 Solution

Put $K=N+s$, the total capacity. The customer count is an M-M-s-K queue, hence a birth-death process on $\{0,1,\dots ,K\}$. Its nonzero transition rates areThe first rate is zero at $K$ because an arrival finding the shop full is rejected. The second rate is the number of busy servers times the individual service rate.

Write $\rho =\lambda /\mu$. The detailed balance for a birth-death process recursion isThereforewhereThis is the stationary distribution of an M-M-s-K queue.

The state space is finite, and every state communicates with every other through successive births and deaths because $\lambda ,\mu >0$. The chain is therefore an irreducible Markov chain, so its invariant distribution is unique. Finally the adjacent-state identitiesare detailed balance, while all nonadjacent rates vanish in both directions. Part a now shows that the queue is reversible in equilibrium.