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The diffusion approximation for tandem queues in heavy traffic

Published online by Cambridge University Press:  01 July 2016

J. Michael Harrison
J. Michael Harrison*
Affiliation:
Stanford University
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Abstract

Consider a pair of single server queues arranged in series. (This is the simplest example of a queuing network.) In an earlier paper [2], a limit theorem was proved to justify a heavy traffic approximation for the (two-dimensional) equilibrium waiting-time distribution. Specifically the waiting-time distribution was shown to be approximated by the limit distribution F of a certain vector stochastic process Z. The process Z was defined as an explicit, but relatively complicated, transformation of vector Brownian motion, and the general problem of determining F was left unsolved.

It is shown in this paper that Z is a diffusion process (continuous strong Markov process) whose state space S is the non-negative quadrant. On the interior of S, the process behaves as an ordinary vector Brownian motion, and it reflects instantaneously at each boundary surface (axis). At one axis, the reflection is normal, but at the other axis it has a tangential component as well. The generator of Z is calculated.

It is shown that the limit distribution F is the solution of a first-passage problem for a certain dual diffusion process Z∗. The generator of Z∗ is calculated, and the analytical theory of Markov processes is used to derive a partial differential equation (with boundary conditions) for the density f of F. Necessary and sufficient conditions are found for f to be separable (for the limit distribution to have independent components). This extends slightly the class of explicit solutions found previously in [2]. Another special case is solved explicitly, showing that the density is not in general separable.

Keywords

DIFFUSION PROCESSTANDEM QUEUEQUEUING NETWORKREFLECTING BARRIERHEAVY TRAFFICMARKOV PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 4 , December 1978 , pp. 886 - 905
Copyright
Copyright © Applied Probability Trust 1978 

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References

[1] Dynkin, E. B. (1965) Markov Processes, Vol. 1. Springer-Verlag, New York.Google Scholar
[2] Harrison, J. M. (1973) The heavy traffic approximation for single server queues in series. J. Appl. Prob. 10, 613–629.CrossRefGoogle Scholar
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[5] Reiman, M. (1977) Queuing Networks in Heavy Traffic. Ph.D. Dissertation, Department of Operations Research, Stanford University.Google Scholar
[6] Stroock, D. W. and Varadhan, S. R. S. (1971) Diffusion processes with boundary conditions. Comm. Pure Appl. Math. 24, 147–225.CrossRefGoogle Scholar
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