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The Longer Queue Model

Published online by Cambridge University Press:  27 July 2009

Leopold Flatto
Leopold Flatto
Affiliation:
AT&T Bell Laboratories Murray Hill, New Jersey 07974
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Abstract

Two queues forming two independent Poisson processes are served by one server with exponential service time. The server always works on the longer queue and, in case that they are of equal length, chooses either one with probability ½. Let πij be the probability that the two queue lengths equal i andj at equilibrium and π(z, w) = ∑πi j Ziwj. We determine π(z, w) and derive from this asymptotic formulas forπij as i, j → ∞. These asymptotic formulas are used to study the interdependence of the queue lengths. In particular, we obtain limit laws for the queue lengths conditioned on each other.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 3 , Issue 4 , October 1989 , pp. 537 - 559
Copyright
Copyright © Cambridge University Press 1989

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References

Cohen, J.W. (1982). The single server queue, revised edition. Amsterdam: North Holland.Google Scholar
Cohen, J.W. & Boxma, O.J. (1983). Boundary-value problems in queueing system analysis. Amsterdam: North Holland.Google Scholar
Cohen, J.W. (1987). A two-queue, one-server model with priority for the longer queue. Queueing Systems 2: 261–283.CrossRefGoogle Scholar
Cohen, J.W. (1988). Boundary-value problems in queueing theory. Queueing Systems 3: 97–128.CrossRefGoogle Scholar
DeBruijn, N.G.Asymptotic methods in analysis, third edition. Amsterdam: North Holland.Google Scholar
Flatto, L. & McKean, H. (1977). Two queues in parallel. Communication in Pure and Applied Mathematics 30: 255–263.CrossRefGoogle Scholar
Handbook of mathematical functions. National Bureau of Standards, Applied Mathematics Service 55, Tenth Printing, 1972.Google Scholar
Littlewood, J.E. (1944). Lectures on the theory of functions. England: Oxford University Press.Google Scholar
Titchmarsh, E.C. (1939). The theory of functions, second edition. England: Oxford University Press.Google Scholar
Whitney, H. (1972). Complex analytic varieties. Reading, Massachusetts: Addison-Wesley Publishing Co.Google Scholar
Zheng, Y.S. & Zipkin, P. (1986). A queuing model to analyze value of centralized inventory information. Research working paper no. 86−6, Columbia University Business School, New York.Google Scholar