Let $b$ be the busy period of the server in the M/G/1 queueing system, that is, the time from a server start up to service a customer (after an idle period) until the server is again idle: Kleinrock [32] shows that
$$
E[b]=\frac{W_s}{1-\rho},
$$
(note that this is the average time a customer spends in an $\mathrm{M} / \mathrm{M} / 1$ system), and
$$
E\left[b^2\right]=\frac{E\left[s^2\right]}{(1-\rho)^3},
$$
so that
$$
\sigma_b^2=\frac{\sigma_s^2+\rho W_s^2}{(1-\rho)^3} .
$$
Find $E[b], E\left[b^2\right]$, and $\sigma_b^2$ in terms of the system parameters for
(a) the $\mathrm{M} / \mathrm{M} / 1$ queueing system,
(b) the $\mathrm{M} / E_k / 1$ queueing system, and
(c) the M/D/1 queueing system.