- #1
ra_forever8
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Comparing Traffic Flow Models: Which One is More Realistic?
In summary, the conversation discusses traffic flow modelling and the equations used to represent traffic density and velocity. Two models are proposed for the velocity, one with a positive constant and the other using logarithmic functions. The realism of each model is compared, with the first model showing a decrease in velocity as density increases and the second model showing a constant velocity until reaching maximum density. The preferred model is not stated.
- #2
CaptainBlack
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grandy said:
For part (a) you need to observe that the flow rate \(f(x,t)\) in vehicles per unit time is \(u(x,t) \rho(x,t)\).
Now you need to show that for (i) and (ii) that \(u(x,t)\le u_{sl}\), then as \(\rho(x,t) \le \rho_{max}\) you will have shown that the flow rate:
\[f(x,t)\le u_{sl}\rho_{max}\]
- #3
CaptainBlack
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grandy said:
The next step is to write down the partial differential equation satisfied by the traffic density. This is derivable from a conservation of mass (or vehicle numbers) argument that you will have seen innumerable times.
CB
- #4
ra_forever8
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CaptainBlack said:
Thank you very much captainBlack. I am very humble with your reply but same time i m lost with your answers provided. Would please go through little deeply and clarify it nicely please. Thnx
- #5
CaptainBlack
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grandy said:
Consider a road element between \(x\) and \(x+\Delta x\) the traffic flow into the element at \(x\) per unit time is \(u(\rho(x,t))\rho(x,t)\) and out at \(x+\Delta x\) is \(u(\rho(x+\Delta x,t))\rho(x+\Delta x,t)\) Therefore the rate of change of car numbers in the element is:
\[\frac{\partial N}{\partial t}=u(\rho(x,t))\rho(x,t)-u(\rho(x+\Delta x,t))\rho(x+\Delta x,t)\]
and so the rate of change of density in the element is:
\[\frac{1}{\Delta x}\frac{\partial N}{\partial t}=\frac{u(\rho(x,t))\rho(x,t)-u(\rho(x+\Delta x,t))\rho(x+\Delta x,t)}{\Delta x}\]
Now take the limit as \(\Delta x \to 0 \) to get:
\[\frac{\partial \rho}{\partial t}=\frac{\partial}{\partial x}u(\rho)\rho\]
CB
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- #6
CaptainBlack
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CaptainBlack said:
In small time interval \(\Delta t\) all the vehicles less than a distance \(u(x,t)\Delta t\) down stream of \(x\) will pass \(x\). The number of vehicles in this stretch of road is \(u(x,t)\rho(x,t)\Delta t\), so \(u(x,t)\rho(x,t)\Delta t\) vehicles pass \(x\) in \(\Delta t\) so the vehicle flow rate at \(x\) is \(u(x,t)\rho(x,t)\) vehicles per unit time.
CB
- #7
ra_forever8
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Thank you very much for your lovely answer. Please explain me what do you mean by Triangle sign x represent. And also in next line below what does Triangle sign t mean?
- #8
ra_forever8
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Would please help me with the last part of this question as well please. At maximum speed limit of 40 m.p.h ...
- #9
ra_forever8
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Traffic flow Modelling
On a stretch of single-lane road with no entrances or exits the traffic density ρ(x,t) is a continuous function of distance x and time t, for all t > 0, and the traffic velocity ) u( ρ) is a function of density alone.
Two alternative models are proposed to represent u:
i)u = u_(SL)*(1- ρ^n/ρ^n_max ), where n is a postive constant
ii) u = u_(SL)* In (ρ_max / ρ)
Where u_SL represents the maximum speed limit on the road and p_max represents maximum density of traffic possible on the road(meaning bumper-to-bumper traffic)
Compare the realism of the 2 models for u above. You should consider in particular the variations of velocity with density for each model, and the velocities for high and low densities in each case. State which model you prefer, giving reasons.
=>
I did for case i) which is u = u_(SL)*(1- ρ^n/ρ^n_max ),
u(ρ) = u_(SL)*(1- ρ^n/ρ^n_max ), for 0<ρ<ρ_max
Since ρ>= 0, cannot exceed u_SL
when ρ= ρ_max , u (ρ_max)= u_SL(1- ρ_max/ρ_max) =0
when ρ=0, u(0)= u_SL(1-0/ρ_max)= u_SL
Also, du/dρ= (- u_SL/ρ_max ) <0, so drivers reduce speed as density increase
Can anyone please help me for case ii) and state which model to choose?
On a stretch of single-lane road with no entrances or exits the traffic density ρ(x,t) is a continuous function of distance x and time t, for all t > 0, and the traffic velocity ) u( ρ) is a function of density alone.
Two alternative models are proposed to represent u:
i)u = u_(SL)*(1- ρ^n/ρ^n_max ), where n is a postive constant
ii) u = u_(SL)* In (ρ_max / ρ)
Where u_SL represents the maximum speed limit on the road and p_max represents maximum density of traffic possible on the road(meaning bumper-to-bumper traffic)
Compare the realism of the 2 models for u above. You should consider in particular the variations of velocity with density for each model, and the velocities for high and low densities in each case. State which model you prefer, giving reasons.
=>
I did for case i) which is u = u_(SL)*(1- ρ^n/ρ^n_max ),
u(ρ) = u_(SL)*(1- ρ^n/ρ^n_max ), for 0<ρ<ρ_max
Since ρ>= 0, cannot exceed u_SL
when ρ= ρ_max , u (ρ_max)= u_SL(1- ρ_max/ρ_max) =0
when ρ=0, u(0)= u_SL(1-0/ρ_max)= u_SL
Also, du/dρ= (- u_SL/ρ_max ) <0, so drivers reduce speed as density increase
Can anyone please help me for case ii) and state which model to choose?
FAQ: Comparing Traffic Flow Models: Which One is More Realistic?
What is traffic flow modelling?
Traffic flow modelling is a mathematical representation of the movement of vehicles on roads or highways. It involves creating a simulation or model that can predict how traffic will behave under different conditions.
Why is traffic flow modelling important?
Traffic flow modelling is important because it helps transportation planners and engineers make informed decisions about road design, traffic management, and transportation policies. It can also be used to identify potential traffic problems and develop solutions to improve traffic flow.
What factors are considered in traffic flow modelling?
There are several factors that are considered in traffic flow modelling, including road geometry, traffic volume, vehicle characteristics, driver behavior, and traffic control devices. Weather conditions and road surface conditions may also be taken into account.
How accurate is traffic flow modelling?
The accuracy of traffic flow modelling depends on the quality of data used and the assumptions made in the model. With accurate data and realistic assumptions, traffic flow modelling can provide reasonably accurate predictions. However, it should be noted that traffic is a complex and dynamic system, so there is always some degree of uncertainty in the predictions.
Can traffic flow modelling be used to predict future traffic patterns?
Yes, traffic flow modelling can be used to predict future traffic patterns. By using historical data and making assumptions about future conditions, traffic flow models can estimate how traffic will behave in the future. However, these predictions should be used with caution as unforeseen events or changes in behavior can affect traffic patterns.