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

该奖项支持建立一个数学框架，用于对具有时变随机和确定性/调度输入过程的服务系统的性能进行建模、评估、近似和优化。两个重要的示例问题类别是(1)在医疗设施中优化效率和利用率，同时提高患者满意度，这些医疗设施既治疗时变的随机到达的患者(例如，急诊或步行)，也治疗预定预约的患者(例如，初级保健医生转诊、学校要求的体检或预定的疫苗接种)；以及(2)优化效率和利用率，同时提高制造设施的灵活性和对全球竞争的响应能力，这些制造设施既在时变的随机(例如，生产)环境中运营，也在确定性/预定(例如，工作发布时间表)环境中运营。解决这两个问题类的统一抽象需要建模和分析方法，这些方法允许模型输入过程和模型逻辑的丰富变化，同时充分捕获所产生的概率网络的依赖于时间的演变。此类网络的传统(精确)依赖于时间的微分-差分方程式建模是不可行的，因为即使描述中等规模的网络的微分-差分方程式的数量也可能是数十万(或更多)的数量级。蒙特卡洛(MC)计算机模拟是自然而然的替代选择，它很方便，但存在收敛速度慢和额外的数学技术效率低下的问题。研究小组调查的方法将有助于医疗保健(以及其他)服务和制造业提高其经济竞争力和患者/客户满意度。研究将产生封闭配备的偏矩微分方程(PMDE)，用于数值逼近具有预定干预的一般随机网络的时间依赖演化。通过利用PMDE的结构，然后战略性地使用闭包近似，研究团队将能够有效地描述非常一般网络的时间依赖演化。初步证据表明，在一台典型的笔记本电脑上，适度随机网络的时间依赖演化可以在几秒钟内接近机器精度。此外，高阶导数在蒙特卡罗背景下通常需要很大的努力，通过利用近似中固有的丰富结构，可以很少甚至不需要额外的努力就能获得。

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.