Examples of PH distributions

We start by providing canonical examples of PH distributions. Here, we provide both pictorial explanation and more formal explanation. Pictorial explanation gives intuitive understanding of the PH distribution, and more formal explanation allows us to get used to the notation that we use later.

First, an exponential distribution is a PH distribution. Figure 2.6(a) illustrates an exponential distribution as the absorption time in a (continuous time) Markov chain^2.2. At time 0, we start at state 1. We stay in this state for a random time having an exponential distribution with rate $\lambda$ , and then transition to state 0, the absorbing state. The time until we enter the absorbing state is, of course, an exponential distribution. More formally, an exponential distribution with rate $\lambda$ is the distribution of the time until absorption into state 0 in a Markov chain on the states $\{0,1\}$ with initial probability vector

and infinitesimal generator:

$\begin{displaymath} \left(\begin{array}{cc} 0 & 0\\ \lambda & -\lambda \end{array}\right). \end{displaymath}$

Figure 2.6: Examples of PH distributions.

$\includegraphics[width=.8\linewidth]{fig/exponential_distribution.eps}$

(a) exponential

$\includegraphics[width=.9\linewidth]{fig/Erlang2_distribution.eps}$

(b) Erlang-2

$\includegraphics[width=.95\linewidth]{fig/H2_distribution.eps}$

Second, a convolution of two independent identical exponential distributions is a PH distribution (i.e., the sum of two i.i.d. exponential random variables has a PH distribution); this distribution is called an Erlang-2 distribution. Figure 2.6(b) illustrates an Erlang-2 distribution as the absorption time in a Markov chain. At time 0, we start at state 1. After a random time having an exponential distribution with rate $\lambda$ , we transition to state 2. We stay in state 2 for a random time having an exponential distribution with rate $\lambda$ , and then transition to state 0, the absorbing state. The time until we enter the absorbing state has an Erlang-2 distribution. More formally, an Erlang-2 distribution with parameter $\lambda$ is the distribution of the time until absorption into state 0 in a Markov chain on the states $\{0,1,2\}$ with initial probability vector

and infinitesimal generator:

$\begin{displaymath} \left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & -\lambda & \lambda\\ \lambda & 0 & -\lambda \end{array}\right). \end{displaymath}$

Third, a mixture of two exponential distributions is a PH distribution; this distribution is called a two-phase hyperexponential distribution, H

. Figure 2.6(c) illustrates an H

distribution as the absorption time in a Markov chain. At time 0, we start at state 1 with probability $\tau_1$ and at state 2 with probability $\tau_2$ . If we start at state 1 (respectively, state 2), we stay there for a random time having an exponential distribution with rate $\lambda_1$ (respectively, $\lambda_2$ ), and then transition to state 0, the absorbing state. The time until we enter the absorbing state has an H

distribution. More formally, an H

distribution with parameter ( $\tau_1$ , $\tau_2$ , $\lambda_1$ , $\lambda_2$ ) is the distribution of the time until absorption into state 0 in a Markov chain on the states $\{0,1,2\}$ with initial probability vector $(0,\tau_1,\tau_2)$ and infinitesimal generator:

$\begin{displaymath} \left(\begin{array}{ccc} 0 & 0 & 0\\ \lambda_1 & -\lambda_1 & 0\\ \lambda_2 & 0 & -\lambda_2 \end{array}\right). \end{displaymath}$