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

Abstract

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. [Probl. Inf. Transm. 32 (1996) 15–29] that the tail of the equilibrium queue size decays doubly exponentially in the limit as $N\rightarrow\infty$. 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 $N\rightarrow\infty$. 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 $N\rightarrow\infty$, 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 $N\rightarrow\infty$, 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.