Gaussian Random Fields Methods for Fork-Join Network with Synchronization Constraints

Abstract : The objective of this research is to develop Gaussian random field methods for the design, modeling, analysis and control of stochastic fork-join networks (FJNs). The synchronization constraints require that tasks can only be synchronized only if all tasks of the same job are completed. The main mathematical challenge lies in the resequencing of arrival orders after service completion at each station, which requires an infinite dimensional state space to track the status of all parallel tasks for each job. It was an extremely difficult open problem. We have developed a novel method using multiparameter sequential empirical processes driven by service vectors of parallel tasks of each job to describe the system dynamics of FJNs. We have proved functional law of large numbers and functional central limit theorems for the service and queueing dynamics for synchronization jointly in an asymptotic regime where the arrivals of jobs and the numbers of servers get large appropriately. This research has produced two research papers, under review in Mathematics of Operations Research (minor revision) and Annals of Applied Probability. One paper was on the finalist of the INFORMS 2014 JFIG Paper Competition. We have given three conference and three seminar presentations for this work.