부산대학교 도서관

In this paper the authors study a queueing system consisting of one batch-service queue with batch size $M$, and $N$ FIFO exponential-service queues. There are two independent Poisson arrival streams, of which one always joins the batch-service queue, and the other one needs to decide which queue to enter according to probabilistic routing. The object of interest is the user equilibria in the system. A user equilibrium is a probabilistic routing, defined according to Wardrop's first principle, which says that at a user equilibrium, the expected delays at all the queues in use are no larger than those at any unused queue. This paper characterizes the user equilibria and shows that there is a unique user equilibrium when $M>2N$, and furthermore, there are at most three user equilibria for the system, and when there are three, two are stable and one is unstable. As observed in examples, multiple equilibria may appear when a FIFO queue is added to the system (the so-called Downs-Thomson paradox), and even when the equilibrium is unique, adding a FIFO queue may lead to greater delays (the so-called Braess paradox). This paper provides a nice extension of the results on user equilibria for a parallel queueing system with one batch-service queue and one FIFO queue by Afimeimounga et al. (2005).