Large Fluctuations in a Deterministic Multiclass Network of Queues

In this paper we investigate a relatively simple deterministic four-class two-queue multiclass open network of single-server FIFO queues with traffic intensity one at each queue. Our purpose is to better understand the effect of feedback with class-dependent service times at the queues. The example is sufficiently tractable that we are able to describe its transient behavior in great detail. The transient behavior depends strongly on the initial conditions and, for some initial conditions, the sample paths of the queue-length processes at individual stations have sudden large fluctuations (a large jump up followed immediately by a large jump down). These large fluctuations occur because batches of customers with short service times build up in the queues. Consistent with recent work by Dai and Wang (1993) on Brownian network models, these fluctuations rule out conventional heavy-traffic limit theorems. We show how to obtain proper heavy-traffic limits for this example by weakening the topology or enlarging the space of prospective limits (and changing the topology). This example also dramatically demonstrates a disadvantage of the FIFO discipline compared to other disciplines like head-of-the-line processor-sharing (HOL-PS) among the classes at each queue (under which, the large fluctuations do not occur). Finally, the critical arrival rate for stability in our example actually depends on the service discipline, being even lower if the classes with longer service times are given high priority at each queue. This phenomenon can occur in the network setting because individual queues can be empty when there is work in the network.