Non-trivial Generalization of the Markov Chains and the Phase-Type Distributions

What is it about?

The work is concerned about non-trivial generalization of the Markov chain and its lifetime distribution - also known as the Phase-Type Distribution. The model represents a continuous-time stochastic process which describes a sequence of events arriving in continuous time for which the probability of occurrence of each event depends not only on the current state, but also, among other things, on the current time (the process age), past information of previous events, inter-arrival time between previous events and the speed at which event occurs or changes. It is a non-Markov chain. Variety of distributional properties of the process are discussed. All identities are explicit in terms of the Bayesian update of switching probability and the speed of the process.

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Why is it important?

The model can be used as an alternative to the widely used Markov chain to analyse complex stochastic systems evolution taking account past information and heterogeneity within the systems being considered. Its availability in closed form offers tractability and more flexibilities in various applications than that of the Markov chain.

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