The Problem of Traffic: A Mathematical Modeling Journey
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- เผยแพร่เมื่อ 14 มิ.ย. 2024
- How can we mathematically model traffic? Specifically we will study the problem of a single lane of cars and the perturbation from equilibrium that occurs when one car brakes, and that braking effect travels down the line of cars, amplifying as it goes along, due to the delayed reaction time of the drivers. The ultimate phenomena we would like to predict is that stop-and-go behaviour where cars don't just travel at a constant speed in rush hour, but alternate between braking and accelerating. However, when building this model our inputs are very microscopic considerations about how an individual car brakes or accelerates based on the following distance and relative velocity of the car ahead of it.
This video also aims to be an introduction to broad themes in mathematical modelling of real world problems, where we define a problem, choose the inputs to the system we will consider, make assumptions, build the model, and finally assess the model. Finally, the real piece of mathematics we are going to get out of this are called differential-delay equations, and I'll show you a bit about how to solve such equations at the end.
This video is part of the second iteration of the Summer of Math Exposition, hosted by @3blue1brown and @LeiosLabs . There are so many great videos in this in this exposition so definitely check them out by using the hashtag #SoME2.
0:00 The Challenge of Traffic
0:28 #SoME2
0:53 The Modelling Process
1:27 Defining the Problem
2:04 Choosing Which Variables to Consider
4:03 Making Assumptions
5:46 Building the Microscopic Model for Each Car
9:56 Macroscopic Equilibrium
10:34 The Relationship between Density and Velocity
16:23 Maximizing Flux and the Optimal Oensity
20:33 Modelling a Sequence of Cars
24:07 Modelling the First Car
26:05 Full Model: A Differential Delay System
27:06 Assessing the Model Graphically
29:33 Assessing the Model Qualitatively
31:45 Solving Differential Delay Systems
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This must be supported. A series on mathematical modelling with different topics would be a great resource, in my opinion. Thank you for making this video.
This is really cool! I like how this gives some variety for people like me who are more pure math inclined, to see from start to finish an example of a model studying something in the real world. Definitely wish I had done at least a little bit of that in school!
I was the same way, pure math first and seeing some applied math later and I really appreciate it now
Really fun video! I feel like mathematical modeling is something that doesn’t get enough respect in the online math community. I personally did a “math modeling competition” (specifically the “M3 challenge”) recently and was shocked at how it’s in many ways a completely different skill set than solving a pure math problem, and in many ways I found it significantly harder.
Make sure you check out the many other cool videos using the #SoME2 hashtag, so many good ones! I had so much fun making this video for the Summer of Math Exposition 2, thanks to everybody who has been part of this experience:)
Errata:
1) at 5:33 the +L in the assumption should be a -L
Video will help me in solving real life problems I saved it .
I teach DE, and I would love to use this as a group project. Is there any way you could post your MatLab code for this project?
Sometimes doing math in traffic is the only way I stay sane also
Cool, you can actually concentrate on something ,usually one gets so frustrated during traffic .
Please come to India
@@whykoks ikr
As a pure mathematician, differential equations as mathematical modeling tools have always been a danger zone for me. I appreciate this breakdown. Great work, and thank you!
I know what you mean, ha! I didn’t say anything about existence and uniqueness so the pure math people won’t be happy lol
I found this video really interesting! Towards the tail end of my degree in mechanical engineering, I actually took a grad course in modeling energy systems and the methodology was very similar to what you describe in the video. The only difference was (because we're engineers, and not necessarily mathematicians) once the math got to the point of having to solve complex systems of differential equations, we typically defaulted to using some sort of numerical method to approximate the solution and develop a program to get a solution (much like you did with MATLAB, except we probably would've done it much sooner. Or I would've. My math skills have waned over the years, admittedly).
