Brumelle [6] proves that, for an $M / G / 1$ queueing system,
$$
E\left[N_q\left(N_q-1\right) \cdots\left(N_q-k+1\right)\right]=\lambda^k E\left[q^k\right],
$$
for $k=1,2,3, \ldots$.
(a) Use Brumelle's result to show that
$$
\sigma_{N_q}^2=\frac{\lambda^3 E\left[s^3\right]}{3(1-\rho)}+\left(\frac{\lambda^2 E\left[s^2\right]}{2(1-\rho)}\right)^2+\frac{\lambda^2 E\left[s^2\right]}{2(1-\rho)} .
$$
(b) Use part (a) and the result of Exercise 40 to show that
$$
\operatorname{Cov}\left[N_q, N_s\right]=\frac{\lambda^2 E\left[s^2\right]}{2},
$$
and thus that
$$
E\left[N_q N_s\right]=\frac{\lambda^2 E\left[s^2\right]}{2(1-\rho)} .
$$
[Hint: Use Theorem 2.7.2(c) and the fact that $\sigma_{N_A}^2=\rho(1-$ $\rho)$.] Note: Since $E\left[N_s N_q\right] \neq E\left[N_s\right] E\left[N_q\right], N_s$ and $N_q$ are not independent random variables (see Theorem 2.7.1(d)). Of course we would not expect them to be because the number in the queue clearly depends on the number in service. However, the random variables $q$ and $s$ are independent by assumption.