Two servers A and B in a fast-food restaurant each start their first customers at the same time. After finishing her second customer, A notices that B has not yet finished his first customer. A then chides B for being slow, and B responds that A just got a couple of easier customers. Suppose that we model all service times, regardless of the server, as i.i.d. random variables having the exponential distribution with parameter 0.4. Let X be the sum of the first two service times for server A, and let Y be the first service time for server B. Assume that you can simulate as many i.i.d. exponential random variables with parameter 0.4 as you wish.
a. Explain how to use such random variables to approximate Pr(X < Y).
b. Explain why Pr(X < Y) is the same no matter what the common parameter is of the exponential distributions. That is, we don’t need to simulate exponentials with parameter 0.4.We could use any parameter that is convenient, and we should get the same answer.
c. Find the joint p.d.f. of X and Y, and write the two-dimensional integral whose value would be Pr(X < Y).