On optimal dynamic scheduling for sum-queue minimization in trees

We investigate the problem of minimizing the sum of the queues 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 primary interference model. We first consider the case where the root has only one child while the rest of the tree is arbitrary, and provide a causal sample-path delay optimal scheduling policy, i.e., at each time slot, for any traffic arrival pattern, the sum of the queues of all the nodes is minimum among all policies. We are able to fully characterize tree structures for which such policies exist. In particular, when the root has multiple children, there exists a causal sample-path delay optimal policy as long as only one child is not a leaf node. We also show that for any other tree structure there exists no causal sample-path delay optimal policy, thus underscoring the inherent limitation of using sample-path optimality as a performance metric and implying that other weaker metrics of delay performance should be investigated.