$(20$p$.)$ For the barber shop in question $4,$ assume that at most $2$ customers can stand inside the shop due to the

limited space. Also, assume that customers seeing that the barber shop is full $($that is$,$ there are $4$ customers

inside the shop$)$ do not enter, and they leave.

a$.$ Construct a CTMC$.$ Draw the state transition diagram. Find ${}_n$ and ${}_n$ for each $n.$

b$.$ Write the balance equations.

c$.$ Is this queuing system stable? If so$,$ find the steady$-$state expected number of customers in the shop and

the expected queue length.