Problem 1.3 (10 points) Customers arrive at a pie shop according to a Poisson process with rate 10 per hour and are served with rate 6 per hour. Suppose that in addition, each customer has an impatience clock and will leave the line if they are not served before a random time, which has independent exponential distribution of mean 20 minutes (a) Suppose that Alice is rst in line and Bob is in line behind her. Find the probability that Bob is served before he loses patience and leaves. Hint: don't forget the possibility that Alice becomes impatient and leaves before she is served. (b) Can the number of customers in the queue be reduced to a M/ M/ 1 queue with only two parameters, the rate at which customers join the queue and the rate at which they leave (either by being served or impatience)? If so, provide the parameters, and if not explain why not. (c) Does the process have a stationary distribution? Please justify your answer, although rigorous proof is not necessary