(a) Give the definition of a birth and death chain in terms of its generator. Show that a measure $\pi$ is invariant for a birth and death chain if and only if it solves the detailed balance equations.

(b) There are $s$ servers in a post office and a single queue. Customers arrive as a Poisson process of rate $\lambda$ and the service times at each server are independent and exponentially distributed with parameter $\mu$. Let $X_{t}$ denote the number of customers in the post office at time $t$. Find conditions on $\lambda, \mu$ and $s$ for $X$ to be positive recurrent, null recurrent and transient, justifying your answers.