00q1sol

# 00q1sol - Urban OR 2000 Quiz 1 Solutions Prepared by JTH...

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Unformatted text preview: Urban OR 2000: Quiz 1 Solutions 11/10/2000 Prepared by JTH Problem 1 a) Î» Â½ Î» Î» Î» 0,1 0,0 1,0 1 taxi 1 passenger waiting waiting Ï€ 0,1 = Ï€ 0,0 1 2 Ï€ 0,0 = Ï€ 1,0 Ï€ 0,1 + Ï€ 0,0 + Ï€ 1,0 = 1 âˆ´ Ï€ 0,1 = 2 5 ; Ï€ 0,0 = 2 5 ; Ï€ 1,0 = 1 5 b) The chance that a passenger arriving at the taxi stand actually leaves it in a cab is: Ï€ 0,1 + 1 2 Ï€ 0,0 = 3 5 c) The chance that a taxi stops is: Ï€ 0,0 + Ï€ 1,0 = 3 5 This is the same as b) because Î» T = Î» P which implies that taxis and customers reach the stand at the same rate. Because, over the long run, the number of cabs that leave with customers is the same as the number of customers who leave in cabs, the FRACTION of arrivals that leave happily must be the same for taxis and customers. d) To find the average waiting time for passengers who leave by taxi, determine the expected waiting time for passengers and divide by the chance that an arriving passenger leaves by taxi: 1 1 1 Ï€ Ï€ 0,1 + Ï€ 1,0 3 3 Î» 5 ( ) Ï€ 0 + 0,1 1,0 Î» Î» 5 1 = = e)...
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00q1sol - Urban OR 2000 Quiz 1 Solutions Prepared by JTH...

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