- #1
ra_forever8
- 106
- 0
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Comparing Traffic Flow Models: Which One is More Realistic?
Click For Summary
Discussion Overview
The discussion revolves around comparing two traffic flow models to determine their realism in representing traffic velocity as a function of density. Participants explore the mathematical formulations of the models and their implications for traffic behavior, focusing on theoretical aspects and conceptual understanding.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- Some participants express confusion about how to start analyzing the traffic flow models and seek clarification on the mathematical formulations.
- One participant notes that the flow rate \(f(x,t)\) can be expressed as \(u(x,t) \rho(x,t)\) and discusses conditions under which this flow rate is bounded by \(u_{sl}\rho_{max}\).
- Another participant describes the derivation of the partial differential equation for traffic density based on conservation of mass, indicating a common understanding of the foundational principles.
- Participants propose two models for traffic velocity as a function of density:
- Model i: \(u = u_{SL}(1 - \frac{\rho^n}{\rho_{max}^n})\)
- Model ii: \(u = u_{SL} \ln(\frac{\rho_{max}}{\rho})\)
- One participant analyzes Model i, noting the behavior of velocity at different densities and the negative derivative indicating that speed decreases as density increases.
- Another participant requests assistance in analyzing Model ii and asks for opinions on which model might be preferred.
Areas of Agreement / Disagreement
Participants express varying levels of understanding and confusion regarding the models and their implications. There is no consensus on which model is more realistic, as some participants have only analyzed one model and seek further input on the other.
Contextual Notes
Some participants have not fully explored the implications of the models, and there are unresolved mathematical steps in the analysis of Model ii. The discussion reflects differing levels of familiarity with the concepts involved.
Who May Find This Useful
Readers interested in traffic flow theory, mathematical modeling in transportation, or those seeking to understand the implications of different traffic density models may find this discussion relevant.
- #2
CaptainBlack
- 801
- 0
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
- 801
- 0
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
- 106
- 0
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
- 801
- 0
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
- 801
- 0
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
- 106
- 0
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
- 106
- 0
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
- 106
- 0
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 positive 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 positive 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?
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