Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 27K / i / Solution

An M-M-1 queue has Poisson process arrivals of rate $\lambda$, independent exponential service times of rate $\mu$, and one server. The two letters $M$ indicate the memoryless property of the interarrival and service-time distributions. Its queue length is a birth-death process with birth rate $\lambda$ and death rate $\mu$ away from zero. When $\rho =\lambda /\mu <1$, its normalized stationary distribution isthe powers ${\rho }^n$ quoted in the question form the corresponding unnormalized invariant measure.