(a) Give the definition of an $M / M / 1$ queue. Prove that if $\lambda$ is the arrival rate and $\mu$ the service rate and $\lambda<\mu$, then the length of the queue is a positive recurrent Markov chain. What is the equilibrium distribution?

If the queue is in equilibrium and a customer arrives at some time $t$, what is the distribution of the waiting time (time spent waiting in the queue plus service time)?

(b) We now modify the above queue: on completion of service a customer leaves with probability $\delta$, or goes to the back of the queue with probability $1-\delta$. Find the distribution of the total time a customer spends being served.

Hence show that equilibrium is possible if $\lambda<\delta \mu$ and find the stationary distribution.

Show that, in equilibrium, the departure process is Poisson.

[You may use relevant theorems provided you state them clearly.]