Only thing I can add for discussion, which I can't remember if you discussed in the video or not: once you reach the limits of your ability to exactly solve the differential equations that "exactly" model reality, there are two approaches: find an exact solution to an approximate model (simply the equations until you can solve them to get an exact numerical answer to equations which may or may not reflect reality), OR find an approximate answer to a more exact model (use numerical methods like finite difference to get an approximate answer to a model which more closely reflects reality but has some error introduced by those numerical methods and not the model itself).
When one's developing a model, they need to be careful which approach they use. In this video, I'd say you demonstrated both approaches very well with the equilibrium and perturbation approaches, and even showed how doing both might give you more insight than just using one or the other. I certainly will be rewatching this video over and over and hope you make many more like it. I'll also send it to my old professor and see what he thinks.
What an elegant model! As an extension, two revisions could be made to make the model capture more realism: (1) instead of assuming all cars have the uniform length, drawing car length from a probability distribution from empirical data, and (2) instead assuming the same car velocity, draw it from a certain distribution.
Absolutely!
Yes, I was thinking the exact same thing. I think the reason overall traffic velocity decreases with increasing density is because of this variance. You're taking the miminum of a larger set of random variables.
This extention to his model would test that hypothesis.
At that point you might as well use distributions to form a queuing model and get a better model in general… You can just have a list of harmonic means of vehicle velocities (i.e. MPH) and randomly select one or just use the mean of the harmonic means (i.e. total avg speed) as the baseline average for the computation
Nice video!! Personally, I prefer the quote "All models are wrong, some models are useful", meaning that no model represents the real world perfectly, but some models can allow us to understand how something works
I used this modeling for my optimization course. I change the dynamical model a bit and solved for optimized initial spacing of cars and green light time to reach maximum flux in a red light traffic.
What an incredible video, earned a subscriber. I love how you're able to take difficult sounding concepts and turn them into easy, understandable material, as well as giving some good general tips
on the practice of mathematical modelling. Not once in the video was I bored, disinterested, or distracted!
As for critiques, I will say that the audio occasionally peaks which makes the sound quality noticeably less smooth, also there's quite a lack of audio other than your voice, adding some extra music in the background or something would make it feel a lot more lively.
We don't need "extra music".
Amazing video! I used to love messing with differential delay models during my EE degree. I never did traffic, but I did some more curiosity-driven equations wrt signal analysis
I used to do timing of packages across a fixed length scale on a conveyor line to maximize weighing time and throughput in packages per minute. Faster belt speeds got more packages through but less time on the scale yielding a sharp peak for optimal speed. I've often thought of this as being analogous to traffic flow. Also electricity is electrons moving to "holes" thus a flow of "holes".
Cars can only move into holes, but speeders fill the holes and cause those amplified ripples that slow traffic or cause accidents. Worse is the speeding of traffic on on ramps that force stops at the end by not matching traffic speed and yielding the right of way to existing traffic. This causes more ramps cars to force their way in. This is a positive feedback loop short circuiting the flow on the main highway.
Interesting timing. This past weekend I was entering a construction area where two lanes merged into one. I was able watch with horror as a great big pickup weaved through traffic as everyone else was slowing down.
Luckily, when (not if) the truck crashed, they swerved to avoid a cement barrier and hit the guard rail. They caused harm to no one but their vehicle.
I spent the rest of the drive thinking about how cars might communicate with each other to eliminate or reduce this sort of behavior. There is the potential for a lot of interesting communication methods... but for they most part they depend on not having any 'bad actors' in the mix.
great analysis, that was a fun watch. I like these sort of videos where we are doing some modeling of a very arbitrary ordinary and typical problem.
hey. this video was really fascinating. i'm an audio person tho so i would like to tell you about compressors. they reduce dynamics in a signal without distorting it too much. cause atm you are basically clipping to prevent some parts of your recording to be too quiet but that creates lots of distortion and distracts audio people like me a bit.
edit: btw i'd like to suggest using a lowpass filter to simulate the delayed response time of human reactions. that is usually a more natural way to soften a signal than just delaying it completely, because it also makes the transition between reacting and not reacting a bit softer
I've heard it said that all models are wrong, but some models are useful.
