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

该合同支持建立一个数学框架，用于建模、评估、近似和优化具有时变随机和确定性/计划输入过程的服务系统的性能。两个重要的示例问题类别是：(1)优化效率和利用率，同时提高医疗保健机构的患者满意度，这些医疗保健机构既治疗时变随机到达的患者（例如，急诊或非预约），也治疗有预约的患者（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）；(2)优化效率和利用率，同时提高在时变随机（如生产）环境和确定性/计划（如工作释放计划）环境中运行的制造设施的灵活性和对全球竞争的响应能力。这两个问题类的统一抽象的解决方案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中的丰富变化，同时充分捕获结果概率网络的时间依赖演化。传统的（精确的）随时间变化的微分-差分方程建模是不可行的，因为描述中等规模网络的微分-差分方程的数量可能达到数十万（或更多）。蒙特卡罗（MC）计算机模拟是一种自然的替代选择，它方便，但具有缓慢的收敛速度和额外的数学技术效率低下的缺点。研究小组调查的方法将有助于医疗保健（和其他）服务和制造业行业提高其经济竞争力和患者/客户满意度。该研究将产生具有关闭装备的偏矩微分方程（pmde），用于数值逼近具有计划干预的一般随机网络的随时间演化。通过利用pmde的结构，然后有策略地使用闭合近似，研究团队将能够有效地描述非常一般的网络的随时间演变。初步证据表明，适度随机网络的随时间演化可以在几秒钟内在一台典型的笔记本电脑上近似达到机器精度。此外，在蒙特卡罗环境中通常需要大量努力的高阶导数可以通过利用近似中固有的丰富结构而无需额外的努力而获得。

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.