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Embedded renewal processes in the GI/G/s queue

Published online by Cambridge University Press:  14 July 2016

Ward Whitt
Ward Whitt*
Affiliation:
Yale University
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Abstract

The stable GI/G/s queue (ρ < 1) is sometimes studied using the “fact” that epochs just prior to an arrival when all servers are idle constitute an embedded persistent renewal process. This is true for the GI/G/1 queue, but a simple GI/G/2 example is given here with all interarrival time and service time moments finite and ρ < 1 in which, not only does the system fail to be empty ever with some positive probability, but it is never empty. Sufficient conditions are then given to rule out such examples. Implications of embedded persistent renewal processes in the GI/G/1 and GI/G/s queues are discussed. For example, functional limit theorems for time-average or cumulative processes associated with a large class of GI/G/s queues in light traffic are implied.

Keywords

QUEUESERVICE SYSTEMCONGESTION THEORYGI/G/s QUEUEMULTI-SERVER QUEUEBUSY PERIODSEMBEDDED RENEWAL PROCESSRENEW AL PROCESSBUSYCYCLE FUNCTIONAL LIMIT THEOREMS FOR QUEUESLIMIT THEOREMS FOR QUEUESBUSY CYCLES AS RECURRENT EVENTS FOR QUEUES
Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 3 , June 1972 , pp. 650 - 658
Copyright
Copyright © Applied Probability Trust 1972 

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