Transportation Polytope and its Applications in Parallel Server Systems

A parallel server system is a stochastic processing network with applications in manufacturing, supply chain, ride-hailing, call centers, etc. Heterogeneous customers arrive in the system, and only a subset of servers can serve any customer type given by the flexibility graph. The goal of the system operator is to minimize the delay that depends on the scheduling policy and the flexibility graph. A long line of literature focuses on designing near-optimal scheduling policies given a flexibility graph. On the contrary, we fix the scheduling policy to be the so-called MaxWeight scheduling given its superior delay performance and focus on designing near-optimal, sparse flexibility graphs. Our contributions are threefold. First, we analyze the expected delay in the heavy-traffic asymptotic regime in terms of the properties of the flexibility graph and use this result to translate the design question in terms of transportation polytope, the deterministic equivalent of parallel server queues. Second, we design the sparsest flexibility graph that achieves a given delay performance and shows the robustness of the design to demand uncertainty. Third, given the budget to add edges arrives sequentially in time, we present the optimal schedule for adding them to the flexibility graph. These results are obtained by proving new results for transportation polytopes and are of independent interest. In particular, translating the difficulties to a simpler model, i.e. transportation polytope, allows us to develop a unified framework to answer several design questions.