On the probability distribution of join queue length in a fork-join model

Authors:
Jun Li;Yiqiang q. Zhao
Affiliations:
Communications research centre (crc) canada, ottawa, on, canadak2h 8s2 e-mail: jun.li@crc.gc.ca;School of mathematics and statistics, carleton university, ottawa, on, canadak1s 5b6 e-mail: zhao@math.carleton.ca
Venue:
Probability in the Engineering and Informational Sciences
Year:
2010

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Abstract

In this article, we consider the two-node fork-join model with a Poisson arrival process and exponential service times of heterogeneous service rates. Using a mapping from the queue lengths in the parallel nodes to the join queue length, we first derive the probability distribution function of the join queue length in terms of joint probabilities in the parallel nodes and then study the exact tail asymptotics of the join queue length distribution. Although the asymptotics of the joint distribution of the queue lengths in the parallel nodes have three types of characterizations, our results show that the asymptotics of the join queue length distribution are characterized by two scenarios: (1) an exact geometric decay and (2) a geometric decay with the prefactor n−1/2.