Asymptotic optimality of maximum pressure policies in stochastic processing networks

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.Comment: Published in at http://dx.doi.org/10.1214/08-AAP522 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Dynamic Control in Stochastic Processing Networks

A stochastic processing network is a system that takes materials of various kinds as inputs, and uses processing resources to produce other materials as outputs. Such a network provides a powerful abstraction of a wide range of real world, complex systems, including semiconductor wafer fabrication facilities, networks of data switches, and large-scale call centers. Key performance measures of a stochastic processing network include throughput, cycle time, and holding cost. The network performance can dramatically be affected by the choice of operational policies. We propose a family of operational policies called maximum pressure policies. The maximum pressure policies are attractive in that their implementation uses minimal state information of the network. The deployment of a resource (server) is decided based on the queue lengths in its serviceable buffers and the queue lengths in their immediate downstream buffers. In particular, the decision does not use arrival rate information that is often difficult or impossible to estimate reliably. We prove that a maximum pressure policy can maximize throughput for a general class of stochastic processing networks. We also establish an asymptotic optimality of maximum pressure policies for stochastic processing networks with a unique bottleneck. The optimality is in terms of minimizing workload process. A key step in the proof of the asymptotic optimality is to show that the network processes under maximum pressure policies exhibit a state space collapse.Ph.D.Committee Chair: Dai, Jim; Committee Member: Ayhan, Hayriye; Committee Member: Foley, Robert; Committee Member: Kleywegt, Anton; Committee Member: Ward, Amy; Committee Member: Xia, Cath

On dynamic crane deployment in container terminals

This thesis studies the problem of scheduling the movements of cranes in a container storage yard so as to minimize the total unfinished workload at the end of each time period. The problem is formulated as a mixed integer linear program, and the computational complexity of the problem is analyzed. A Lagrangian decomposition solution procedure is described. A new solution approach, called the successive piecewise-linear approximation method, is also developed. Through computational experiments, we show that our proposed solution methods are both efficient and effective for large size problems