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| {{context|date=January 2011}}
| | Hi, everybody! <br>I'm Norwegian female ;=). <br>I really like Supernatural! |
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| [[File:TFD.PNG|300px|thumb| border|right|text-top|Moving Bottleneck in Triangular Fundamental Diagram.]]
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| In [[traffic flow]] theory, the impact of [[freeway]] [[truck lane]] restrictions is an interesting topic. Intuitively, slow vehicles (e.g. [[truck]]s) will cause queues behind them, but how it relates to the kinematic [[wave theory]] was not revealed until Newell.<ref>{{cite journal|title=A moving bottleneck.|author=Newell, Gordon |journal=Transportation Research Part B: Methodological|volume= 32|issue=8 |pages=531–537 |year=1998}}</ref> Leclercq ''et al'' <ref>{{cite journal |title=Moving bottlenecks in the LWR model: a unified theory.|author=Leclercq, L., Chanut, S., Lesort, J.|journal=Transportation Research Record|volume= 1883|pages=3–13|year=2004}}</ref> did a complete review of Newell's theory. In addition to the [[Simulation modeling|simulation models]] developed by Laval and Daganzo <ref>{{cite journal|title=Lane-changing in traffic streams.|author=Laval, J. A., Daganzo, C. F.|journal=Transportation Research Part B: Methodological|volume= 40|issue=3 |pages=251–264 |year=2006}}</ref> on the basis of [[numerical solution]] methods for Newell's theory to capture the impacts of slow vehicle, Laval <ref>{{cite journal |title=Effects of Geometric Design on Freeway Capacity: Impacts of Truck Lane Restrictions.|author=Laval, Jorge |journal=Transportation Research Part B: Methodological|volume= 43|issue=6 |pages=720–728 |year=2009 |doi=10.1016/j.trb.2009.01.003}}</ref> also mathematically derived the analytical capacity formulas for [[Bottleneck (traffic)|bottlenecks]] caused by single-type of trucks for multi-lane freeway segments.
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| ==Analytical Solution of the Single-type Truck Problem==
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| [[File:MB1.PNG|300px|thumb| border|right|text-top|Truck Trajectory and Induced Traffic States]]
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| Laval's solution could be summarized as follows: Assuming a one-lane freeway segment obeying the triangular [[Fundamental diagram of traffic flow|fundamental diagram]] defined in the figure to the right with free-flow speed ''u'', [[wave velocity]] ''w'' and jam density ''k<sub>j</sub>''. Only one truck type is considered. In this scenario, the normalized capacity ''I'' of the freeway segment is given as：
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| <math>I=\frac{1}{rHC}</math>
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| where ''r'' is the time-mean proportion of trucks in the traffic stream,''C = uwnk<sub>j</sub>/(w+u)''is the capacity of the freeway lane without trucks and ''H'' is the expected value of headway between two consecutive trucks at the location where trucks begin to slow down
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| It can be shown that, by approximating truck arrivals with [[Poisson process]]es, the [[probability density function]] (PDF) of ''H'' is the equation below, in which ''τ'' is defined as the clearance time of the queue induced by the slow-moving truck, ''λ<sub>0</sub>''=''rC'', ''λ<sub>1</sub>''=''rU'' and ''τ''=''L''(''w''+''v'')/''wv''. Note that ''λ<sub>0</sub>'' and ''λ<sub>1</sub>'' refer to the mean truck arrival rate at traffic state ''C'' or ''U'', respectively. In particular, traffic state D, which corresponds to the downstream of the moving bottleneck, is assumed to be equal to the capacity of the unblocked lanes.
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| <math>f_H(h)=\begin{cases}\lambda_1 e^{-h\lambda_1}, & h\le\tau \\ e^{\tau(\lambda_0-\lambda_1)}\lambda_0 e^{-h\lambda_0}, & h>\tau\end{cases}</math>
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| According to Newell's moving bottleneck theory, we have:
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| <math>U=D+(\frac{wvkj}{w+v})</math>
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| Given all the above information, we can conclude that the average truck headway H is
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| ''H=(1-e<sup>-λ<sub>1</sub>τ</sup>)/(λ<sub>1</sub>)+(e<sup>-λ<sub>1</sub>τ</sup>)/(λ<sub>0</sub>)''
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| And the above [[equation]] gives us all the necessary information to solve the normalized capacity ''I''.
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| == References ==
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| {{Reflist}}
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| [[Category:Road traffic management]]
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Hi, everybody!
I'm Norwegian female ;=).
I really like Supernatural!