Re: Elementary Math for the Traffic Problems

[Sonnet Nguyen <sonnet@white.cft.edu.pl>]

Hi d/c AV & lovers of Games,

I'd like to infor that Mr. Aiviet has gived a constructive model 
(plus speculation for the next) /quoted below/ and he wins 1/2 
champagne bottle plus a title "smart guy" from me :-))  Why only 
1/2 bottle ? 
Not only because his model is still too primitive, requires further 
developments, but also his designing style is unproper :-))
  Speculations of TA are out of the context, so no further comments 
of mine on it are needed :-)
 Only Aiviet's second model is worth for my comments because the first 
model (namely social, V.H.L, banana, B.A.V. or whatever you want:-) 
is impractical and ... very dangerous. Why it is dangerous? 
Well, there always exists something called "reaction time"
(I define it as a necessary time for a dviver to notice the break 
lights of the car in the front of him), so for the safety and
INDEPENDENTLY OF MODELS, the safe distance between cars always 
must be larger than : 
      (Reation Time) * (Your Actual Velocity).  
It means that the safe distance must increase quickerly (or equally) 
than linearly to the velocity.  So for a flow of limit velocity 
moving cars (typically: 65 mph in USA, 130 kmph in Europe) the 
public safety requires that the car's density must be very small 
(it's true, you can drive very quickely & safely when you do not 
see any car in the front of you) and therefore, the road's
passibility does not increase when (almost) each driver on the road 
increases his/her car's velocity, as at the first sight.    
  The main idea in the second AV's model is based on the simple 
obsevation that car's flow could be regarded as ... laminar 
car-liquid flow.   Obviously there are also few natural 
requirements on this car-liquid :

 1) Conservative law: The number of cars is conserved.
 2) No car-car collision: There exists a characteristic safe 
    distance between cars.

Reading 1) people with good back ground in Physics & Engineering 
must remind themself about famous continuity equation and here
in the simple day-life's problem, they could learn how to apply
effectively it into the life.  Life is too boring if there are 
only some poor applications in the Univ. Text-Books !! :-)
 
However AV's model is unrealistic, because he did know how to 
implement 2) properly.  Anyway he could implement 2) a bit better, 
make his model becoming more realistic and helpfull, if he knows 
at least which parameters are important (consider time reaction, 
car's technical data: like car's acceleration, etc.) and ready to 
pay a little more time "playing" on it.  Hope that some of you can 
improve the model or develope completely new one.  Use a chance 
to learn (freely) sth news from others.  It's boring game that 
I post a simple problems first and later Mrs Iga post a simplest 
solutions :-)

Cheers,
SN  
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On Tue, 26 Oct 1999, Aiviet Nguyen wrote:

> Tu+` model 2 tro+? ddi ba`i toa'n thu+.c cha^'t la` 
> set speed limit cho DMV.
> 
> Model 1 (For VHL) Speed limit la` 55 mile/h.
>    Solution: cha.y vo+'i va^.n to^'c 55miles/h. Lo+`i
> gia?i kho^ng phu. thuo^.c va`o model va` ra^'t
> realistic.
> 
> Model 2 ( Elementary problem for 8th grade)
>      Model high way gio^'ng nhu+ mo^.t ca'i o^'ng
> mo^.t dda^`u va`o va` mo^.t dda^`u ra. Ca^u ho?i ta
> co' the^? cho dda^`u va`o vo+'i rate nhu+ the^' na`o
> dde^? so^' lu+o+.ng dda^`u ra tre^n ddo+n vi. tho+`i
> gian la` max.
> Simplification: kho^ng co`n  mo^.t dda^`u na`o kha'c
> tre^n o^'ng. Gia? thie^'t kho^ng co' hie^.u u+'ng skin
> (va^.n to^'c o+? ca'c la`n kha'c nhau la` nhu+ nhau
> va` kho^ng co' ddo^?i lane...stb....). Ca'c
> simplication kha'c thi` student co' the^? gia? ddi.nh
> the^m mie^~n la` gia?i ddu+o+.c
> 
>    Solution:
>             Lu+u lu+o+.ng  va`o  \rho v  ( v la` va^.n
> to^'c  cu?a  ca'c xe , \rho la` ma^.t ddo^. xe trong
> o^'ng =N/l ). Lu+u lu+o+ng ra to^'i dda ba(`ng lu+u
> lu+o+ng va`o va` ta chi? ca^`n ddo`i ho?i kho^ng co'
> collision. Quy ta('c  na`y co' the^? la^'y tu+` thumb
> rules: dde^? an toa`n ca^`n khoa?ng ca'ch mo^.t tha^n
> xe cho va^.n to^'c 10 miles/h.  Tu+'c la` cha.y vo+'i
> va^.n to^'c v ta ca^`n (v/10 +1) X h cho mo^~i xe ( h
> la` chie^`u da`i cu?a xe). Ma^.t ddo^.  la` nghi.ch
> dda?o cu?a dda.i lu+o+.ng na`y
>          Va^.y lu+u lu+o+.ng va`o dde^? kho^ng co'
> collision ( = LLl ra) la`
> v/ h(v/10+1). Lu+o+.ng na`y dda.t to^'i dda thi` pha/i
> ho?i ho.c sinh lo+'p 10 :-))
> 
> Model 3: Should be more difficult with a lot of exit
> and entrance. It should be model as nested queues.....
> (need further realistic formulation)