Mathematical Sciences: Stochastic Modelling with Applications to Markov Chain Monte Carlo Methods and Design and Analysis in Systems Engineering

摘要本项目旨在扩展马尔可夫过程的稳定性理论，并考虑其在两个特定领域的应用，即马尔可夫链蒙特卡罗（MCMC）方法的行为和复杂网络系统的分析。理论工作将涉及(a)评估模型稳定性和不稳定性的方法，这将包括离散和连续时间马尔可夫模型；(b)制订稳定模式的绩效衡量标准，特别是比较这些模式达到稳定制度的比率；(b)随机有序或部分随机有序模型的详细行为；(d)系统设计对假设扰动的鲁棒性。这些方法的一个应用将是MCMC方法，它目前正在彻底改变贝叶斯和计算统计方法，在这个项目下，希望开发在线收敛标准，评估收敛率的方法，以及加速这些算法的方法。另一个应用领域是分析复杂的网络、排队和存储系统。所设想的这种应用的具体类型包括目前制造系统的高优先级领域，以及更一般的具有反馈的排队系统，其稳定性和性能结果将非常适合。在评估许多复杂系统时，数学模型，特别是概率和统计中的数学模型，在两个步骤中起着关键作用。第一步是描述系统。本研究解决了“制造网络”的建模，目前在高科技制造业中具有核心重要性。这里的目的是描述诸如半导体组件之类的物品在各种机器之间移动的方式，这些机器可以工作并重新工作组件以获得最终产品。与许多模型一样，描述这种制造系统的数学也可以描述其他系统，比如信息网络上比特的移动。这里提出的工作将解决一个至关重要的问题：这样一个系统何时是稳定的，在平衡运行的意义上，而不是产生导致关闭、积压或溢出的问题。这个建模步骤有助于定义描述系统可控制范围的关键参数：第二步是评估这些参数，这里建议研究新的仿真方法，使此类评估能够发生。这些“马尔可夫链蒙特卡罗（MCMC）算法”具有非常广泛的适用性，但它们本身是“复杂的系统”，就像在实际系统中一样，它们在使用中可能会出现溢出和积压：这里的工作应该为MCMC算法提供控制方法，从而实现更快、更准确的模拟方法。这些模拟方法已在广泛的领域得到应用，从模式识别到农业实验，到临床和流行病学领域，再到环境评估：因此，本提案所支持的研究实际上已推广到许多领域。

9504561 Tweedie Abstract This project aims to extend the stability theory mf Markov processes, and consider its application to two specific areas, namely the behavior of Markov chain Monte Carlo (MCMC) methods and the analysis of complex network systems. The theoretical work will involve (a) methods of evaluating model stability and instability, which will include both discrete and continuous time Markov models; (b) development of performance measures for stable models, and specifically comparison of rates at which such models achieve a stable regime; (b) detailed behavior of stochastically ordered or partially stochastically ordered models; and (d) robustness of system design against perturbations in assumptions. One application of these methods will be to MCMC methods, which are currently revolutionizing Bayesian and computational statistics methods, and under this project it is hoped to develop on-line convergence criteria, methods of evaluating convergence rates, and ways of speeding up these algorithms. The other area of application is in the analysis of complex network, queueing and storage systems. The specific types of such application envisaged include the currently high-priority area of manufacturing systems, and more general queueing systems with feedback, for which the stability and performance results to be developed are well suited. In evaluating many complex systems, there are two steps where mathematical models, especially those in probability and statistics, play a critical role. The first step is in describing the system. This research addresses the modeling of ``manufacturing networks'', currently of central importance in the high-technology manufacturing sector. Here the aim is to describe the way in which items such as semiconductor components move around the various machines which work and rework components to get to a final product. As with many models, the mathematics that describes such manufacturing systems also describes other systems, such as the movement of bits over an information network. The work proposed here will attack the vital question: when is such a system stable, in the sense of running in equilibrium rather than developing problems which cause shutdowns, backlogs or overflows. This modeling step helps define the crucial parameters that describe ranges when systems can be controlled: the second step is then in estimating such parameters, and here it is proposed to work on new simulation methods that enable such estimation to take place. These " Markov chain Monte Carlo (MCMC) algorithms" have very wide applicability, but are themselves "complex systems", and just as with real systems, they can develop overflows and backlogs in usage: the work here should provide methods of control for MCMC algorithms leading to faster and more accurate methods of simulation. These simulation methods have found application in a huge range of areas, from pattern recognition, to agricultural experiments, to clinical and epidemiological areas, to environmental assessments: the research supported in this proposal therefore propagates to usage in very many areas indeed.