Stability analysis of parallel server systems under longest queue first

Stability analysis of parallel server systems under longest queue first

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Article ID: iaor201110538
Volume: 74
Issue: 2
Start Page Number: 257
End Page Number: 279
Publication Date: Oct 2011
Journal: Mathematical Methods of Operations Research
Authors: ,
Keywords: heuristics, markov processes
Abstract:

We consider the stability of parallel server systems under the longest queue first (LQF) rule. We show that when the underlying graph of a parallel server system is a tree, the standard nominal traffic condition is sufficient for the stability of that system under LQF when interarrival and service times have general distributions. Then we consider a special parallel server system, which is known as the X‐model, whose underlying graph is not a tree. We provide additional ‘drift’ conditions for the stability and transience of these queueing systems with exponential interarrival and service times. Drift conditions depend in general on the stationary distribution of an induced Markov chain that is derived from the underlying queueing system. We illustrate our results with examples and simulation experiments. We also demonstrate that the stability of the LQF depends on the tie‐breaking rule used and that it can be unstable even under arbitrary low loads.

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