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，用于建模，评估，近似和优化服务系统的性能，这些系统具有时变随机以及确定性/计划输入过程。 两个重要的示例问题类是（1）优化效率和利用率，同时提高医疗机构中的患者满意度，该医疗机构治疗时变随机到达的患者（例如，急诊或无预约）以及具有预定预约的患者（例如，初级保健医生转诊、学校要求的身体检查或预定的疫苗接种），以及（2）优化效率和利用率同时提高在时变随机（例如，生产）环境以及确定性/调度的（例如，作业发布调度）环境。 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，允许丰富的变化，在模型输入过程中，模型逻辑，同时充分捕获的概率网络的时间依赖性的演变。传统的（精确的）时间相关的微分-差分方程建模这样的网络是不可行的，因为微分-差分方程的数量描述甚至中等规模的网络可以是几十万（或更多）的顺序。蒙特卡罗（MC）计算机模拟，自然的替代选择，是方便的，但负担缓慢的收敛速度和额外的数学技术效率低下。 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，以提高他们的经济竞争力和病人/客户，satisfactions.The研究将导致封闭装备偏矩微分方程（PMDEs）的数值近似的时间依赖性演化的一般随机网络有预定的干预措施。通过利用PMDE的结构，然后策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的网络的时间依赖性演化。 初步证据表明，适度随机网络的时间依赖性演化可以在典型的笔记本电脑上在几秒钟内近似于机器精度。 此外，高阶导数，这往往需要显着的努力，在蒙特卡洛的情况下，可以得到很少或没有额外的努力，利用丰富的结构固有的近似。

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.