The Simple solution

We are now ready to present the Simple solution. The Simple solution assumes that $G\in {\cal PH}_3^-$, where ${\cal PH}_3^{-} = {\cal U} \cup {\cal M} \cup {\cal L}$. Observe ${\cal PH}_3^{-}$ includes almost all distributions in ${\cal PH}_3$. Only the borders between the ${\cal U}_i$'s, ${\cal M}_i$'s, and ${\cal L}_i$'s are not included. Figure 2.11 shows an implementation of the Simple solution. The solution differs according to the classification of the input distribution $G$. When $G\in {\cal U}_0\cup {\cal M}_0$, a two-phase Coxian$^+$ PH distribution suffices to match the first three moments. When $G\in {\cal U}^+\cup {\cal M}^+$, $G$ is well-represented by an EC distribution with $p=1$. When $G\in {\cal L}$, $G$ is well-represented by an EC distribution with $p<1$.

Figure 2.12: Ideas in the Simple solution. Let ${G}$ be the input distribution. (i) If ${G}\in {\cal U}_0\cup {\cal M}_0$, ${G}$ is well-represented by a two-phase Coxian$^+$ PH distribution ${X}$. (ii) If ${G}\in {\cal U}^+\cup {\cal M}^+$, ${G}$ is well-represented by $\phi^N({X})$, where ${X}$ is a two-phase Coxian$^+$ PH distribution. (iii) If ${G}\in {\cal L}$, ${G}$ is well-represented by ${Z}$, where $Z$ has a distribution function $p\phi^N(X)(\cdot)+(1-p)O(\cdot)$.

(i)

(ii)

(iii)

(i) If $G\in {\cal U}_0\cup {\cal M}_0$ (see Figure 2.12(i)), then a two-phase Coxian$^+$ PH distribution suffices to match the first three moments, i.e., $p=1$ and $n=2$ ($N=0$). The parameters ($\lambda_{X1}$, $\lambda_{X2}$, $p_X$) of the two-phase Coxian$^+$ PH distribution are chosen as follows [144,190]:

$$\lambda_{X1} = \frac{u+\sqrt{u^2-4v}}{2\mu_1^{G}},\quad \lam... ... = \frac{\lambda_{X2}(\lambda_{X1}\mu_1^{G}-1)}{\lambda_{X1}},$$ (2.4)

where

$$u = \frac{6-2m_3^{G}}{3m_2^{G}-2m_3^{G}} \quad\mbox{and}\quad v = \frac{12-6m_2^{G}}{m_2^{G}(3m_2^{G}-2m_3^{G})}.$$ (2.5)

(ii) If $G\in {\cal U}^+\cup {\cal M}^+$ (see Figure 2.12(ii)), Corollary 1 specifies the number of phases needed:

$$n = \min\left\{ k \Big\vert m_2^{G} > \frac{k}{k-1} \right\} = \left\lfloor \frac{m_2^{G}}{m_2^{G}-1}+1 \right\rfloor.$$ (2.6)

Let $N=n-2$. Next, we find the two-phase Coxian$^+$ PH distribution $X\in {\cal U}_0\cup {\cal M}_0$ such that $G$ is well-represented by $Z=\phi^N(X)$. By Theorem 3, this can be achieved by setting

$$m_2^{X} = \frac{(n-3)m_2^{G}-(n-2)}{(n-2)m_2^{G}-(n-1)}, \qu... ...ox{and}\quad \mu_1^{X} = \frac{\mu_1^{G}}{(n-2)m_2^{X}-(n-3)},$$

where

$$\begin{eqnarray*} \alpha & = & (n-2)(m_2^{X}-1)\left(n(n-1)(m_2^{X})^2-n(2n-5)m_... ...left((n-1)m_2^{X}-(n-2)\right)\left((n-2)m_2^{X}-(n-3)\right)^2. \end{eqnarray*}$$

Thus, we set $p=1$, and the parameters ($\lambda_{X1}$, $\lambda_{X2}$, $p_X$) of ${X}$ are given by case (i), using $m_2^X$, $m_3^X$, and $\mu_1^X$, specified above.

(iii) If $G\in {\cal L}$ (see Figure 2.12(iii)), then let

$$p = \frac{1}{2m_2^{G}-m_3^{G}},\quad m_2^{W} = p m_2^{G},\qu... ... m_3^{G},\quad\mbox{and}\quad \mu_1^{W} = \frac{\mu_1^{G}}{p}.$$

Observe that $p$ satisfies $0\leq p < 1$. Also, since $W$ is in ${\cal M}$, $W$ can be chosen as an EC distribution with no mass probability at zero. If ${W}\in {\cal M}_0$, the parameters of ${W}$ are provided by case (i), using $m_2^W$, $m_3^W$, and $\mu_1^W$, specified above. If ${W}\in {\cal M}^+$, the parameters of ${W}$ are provided by case (ii), using $m_2^W$, $m_3^W$, and $\mu_1^W$, specified above. ${G}$ is then well-represented by distribution $Z$, where $Z(\cdot)=pW(\cdot)+(1-p)O(\cdot)$.