Constant Time per Edge Is Optimal in Rooted Tree Networks

We analyze how the expected packet delay in rooted tree networks is aaected by the distribution of time needed for a packet to cross an edge using stochastic comparison methods. Our result generalizes previously known results that the delay when the crossing time is exponentially distributed yields an upper bound for the expected delay when the crossing time is constant on this class of of networks. Unlike previous work, we do not assume Poisson arrivals. Our result also extends to a variety of service distributions, and it can be used to bound the expected value of all convex, increasing functions of the packet delays. An interesting corollary of our work is that in rooted tree networks, if one can change the distribution of the time to cross an edge while keeping the expectation xed, the strategy that minimizes the expected delay and the expected maximum delay is to make the time to cross an edge constant. Our result also holds in mulitcasting rooted tree networks, where a single message can have several possible destinations.