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# Carrboro High School is having a car wash to raise money for a Spring Break trip to Amsterdam….

Carrboro High School is having a car wash to raise money for a Spring Break trip to Amsterdam. Eight students are working at the car wash. Jill and Kelly are on the sidewalk holding signs advertising the car wash. If there are one, two, or three cars getting washed, then the other six students divide themselves up between the cars. If there are four cars, then Jill and Kelly also help, so four cars are being washed by two people each. A car being washed by m students departs at rate m per hour. Cars arrive at rate 12 per hour and enter service with probability 1 – k/4 if k cars are being washed. In short we have the following Markov chain for the number of cars being washed:

(a) Find the stationary distribution. (b) Compute the average number of cars in the system L, and the average time a car takes to get washed W. (c) Use π(0) to compute the expected duration of the busy period E1T0. (d) Find E1T0 by computing ExT0 for x = 1, 2, 3.