Question

(10 points) Ben, Max, and Yolanda are at the front of three separate lines in the cafeteria waiting to be served. The serving times for the three lines follow independent Poisson processes with respective parameters 1, 2, and 3. (a) Find the probability that Yolanda is served first. (b) Find the probability that Ben is served before Yolanda. (c) Find the expected waiting time for the first person served.

Answer 1

1/2 ; 1/3 ; 1/6

Explanation:

Given :

Ben ~ exp(1)

Max ~ exp (2)

Yolanda ~ exp(3)

Ben :

X ~ exp(1)

Max:

Y ~ exp(2)

Yolanda :

Z ~ exp(3)

Probability that Yolanda is served first :

P(Z < (X, Y)) = Z /(x + y + z) = 3 /(1 + 2 + 3) = 3/6 = 1/2

Probability that Ben is served before Yolanda :

P(X < Z) = 1/ (1 + 2) = 1/3

Expected waiting time for first person served :

First person served = 1

Total = (1 + 2 + 3) = 6

Expected waiting time; E(x) :

E(x) : 1 / 6