Prove that $w$ for the $\mathrm{M} / \mathrm{H}_2 / 1$ queueing system has the twostage hyperexponential distribution function
$$
W[t]=P[w \leq t]=1-\pi_a e^{-\mu_a t}-\pi_b e^{-\mu_b t}, t \geq 0 .
$$
In (5.296) the constants are given by
$$
\begin{gathered}
\pi_a=\frac{C_1 z_1}{z_1-1}, \\
\pi_b=\frac{C_2 z_2}{z_2-1}, \\
\mu_a=\lambda\left(z_1-1\right),
\end{gathered}
$$
and
$$
\mu_b=\lambda\left(z_2-1\right),
$$
where the constants in (5.297) through (5.300) are those used in the formula for $g_N(z)$ derived in Example 5.3.2; that is,
$$
g_N(z)=\frac{C_1 z_1}{z_1-z}+\frac{C_2 z_2}{z_2-z} .
$$