(i) Give the definition of the time-reversal of a discrete-time Markov chain $\left(X_{n}\right)$. Define a reversible Markov chain and check that every probability distribution satisfying the detailed balance equations is invariant.

(ii) Customers arrive in a hairdresser's shop according to a Poisson process of rate $\lambda>0$. The shop has $s$ hairstylists and $N$ waiting places; each stylist is working (on a single customer) provided that there is a customer to serve, and any customer arriving when the shop is full (i.e. the numbers of customers present is $N+s$ ) is not admitted and never returns. Every admitted customer waits in the queue and then is served, in the first-come-first-served order (say), the service taking an exponential time of rate $\mu>0$; the service times of admitted customers are independent. After completing his/her service, the customer leaves the shop and never returns.

Set up a Markov chain model for the number $X_{t}$ of customers in the shop at time $t \geq 0$. Assuming $\lambda<s \mu$, calculate the equilibrium distribution $\pi$ of this chain and explain why it is unique. Show that $\left(X_{t}\right)$ in equilibrium is time-reversible, i.e. $\forall T>0,\left(X_{t}, 0 \leq t \leq T\right)$ has the same distribution as $\left(Y_{t}, 0 \leq t \leq T\right)$ where $Y_{t}=X_{T-t}$, and $X_{0} \sim \pi$.