The Asymptotic Workload Behavior of Two Coupled Queues

We consider a system of two coupled queues Q1 and Q2. When both queues are backlogged, they are each served at unit rate. However, when one queue empties, the service rate at the other queue increases. Thus, the two queues are coupled through the mechanism for dynamically sharing surplus service capacity. We derive the asymptotic workload behavior at Q1 for various scenarios where at least one of the two queues has a heavy-tailed service time distribution. First of all, we consider a situation where the traffic load at Q1 is below the nominal unit service rate. We show that if the service time distribution at Q1 is heavy-tailed, then the workload behaves exactly as if Q1 is served in isolation at a constant rate, which only depends on the service time distribution at Q2 through its mean. In addition, we establish that if the service time distribution at Q1 is exponential, then the workload distribution is either exponential or semi-exponential, depending on whether the traffic load at Q2 exceeds the nominal service rate or not. Next, we focus on a regime where the traffic load at Q1exceeds the nominal service rate, so that Q1 relies on the surplus capacity from Q2 to maintain stability. In that case, the workload distribution at Q1 is determined by the heaviest of the two service time distributions, so that Q1 may inherit potentially heavier-tailed characteristics from Q2.