Decay of Tails at Equilibrium for FIFO Join the Shortest Queue Networks

Details Decay of Tails at Equilibrium for FIFO Join the Shortest Queue Networks Decay of Tails at Equilibrium for FIFO Join the Shortest Queue Networks

2011-06-22

arxiv.org/abs/1106.4582v1

Mathematics

Probability

Scientific paper

In join the shortest queue networks, incoming jobs are assigned to the shortest queue from among a randomly chosen subset of D queues, in a system of N queues; after completion of service at its queue, a job leaves the network. We also assume that jobs arrive into the system according to a rate-\alpha N Poisson process, \alpha < 1, with rate-1 service at each queue. When the service at queues is exponentially distributed, it was shown in Vvedenskaya et al. [16] that the tail of the equilibrium queue size decays doubly exponentially in the limit as N goes to infinity. This is a substantial improvement over the case D=1, where the queue size decays exponentially. The reasoning in [16] does not easily generalize to jobs with nonexponential service time distributions. A modularized program for treating general service time distributions was introduced in Bramson et al. [4]. The program relies on an ansatz that asserts, in equilibrium, any fixed number of queues become independent of one another as N goes to infinity. This ansatz was demonstrated in several settings in Bramson et al. [5], including for networks where the service discipline is FIFO and the service time distribution has a decreasing hazard rate. In this article, we investigate the limiting behavior, as N goes to infinity, of the equilibrium at a queue when the service discipline is FIFO and the service time distribution has a power law with a given exponent -\beta, for \beta > 1. We show under the above ansatz that, as N goes to infinity, the tail of the equilibrium queue size exhibits a wide range of behavior depending on the relationship between \beta and D. In particular, if \beta > D/(D-1), the tail is doubly exponential and, if \beta < D/(D-1), the tail has a power law. When \beta = D/(D-1), the tail is exponentially distributed.

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