Stability of multiclass queueing networks under longest-queue and longest-dominating-queue scheduling

We consider the stability of robust scheduling policies for multiclass queueing 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. 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.