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ps3sol - Logistical and Transportation Planning Fall 2006...

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Logistical and Transportation Planning _ Fall 2006± Problem Set 3 Due: Thursday, October 19 Problem 1 (a) If Bo did not have to wait, then he arrived while the system was empty. For an M/M/1 queueing system, the probability to be in the state with no customer is: λ P = 1 ρ with ρ = . 0 µ If Alvin had to wait, then Bo was still in service when he arrived, since no customer can go into the building between Bo and Alvin’s arrivals. The probability that Bo’s service is longer than 2 minutes is e 2 µ . Therefore, the probability that Bo did not have to wait at all for the service but that Alvin did have to wait is: λ 2 µ = 0.2696± = µ (b) The probability that Bo finds 8 customers is the steady state probability of their being 8 customers in an M/M/1 system at a random point in time. This is just ρ 8 ( 1 ρ ) . If only one customer is in service when Alvin arrives, then it must be Bo. This means that in the two minutes between Bo and Alvin’s arrivals exactly 8 departures must have occurred from the system. Since we are talking about the departure process of customers already in the system we can think of them as departing according to a Poisson process of rate µ . Therefore the probability of exactly eight customers leaving during the two minutes is the probability of having 8 poisson occurances in two minutes which is given by: ( 2 µ ) 8 e 2 µ . Plugging in our known values of λ = 0.4 customers per minute and 8! µ = 1 customer per minute. We get that the probability of the event described in part (b) is: ρ 8 ( 1 ρ ) ( 2 µ ) 8 e 2 µ = 3.39 10 7 8! Page 1 of 12±
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Logistical and Transportation Planning - Fall 2006 Problem 2 By Arwxiniz Ifigol$sso~ '93 The stale lransitiflth diagram is I [in# . i taxi; and j pasengors waiting 1 1 Note that, if there are taxis aait.ing. t,hen no pItSseagers will have to wait, arrd if there are passerlgers waiting, it m ~ a t l ~ t f because there are no taxis available. Thl~s, states that have 11ol.h i > 0 ant1 j > O are not. po~;.-ihlt. The balaric~ erll~a t ions are 0.2, ~ 0 , 2 (0.8)(0.2), TO,, 2 /0.8)~~(0.2) lio,~ = = = (0.8) (0.2), q~ = for i = 0,1,2,. . . lJO w m E[# of taxis waiting] = = ri(0.83''"0.2) = (0.8)'' 6(0.8)"'(0.2) x i ~ i , ~ Page 2 of 12
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Lo~istical and Transportation Planning - Fall 2006 (c) k t W he: the number of pasMiengm that leave in one hour bmsuse iel~eg arriue when there is no more mom.
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