Chapter 5 _2_

# Chapter 5 _2_ - Queuing Theory and Flow Analysis The...

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Queuing Theory and Flow Analysis The formation of traffic queues during congested periods is a source of considerable time delay and results in a loss of highway performance.

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2 Outline s Traffic Flow Models s Arrival/departure Patterns s Poisson Model s Queuing Theory s D/D/1 Queuing s M/D/1 Queuing s M/M/1 Queuing
3 Queuing Theory

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4 Queuing Theory Source: http://www.sierraclub.org/sprawl/report04/commuting.asp
5 s Flow arrival pattern in time: s Uniform (like e.g. 360 veh/h) Arrival/Departure Patterns Time Uniform Distribution f(t)

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6 s Flow arrival pattern in time: s Nonuniform (random process e.g. Poisson model) Arrival/Departure Patterns Time f(t)
7 Poisson Model ! ) ( ) ( n e t n P t n λ - = P(n) Volume (# of vehicles) P(n): probability of having n vehicles λ : average vehicle flow per unit time t= duration of the time interval

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8 Example The average arrival rate is 240 vehicles/hr at a roadway section. Calculate the probability of having exactly 0, 4, 8, and 12 vehicles in a 60- second interval.
9 3600 q = λ Using Poisson model for distribution of the time intervals between the arrivals of vehicles: ! ) 3600 / ( ) ( 3600 / n e qt n P qt n - = Veh/hr Veh/s Poisson Model

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Probability of having no vehicles arrive in time interval “t”: P(0) 3600 / ) 0 ( qt e P - = This is equivalent to the probability of a vehicle headway (h) being greater than or equal to the time interval “t”. 3600
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Chapter 5 _2_ - Queuing Theory and Flow Analysis The...

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