Collaborative Research: A Framework for Evaluation, Approximation, and Optimization of Time-Dependent Stochastic Service System Models having Deterministic/Scheduled Interventions

Purdue University, West Lafayette IN

Investigators

Abstract

This award supports establishing a mathematical framework for modeling, evaluating, approximating, and optimizing the performance of service systems featuring time-varying random as well as deterministic/scheduled input processes. Two important example problem classes are (1) optimizing efficiency and utilization while improving patient satisfaction in healthcare facilities that treat both time-varying randomly-arriving patients (e.g., emergent or walk-in) as well as patients having scheduled appointments (e.g., primary-care-physician referrals, school-required physical exams, or scheduled vaccinations), and (2) optimizing efficiency and utilization while improving flexibility and responsiveness to global competition in manufacturing facilities that operate in both a time-varying stochastic (e.g., production) environment as well as a deterministic/scheduled (e.g., job-release schedule) environment. The solution to a unified abstraction of both problem classes requires modeling and analysis methods that allow rich variations in model-input processes, and model logic, while adequately capturing the time-dependent evolution of the resulting probabilistic network. Traditional (exact) time-dependent differential-difference equation modeling of such networks is infeasible since the number of differential-difference equations describing even modest-sized networks can be of the order of hundreds of thousands (or more). Monte Carlo (MC) computer simulation, the natural alternative choice, is convenient but burdened with slow convergence rates and additional mathematically technical inefficiencies. Methods investigated by the research team will assist healthcare (and other) service and manufacturing sector industries to increase their economic competitiveness and patient/customer, satisfaction. The research will result in closure-equipped partial moment differential equations (PMDEs) for numerically approximating the time-dependent evolution of general stochastic networks having scheduled interventions. By exploiting the structure of PMDEs, and then strategically using closure approximations, the research team will be able to efficiently describe the time-dependent evolution of very general networks. Preliminary evidence indicates that the time-dependent evolution of modest stochastic networks can be approximated to machine accuracy within a few seconds on a typical laptop computer. Moreover, higher order derivatives, which often require significant effort in the Monte Carlo context, can be obtained with little to no extra effort by exploiting the rich structure inherent in the approximations.