Distribution Solutions.pdf

# Distribution Solutions.pdf - Examples Flow/headway...

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Page 1 of 8 Examples – Flow/headway distribution Problem 1 An observer counts 360 veh/h at a specific highway location. Assuming that the arrival of vehicles at this highway location is Poisson distributed, estimate the probabilities of having 5 or more vehicles arriving over a 20-second time interval. Solution: ࠵? = 360 3600 = 0.1 ࠵?࠵?ℎ/࠵? ࠵?࠵? = 0.1×20 = 2 Poisson distribution: ࠵? ࠵? = ࠵? 4 ࠵? 56 ࠵?! In here, ࠵? = ࠵?࠵? = 2 ; So, the probability for a certain number of vehicle arriving using Poisson distribution: ࠵? 0 = 2 8 ࠵? 59 0! = 0.135 ࠵? 1 = 2 ; ࠵? 59 1! = 0.271 ࠵? 2 = 2 9 ࠵? 59 2! = 0.271 ࠵? 3 = 2 = ࠵? 59 3! = 0.180 ࠵? 4 = 2 @ ࠵? 59 4! = 0.090 For 5 or more vehicles arrive over a 20s: ࠵? ࠵? ≥ 5 = 1 − ࠵? ࠵? < 5 = 1 − ࠵? 0 − ࠵? 1 − ⋯ − ࠵? 4 = 1 − 0.135 − 0.271 − 0.271 − 0.180 − 0.09 = 0.053 Problem 2 Consider the traffic situation in Problem 1 (360veh/h). Again assume that the vehicle arrivals are Poisson distributed. What is the probability that the headway between

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Page 2 of 8 successive vehicles will be less than 8 seconds, and what is the probability that the headway between successive vehicles will be between 8 and 10 seconds? Solution: If vehicle arrival follows Poisson distribution, then their headway follows Exponential distribution: ࠵? ℎ ≤ ࠵? = 1 − exp(−࠵?࠵?) So for a headway < 8s when λ = 0.1 : ࠵? ℎ ≤ 8 = 1 − exp −0.1×8 = 0.551 For 8s < headway < 10s, we can do either (1) 1 − [࠵? ℎ > 10 + ࠵? ℎ < 8 ] ; or (2) ࠵? ℎ > 8 − ࠵?(ℎ > 10) ; or (3) ࠵? ℎ < 10 − ࠵?(ℎ < 8) We demonstrate (3): ࠵? ℎ < 10 = 1 − exp −0.1×10 = 0.632 ࠵? 8 < ℎ < 10 = ࠵? ℎ < 10 − ࠵? ℎ < 8 = 0.632 − 0.551 = 0.081 Problem 3 An observer has determined that the time headways between successive vehicles on a section of highway are exponentially distributed and that 65% of the headways between vehicles are 9 seconds or greater. If the observer decides to count traffic in 30-second time intervals, please estimate the probability of the observer counting exactly 4 vehicles in an interval. Solution: The headway follows Exponential distribution, which indicates that the vehicle arrival follows Poisson distribution: ࠵? ℎ ≤ ࠵? = 1 − exp(−࠵?࠵?) ࠵? ࠵? = 6 Q R ST 4!
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