Asymptotic optimality of maximum pressure policies in stochastic processing networks

TL;DR

This paper proves that maximum pressure policies are asymptotically optimal in stochastic processing networks satisfying certain conditions, minimizing workload and holding costs in heavy traffic through state space collapse and limit theorems.

Contribution

It extends existing frameworks to stochastic processing networks, establishing asymptotic optimality of maximum pressure policies under heavy traffic conditions.

Findings

Maximum pressure policies asymptotically minimize workload.

Under quadratic costs, they also minimize holding costs.

The framework extends prior queueing network results.

Abstract

We consider a class of stochastic processing networks. Assume that the networks satisfy a complete resource pooling condition. We prove that each maximum pressure policy asymptotically minimizes the workload process in a stochastic processing network in heavy traffic. We also show that, under each quadratic holding cost structure, there is a maximum pressure policy that asymptotically minimizes the holding cost. A key to the optimality proofs is to prove a state space collapse result and a heavy traffic limit theorem for the network processes under a maximum pressure policy. We extend a framework of Bramson [Queueing Systems Theory Appl. 30 (1998) 89--148] and Williams [Queueing Systems Theory Appl. 30 (1998b) 5--25] from the multiclass queueing network setting to the stochastic processing network setting to prove the state space collapse result and the heavy traffic limit theorem. The extension can be adapted to other studies of stochastic processing networks.