Decay of tails at equilibrium for FIFO join the shortest queue networks

摘要

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-αN Poisson process, α < 1, with rate-1 service at each queue. When the service at queues is exponentially distributed, it was shown in Vvedenskaya et al. [Probl. Inf. Transm. 32 (1996) 15-29] that the tail of the equilibrium queue size decays doubly exponentially in the limit as N → ∞. This is a substantial improvement over the case D = 1, where the queue size decays exponentially. The reasoning in [Probl. Inf. Transm. 32 (1996) 15-29] 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. [In Proc. ACM SIGMETRICS (2010) 275-286]. The program relies on an ansatz that asserts, in equilibrium, any fixed number of queues become independent of one another as N → ∞. This ansatz was demonstrated in several settings in Bramson et al. [Queueing Syst. 71 (2012) 247-292], 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 → ∞, 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 -β, for β > 1. We show under the above ansatz that, as N → ∞, the tail of the equilibrium queue size exhibits a wide range of behavior depending on the relationship between β and D. In particular, if β > D/(D-1), the tail is doubly exponential and, if β < D/(D-1), the tail has a power law. When β = D/(D - 1), the tail is exponentially distributed.