Characterization of Phase-type Distributions and their Applications

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Phase-type (PH) distributions were introduced by Marcel Neuts in 1975 and have been widely used in science and engineering. PH-distributions are useful since i) they can approximate probability distributions with nonnegative support; ii) they are amenable for building mathematical models; and iii) they lead to an algorithmic approach for stochastic analysis. PH-distributions are suitable for stochastic modeling and statistical analysis of stochastic systems such as queueing models, queueing networks, reliability models, inventory models, supply chains, risk/insurance models, etc.  As the application of PH-distributions expands, there come higher requirements and standards on the accuracy in data fitting and the speed in computation in stochastic modeling. Two problems that are commonly encountered by researcher and practitioners are a) the selection of the proper type of PH-distributions, and b) the selection of proper approximation methods using PH-distributions. Motivated by their applications and inspired by recent progress in the area of matrix-analytic methods and machine learning, we plan to explore three issues on PH-distributions to address the accuracy and speed problems. Objective 1 (Obj1): Characterization of the MMPP (Markov modulated Poisson process) type phase-type distributions. We plan to prove that the squared coefficient of variations (SCV) of all MMPP type PH-distributions are greater than or equal to one, and to investigate further properties, e.g., bounds on the moments, by using stochastic comparison methods and martingales. The results are useful for selecting the proper PH-distributions and Markov modulated Poisson processes and, consequently, improving the accuracy and speed of data fitting. Objective 2 (Obj2): Approximation of discrete probability distributions by continuous PH-distributions. We plan to use the Erlangization method to construct continuous PH-distributions to approximate discrete probability distributions. We expect that the Erlangization based approximation not only has the same mean as the original distribution, but also ensures that the variance, SCV, and distribution function are close, which may improve the accuracy in data fitting. Objective 3 (Obj3): Characterization of finite support PH-distributions. We plan to investigate the ranges of the moments, variances, and SCVs of finite support phase-type distributions under certain conditions (e.g., with a given mean), which may improve the accuracy and speed in parameter estimation. The proposed research, on one hand, advances the theory of PH-distributions, and, on the other hand, makes it possible for researchers and practitioners to use PH-distributions and Markov modulated Poisson processes effectively and efficiently. The proposed research offers opportunities to train HQP for data fitting, parameter estimation, and stochastic modeling, which play an important role in machine learning and data analytics.