The MacLaurin series for the GI/G/1 queue

Weibo Gong · Jian-Qiang Hu

63 citations0/0 files storedJournal of Applied Probability

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

Abstract

We derive the MacLaurin series for the moments of the system time and the delay with respect to the parameters in the service time or interarrival time distributions in the GI/G/1 queue. The coefficients in these series are expressed in terms of the derivatives of the interarrival time density function evaluated at zero and the moments of the service time distribution, which can be easily calculated through a simple recursive procedure. The light traffic derivatives can be obtained from these series. For the M/G/1 queue, we are able to recover the formulas for the moments of the system time and the delay, including the Pollaczek–Khinchin mean-value formula.

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References

1987Applied Probability and Queues 1984Matrix-Geometric Solutions to Stochastic Models 1952The theory of queues with a single server 1985Matrix-geometric solutions in stochastic models 1966Mathematical Theory of Connecting Networks and Telephone Traffic 1989Open Queueing Systems in Light Traffic 1988Power Series for Stationary Distributions of Coupled Processor Models 1989Measurements and approximations to describe the offered traffic and predict the average workload in a single-server queue

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2001Queueing Models with Multiple Waiting Lines 1996Taylor series expansions for Poisson-driven $(\max,+$)-linear systems 1995Pade approximation for stochastic discrete-event systems 2001Laplace Transform and Moments of Waiting Times in Poisson Driven (max,+) Linear Systems 1997Transient and stationary waiting times in (max,+)-linear systems with Poisson input 1993The MacLaurin expansion for a G/G/1 queue with Markov-modulated arrivals and services

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