lect0414

# lect0414 - IEOR 4106: Introduction to Operations Research:...

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IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2009, Professor Whitt, More Renewal Theory, Chapter 7, Tuesday, April 14 1. Driving Back and Forth A truck driver continually drives from A to B and then immediately back from B to A . Each time he drives from A to B , he drives at a ﬁxed speed that (in miles per hour) is randomly chosen, uniformly in the interval [40 , 60]. Each time he drives from B to A , he drives at a ﬁxed speed that (in miles per hour) is randomly chosen, being either 40 or 60, each with probability 1 / 2. (Going from B to A the random speed has a Bernoulli distribution.) The successive random experiments at the beginning of each trip are mutually independent. (a) In the long run, what proportion of his driving time is spent driving from A to B ? (b) In the long run, what proportion of his driving time is spent driving at a speed of 40 miles per hour? ANSWERS The key idea has nothing to do with this course: You should remember dirt . If you can’t remember this, you are “ dumb as dirt” , to quote Robin Williams in his recording of Pecos Bill (another weird reference). Dirt means D = RT , distance equals rate times time . Here we are given random speeds or rates, but we ask about times. So whatever the distance D , the time T is T = D/R . We must thus average the reciprocal of the rate. That explains the solution. Note that E [1 /R ] is not 1 /E [ R ]. That explains why the two expected times are not equal.

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## This note was uploaded on 01/26/2011 for the course IEOR 4106 taught by Professor Whitt during the Spring '08 term at Columbia.

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lect0414 - IEOR 4106: Introduction to Operations Research:...

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