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Response times in M/M/s fork-join networks

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Sung-Seok Ko  and
Richard F. Serfozo
Sung-Seok Ko*
Affiliation:
Samsung SDS
Richard F. Serfozo*
Affiliation:
Georgia Institute of Technology
*
∗ Postal address: Samsung SDS Co. Ltd, 707-719 Yoksam-2 Dong, Kangnam-Gu, Seoul, Korea 135-918. Email address: sungseok@hotmail.com
∗∗ Postal address: School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA. Email address: rserfozo@isye.gatech.edu
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Abstract

We study a fork-join processing network in which jobs arrive according to a Poisson process and each job splits into m tasks, which are simultaneously assigned to m nodes that operate like M/M/s queueing systems. When all of its tasks are finished, the job is completed. The main result is a closed-form formula for approximating the distribution of the network's response time (the time to complete a job) in equilibrium. We also present an analogous approximation for the distribution of the equilibrium queue length (the number of jobs in the system), when each node has one server. Kolmogorov-Smirnov statistical tests show that these formulae are good fits for the distributions obtained from simulations.

Keywords

Fork-join networkqueueingcommunications networkassembly networkparallel processingsynchronizationsupply chainlittle law for distributions

MSC classification

Primary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.)
Secondary: 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 3 , September 2004 , pp. 854 - 871
Copyright
Copyright © Applied Probability Trust 2004 

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