RE: [MLUG - DISCUSSION] Offtopic - speeding now

[Jonathan King <EMAIL:PROTECTED>]

On Thu, 4 Dec 2003, Russell Horn wrote:

> > Something to think about:  If the average car is traveling at 50
> > mph, it will spend twice as much time on the road as it would if
> > it were traveling 100 mph.  Therefore, slower speed limits mean
> > more cars on the road and more crowded highways.  Crowded roads
> > may increase accident risk.
> 
> Actually I've seen a few studies showing the most efficient (in
> terms of getting traffic to move) speed is just above 50mph. Where
> traffic is going faster than that, if one car breaks, meets a
> truck or something and brakes, the effects continue for several
> miles behind and become exagerated because of the high speeds.

OK, so there are competing views here that I think seem inconsistent 
because people are looking at different things.  The traffic geeks 
generally are interested in maximizing the volume of traffic (flow 
rate) along a stretch of road; they always call this "q".  Density 
(concentration) is the number of cars per unit length of roadway, 
and v is velocity.  So we have:

   q = k v_s

where v_s is the average "space speed" (let's skip the details on
that).  Intuitvely, and in reality, high traffic densities are
associated with lower speeds, while at very low densities you can
drive as fast as you like (the so-called "free speed" v_f).  Even
more conveniently, this relationship is almost linear in practice,
so that:

   v = v_f * (1 - k / k_max).

In other words, when density (k) is essentially zero, you move at 
the free speed, while when k is the maximum density k_max, you don't 
move at all.  Substituting the second equation into the first, you 
get:

   q =  v_f k * (1 - k / k_max)

Which again tells you that the maximum flow_rate is achieved at a 
speed less than the one you would choose if the road were empty.  To 
get a crude idea about what that speed is, look back at:

   v = v_f * (1 - k / k_max).

k_max is the density of bumper-to-bumper, so you can compute k in 
terms of how many car lengths there are between you and the other 
guy.  If you allow 200 feet between cars (for breaking), then 
that's about 11 car lengths between cars, so k / k_max is like 1/12.
So you can go very close to your free speed.

The problem from the point of view of speed, of course, is that
roads tend to be more congested than that.  If you can only give 3
car lengths between you and the next guy, then k / k_max is 1/4, and
you are at 75% of the free speed.  That's going to be pretty close
to 50 mph if you assume that the free speed is the speed limit of 70
mph.  And, sure enough, the flow rate here is more than twice as
high.  

Now, the above analysis blows off an explicit analsysis of passing
and related stuff, but this turns out to be captured decently in the
trade-off between speed and density: people do drive more slowly
when densities are high.  And when somebody forces you to change 
lanes because they are tail-gating, you increase the density of the 
other lane, and things slow down...

So what really causes problems is *variation* in speed, either in a
single vehicle or across vehicles.  A moment's thought indicates
that the only way collisions can happen is when the velocity of the
hitter and the hittee differ.

So, in the end, though, you are both right.

When Mike is driving his car across a deserted Minnesota highway in
good conditions, road capacity is maximized and Mike's driving time
is minimized when he goes as fast as he likes while retaining
control of his vehicle and keeping his speed variance to a minimum
by not hitting a bridge pylon or a deer. :-)

When Russell is driving his car on a crowded motorway, road capacity 
is maximized when the overall speed is chosen to be as fast as is 
safe, which is going to be rather slower than the free speed, 
but which could be faster than it currently is if people would not 
speed up and slow down so much.

And this is why rush hour is such a time and money sink.  If we
could spread the traffic of one rush hour over two hours, then the
traffic density at any time would be halved, and average speeds
would go up a *lot*.  If traffic density at rush hour gives you only 
one car length in front of you, then doubling that space would allow 
you go about 50% faster.  This is why congestion pricing is in 
prcinciple such a good idea.

Now, if you really wanted to improve road safety via ticketing
people, you would still ticket people who were driving irresponsibly
fast (80+ on a rain-slicked road), but you would concentrate more on
the lane-weaving speed-varying maniacs we all know too well.  And,
alas, also on those people who cannot any longer drive safely at
highway speeds.  A big hidden problem coming up for us will be how
we can simultaneously keep the highways safe and manage to keep the
elderly independently living in an age where there are going to
many, many more people who "need" to drive but really can't hack it.

jking


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