2.6 Phase-type distributions

Phase-type (PH) distributions are based on the method of stages technique that was introduced by Erlang and generalized by Neuts [67]. PH distributions consist of a ``general'' mixture of exponentials and are characterized by a finite and absorbing Markov chain. The number $n$ of phases in the PH distribution is equal to the number of transient states in the associated (underlying) Markov chain. A PH distribution represents random variables that are measured by the time $X$ that the underlying Markov chain spends in its transient portion till absorption. From this perspective, a row vector $\mbox{\boldmath {$\tau$}}$ of size $n$ is associated with the underlying Markov chain of a PH distribution and represents the initial probability vector for each of its transient states. The infinitesimal generator of the underlying Markov chain of a PH distribution is shown in Eq.(2.24). The zero row represent the absorbing state of the chain.

$${\mathbf{Q}}=\left[ \begin{array}{c c} 0 & {\mathbf{0}} \\ {\mathbf{t}}& {\mathbf{T}} \end{array}\right]$$ (2.24)

${\mathbf{T}}$ is an $n \times n$ matrix representing the transitions among the transient states and ${\mathbf{t}}$ is a column vector of size $n$ representing the transitions from the transient states to the absorbing state of the underlying Markov chain. Matrix ${\mathbf{T}}$ and vector ${\mathbf{t}}$ relate to each other as ${\mathbf{t}}= -{\mathbf{T}}\cdot {\mathbf{e}}$. The PH distribution is fully represented by the vector $\mbox{\boldmath {$\tau$}}$ and the matrix ${\mathbf{T}}$. The basic characteristics of a PH distribution are

• the cumulative distribution function: $F(x) = 1 - \mbox{\boldmath {$\tau$}}{e}^{{\mathbf{T}}x} {\mathbf{e}}$,
• the density function: $f(x) = \mbox{\boldmath {$\tau$}}{e}^{{\mathbf{T}}x} (-{\mathbf{T}}\cdot {\mathbf{e}})$,
• the n$^{\rm th}$ moment: $m_n= (-1)^n\cdot n! \mbox{\boldmath {$\tau$}}{\mathbf{T}}^{-n} {\mathbf{e}}$,

where ${\mathbf{e}}$ is a column vector of ones with the appropriate dimension. Observe that in the PDF and CDF of a PH distribution, the matrix ${\mathbf{T}}$ appears as an exponential coefficient. As such, PH distributions are called matrix-exponential distributions. The term $e^{{\mathbf{T}}x}$ is defined as

$$e^{{\mathbf{T}}x} = \sum_{n \geq 0} \frac{1}{n!} ({\mathbf{T}}x)^n.$$

Obviously the number of parameters that define a PH distribution is $n(n+1)$, however the cases of PH distributions that are used in practice have the majority of the values in $\mbox{\boldmath {$\tau$}}$ and ${\mathbf{T}}$ equal to zero. For the hyperexponential, hypoexponential, and Coxian^2.6 distributions, special cases of PH distributions, the definitions of $\mbox{\boldmath {$\tau$}}$ and ${\mathbf{T}}$ are as follows (assuming $n=4$).

Table: Structure of $\mbox{\boldmath{$\tau$}}$ and ${\mathbf{T}}$ for the Hypoexponential, Coxian, and Hyperexponential distributions.

	Hypoexponential	Coxian	Hyperexponential
$\mbox{\boldmath {$\tau$}}$	$[1,~0,~0,~0]$	$[1,~0,~0,~0]$	$[p_1,~p_2,~p_3,~p_4]$
${\mathbf{T}}$	$\left[ \begin{array}{c c c c} -\lambda_1 & \lambda_1& 0 & 0 \\ 0 & -\lambda_... ...& 0 & -\lambda_3 & \lambda_3 \\ 0 & 0 & 0 & -\lambda_4\\ \end{array}\right]$	$\left[ \begin{array}{c c c c} -\lambda_1& \lambda_1^1 & 0 & 0 \\ 0 & -\lambd... ... & -\lambda_3 & \lambda_3^1 \\ 0 & 0 & 0 & -\lambda_4 \\ \end{array}\right]$	$\left[ \begin{array}{c c c c} -\lambda_1 & 0 & 0 & 0 \\ 0 & -\lambda_2 & 0 & 0 \\ 0 & 0 & -\lambda_3 & 0 \\ 0 & 0 & 0 & -\lambda_4 \\ \end{array}\right]$

Since the structure of the underlying Markov chain is arbitrary, a PH distribution captures a wide range of characteristics including high variability. The PH random variables are independently identically distributed because the initial probability vector $\mbox{\boldmath {$\tau$}}$ is the same every time the Markov chain starts in its transient portion in the corresponding renewal process. A PH distribution with infinite number of phases captures exactly a power-tailed distribution [71]. There is another set of processes known as Markovian Arrival Processes (MAPs) that are still based on the method of exponential stages and capture complex characteristics such as short- and long-range dependence. PH distributions are a special case of MAPs [69].