I did a couple of classes on computational modelling for physics, so this felt very familiar. When you got to the table at the end, I found myself saying, "Well, this is just Euler's method" just before you said it. Clearly I at least remembered something!
(Though given the oscillatory behaviour, this might lend itself better to something like a leapfrog.)
Yes. Same in molecular modelling. Problems arise when people either forget, or just ignore, the initial assumptions. Once someone has built a model it can become very easy to ignore the fact that the initial assumptions are simply rendering the model not useful. Problems can multiply when some influential human (the boss or a politician) decides they are going to use the model to win an argument and just do what they already wanted to anyway. A competent modeller can easily tweak the model to produce a desired outcome.
To be fair, he does draw attention to the problem with assumptions.
Thank you. I’ve been wondering about this for a long time but never taken the trouble to model it.
Best professor I’ve ever had. Great video!
So you confirm my ideas how every driver could make less traffic problems: reduce reaction time (control situation, react proactively) and at the same time avoid big accelerations, which result into oscillations and crashes.
Thanks Dr. Trefor, that was truly what we can call a masterpiece. Presenting a relatively complicated cases study in a such oversimplified way is a thing to be proud for. Thanks a lot and keep it up for such interesting videos.
You're very welcome!
This reminded me of my undergraduate project, it was nice to see that the steps you followed start to finish was similar to what I had done at the time!
This is exciting! I've thought about this exact problem before when commuting to work, and even sat down with a notebook one time to try and figure it out, nice to see my idea actually fleshed out
I have a modeling project due for one of my courses and I struggled to wrap my head around what I needed to do. 😣 This video helped clear up my confusion and give me a much better idea of how to do the project. 👍 Thank you so much! 🥰 I feel much more confident that I can complete my project correctly thanks to your video! 😄
Brilliant. This is the kind of modeling and problem solving and I love doing as an engineering student
Can’t wait to show this around in rushour later
My roommate in college specialized Transportation Engineering. I always wondered what type of math models could be used in that field. Really enjoyed the video.
I also submitted a SoME2 and hope that I can get as good at presenting math topics as you are. I think it's great that some of the bigger math youtubers are bringing more awareness to it this year.
Its just awesome , ive been reading on this exact eg in a book not more than an hr earlier to having this vid pop up :D
Thank you for having it online
Interesting. This topic was treated in a Scientific American article some years ago, perhaps in a Martin Gardner column. A real-world example was given, of optimizing traffic throughput. New York traffic engineers improved the flow rate through the Holland Tunnel, by reducing the speed limit. Prior to this, traffic was operating to the right of the optimal density curve. When speed limit was reduced, following distances shrank, and the net flow rate improved. Not an obvious solution, without a traffic model.
Also discussed was the existence of standing waves in density and velocity, as initial perturbations cause persistent interruptions that occur long after the initial perturbation occurs. This explains the periodic stop-and-go behavior in some traffic jams, long beyond the initial problem.
It's not the separation distance that increases the flow, it's the elimination in the indiviidual changes in speed, this creates a higher over all average speed, the increased density is a bonus.
I think the main thing you need to work on is checking your levels so you don't oversaturate your mic and you keep the same loudness level throughout the video. But otherwise, it was great, I loved the engaging style of your narration and the video of you which helps us ride the ups and downs of your dynamic narration style. :)
Thanks for the note!
Hey Dr. Trevor, you should really look into better audio recording. The voice is very distorted, like it is overdriving your mic or being clipped somewhere in the signal chain.
Love your videos.
Hello! Thanks for all your input! I solved it but I had to change something in the final equation (26:31) because it did not reproduce the trivial solution. If you impose the disturbance z1(t)=0, the solution zi(t)=0 (or vi(t)=ve) is not achieved with the proposed equation. I think the "-v" term should be removed and rho_max is actually rho_equilibrium =(1/(L+d)).
