On optimal routing in overloaded parallel queues

We consider the problem of routing Bernoulli arrivals to parallel queues, where each queue provides service according to an independent Bernoulli process. We assume that the total arrival rate exceeds the sum of the service rates of the queues. Since such a queueing system is unstable, the vector of queue lengths does not have a well-defined stationary distribution. However, one metric which can be used to compare routing policies is the amount of unused service in the system. To lower-bound the cumulative unused service in the system, we present a "queue reversal" theorem for a single-server queue with independent and identically distributed (i.i.d.) arrivals and i.i.d. services: Assuming that the queue is initially empty, the expected cumulative unused service is equal to the expected queue length in a queue where the arrivals and services are reversed. Thus, the expected cumulative unused service in the unstable system is equal to the expected queue length in a stable system, which can be calculated. Using this result for a single-server queue, we obtain a lower bound on the expected unused service in the parallel queueing system for any feasible routing policy.We then compare this lower bound to the performance of two simple routing policies: Randomized and Join-the-Shortest Queue Routing.