Martingales and buffer overflow for the symmetric shortest queue model

Abstract

A variant of the standard symmetric system of two parallel queues under the join-the-shortest-queue policy is introduced. Here, the shortest queue has service rate $\mu _1$, while the longest queue has rate $\mu _2$, where $\mu _1 + \mu _2 = 1$. In the case of equality, both queues are served at rate 1 / 2. Each queue has capacity K, which may be finite or infinite, and the global Poisson arrival rate is $\rho$. Couplings show that, as $\mu _2$ varies, the model is totally ordered, in terms of both total number N(t) of customers in the system and longest queue length. The two extreme cases $\mu _2= 0$ and $\mu _2= 1$ then provide simple stochastic bounds for N(t) for arbitrary $\mu _2$. The ordering partially extends to the enlarged model in which, whenever the shortest queue is empty, the idle server at that queue helps—or on the contrary disturbs—the other server. The previous bounds remain valid. In the extended setup, different martingales are next built from the infinite capacity process. Those lead to simple explicit formulations of the means and Laplace transforms of the hitting time of saturation, for the process with finite K started from any initial state. Asymptotics are then derived, as K gets large. Using one particular mean time, the stationary blocking probability is explicitly obtained, extending the result by Dester et al. regarding the standard symmetric model. Finally, the joint distribution of the time and state of the system at the first queue-length equality is expressed as a function of the inverse of a simple explicit matrix.