This is great...thanks for sharing this. I am linking your Calc 1 videos to my course.
man I always wondered how to mathematically model traffic!
Driving wise, the trick is to change speed as little as possible.: Accelerate slowly and start braking lightly early also. This kind of driving also progresses backward and the car behind you, can drive even more smoothly.
I feel, one of the powerful uses of a model like this is to explore what happens when human behaviour which can be changed, or trained, *is* changed. The observation in the video is that reaction time is a big factor in accidents happening: educating people not to drive drunk, or tired, or distracted on the phone is most definitely a public good (not to mention personally good for those who avoid getting injured because they paid attention to such warnings).
I was about to make the same observation as this commenter: when the car in front accelerates, accelerate slower, let the following distance get larger. When the car in front brakes, start braking - but you can afford to do it more gently, because you have lots of space to brake into. Keep the velocity smooth, only gentle accelerations and decelerations. This makes for smoother traffic flow.
What a model, with tunable parameters, can do is, it can allow for comparisons between different driver behaviours. You might be in charge of publishing the next student driver's manual. You're choosing between two candidate texts; one has two paragraphs on reaction time and thirty pages on following distance, the other has the reverse, two paragraphs on following distance and thirty pages on reaction time. You'll probably reject them both, but running different models may give you an idea which is the one that would be a better starting point.
Playing with the model is also very important. Play. Try a formula describing how someone brakes for a squirrel crossing the road, try another formula. It may turn out that you can learn something valuable. A shape more like half a cycle of a sine wave causes accidents much more seldom than a shape dominated by logarithmic shapes, I don't know. And no-one knows until a model is run by someone who says "it might be interesting with this parameter, let's take a look". And then there is yet more insight into how to train drivers to drive safely. Because a person with a model... played.
@@hughobyrne2588 Whenever the gasoline price goes up, as right now, drivers are more inclined to eco drive. These days it is not uncommon for me to be in a row of 3 or 5 cars that drive like that.
@@hughobyrne2588 you cannot educate people to not drive home after drinking with their friends, if the only way to get home is to drive
@@jan-lukas You can educate people that there are other people who drive. A car which is both being driven, and simultaneously has a drunk person being carried in it, is not necessarily a car being driven by a drunk person. You can educate people about decision-making skills that help them avoid getting into situations so bad that drunk driving starts to look not-so-bad in comparison to the alternatives.
You are absolutely right! If you add following distance with the limited speed changes you increase the ability to maintain an average speed. If everyone left 5 to 10x the space they do now, there would be little to no traffic except in extreme conditions.
Cool video, traffic engineers use similar results as the basis of microsimulation software and in the early 20th c. empircal observations gave macroscopic results between the three "fundamental variables" flow q, speed v, and density k as your model produced.
Great video!
As a physicist I kept wondering if 1) can you define a field as the local displacement? and 2) will that field hold the wave equation? Probably need to take the continuum limit and expand the log to find that. Either way it looks soooo similar to the longitudinal sound waves
Really enjoyable way to see how math is all around us and how different areas are so useful when combined
Glad you enjoyed!
I bet designing some velocity controllers and running simulations with those could be a lot of fun.
Awesome video! Thank you!
Thanks so much sir you are increasing my way of reasoning
Sir, at everytime I learn something new
Love from INDIA 🇮🇳
Respect to you sir
Such an excellent explanation...
so happy to see you make a submission to SoME2
This is freaking awesome! Nice video!
Glad you enjoyed!
Wow very interesting! Great Job!
Interesting, I was expecting queuing theory from the title, but this is nice too! I've heard and used a preferred minimum following distance of 2 sec * v, which presumes tau 3×L . 🤔)
Love your t-shirt and your lecture!
Brilliant!!
