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(https://www.cs.cmu.edu/~osogami/thesis/html/node39.html)
An [[Exponential Distribution]] is a PH distribution. Second a [[Convolution]] of two independent identical exponential distributions is a PH distribution. This is called an Erlang-2 distribution. Third a mixture distribution between 2 exponential distribution is also a PH distribution
In general, a PH distribution is the distribution of the time until absorption into state 0 in a Markov chain.
By restricting the structure of the Markov chain , we can define a subclass of the PH distribution.
First if the Markov chain whose absorption time defines a PH distribution is acyclic the PH distribution is called an acyclic PH distribution. ![[Pasted image 20230603160804.png]]
An acyclic PH distribution is called Coxian PH distribution if the Markov chain, whose absorption time defines the acyclic PH distribution, has the following two properties:
- the initial non-absorbing state is unique (i.e. the initial state is either the unique non absorbing state or the absorbing state)
- for each state, the next non-absorbing state is unique (i.e. the next state is the unique non-absorbing state or the absorbing state) ![[Pasted image 20230603160817.png]] When a Coxian PH distribution does not have mass probability at zero(i.e. the initial state is not the absorbing state), we refer to the Coxian PH distribution as a Coxian$^+$ PH distribution.
AN acyclic PH distribution is called a hyperexponential distribution if the Markov chain whose absorption time defines the acyclic PH distribution has the following property: for any state, the next state is the absorbing state. That is, a mixture of exponential distributions is a hyperexponential distribution.
An acyclic PH distribution is called an Erlang distribution if the Markov chain whose absorption time defines the acyclic distribution has following properties:
- the initial state is a unique non-absorbing state
- for each state the next state is unique
- the [[Sojourn Time]] distribution at each state is identical
That is, the sum of
$n$ independent and identically distributed exponential random variables has an$n$ -phase Erlang distribution. An$n$ -phase Erlang distribution is also called an Erlang-$n$ distribution.
An Erlang distribution is generalised to a generalised Erlang distribution by allowing a transition from the initial state to the absorbing state. ![[Pasted image 20230603164332.png]]
A mixture of Erlang distributions is called a mixed Erlang distribution. A mixture of Erlang distributions with the same number of phases is called a mixed Erlang distribution with common order. ![[Pasted image 20230603164344.png]]
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