How Does Traffic Flow Modeling Impact Single-Lane Road Traffic Dynamics?

[ra_forever8]

On a stretch ofsingle-lane road with no entrances or exits the traffic density r(x,t)is a continuous function of distance xand time t, for all t > 0, and the traffic velocity u(r) is a function of density alone.
Two alternative models are proposed to represent u:

(i) u= uSL (1 – r^n/ r^nmax)
where n is a positive constant
(ii) u= 1/3 uSL In (rmax/r)where uSLis the maximum speed limit on the road and rmax is themaximum density of traffic possible on the road (corresponding tobumper-to-bumper traffic).

a) Evaluate the maximum rate of traffic flow for cases (i)and (ii) above. Show that for both cases the maximum rate of traffic flow isless than uSLrmax and that for case (i)it increases towards this upper limit as nbecomes very large.
b) It is assumed that a model of the form given in case (i)is a reasonable representation of actual traffic behaviour. Apply the method ofcharacteristics to analyse the following situation. A queue of cars is stoppedat a red traffic light on a road for which the maximum speed limit is 40 m.p.h.It may be assumed that the queue is very long and that the road ahead of thelight is empty of traffic. The light turns green and remains green for only 45seconds. If a car which is initially a quarter of a mile behind the light getsthrough before the light changes back to red determine the smallest integervalue that n can have. (Hint: showinitially that the car will not start moving until a certain time after thelight has turned green, and then solve the appropriate differential equationfor the position of the car.)



=> The attempt at a solution

For part (a) you need to observe that the flow rate f(x,t) in vehicles per unit time is u(x,t)ρ(x,t) .

Mow,to show (i) and (ii) that u(x,t)≤u sl , then as ρ(x,t)≤ρ max I will have shown that the flow rate: f(x,t)≤u sl ρ max .

b)Consider a road element between x and x+Δx the traffic flow into the element at x per unit time is u(ρ(x,t))ρ(x,t) and out at x+Δx is u(ρ(x+Δx,t))ρ(x+Δx,t) Therefore the rate of change of car numbers in the element is:


∂N ∂t =u(ρ(x,t))ρ(x,t)−u(ρ(x+Δx,t))ρ(x+Δx,t)


and so the rate of change of density in the element is:


1 Δx ∂N ∂t =u(ρ(x,t))ρ(x,t)−u(ρ(x+Δx,t))ρ(x+Δx,t) Δx


Now take the limit as Δx→0 to get:


∂ρ ∂t =∂ ∂x u(ρ)ρ


(Could please confirm with my answer, I am not sure and unable to move on from there. Please help me)

 

[haruspex]

It's not clear to me whether you have solved part (a).
For part (b), your differential eqn looks ok (apart from some missing division symbols and parentheses). You can substitute for u using the model and expand the derivative using the chain rule.

 

[ra_forever8]

I think I have solved the qs part b). Please don't worry about part b now because i have done it. Would please help with the qs part a).

As i mentioned for part a)
To observe that the flow rate f(x,t) in vehicles per unit time is u(x,t)ρ(x,t) .
Now,to show (i) and (ii) that u(x,t)≤u sl , then as ρ(x,t)≤ρ max
I will have shown that the flow rate: f(x,t)≤u sl ρ max
Would please help to complete this qs part a after that.

 

[haruspex]

For max flow rate, you can treat ρ (=r) as the same for all x and t. So it simply becomes a matter of maximising f wrt r, given f(r) = r*u(r). That's enough to find the max flow rate as a function of rmax and n in case (i). But I don't understand the equation for u(r) in case (ii). Is In() supposed to be ln()? But that doesn't make sense because as r tends to zero u will tend to infinity instead of levelling out at uSL. And is that really a factor of 1/3 out the front?

 

[ra_forever8]

For case ii) Yes its correct, it is supposed to be In (r_max/r) and it is factor of 1/3 infront. Would please explain me clearly for case (i) again as i confused. Please try to solve the solution for case i) , it will easier for me to understand.

 

[haruspex]

For (i), f(r) = r*u(r) = r*u_SL (1 – (r/ r_max)ⁿ)
How do you find the r that maximises f? Find where df/dr = 0, right?
For (ii), I'm not familiar with any standard function In(). What is it?

 

[ra_forever8]

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?

 

[TSny]

For case (ii), what do you get for u as ρ→0? (Is that the natural log function in case ii?)

 

[haruspex]

This appears to be a continuation (different question, same scenario) of unfinished thread https://www.physicsforums.com/showthread.php?t=662238.
ra_forever8, I already asked you to explain this "In()" function on that thread. You said it was not ln() (and I agree that wouldn't make sense), but still haven't told me what it is. Do you know? If not, it's unlikely anyone will be able to help you.

 

[ra_forever8]

Sorry,it is In() something there.
For case ii) u = u_(SL)* In(ρ_max /ρ)
I know for case i) now. please help me for case ii)

 

[ra_forever8]

For Tsny,
In case ii) u = u_(SL)* In (ρ_max / ρ) ,if ρ→ 0, then u = u_(SL)* In (ρ_max/0) which gives math error.

 

[haruspex]

Sorry,it is In() something there.

I have no idea what you mean by that.

 

[ra_forever8]

it is natural logarithm like log

 

[haruspex]

it is natural logarithm like log

OK, I thought you denied that before.
But as you can see it doesn't make sense. It will exceed the speed limit whenever r_max > e*r.
Something like this would work: u = u_SL - ln(1+(e^u_SL-1)r/r_max)

 

[ra_forever8]

what e there? is the natural logarithm like log?
rmax is same as ρ_max and r as ρ.
if it will exceed the speed limit whenever rmax > e*r then density will decrease right?

 

[haruspex]

what e there? is the natural logarithm like log?
rmax is same as ρ_max and r as ρ.

Yes, I mean e as in the base of natural logarithms. Btw, where you wrote In() in your posts, if you mean natural logarithm it's written ln() - that's a lowercase L not an uppercase I. Writing In() confuses everyone.
Before I go on, I want to correct what I wrote in may last post. In the second thread you started on this you wrote u= u_SL ln (r_max/r). In that case, what I wrote about exceeding speed limit was correct: when r_max > e*r, u > u_SL ln (e) = u_SL. I.e. u will exceed the speed limit. But in this, the original thread you specified a factor one third: u= 1/3 uSL ln (r_max/r). That changes things, but not much. Now the speed limit will be exceeded if r_max > e³*r. Maybe this is just part of the answer to "is the model realistic?"
So let's just accept u= 1/3 uSL ln (r_max/r) and see where it takes us.
In the OP you had questions (a) and (b) to answer. In your repost (now a post within this thread) you asked a different question. Does that mean you have completed (a)(ii) and (b)(i)? I will assume so.

if it will exceed the speed limit whenever rmax > e*r then density will decrease right?

Do you mean, given a a running model in which the speed limit is being exceeded, will the speed decrease over time? Not necessarily. If the density is the same everywhere then it will stay that way. But that's not what this question is asking. It is asking whether increasing r decreases speed.
So, what do you get for du/dr in model (ii)?