Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 27K (Applied Probability)

Let regeneration cycles have lengths $C_i$, contain $A_i$ arrivals, and accrue queue-length rewardFubini's geometric identity writes this area as the sum, over customers in the cycle, of their time in the system, up to boundary terms whose contribution vanishes over many cycles. The renewal-reward theorem therefore givesHence Little law is

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.

For state $n\ge 1$, births occur at rate $\lambda$ and departures at rateThe birth--death detailed-balance weights areTheir ratios tend to zero, so the series is summable and normalization gives an invariant probability distribution.

The chain is irreducible and nonexplosive, and it has an invariant probability distribution. Therefore every state is positive recurrent. The increasingly strong abandonment rate is what stabilizes the system even when $\lambda \ge \mu$.