Dieker, A. B. and Warren, Jon. (2010) Series Jackson networks and noncrossing probabilities. Mathematics of Operations Research, Vol.35 (No.2). pp. 257-266. ISSN 0364-765X

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Abstract

This paper studies the queue-length process in series Jackson networks with external input to the first station only. We show that its Markov transition probabilities can be written as a finite sum of noncrossing probabilities, so that questions on time-dependent queueing behavior are translated to questions on noncrossing probabilities. This makes previous work on noncrossing probabilities relevant to queueing systems and allows new queueing results to be established. To illustrate the latter, we prove that the relaxation time (i.e., the reciprocal of the "spectral gap") of a positive recurrent system equals the relaxation time of an M/M/1 queue with the same arrival and service rates as the network's bottleneck station. This resolves a conjecture of Blanc that he proved for two queues in series.

Item Type:	Journal Article
Subjects:	H Social Sciences > HD Industries. Land use. Labor > HD28 Management. Industrial Management Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Journal or Publication Title:	Mathematics of Operations Research
Publisher:	Informs
ISSN:	0364-765X
Official Date:	May 2010
Volume:	Vol.35
Number:	No.2
Number of Pages:	10
Page Range:	pp. 257-266
Identification Number:	10.1287/moor.1090.0421
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	IBM T.J. Watson Research Center, Yorktown Heights, New York
URI:	http://wrap.warwick.ac.uk/id/eprint/5812

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