| We consider the closed Lu-Kumar network with independent, exponentially distributed service times and the Balanced Loading Condition. The behavior of this network can be classified into three different cases according to the mean service times for each queue: the supercritical case, the subcritical case, and the critical case. We prove heavy traffic limit theorems for queue length and cumulative server idleness processes for the closed Lu-Kumar network in each of these cases.; In the supercritical case, the mean service times balance with the server discipline; each queue length process has a nontrivial fluid limit. In the subcritical case, the priority queues have relatively low mean service times, so the non-priority queues have trivial limits, even under diffusion scaling. In the critical case, a combination of fluid and diffusion scaling for the queue length process produces a nontrivial limit. The heavy traffic limit theorem in the critical case is a generalization of "A multiclass closed queueing network with unconventional heavy traffic behavior" by Harrison and Williams.; We also prove a functional central limit theorem for Markov additive processes. This theorem is used to produce the heavy traffic limit theorem for the queue length process in the subcritical case. |
