Definition of PH distribution

In general, a PH distribution is the distribution of the time until absorption into state 0 in a Markov chain.

Figure 2.7: A three-phase PH distribution.

Definition 8   A PH distribution with parameter $(\Vec{\tau},\mathbf{T})$, PH( $\Vec{\tau},\mathbf{T}$), is the distribution of the time until absorption into state 0 in a Markov chain on the states $\{0,1,...,n\}$ with initial probability vector $(\tau_0,\Vec\tau)$ and infinitesimal generator

$$\left(\begin{array}{cc}0 & \Vec{0}\\ \Vec{t} & \mathbf{T} \end{array}\right),$$

where $\Vec{t}=-\mathbf{T}\Vec{1}$ and $\tau_0 = 1 - \Vec\tau\Vec{1}$, where $\Vec{1}$ is a column vector of 1's.

Figure 2.7 illustrates a three phase PH distribution having parameters

$$\Vec{\tau} = (\tau_1,\tau_2,\tau_3) \quad\mbox{and}\quad \... ...\ \lambda_{31} & \lambda_{32} & -\sigma_3 \end{array}\right),$$

where $\sigma_i=\sum_{j\neq i} \lambda_{ij}$ for $i=1,2,3$. Note that the initial state can be the absorption state (state 0) with positive probability. If the initial state is the absorption state, the absorption time is zero.