Given that, for the GI/M/1 queueing system,
$$
W_q[t]=P[q \leq t]=1-\left(1-\pi_0\right) e^{-\pi_0 t / W_s}, t \geq 0,
$$
prove the following:
(a)
$$
W_q=\left(1-\pi_0\right) \frac{W_s}{\pi_0} .
$$
(b)
$$
E\left[q^2\right]=2\left(1-\pi_0\right)\left(\frac{W_s}{\pi_0}\right)^2 .
$$
(c)
$$
\sigma_q^2=\left(1-\pi_0^2\right)\left(\frac{W_s}{\pi_0}\right)^2 .
$$