On a Generalization of Join-the-Shortest-Queue Scheduling with a Bias

We consider a system of two parallel queues sharing a common arrival process, where the arriving customers select which queue to join based on a generalized Join the Shortest Queue ( ) policy. We assume that both servers run at the same speed. However, we assume that departing customers face some additional delay after leaving the system, and that this downstream delay is different between the two queues. Therefore, we introduce a bias into the routing policy, such that customers prefer the path with lower downstream delay unless the current queue length for the fast path exceeds the current queue length for the slow path by at least customers. We model the number of customers in the two-queue system as a two-dimensional Markov chain, and present a closed-form solution for its steady-state distribution for the case of heavy traffic. When , our results for the policy can be applied to , for which no closed-form solution is currently available.