# Discrete Phase-Type Distribution

< List of probability distributions < Discrete Phase-Type Distribution

A discrete phase-type distribution is a general class of distributions that models the time until a discrete event occurs; The probability of moving from one state to another depends only on the current state and not on past events. These distributions are often used to approximate continuous distributions; one advantage with using discrete, rather than continuous, phase-type distributions is that a lower coefficient of variation can be obtained with the same number of phases .

## Phase-type vs discrete phase-type distribution

A phase-type (PH) distribution models the time it takes for a system to transform from one state to another. It is widely used for complex systems found in areas such as reliability analysis and queueing theory. On the other hand, a discrete phase-type distribution is where the system is in one of a finite number of states; the system moves between these states at specific rates. The “phases” of the distribution refer to the different periods of time that it takes for the system to move from one state to another.

The distribution is typically represented by a matrix, called a transition rate matrix, that specifies rates of movement between states. The matrix can be used to calculate the probability of staying in a certain state or moving from one state to another, and the expected time of moving from state to state. The matrix form of the phase-type distribution is simpler to use and understand, in particular when it comes to find the moments of time before reaching the absorbing state .

## Discrete phase-type distribution and Markov chains

A discrete phase-type distribution is a type of Markov chain, where each state represents a phase and the transition between phases is described by a set of transition rates. In a Markov chain, the transition probabilities between the states are usually specified, while in a discrete phase-type distribution the transition probabilities are derived from transition rates. In addition, the duration of each phase in a discrete phase-type distribution is modeled using an exponential distribution, while in a general Markov chain the duration of each state can be modeled using any arbitrary distribution.

More specifically, a discrete phase–type distribution is the distribution of time to absorption in a finite discrete time Markov chain with transition matrix P of dimension m + 1 

The Markov chain has m transient states and 1 absorbing state. The initial probability vector is (α, αm+1). The pair (α,S) is a representation for the phase.

Any distribution that can be described as the time to absorption of a discrete-time Markov chain on a finite state space, with substochastic transition matrix P and initial distribution α, is a discrete phase-type distribution . A substochastic matrix is a square matrix with nonnegative entries where every row adds up to at most 1.

## Uses

Phase-type distribution are often used to approximate general distributions because they are often analytically tractable; Any positive-valued discrete distribution or continuous distribution can, theoretically, be approximated with a phase-type distribution to arbitrary precision.

The phase-type distribution is flexible in that it can approximate a wide range of probability distributions with varying numbers of phases, making it a useful tool in modeling complex systems. It can also be easily combined with other distributional models, such as the exponential distribution or gamma distribution, to provide even greater flexibility in modeling.

Discrete phase-type distributions have applications in fields such as insurance mathematics, queueing theory, population genetics, reliability analysis, and machine learning. They are useful in modeling systems that have multiple possible states and can be used to predict various outcomes, such as the time it takes for a machine to fail or the time it takes for a customer to be served in a queue.

## References

 Hakan Lorens Samir Younes (2005). Verification and Planning for Stochastic Processes with Asynchronous Events.

 Oregon State University. PhaseTypeR.

 Nielsen, B. Lecture notes on phase–type distributions for 02407 Stochastic Processes. October 2022. Retrieved May 7, 2023 from: http://www2.imm.dtu.dk/courses/02407/lectnotes/ftf.pdf

 Bean, N. & Nielsen, B. Decay rates of discrete phase-type distributions with infinitely-many phases. Matrix-analytic Methods Theory and Applications : Proceedings of the Fourth International Conference : Adelaide, Australia, 14-16 July 2002

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