Putting numbers into this, for vehicles of length 5m, the optimal interval is just under 14m (following distance like 9m) and corresponding speed allowing a 2s interval is under 7km/hr or just over 4mph. This sounds painfully slow, but accords with experience. If only it was a civic virtue to creep along cheerfully until reaching a point where travel demand eases. (And if only transport were electrified so not breathing one's own exhausts.)
This is something our type II rational thinking should be able to train our type I intuitive thinking.
Shout out to our governmental systems for providing transport that lets us move much faster than optimal so much of the time.
OMG THANKS SO SO MUCH THIS HELPED!!!
Quite excellent💯👍.
BEEN WAITING FOR THISSSSSS
This is a great video for sure!
I'd like to make a few comments, both on the model and on the discipline of traffic simulation.
Model: a ~ ∂v/∂x (where ∂x is x_(i+1)-x_i etc.) seems a bit too naive. We should at least satisfy v=at and x=1/2at^2 which gives us: a ~ (∂v)^2/∂x. Dunno if this makes any difference to the conclusions drawn in the video. Maybe not. Of course, more mods to the model are possible, but confusing v for v^2 may be significant enough to mention here.
Then I don't know if an optimization of flux is a good cost function. When in reality one should minimize the average travel time (optimize average v^-1). Your very own example with the two lanes nicely illustrates that the two cost functions do differ significantly.
From personal experience driving on German autobahns: there often is traffic-dependent speed limit, the more traffic, the slower. This is certainly influenced by models like the one presented here. However, I routinely observe how the very signs actually induce the traffic jams they are meant to avoid. This sound also very intuitive to me: if the flux has to increase (because of traffic) then so has the speed. Exactly like water flows faster through a thinner pipe. So, better measures against traffic jams would be to increase a minimal speed and divert slow cars off, e.g., by a special high traffic permit. Of course, the opposite is what is politically correct.
However, let me add another observation: In Germany, there often is widening of two lane highways into 3 lane highways, often with an end to a speed limit. In practice, this means that cars take more lanes (2/3 density) and speed up. The end effect is that while the 2 lane portion feels congested and almost stop and go, the 3 lane portion feels almost empty and deserted. This drastic change is a 'state transition'. Something maybe no as obvious to be observed in the US because of a 55+mph speed limit. It exemplifies my point why I think that "slower speed yields higher flux" wisdom by traffic research is deeply flawed. Certainly a point for a math thesis ;)
One more point against a slow traffic policy: driving in slow traffic reduces awareness, increases reaction time. Which induces traffic jams and accidents as you rightfully point out. There is a reason why the rate of accidents or fatalities on German autobahn is surprisingly low: drivers stay focused ... For the same reason, long straight routes are avoided as well.
So as a mathematician I love this. I have a VERY Simple driving style. #stayrightstayreasonable. 98% of my driving is in the right lane, and I NEVER speed, even when people around me do. When I tell people I often drive 5 mph below the speed limit (typically in 'rush hour' traffic they tell me I'm causing accidents. I remind them that if I am driving 5 below and everyone else is speeding 10 mph over: 1. I am only responsible for 1/3 of the variance, and 2, I am NOT breaking the law. 3. 15 mph = 22 ft/sec. And since you can typically see a full .25 mile down any stretch of highway it will take someone 60 second to close that distance. and if you can't determine you are approaching me in 60 seconds, hey, maybe you shouldn't be on the highway. I've figured that it takes approximately a half a mile to naturally decelerate from 60 to 40, and that it only takes an additional 6 second to travel that distance. So, please, can you model what happens when a car 'refuses' to speed when everyone else does? Also, I have hypothesized that it would only take about 10% of vehicles to follow #stayrightstayreasonable to nearly eradicate accidents and increase fuel efficiency across the system.
My driving style is based on the fundamental principles of physics and mathematics coupled with a desire to maximize safety, courtesy, and efficiency, in that order of precedence.
check me out on TIK TOK, #STAYRIGHTSTAYREASONABLE. I have a couple excel graphs that show this.
