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The heavy traffic approximation for single server queues in series

Published online by Cambridge University Press:  14 July 2016

J. Michael Harrison
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Abstract

A tandem queue with K single server stations and unlimited interstage storage is considered. Customers arrive at the first station in a renewal process, and the service times at the various stations are mutually independent i.i.d. sequences. The central result shows that the equilibrium waiting time vector is distributed approximately as a random vector Z under traffic conditions (meaning that the system traffic intensity is near its critical value). The weak limit Z is defined as a certain functional of multi-dimensional Brownian motion. Its distribution depends on the underlying interarrival and service time distributions only through their first two moments. The outstanding unsolved problem is to determine explicitly the distribution of Z for general values of the relevant parameters. A general computational approach is demonstrated and used to solve for one special case.

Keywords

TANDEM QUEUEHEAVY TRAFFIC APPROXIMATIONDIFFUSION APPROXIMATIONWEAK CONVERGENCE
Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 3 , September 1973 , pp. 613 - 629
Copyright
Copyright © Applied Probability Trust 1973 

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