Show that for an $\mathrm{M} / \mathrm{M} / \mathrm{c}$ queueing system in equilibrium, the following are true:
(a) If $n<c$,
$$
P[N \geq n]=p_0\left\{\sum_{k=n}^{c-1} \frac{a^k}{k !}+\frac{a^c}{c !(1-\rho)}\right\} .
$$
(b) If $n \geq c$,
$$
P[N \geq n]=C[c, a] \rho^{n-c} .
$$
(c) If $c=2$, (a) and (b) reduce to
$$
P[N \geq n]=\frac{2 \rho^n}{1+\rho}, \quad n=1,2, \ldots .
$$
(d) If $c=1$, (a) and (b) reduce to
$$
P[N \geq n]=\rho^n, \quad n=0,1,2, \ldots
$$