Asymptotic expansion for large closed queuing networks

Charles Knessl · Charles Tier

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Advanced Queuing Theory Analysis Probability and Statistical Research Probability and Risk Models

Abstract

In this paper, a new asymptotic method is developed for analyzing closed BCMP queuing networks with a single class (chain) consisting of a large number of customers, a single infinite server queue, and a large number of single server queues with fixed (state-independent) service rates. Asymptotic approximations are computed for the normalization constant (partition function) starting directly from a recursion relation of Buzen. The approach of the authors employs the ray method of geometrical optics and the method of matched asymptotic expansions. The method is applicable when the servers have nearly equal relative utilizations or can be divided into classes with nearly equal relative utilizations. Numerical comparisons are given that illustrate the accuracy of the asymptotic approximations.

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References

1975Open, Closed, and Mixed Networks of Queues with Different Classes of Customers 1980Mean-Value Analysis of Closed Multichain Queuing Networks 1963Jobshop-Like Queueing Systems 1973Computational algorithms for closed queueing networks with exponential servers 2004Jobshop-Like Queueing Systems 1967Closed Queuing Systems with Exponential Servers 1978The Operational Analysis of Queueing Network Models 1984Open and Closed Models for Networks of Queues

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1996Condensation in large closed Jackson networks 2005A survey of bottleneck analysis in closed networks of queues 1992Asymptotic analysis of closed queueing networks with bottlenecks 2013Closed Queueing Networks Under Congestion: Nonbottleneck Independence and Bottleneck Convergence 1992Asymptotic expansions for large closed queueing networks with multiple job classes 1998Asymptotic approximations and bottleneck analysis in product form queueing networks with large populations

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