This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: IEOR 4701: HMWK 4 1. You arrive at the West 96th Street Subway station to go Downtown. Suppose Express trains arrive according to a Poisson process at rate 4 (per hour), and independent of this, Local trains arrive according to a Poisson process at rate 7 (per hour). You decide to take the first train (whatever it is) that arrives. (a) What is the expected number of express trains to arrive during the 2 hour interval 2 4PM? SOLUTION: E ( N 1 (4) N 1 (2)) = E ( N 1 (2)) = 1 2 = 4(2) = 8. (Stationary increments.) (b) What is the variance of the total number of trains (regardless of type) to arrive during the 2 hour interval 2 4PM? SOLUTION: The superposition is a Poisson process at rate = 1 + 2 = 11. V ar ( N (2)) = E ( N (2)) = 2 = 22. (c) What is the probability that you take an Express train? SOLUTION: 1 / ( 1 + 2 ) = 4 / 11. (d) What is the expected amount of time you wait for a train to arrive? SOLUTION: Z = min { X,Y } where X exp ( 1 ) (first arrival time of an express) and (indepen dently) Y exp ( 2 ) (first arrival time of a local). Thus Z exp ( ); E ( Z ) = 1 / = 1 / 11. (e) Given that you took an Express train (it arrived first), what was the expected amount of time you waited for it to arrive? SOLUTION: E ( Z  X < Y ) = E ( Z ) = 1 / 11. ( Z is independent of which of the two is the minimum.) (f) What is the probability that you must wait longer than 12 minutes? SOLUTION: 12 minutes is 1 / 5 an hour. P ( N (12) = 0) = e (1 / 5) = e 11 / 5 . (g) What is the probability that exactly 3 express trains arrive before the first local?...
View Full
Document
 Spring '10
 KarlSigma

Click to edit the document details