Routing and scheduling in heterogeneous systems: a sample path approach

Consider the problem of routing customers to a set of K parallel servers that have different rates. Each server has infinite capacity. The arrival process is general and the service times are assumed to be i.i.d. exponential random variables. Using sample path arguments, we show that, given any Bernoulli policy π, there exists another policy ρ which out-performs π by using a randomized version of a round-robin policy. Moreover, ρ is easily specified and implemented. We present extensions of our results to systems with finite capacities and service times that have an increasing hazard rate. Finally, a similar result is shown to hold in the context of scheduling customers from a set of K parallel queues