Traffic, roads, and travel time

[Genady]

There are two traffic related threads going on right now, and here is a traffic related question which would've been OT in both, so I post it here.

The points A and B are connected by two roads, APB and AQB.

The section AP takes 20 minutes regardless of traffic. The section PB takes time equal to T/10, where T is the number of cars on the road. For example, if there are 200 cars on it, it takes 200/10 = 20 minutes.

Similarly, AQ takes T/10 minutes and QB takes 20 minutes regardless of traffic.

The city planners came up with an ingenious way of connecting points P and Q with a road that does not take any time at all regardless of traffic:

How will it affect the average time of travel from A to B?

[TheVat]

Well, minimum travel time, by any route, will still be 20.1  minutes (assuming your own vehicle counts as 1 car on the road, so t/10 will be O.1 minute for just you).  

A driver who starts on a t/10 road and encounters heavy traffic (where t >200) will assume they should not take the PQ instant road because that would dump them in more heavy traffic that will take more than 20 minutes.  So they will continue on the 20 road.

A driver who starts on a 20 road cannot predict traffic on the t/10 leg, so they will have to select the path based on past experience of it.  If they anticipate heavy traffic (>200), then they would take PQ, knowing they are certain to only take 20 more minutes.  If not, they would take the t/10 road, hoping to take less total time.

So the optimal route really depends on the driver's knowledge of traffic conditions on those roads, it seems to me.  

[Genady]

1 hour ago, TheVat said:

minimum travel time, by any route, will still be 20.1  minutes (assuming your own vehicle counts as 1 car on the road, so t/10 will be O.1 minute for just you)

If you are the only car on the road, the minimum travel time is .2 minutes, by the route AQPB, isn't it?

 

1 hour ago, TheVat said:

A driver who starts on a t/10 road and encounters heavy traffic (where t >200) will assume they should not take the PQ instant road because that would dump them in more heavy traffic that will take more than 20 minutes.  So they will continue on the 20 road.

OTOH, when they get to Q, they might expect that other drivers will think this way and thus it is smart to take QPB instead. If half drivers think so, in case T=200, then 100 drivers will have total time 20+10=30, another half will have 20+20=40, with the average of 35 minutes.

 

1 hour ago, TheVat said:

A driver who starts on a 20 road cannot predict traffic on the t/10 leg, so they will have to select the path based on past experience of it.  If they anticipate heavy traffic (>200), then they would take PQ, knowing they are certain to only take 20 more minutes.  If not, they would take the t/10 road, hoping to take less total time.

Like before, if they split half and half, then the average will be 35 minutes.

Without the extra road, for T=200, half take APB, half take AQB, and the average is 30 minutes.

Adding the road, in this case, increases the average time.

Edited August 14, 2023 by Genady

[exchemist]

3 hours ago, Genady said:

There are two traffic related threads going on right now, and here is a traffic related question which would've been OT in both, so I post it here.

The points A and B are connected by two roads, APB and AQB.

The section AP takes 20 minutes regardless of traffic. The section PB takes time equal to T/10, where T is the number of cars on the road. For example, if there are 200 cars on it, it takes 200/10 = 20 minutes.

Similarly, AQ takes T/10 minutes and QB takes 20 minutes regardless of traffic.

The city planners came up with an ingenious way of connecting points P and Q with a road that does not take any time at all regardless of traffic:

How will it affect the average time of travel from A to B?

Don't tell me, the road PQ goes over a bridge, at a place called Wheatstone. 😀

[Genady]

4 minutes ago, exchemist said:

Don't tell me, the road PQ goes over a bridge, at a place called Wheatstone. 😀

LOL. (had to google it to appreciate the joke, and learned something)

[TheVat]

4 hours ago, Genady said:

If you are the only car on the road, the minimum travel time is .2 minutes, by the route AQPB, isn't it?

Yes, I didn't write clearly.  I meant without the "Wheatstone" pathway.  And plus one to @exchemist for that clever joke.

As for the rest, I misunderstood the game theory aspect of what the driver, or we, would assume about other drivers.  I will chew on this a little more.  

Hopefully I won't drive in circles!

[TheVat]

There may be a Bayesian analysis that escapes me here.

For one thing, if these are chronic high traffic roads then it may always be faster to use the 20 roads.  If the t10 legs always take at least 30 minutes (with at least 300 cars at any time on road), for example, then you will always do better to take only the 20 legs.  The problem is rather unreal, given there are no roads that "take only 20 minutes regardless of traffic," and no actual driver would believe such a thing.  If everyone took the 20 roads, then they would by normal human expectations cease to be 20 roads.  

[Genady]

1 minute ago, TheVat said:

There may be a Bayesian analysis that escapes me here.

For one thing, if these are chronic high traffic roads then it may always be faster to use the 20 roads.  If the t10 legs always take at least 30 minutes (with at least 300 cars at any time on road), for example, then you will always do better to take only the 20 legs.  The problem is rather unreal, given there are no roads that "take only 20 minutes regardless of traffic," and no actual driver would believe such a thing.  If everyone took the 20 roads, then they would by normal human expectations cease to be 20 roads.  

Yes, the problem is not very good, perhaps. It supposed to illustrate the thing I've linked today in another thread: Braess's paradox - Wikipedia.

[TheVat]

Ahh.  Drivers keep switching until the system reaches a Nash equilibrium, which is not optimal for flow.  That helps, thanks.