These are some ideas for modeling, I'm not quite as savvy, but hey, instead of every car acting the same, throw some variance into it. Here are some 'assumptions' of modelling I have considered:
Car object has several attributes:
1. Top speed (TS)
2. Follow Distance (FD)
3. Acceleration rate (AR)
4. Lane changing (to be figured out later) (LC)
All these attributes will be dependant variables based off of 1 independent variable.
Aggression factor 1-10 (AF)
Top speed = speed limit -1 + aggression factor mph
Follow distance = (11-aggression factor)*10 feet
Acceleration = (0-topspeed) in 16-aggression factor seconds
Aggression factor ranges
1 1% 1
2 1% 2
3 2% 3-4
4 2% 5-6
5 10% 7-16
6 20% 17-36
7 30% 37-66
8 28% 67-94
9 3% 95-97
10 3% 98-100
Please, have at it!!!!
This was a really interesting video. I was thinking about modelling traffic as a liquid but never really got to it lol
Great mathematical modelling! 👍🏼 Would be also interesting to derive rules for smoother traffic flow.
One rule, I heard somewhere, is, to keep the distance (d1) to the car in front of you the same as the distance (d2) to the car in the back of you. Maybe d1 = d2*0.7 > d_min would be more optimal, while always keeping the safe-breaking-distance, which is d_min=v*2seconds
Dr. Trefor Bazett, I drive for 35 years and from my experience I know one thing.
The best way to avoid traffic jams, even for a small or a large area, is every car to keep large distance between the next one.
Large distance means at least 5 car's lengths for speeds around 25 km/h (15 miles/h) and that must increases
accordingly of the square of speed. So if you double the speed you must quadruple the distance between the next car.
That has as result two mainly things:
First, in cross-roads without traffic-lights each car has the time to pass safe between the cars of other direction,
so one road does not delays the other.
Second, if something happen in the car in front of you there is time to make some adjustment without block
the other cars coming.
Ofcourse, there are many other reasons that affect traffic such as the lack of driver training or
the very large number of vehicles in some cities or the bad roads design.
Fascinating video! I have used cellular automata to study traffic dynamics.
That's cool!
@19:44 just one thing: If there's less than two car-lengths between you and the next, then you also move into the turbulent flow regime. Watching further ...
I love this. . .I work from home.
Nice video!!!!!
Was surprised at the step of integration. My assumption, at that point in the video, was that the model was discrete. (Btw, think this video is great and am going to use it at my company as a tutorial of how to build a relatively sophisticated math model.)
Didn't expect you to make a video for the SoME2 competition.
I don't expect to win by any stretch, but I enjoyed the process nevertheless:)
27:59 Water hammer phenomena when valve suddenly shut in water supply pipeline additionally with rubber balloon for visual oscillation phenomena proof for students. Traffic is also similar breaking and sudden acceleration create the same wavy pattern.
Amazing!
This is gold
Amazing
love it!
Hey Trefor the term is typically called offensively
Eg offense v defense
Aggressive vs passive
Fun fact: the average traffic jam is long enough to figure out all this
29:13 It looks like we are describing a train in here... "tiny following distances", "series of (ai driven) cars". Trains are the best transportation machines hahahahaha
This video is great! is there any relevant bibliography adopting such an approach?
One point to be careful about in the breaking function. Velocity and Acceleration as differential position functions produce average values. Since the formula was integrated and the apostrophes were dropped from notation, it may be easy to miss.
Superb video! Please include me to one car looking not only to the driver before me but also to few more ahead🎉. Unfortunately I believe few good drivers cannot help much overall e.g. on throughput or crash avoidance.
Respect bro
Do perturbations always grow over time, or is there a small enough reaction delay, tau, below which the system becomes stable and perturbations start to decrease over time instead?
Great demonstration of the modelling process and an elegant model. Just one ‘issue’ for my pedantic nature…BRAKING distance not breaking (the revenge of the auto-complete?!)
