Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 27J (Applied Probability)

The queue length is an M-M-1 queue and hence a birth-death process with birth rate $\lambda$ and death rate $\mu$ away from zero. The detailed-balance equations areWriting $\rho =\lambda /\mu$, they give ${\pi }_n={\pi }_0{\rho }^n$. Since $\rho <1$, this measure is summable and normalization yieldsThe existence of this invariant probability distribution for the irreducible nonexplosive chain proves positive recurrence.

With state-dependent admission, the effective birth rate in state $n$ iswhile the death rate is $\mu$ for $n\ge 1$. The stationary ratios for an M-M-1 queue with state-dependent admission are thereforeInduction gives ${\pi }_n={\pi }_0{\rho }^n/n!$, and normalization identifies a Poisson distribution of mean $\rho$:

Here $b_n=\lambda 2^{-n}$, so detailed balance givesConsequentlyThe series converges for every $\rho >0$, so this defines the unique invariant distribution.

By the PASTA property, an arrival joins with probability ${\sum }_{n\ge 0}{\pi }_{n}p(n)$. In equilibrium the accepted-arrival rate must also equal the departure rate. Departures occur at rate $\mu$ precisely when the queue is nonempty, henceThe required joining probability is therefore