(i) Explain the notation $\mathrm{M}(\lambda) / \mathrm{M}(\mu) / 1$ in the context of queueing theory. [In the following, you may use without proof the fact that $\pi_{n}=(\lambda / \mu)^{n}$ is the invariant distribution of such a queue when $0<\lambda<\mu$.

(ii) In a shop queue, some customers rejoin the queue after having been served. Let $\lambda, \beta \in(0, \infty)$ and $\delta \in(0,1)$. Consider a $\mathrm{M}(\lambda) / \mathrm{M}(\mu) / 1$ queue subject to the modification that, on completion of service, each customer leaves the shop with probability $\delta$, or rejoins the shop queue with probability $1-\delta$. Different customers behave independently of one another, and all service times are independent random variables.

Find the distribution of the total time a given customer spends being served by the server. Hence show that equilibrium is possible if $\lambda<\delta \mu$, and find the invariant distribution of the queue-length in this case.

(iii) Show that, in equilibrium, the departure process is Poissonian, whereas, assuming the rejoining customers go to the end of the queue, the process of customers arriving at the queue (including the rejoining ones) is not Poissonian.