Stationary Distribution Analysis of a Queueing Model with Local Choice

Abstract

The paper deals with load balancing between one-server queues on a circle by a local choice policy. Each one-server queue has a Poissonian arrival of customers. When a customer arrives at a queue, he joins the least loaded queue between this queue and the next one, ties solved at random. Service times have exponential distribution. The system is stable if the arrival-to-service rate ratio called load is less than one. When the load tends to zero, we derive the first terms of the expansion in this parameter for the stationary probabilities that a queue has 0 to 3 customers. We investigate the error, comparing these expansion results to numerical values obtained by simulations. Then we provide the asymptotics, as the load tends to zero, for the stationary probabilities of the queue length, for a fixed number of queues. It quantifies the difference between policies with this local choice, no choice and the choice between two queues chosen at random.