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 non-causal 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 non-existence 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.