ps3sol - Logistical and Transportation Planning _ Fall 2006...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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 = . 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 . Therefore, the probability that Bo did not have to wait at all for the service but that Alvin did have to wait is: 2 = 0.2696± = (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 8 ( 1 ) . 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 . 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 ) 8 e 2 . Plugging in our known values of = 0.4 customers per minute and 8! = 1 customer per minute. We get that the probability of the event described in part (b) is: 8 ( 1 ) ( 2 ) 8 e 2 = 3.39 10 7 Page 1 of 12±
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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~at l~tf 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 waiting] = = ri(0.83''"0.2) = (0.8)'' 6(0.8)"'(0.2) xi~i,~ Page 2 of 12
Background image of page 2
Lo~istical and Transportation Planning - Fall 2006 (c) kt W he: the number of pasMiengm that leave in one hour bmsuse iel~egarriue when there is no more mom.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 12

ps3sol - Logistical and Transportation Planning _ Fall 2006...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online