The Proof of a Folk Theorem on Queuing Delay with Applications to Routing in Networks

It is shown that among all arrival processes (not necessarily stationary or renewal type) for an exponential server queue with specified arrival and service rates, that the arrival process which mmimizes the average delay and related quantities is the process with constant interarrival times. The proof is based on a convexity property of exponential server queues which is of independent interest. The folk theorem provides a lower bound, which is readily computable by existing methods, to the average delay in a network of queues under rather general routing disciplines. A sharper lower bound on average delay is provided for the special case of Generalized Round Robin routing for a Poisson arrival process.