There are two tennis courts. Pairs of players arrive at rate 3 per hour and play for an exponentially distributed amount of time with mean one hour. If there are already two pairs of players waiting, new arrivals will leave. (a) Find the stationary distribution for the number of course occupied. (b) Find the rate, at which customers enter the system. (c) Find the expected amount of time a pair has to wait before they can begin playing.
I was able to solve part a and get π(4) = 81/203, π(3) = 54/203, π(2) = 36/203, π(1) = 24/203, π(0)= 8/203.
After that, I got stuck on parts b and c. If someone could help me that would be great.
Try Little's Formula. This is 4.23 from Durrett's book.
For part b, we already got the stationary distribution, and players enter the system only when there are less than 4 people in the system. Conditioned on that there are less than 4 people in the system, we can multiply the this probability with the rate of arrival to obtain the rate of influx into the system. For part c, we can use previous results and Little's Formula to derive the waiting time.