On Sample-Path Optimal Dynamic Scheduling for Sum-Queue Minimization in Trees Under the $K$ -Hop Interference Model

We investigate the problem of minimizing the sum of the queue lengths of all the nodes in a wireless network with a tree topology. Nodes send their packets to the tree's root (sink). We consider a time-slotted system and a K-hop interference model. We characterize the existence of causal sample-path optimal scheduling policies in these networks, i.e., we wish to find a policy such that at each time-slot, for any traffic arrival pattern, the sum of the queue lengths of all the nodes is minimum among all policies. We provide an algorithm that takes any tree and K as inputs, and outputs whether a causal sample-path optimal policy exists for this tree under the K-hop interference model. We show that when this algorithm returns FALSE, there exists a traffic arrival pattern for which no causal sample-path optimal policy exists for the given tree structure. We further show that for certain tree structures, even noncausal sample-path optimal policies do not exist. We provide causal sample-path optimal policies for those tree structures for which the algorithm returns TRUE. Thus, we completely characterize the existence of such policies for all trees under the K-hop interference model. The nonexistence of sample-path optimal policies in a large class of tree structures implies that we need to study other (relatively) weaker metrics for this problem.