We have a queuing system. $($Assume an exponential model.$)$ Its system capacity is two. When there is

no customer in the system, the arrival rate is $10/$min$.$ When there is one customer in the system, the

arrival rate is $8/$min$.$ When there is one customer in the system, the service rate is $10/$min$,$ and when

there are two customers in the system, the service rate is $15/$min$.$

a$)$ Make a transition rate diagram showing the service and arrival rates.

b$)$ Write balance equations for all states.

c$)$ Determine the steady state probability distribution.

d$)$ Determine the average system size and average queue size.