Attempting to model traffic flow after front car decelerates

[cmkluza]

So I'm trying to create a very simple mathematical model for a very idealized theoretical situation of traffic flow. In this situation I'm considering vehicles to be points (no length) each going the same speed with the same distance between them. The vehicle in front (vehicle 1) slows down and as a result each vehicle behind it slows down by the same amount, however there is a lag/reaction time in-between each consecutive vehicle slowing down. All I've got right now is a jumbled mess of variables in my head as I try to create a general model for this. I'll define them as follows:

• Vehicle Speed: $v = \frac{dx}{dt}$
• Vehicle Distance: $x$
• Vehicle Speed after Deceleration: $v - Δv = \frac{dx}{dt} - \frac{Δx}{Δt}$ (Is there perhaps a different way of writing this that's more useful? Not sure why but I feel like there is. Probably not.)
• Lag/Reaction Time: $t$

I feel like some of the variables I've declared could be declared in a different more comprehensible way, and if so, please suggest how I should do it.

That aside, does anyone have any ideas for how I can start to model what I'm trying to model? I've also created a simple graphic that might be helpful, you can ignore the subscript on the vehicle distance variable.

I feel like I might just be lacking the initial intuition, as might be obvious by the jumbled manner in which I've put this post together. If anyone can give me any suggestions on creating this initial model I'd appreciate it!

 

[billy_joule]

What exactly are you trying to find out?

https://en.m.wikipedia.org/wiki/Traffic_wave

 

[cmkluza]

What exactly are you trying to find out?

https://en.m.wikipedia.org/wiki/Traffic_wave

It seems that I never specified what my end goal was, so thanks for pointing that out. I am currently trying to model the traffic wave of this situation, the wave that propagates backwards as a result of the initial slowing down of a vehicle towards the front.

That link didn't seem to have any of the mathematical development for models of traffic waves, so do you know of any other sources I could use to gain an understanding of traffic waves and the mathematical development for models of them? Or in a more limited sense, do you have any clues as to how I could develop such a model for the theoretical situation I have here?

Thanks for your response!

 

[S.G. Janssens]

In the topic How do differential equations relate to traffic flow? @lavinia mentioned the book by Whitham. Was that not useful? (This is not meant as criticism, but I was just curious whether you managed to have a look at it.) In the same topic @pasmith also gave a link to a book by the highway administration, but at the moment that link doesn't seem to work.

 

[cmkluza]

In the topic How do differential equations relate to traffic flow? @lavinia mentioned the book by Whitham. Was that not useful? (This is not meant as criticism, but I was just curious whether you managed to have a look at it.) In the same topic @pasmith also gave a link to a book by the highway administration, but at the moment that link doesn't seem to work.

I was actually able to find a PDF of the Whitham book, and it definitely looks helpful and like what I'm looking for. Perhaps it's a testament to my current skill (or rather lack thereof) in mathematics, but I couldn't quite understand Whitham's developments in Chapter 3.

I'm still reading through the highway administration's work, and I've just gotten around to Chapter 5, which seems promising. I think I'll be using ideas from this chapter on Simple Continuum Models for my paper. Evidently I'm stuck on a bit of it at the moment. Hate to ask unrelated things here, but on the second page while deriving the conservation equation, the monograph states that density is equivalent to the negative of change in vehicles over a segment of time:

\Delta k = \frac{-\Delta N}{\Delta x}

Could you possibly explain why it is that they put this value as negative?

 

[S.G. Janssens]

Also, I'm not familiar with traffic flow modelling, but in

Vehicle Speed after Deceleration: $v - Δv = \frac{dx}{dt} - \frac{Δx}{Δt}$ (Is there perhaps a different way of writing this that's more useful? Not sure why but I feel like there is. Probably not.)

I find it confusing that you seem to use $\Delta v = \frac{\Delta x}{\Delta t}$, because when $\Delta t$ is small, this fraction just approximates $v(t)$.

I would say that if the velocity at time $t$ is $v(t)$, after which the car is subject to deceleration $-|a|$ over a time interval $[t, t + \Delta t]$, then the velocity at time $t + \Delta t$ is
$$
v(t + \Delta t) = v(t) - \int_t^{t + \Delta t}{|a(s)|\,ds}
$$
Now, when $\Delta t$ is small, $v(t) \approx \frac{\Delta x}{\Delta t}$ and $\Delta v = v(t + \Delta t) - v(t) \approx -|a(t)| \Delta t$.

 

[S.G. Janssens]

I was actually able to find a PDF of the Whitham book, and it definitely looks helpful and like what I'm looking for. Perhaps it's a testament to my current skill (or rather lack thereof) in mathematics, but I couldn't quite understand Whitham's developments in Chapter 3.

No problem, you are just learning and finding your way around. Were you already familiar with the method of characteristics?

Hate to ask unrelated things here, but on the second page while deriving the conservation equation, the monograph states that density is equivalent to the negative of change in vehicles over a segment of time:

\Delta k = \frac{-\Delta N}{\Delta x}

Could you possibly explain why it is that they put this value as negative?

I think it is just because it is assumed earlier that $N_1 > N_2$, so as cars move from station 1 to station 2 there is an increase in the density, i.e. $\Delta k > 0$.

Incidentally, in post #6 of the other topic the same continuity equation is derived in a way that I find somewhat clearer.