(a) What does it mean to say that a continuous-time Markov chain $X=\left(X_{t}: 0 \leqslant\right.$ $t \leqslant T$ ) with state space $S$ is reversible in equilibrium? State the detailed balance equations, and show that any probability distribution on $S$ satisfying them is invariant for the chain.

(b) Customers arrive in a shop in the manner of a Poisson process with rate $\lambda>0$. There are $s$ servers, and capacity for up to $N$ people waiting for service. Any customer arriving when the shop is full (in that the total number of customers present is $N+s$ ) is not admitted and never returns. Service times are exponentially distributed with parameter $\mu>0$, and they are independent of one another and of the arrivals process. Describe the number $X_{t}$ of customers in the shop at time $t$ as a Markov chain.

Calculate the invariant distribution $\pi$ of $X=\left(X_{t}: t \geqslant 0\right)$, and explain why $\pi$ is the unique invariant distribution. Show that $X$ is reversible in equilibrium.

[Any general result from the course may be used without proof, but must be stated clearly.]