A Generalization of L = λW to Moments of Queue Length and Waiting Times

Shelby L. Brumelle

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Advanced Queuing Theory Analysis Probability and Statistical Research Age of Information Optimization

Abstract

The well known formula L = λW relates the time-average number in queue to the expected wait in queue of a customer. This paper specializes a more general formula, denoted by H = λG, in order to obtain relations between moments of L and W other than the first. The basic queue considered is G/G/k with stationary input. The special case where the arrival times form a renewal process and the more special case where they are a Poisson process are also discussed.

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References

1961A Proof for the Queuing Formula: L = λ W 1971On the relation between customer and time averages in queues

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2005Analysis of multi-server systems via dimensionality reduction of markov chains 1980Note—Simple Inequalities for Multiserver Queues 1989A generalization of little's law to moments of queue lengths and waiting times in closed, product-form queueing networks 1996Deriving delay characteristics from queue length statistics in discrete-time queues with multiple servers 1983A Survey of J. Little's Formula 2013with staggered setup

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