부산대학교 도서관

In this paper the authors study a queueing system consisting of onebatch-service queue with batch size $M$, and $N$ FIFOexponential-service queues. There are two independent Poisson arrivalstreams, of which one always joins the batch-service queue, and theother one needs to decide which queue to enter according toprobabilistic routing. The object of interest is the user equilibria inthe system. A user equilibrium is a probabilistic routing, definedaccording to Wardrop's first principle, which says that at a userequilibrium, the expected delays at all the queues in use are no largerthan those at any unused queue. This paper characterizes the userequilibria and shows that there is a unique user equilibrium when$M>2N$, and furthermore, there are at most three user equilibria forthe system, and when there are three, two are stable and one isunstable. As observed in examples, multiple equilibria may appear whena FIFO queue is added to the system (the so-called Downs-Thomsonparadox), and even when the equilibrium is unique, adding a FIFO queuemay lead to greater delays (the so-called Braess paradox). This paperprovides a nice extension of the results on user equilibria for aparallel queueing system with one batch-service queue and one FIFOqueue by Afimeimounga et al. (2005).