Routing Heterogeneous Jobs to Heterogeneous Servers: A Global Optimization-Based Approach

We study the problem of routing heterogeneous jobs to heterogeneous servers, where the service rate depends both on the job type and the server type. We employ a math programming approach which is desirable since it can easily be embedded within larger planning problems. While convex objectives which minimize expected waiting time or expected sojourn time are well-studied in this context, we study a particular class of nonconvex objective functions that can be factored by arrival rate and workload. These include maximizing the service level, i.e., the probability that a job is served within an acceptable waiting time, and minimizing the expected queue length. We develop an optimization algorithm called fixed-ratio shifting envelopes to find near-optimal solutions to our math program, and design online routing policies which make use of these solutions. To showcase our model, we apply our model and solution method to the problem of assigning Irvine census block groups to fire stations.