Representations of phase type distributions and their applications in stochastic modelling

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.