It is shown in Section 5.2.4 that the probability that all $c$ servers are busy in an $\mathrm{M} / \mathrm{M} / \mathrm{c} / \mathrm{c}$ queueing system $(\mathrm{M} / \mathrm{M} / \mathrm{c}$ loss system) is given by Erlang's $B$ formula, $B[c, a]$, defined by
$$
B[c, a]=\frac{\frac{a^c}{c !}}{1+a+\frac{a^2}{2 !}+\cdots+\frac{a^c}{c !}} .
$$
Consider an $\mathrm{M} / \mathrm{M} / \mathrm{c}$ queueing system in the steady state. Show that the following are true:
(a)
$$
\frac{1}{C[c, a]}=\rho+\frac{(1-\rho)}{B[c, a] .}
$$
(b)
$$
\frac{1}{B[1, a]}=1+\frac{1}{a}
$$
(c)
$$
\frac{1}{B[n, a]}=1+\frac{n}{a} \times \frac{1}{B[n-1, a]} \quad \text { for } n=2,3, \ldots, c \text {. }
$$