Two Coupled Queues with Heterogeneous Traffic

We consider a system with two heterogeneous traffic classes, one having light-tailed characteristics, the other one exhibiting heavy-tailed properties. When both classes are backlogged, the two corresponding queues are each served at a certain nominal rate. However, when one queue empties, the service rate for the other class increases. This dynamic sharing of surplus service capacity is reminiscent of the Generalized Processor Sharing (GPS) discipline. GPS-based scheduling algorithms, such as Weighted Fair Queueing, provide a candidate implementation mechanism for achieving differentiated Quality-of-Service in a DiffServ architecture. We characterize the asymptotic workload behavior of both traffic classes. The tail of the workload distribution of the heavy-tailed class is asymptotically equivalent to that of the heavy-tailed class in isolation – but with its nominal service rate inflated by the slack capacity of the light-tailed class. For the light-tailed class, we show a sharp dichotomy in the qualitative behavior, depending on whether its load exceeds its nominal service rate or not. In underload scenarios, the tail of its workload distribution is equivalent to that of the light-tailed class in isolation, multiplied with a certain pre-factor. The pre-factor represents the probability that the heavy-tailed class is backlogged long enough for the light-tailed class to build up a large workload. This provides a measure for the extent to which the light-tailed class benefits from sharing surplus capacity with the heavy-tailed class. In contrast, in overload situations, the light-tailed class is adversely affected by the heavy-tailed class, and inherits its traffic characteristics. 2000 Mathematics Subject Classification: 60K25 (primary), 68M20, 90B18, 90B22 (secondary).