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The Problem of Traffic: A Mathematical Modeling Journey

Dr. Trefor Bazett 110 views

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Great video I really would like to dive into this kind of analysis Does anyone know a good book on this subject (mathematical modelling of real world problems)?
Comment from : @RainierVerschuren

trains trains trains
Comment from : @user-kv4gj9jp4h

This guy has an irritating voice
Comment from : @kafiruddinmulhiddeen2386

@DrTrefor I'm working on a project: optimizing urban traffic flow using linear equations I need your assistance
Comment from : @victorosegbo

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 brThank you for having it online
Comment from : @AbrahamNoble-e6o

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 brbrAnd the industry word for flux is throughput
Comment from : @Marko-qy5eg

This video is great! is there any relevant bibliography adopting such an approach?
Comment from : @xrhsthsuserxrhsths

Great video! Traffic flow theory is very interesting! br br1:58: The shockwaves you show can also be seen in actual videos of traffic Here's an example from a highway: br youtube/BKQUk-9vC4s br brYou can also see it in action with large groups of pedestrians: youtube/OYJwPJOQqQs br brThe basics of traffic flow from a traffic engineering perspective are explained here (flow, density, and speed): youtube/DmrmtYLabrI br brYou might also be interested in the saturation flow rate: youtube/sUXr9j9mbw8
Comment from : @FindleyDaniel

They need to hire you down here in Miami Florida, I'm fairly certain the engineers here are using sticks and stones to fix traffic problems
Comment from : @jorgevaldes6160

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?
Comment from : @huyhuynh7917

Is uncaused traffic caused by people having bad reaction times?
Comment from : @xoso599

Love this video, but please pronounce correctly the greek letters You can read τ as /taph/ or /taf/, but not /tau/ brFun fact: Worst pronounce of greek letter is when I hear "mew" for μ, while the correct pronounce is /me/
Comment from : @alex_1999

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 eg on throughput or crash avoidance
Comment from : @cantkeepitin

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
Comment from : @cordelia358

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
Comment from : @frozenkingfrozenking6989

Why does that #Some2 logo look like 3 Blue 1 Brown?
Comment from : @nkanyisonyawose1025

Mathematical model approximates reality But some physicists and biologists deny this approximation and are so confident about their models for evolution (cosmological or biological),
Comment from : @jbangz2023

As a psychologist that uses mathematical modelling for learning, I laughed for the entire video Keep trying, bud You've got a better chance at success than I do OMGassumptions Hilarious
Comment from : @faupsy1012

One more detail you could add to your model to make it more realistic would be to include some suicidal deer
Comment from : @perrygershin3946

The problem of traffic is cars Boom Done
Comment from : @VeteranVandal

When you integrate and get ln|x| is that because the relative velocities remain constant? *Edit: Oh never mind its because the top numerator is the derivative of the denominator
Comment from : @ytbasketball101

This makes me think of a piece of a square wave
Comment from : @Apophlegmatis

This was a really interesting video I was thinking about modelling traffic as a liquid but never really got to it lol
Comment from : @CarlosFloresP

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
Comment from : @ipcheng8022

The response time is called a dead time in a FOPDT model I would maximize the number of cars passing a point Similar to flux The problem is getting the cars in front to accelerate faster than the car behind must decelerate Need to solve a system of differential equations
Comment from : @pnachtwey

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
Comment from : @devanandannu305

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?
Comment from : @lasmanki1980

perfect
Comment from : @maloukemallouke9735

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
Comment from : @rickhodges4808

Respect bro
Comment from : @rotsenlavinecrow9520

Awesome video! Thank you!
Comment from : @AJ-et3vf

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! 😄
Comment from : @Zappe435

Wish i learned math like this!!!
Comment from : @phos5990

Your mic is clipping
Comment from : @yasoomorimoto8474

Great video!brAs 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
Comment from : @TheOoorrrr

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)brbrOnly 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)brbrWhen 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
Comment from : @Salien1999

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
Comment from : @Konchok_Dawa

Solving the differential delay system is quite similar to how computer solve pde which I find quite interesting
Comment from : @soldenstoll8495

so happy to see you make a submission to SoME2
Comment from : @kono152

On some highways, there's an accident almost every day So maybe the model prediction that there's a guarantee for an accident is not so far-fetchedbrSo the accident experienced by the n-th car was actually caused by the first Therefore you ishould not/i break on the highway brbIf you are breaking on the highway, something is wrong!/b
Comment from : @richardcoppin5332

