Representations of phase type distributions and their applications in stochastic modelling

状态空间具有大量状态的随机系统的性能分析和优化设计是一个具有挑战性的问题。这就是排队模型、库存模型和供应链中常见的“维度诅咒”问题。面临风险的主要问题是计算效率。为了解决这些问题，人们提出了状态空间归约、近似、分解等方法，但目前还没有一种普遍有效的方法。因此，有必要寻找新的解决方法。按照状态空间归约的方法，我们建议研究相型(PH)分布的PH表示，并开发有效的算法来计算PH分布的最小、简单或特殊结构的PH表示。自从1975年Marcel Neuts提出了最新的PH分布以来，PH分布已经被用于科学和工程的各个分支，包括运筹学、管理学、电信、风险和保险分析以及生物统计学。特别是PH分布被用于我们对排队模型、库存模型、PH分布的一个基本问题是为PH分布找到一个状态数最少的PH表示。如果存在最小PH表示，则具有给定结构的随机系统的状态空间可以保持尽可能小。因此，寻找最小PH表示对随机系统性能分析的有效性和效率具有深远的影响。与最小表示问题密切相关的是为PH分布寻找简单的或特殊结构的PH表示的问题。同样，如果有简单的或特殊结构的PH表示可用，可以更有效地进行随机系统的性能分析。所提出的研究结果在解决控制理论中的正实现问题以及PH分布的参数估计和拟合方面具有应用价值。

Performance analysis and optimal design of stochastic systems whose state space has a large number of states can be challenging problems. This is known as the "curse of dimensionality" problem common to queueing models, inventory models, and supply chains. The main issue at stake is computational efficiency. A number of approaches such as the state space reduction, approximation, and decomposition have been introduced to solve the problems.  However, there is no universally effective method. Thus, there is a need to look for new solution methods. Following the state space reduction approach, we propose to study PH-representations of phase-type (PH) distributions and to develop efficient algorithms for computing minimal, simple, or specially structured PH-representations for PH-distributions.    PH-distributions were introduced by Marcel Neuts in 1975.  Since then, PH-distributions have been used in various branches of science and engineering, including operations research, management science, telecommunications, risk and insurance analysis, and biostatistics.  In particular, PH-distributions are used in our studies of queueing models, inventory models, and supply chains.  A fundamental problem of PH-distributions is to find a PH-representation with the minimum number of states for a PH-distribution.  If minimal PH-representations are available, the state space of stochastic systems with a given structure can be kept as small as possible.  Thus, finding minimal PH-representations has far-reaching impact on the effectiveness and efficiency of the performance analysis of stochastic systems.  Closely related to the minimal representation problem are problems of finding simple or specially structured PH-representations for PH-distributions.  Likewise, if simple or specially structured PH-representations are available, performance analysis of stochastic systems can be done more efficiently. The results of the proposed research have applications in solving the positive realization problem in control theory and in parameter estimation and fitting of PH-distributions as well.