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Large-scale join-idle-queue system with general service times

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  30 November 2017

S. Foss  and
A. L. Stolyar
S. Foss*
Affiliation:
Heriot-Watt University and Novosibirsk State University
A. L. Stolyar*
Affiliation:
University of Illinois at Urbana-Champaign
*
* Postal address: Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh EH14 4AS, UK. Email address: s.foss@hw.ac.uk
** Postal address: Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, 104 S. Mathews Avenue, Office 201C, Urbana, IL 61801, USA. Email address: stolyar@illinois.edu
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Abstract

A parallel server system with n identical servers is considered. The service time distribution has a finite mean 1 / μ, but otherwise is arbitrary. Arriving customers are routed to one of the servers immediately upon arrival. The join-idle-queue routeing algorithm is studied, under which an arriving customer is sent to an idle server, if such is available, and to a randomly uniformly chosen server, otherwise. We consider the asymptotic regime where n → ∞ and the customer input flow rate is λn. Under the condition λ / μ < ½, we prove that, as n → ∞, the sequence of (appropriately scaled) stationary distributions concentrates at the natural equilibrium point, with the fraction of occupied servers being constant at λ / μ. In particular, this implies that the steady-state probability of an arriving customer waiting for service vanishes.

Keywords

Large-scale service systempull-based load distributionjoin-idle-queueload balancingfluid limitstationary distributionasymptotic optimality

MSC classification

Primary: 90B15: Network models, stochastic
Secondary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 4 , December 2017 , pp. 995 - 1007
Copyright
Copyright © Applied Probability Trust 2017 

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