Optimal probabilistic routing in distributed parallel queues

We consider the fundamental problem of routing customers among distributed parallel queues to minimize an objective function based on equilibrium sojourn times under general assumptions for the arrival and service processes and under the assumption that customers are routed to the parallel queues in a probabilistic manner. More specifically, we derive explicit solutions for the asymptotically optimal vector of probabilities that control the routing of customers upon arrival among a set of heterogeneous general single-server queues through stochastic-process limits. Our assumption of probabilistic routing is consistent with previous theoretical studies of this optimization problem, and our solutions can be used for the parameter settings of other routing mechanisms found in practice. Stochastic-process limits are exploited in order to be able to handle general arrival and service processes and obtain explicit solutions to the scheduling optimization problems of interest.