Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 27K / b

An M-G-1 queue has Poisson arrivals of rate $\lambda$, independent identically distributed service times with a general law of mean $m$, and one server. During the first service time $S$, a mean $\lambda \mathbb{E}S=\lambda m$ customers arrive, and each initiates an independent descendant busy-period contribution. Thusso for traffic intensity $\rho =\lambda m<1$,For $\rho \ge 1$ the expected busy period is infinite.