Consider the steady state $\mathrm{M} / \mathrm{H}_2 / 1$ queueing system discussed in Example 5.3.3. All parameters in this exercise are defined there. Assume $z_1>z_2>1$.
(a) Prove that
$$
P[N \geq n]=C_1 \frac{z_1^{-n+1}}{z_1-1}+C_2 \frac{z_2^{-n+1}}{z_2-1}, \quad n=0,1, \ldots
$$
(b) Prove, by using part (a), that $P[N \geq 1]=\rho$ and $P[n=0]=1-\rho$.