Consider an $M / G / 1$ queue with arrival rate $\lambda$ and traffic intensity less

than 1. Prove that the moment-generating function of a typical busy period, $M_{B}(\theta)$, satisfies

$$M_{B}(\theta)=M_{S}\left(\theta-\lambda+\lambda M_{B}(\theta)\right),$$

where $M_{S}(\theta)$ is the moment-generating function of a typical service time.

If service times are exponentially distributed with parameter $\mu>\lambda$, show that

$$M_{B}(\theta)=\frac{\lambda+\mu-\theta-\left\{(\lambda+\mu-\theta)^{2}-4 \lambda \mu\right\}^{1 / 2}}{2 \lambda}$$

for all sufficiently small but positive values of $\theta$.