Gah! What happened lol
This is mindbending 😂
Being a control engineering student, I want to ask you: can this model be seen as a multi-agent dynamic system? Can it possibly reach again the former equilibrium after the initial braking (maybe considering a damping term so that the backpropagation will eventually stop growing)? Sorry about the messy comment but this video has blown my mind and now I feel very ignorant 😂
It literally all boils down to following the left-hand rule: always have traffic spiralling in a clockwise motion of faster traffic in the left lane always being supplanted by faster car behind, and traffic will always move smoother on average. It is, unsurprisingly, always the pride factor of moving right and conceding as "slower" traffic or one that is already completed a pass that hinders traffic in >90% of cases. This mindset DIRECTLY influences bad traffic progression and is major contributing factor behind jams. It is exacerbated by poor highway infrastructure that increases local absolute density of traffic at peak rush hours.
EDIT: This is when considering a three-lane highway, where one the most common forms of traffic issues are experienced. Also, these aforementioned principles are symmetric for lane systems.
Wish i learned math like this!!!
this is a very elegant model. I believe it is possible to implement Reinforcement Learning for parameter optimization, and maybe one day it can be implemented in a city with all cars are autonomous.
perfect
Have you considered taking τ and turning it into a δ? So instead of response time we talk in terms of response distances? Then you can include this delta function as a part of your density.
Would you please make a video on real life implications of integration for better idea of understanding it.
Wow.great , ;)
Usually when there’s traffic and there’s no apparent cause it’s usually because someone changes lanes. That randomly adds a car and their braking distance.
And the industry word for flux is throughput.
Your teaching was awesone so can you start a series for iit jee exam prepeartion including for calculus algebra etcc if you start most of the iit aspirants should watch your lectures like the way walter lewin sir youtube channel
I don't understand the delay differential part. Is there any way you can share the code or the program you use to calculate it please?
At 08:33, you were mentioning the breaking force, which means the velocity (when taking integral after that) should be the velocity during the breaking down. Why do you use that velocity in the general case later in the video? I mean it supposed to only works when we talk about the breaking down process of a car, but, a car is not always breaking down, right?
Love the vid. Btw do you teach at UVIC?
I do!
This is what I need, how to actually build math models in real life situations. No point having knowledge if you don't know how to use it!
After about 6 months of using local transport and getting frustrated by traffic ,I had always wondered about how can the problem of traffic be solved and whether is it even solvable?
Today, while on the bus and being stuck in traffic for about half an hour ,I pondered about cyclic traffic and immediately googled it and found some interesting papers modelling traffic!
I haven't watched this video yet but I was 'extremely' happy to see this video's notification pop up:)
What did you use to graph this? I can't figure out how you got the curve at 16:23 without a quantity for roe max
0:05 last week i was stuck in this traffic jam where we were alternating between being completely stopped and crawling along but why? because you were to lazy to walk or to impatient to use public transit
Fluid dynamics is everywhere, probably
Those equations have been in use for a while: all the assumptions are trains. AI driven cars will not help because the safety depends on position and speed sensors precision and reliability. Up to now (and to my knowledge) no one has been able to proof (mathematically speaking) that an connectionist AI is safe at the level required (i.e. SIL4)
Are humans drivers near the SIL4 level though ?
Self driving cars are personal rapid transit systems in a non controlled environnement (except maybe highways), an idea that date back to the 50s, and the few of those that got beyond the drawing board stage use pretty low speeds and headways of one second, and that is on pedestrian separated tracks.
So yeah, it seems like the self driving cars promises will be hard to meet.
Can anyone explain where numerator disappeared when we "integrate" at 9:15 ?
I'd like to see a model of car-oriented development. I think if you look at the density of people carried per mile of road and compare it to the density of car-oriented neighborhoods built to serve cars (suburbs), you'd find that it's unsustainable, meaning roads are destined to be congested no matter what because they're fundamentally inefficient with space.