Decay of tails at equilibrium for FIFO join the shortest queue networks (Q373835)

The aim of the present paper is to investigate a special join the shortest queue problem. 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.NEWLINENEWLINEIn this article, the authors investigate the limiting behavior, as \(N\to\infty\), 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 \(\beta>1\). They show under the above conditions that, as \(N\to\infty\), 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.