM J. Computing 14, 416–425.CrossRefGoogle Scholar
Cooper, R. B. (1981) Introduction to Queueing Theory. 2nd edn. Edward Arnold, London.Google Scholar
Embrechts, P. and Goldie, C. M. (1994) Perpetuities and random equations. In Asymptotic Statistics: Proc. Fifth Prague Symp., Sept. 1993. ed. Mandl, P. and Hušková, M. Physica, Heidelberg. pp. 75–86.CrossRefGoogle Scholar
Kosten, L. (1937) Uber Sperrungswahrscheinlichkeiten bei Staffelschaltungen. Electra Nachrichten – Technik 14, 5–12.Google Scholar
Newall, G. F. (1984) The M/M/8 Service System with Ranked Servers in Heavy Traffic. (Lecture Notes in Econ. and Math. Systems 231.) Springer, Berlin.Google Scholar
Saaty, T. L. (1961) Elements of Queueing Theory. McGraw-Hill, New York.Google Scholar
Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
Vervaat, W. (1979) On a stochastic difference equation and a representation of non-negative infinitely divisible variables. Adv. Appl. Prob. 11, 750–783.CrossRefGoogle Scholar