The effect of variability in the GI/G/s queue

Ward Whitt

40 citations0/0 files storedJournal of Applied Probability

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

Abstract

In 1969 H. and D. Stoyan showed that the stationary waiting-time distribution in a GI/G/1 queue increases in the ordering determined by the expected value of all non-decreasing convex functions when the interarrival-time and service-time distributions become more variable, as expressed in the ordering determined by the expected value of all convex functions. Ross (1978) and Wolff (1977) showed by counterexample that this conclusion does not extend to all GI/G/s queues. Here it is shown that this conclusion does hold for all GI/G/s queues for several other measures of congestion which coincide with the waiting time in single-server systems. One such alternate measure of congestion is the clearing time, the time required after the arrival epoch of the nth customer for the system to serve all customers in the system at that time, excluding the nth customer. The stochastic comparisons also imply an ordering for the expected waiting times in M/G/s queues.

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References

1971Increasing risk II: Its economic consequences 1962The stability of a queue with non-independent inter-arrival and service times 1955On the theory of queues with many servers 1968Some Inequalities in Queuing 1981Comparing counting processes and queues 1971On the relation between customer and time averages in queues 1966Optimal policy for a dynamic multi‐echelon inventory model 1978Average delay in queues with non-stationary Poisson arrivals

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1983Comparison conjectures about the M/G/s queue 1984Minimizing Delays in the GI/G/1 Queue 2019Serial production line performance under random variation: Dealing with the ‘Law of Variability’1984On the M/G/1 queue by additional inputs 1982Existence of Limiting Distributions in the GI/G/s Queue 1982The marshall and stoyan bounds for IMRL/G/1 queues are tight

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