Quite excellent💯👍
Comment from : @solapowsj25

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 😂
Comment from : @GdeJ

1D Gas flow but not liquid flow
Comment from : @lt4376

Model vs 🆚 moral compass for🤣🤣🤣💪 the universe🌌 and million of chance😂😂😂😂 to🤣🤣🤣 get it👌👌👌 right now 😆😆😆😆😆😆
Comment from : @anilkumarsharma8901

Birds eye👀 view of the traffic🚦 problem with probability😅😅😅😅 theory😂😂😂😂brAll the👌 time⌚ we know that total area divided by the vehicles allowed are the limiting beliefs🙏 of the👌 common sense of humor😆😆😆😆😆😆
Comment from : @anilkumarsharma8901

Get real Self driving is a ploy to get money from idiotic sucker investors It is utterly trivial to make software that almost always does the right thing, and utterly impossible to make software that always does the right thing Self driving is a lie You are one stuck-at-magenta pixel away from a head-on collision Thankfully, you only briefly mentioned self driving
Comment from : @douggale5962

I cite this video as a model to follow to have accidents so humans, please turn in your licenses
Comment from : @AspartameBoy

Dr Trefor Bazett, I drive for 35 years and from my experience I know one thing brThe best way to avoid traffic jams, even for a small or a large area, is every car to keep large distance between the next onebrLarge distance means at least 5 car's lengths for speeds around 25 km/h (15 miles/h) and that must increases braccordingly of the square of speed So if you double the speed you must quadruple the distance between the next car br brThat has as result two mainly things: brFirst, in cross-roads without traffic-lights each car has the time to pass safe between the cars of other direction,brso one road does not delays the other brSecond, if something happen in the car in front of you there is time to make some adjustment without block brthe other cars comingbrbrOfcourse, there are many other reasons that affect traffic such as the lack of driver training orbrthe very large number of vehicles in some cities or the bad roads design
Comment from : @Antuan2911

This is greatthanks for sharing this I am linking your Calc 1 videos to my course
Comment from : @jongraham8866

At 5:43 shouldn't that be x_i < x_(i-1) - L?
Comment from : @txikitofandango

Hey! I think you have to add #SoME2 in the video title to participate!
Comment from : @gbpferrao

I love this I work from home
Comment from : @NathanHedglin

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 << 2 sec (Which is hard to maintain when lane changers will drive into any gap > 3×L 🤔)
Comment from : @BillRicker

Sir, at everytime I learn something new brLove from INDIA 🇮🇳 brRespect to you sir
Comment from : @rb6388

Thank you I’ve been wondering about this for a long time but never taken the trouble to model it
Comment from : @RobinHillyard

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"brCars 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
Comment from : @peterdavis9403

The solution is public transportation, bikes, and mixed use density
Comment from : @brittanyfriedman5118

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!
Comment from : @Daniel-ug3ie

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
Comment from : @RMDragon3

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!
Comment from : @scaratb8810

Such an excellent explanation
Comment from : @icenarsin5283

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 (ie SIL4)
Comment from : @reynaldorosas6373

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
Comment from : @PiEndsWith0

LOL the biggest thing I got out of this is that Traffic engineers don't care how long it takes me to get home I could take 4 times longer to get home and if they get 5 times more traffic flux, they are happy and I am sad
Comment from : @lidarman2

This is gold
Comment from : @andrewberardi6158

You can try to solve traffic with math as much as you want, but traffic is only 5 math/physics and 95 psychology
Comment from : @Bendigo1

Great video, you’re making things clear about math modeling in my mind, each step is well explained thank you for that !! brbrCan i use it as a support for a school project?
Comment from : @ganeshbemene7822

Aesome , Just one thing the quality of the audio sometimes drops
Comment from : @alikhatami6610

Love your t-shirt and your lecture!
Comment from : @handhikaramadhan

Best professor I’ve ever had Great video!
Comment from : @nahleaky586

Amazing!
Comment from : @klevisimeri607

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 modelbrbrAlso 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
Comment from : @greese007

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 bitbrbredit: 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
Comment from : @Beatsbasteln

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
Comment from : @phyarth8082

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
Comment from : @absolutelyuniformlyconfuse4176