Stability of Lu-Kumar Networks under Longest-Queue and Longest-Dominating-Queue Scheduling

We consider the stability of robust scheduling policies for Lu-Kumar networks. These are open networks with arbitrary routing matrix and several disjoint groups of queues in which at most one queue can be served at a time in each group. The arrival and potential service processes and routing decisions at the queues are independent, stationary and ergodic. A scheduling policy is called robust if it does not depend on the arrival and service rates nor on the routing probabilities. A policy is called throughput-optimal if it makes the system stable whenever the parameters are such that the system can be stable. We propose two robust polices: longest-queue scheduling and a new policy called longest-dominating-queue scheduling. We show that longestqueue scheduling is throughput-optimal for two groups of two queues. We also prove the throughput optimality of longest-dominating-queue scheduling when the network topology is acyclic, for an arbitrary number of groups and queues.