Two Parallel Queues with Infinite Servers and Join the Shortest Queue Discipline

• Published in: Stochastic models Vol. 31; no. 4; pp. 636 - 672
• Main Authors: Guillemin, F., Olivier, P., Simonian, A., Tanguy, C.
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 02-10-2015; Taylor & Francis Ltd
• Subjects: Fredholm integral equations; Integral equations; Markov analysis; Markov processes; Network performance evaluation; Orthogonal polynomials; Queuing Theory; Scheduling; Stochastic models
• ISSN: 1532-6349; 1532-4214
• DOI: 10.1080/15326349.2015.1061440

We study the stationary distribution of a system of two parallel M/M/∞ queues managed by the Join the Shortest Queue load balancing policy; one motivation for characterizing the efficiency of that policy is its potential application to resource allocation issues in cloud computing. For the general system with distinct service rates, we first show that the tail of each marginal queue length distribution exhibits a much faster decay than that of a Poisson distribution. Second, the determination of the joint stationary distribution is shown to reduce to the resolution of a pair of linear integral equations. In the case when service rates are identical ("symmetric case"), that pair of integral equations simplifies to a single Fredholm integral equation of the first kind whose solution is explicitly given in terms of Legendre polynomials; this enables us to entirely determine the stationary distribution of the system. We provide, in particular, asymptotics for the second moment and the tail of the queue length distribution.