Consider the $\mathrm{M} / \mathrm{H}_2 / 1$ queueing system. Invert the LaplaceStieltjes transform of the queueing time, $q$, by the method of partial fractions to obtain the density function
$$
f_q(t)=(1-\rho) \delta(t)+C_3 e^{-a t}+C_4 e^{-b t}, \quad t \geq 0,
$$
where $a=-z_1$ and $b=-z_2$ for the zeroes $z_1$ and $z_2$ of the polynomial
$$
\theta^2+\left(\mu_1+\mu_2-\lambda\right) \theta+\mu_1 \mu_2(1-\rho)
$$
The parameters $\mu_1$ and $\mu_2$ in (5.303) are the parameters for the distribution of $s$; that is,
$$
W_s=\frac{q_1}{\mu_1}+\frac{q_2}{\mu_2} .
$$
The constants $C_3$ and $C_4$ in (5.302) are given by
$$
C_3=\frac{\lambda(1-\rho) z_1+\rho(1-\rho) \mu_1 \mu_2}{z_1-z_2},
$$
and
$$
C_4=\frac{\lambda(1-\rho) z_2+\rho(1-\rho) \mu_1 \mu_2}{z_2-z_1},
$$
Now integrate (5.302) to show that
$$
W_q[t]=P[q \leq t]=1-\frac{C_3}{a} e^{-a t}-\frac{C_4}{b} e^{-b t}, \quad t \geq 0 .
$$
As part of deriving (5.307), you will need to show that
$$
\frac{C_3}{a}+\frac{C_4}{b}=\rho \text {. }
$$