A forkjoin queueing model: Diffusion approximation, integral representations and asymptotics
 Xiaoming Tan,
 Charles Knessl
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We consider two parallel M/M/1 queues which are fed by a single Poisson arrival stream. An arrival splits into two parts, with each part joining a different queue. This is the simplest example of a forkjoin model. After the individual parts receive service, they may be joined back together, though we do not consider the join part here. We study this model in the heavy traffic limit, where the service rate in either queue is only slightly larger than the arrival rate. In this limit we obtain asymptotically the joint steadystate queue length distribution. In the symmetric case, where the two servers are identical, this distribution has a very simple form. In the nonsymmetric case we derive several integral representations for the distribution. We then evaluate these integrals asymptotically, which leads to simple formulas which show the basic qualitative structure of the joint distribution function.
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 Title
 A forkjoin queueing model: Diffusion approximation, integral representations and asymptotics
 Journal

Queueing Systems
Volume 22, Issue 34 , pp 287322
 Cover Date
 19960901
 DOI
 10.1007/BF01149176
 Print ISSN
 02570130
 Online ISSN
 15729443
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 Forkjoin queue
 heavy traffic
 diffusion approximation
 asymptotics
 Industry Sectors
 Authors

 Xiaoming Tan ^{(1)}
 Charles Knessl ^{(2)}
 Author Affiliations

 1. Medical Research, HoechstRoussel Pharmaceuticals, Room 108, P.O. Box 2500, 088761258, Somerville, NJ, USA
 2. Department of Mathematics, Statistics and Computer Science (M/C 249), University of Illinois at Chicago, 851 South Morgan Street, 606077045, Chicago, IL, USA