http://path.berkeley.edu/topl/docs.html
Gabriel's comment:
On the detector model, I dont think that headway = 1/flow is the right macroscopic variable for generating stochastic vehicle actuations. This is because a small flow could generate large headways, in freeflow, or small headways in congestion. It is argued in the attached note that the"smeared occupancy" lambda is what we want.
See http://path.berkeley.edu/topl/docs.html for Gabriel's detector model.
Andy's comment on Gabriel's idea:
1. I am still trying to understand 'lamda'. One quick observation is that the 'dimension' doesn't match: 'lamda' is dimensionless, while it is supposed to be the expect number of vehicles (instead of occupancy).
2. When it's congested and there are a lot of vehicles on the road, you will get a small spacing between successive vehicles since spacing is the reciprocal of density (Equation 2 in my note).
However, it appears to me that it doesn't necessarily imply you are going to have frequent actuations by detectors installed on the road.
It is because the headway of vehicles (and hence actuations) is going to be pretty long as well since the vehicles are moving slowly (speed is low in congested state..) and it takes longer time for them to reach the detector..
6 comments:
On Andy's comments:
1. Lambda is interpreted as a "smeared occupancy", ie. smeared over the time interval. This seems to me should be the same as the average number of vehicle actuations in the link. So if lambda=2.3, we expect 2.3 vehicle actuations on average (over all possible detector positions) in that link over a time interval \Delta t.
2. I do not agree with this. When density is high you should get a very high probability of at least one actuation. In the extreme case of jam density, zero speed, the average number of actuations (again over all loop positions) should reach 1, or very near 1. More precisely, you should get l/(l+h), which is lambda with v=0.
Hi Gabriel and all,
Thank you for your note. Some further comments from me:
1. I suppose 'h' actually is 'spacing' (i.e. physical distance between two vehicles) instead of 'headway' (time lag between the vehicles), right? You should not be able to add 'h' to 'l' if it's a 'time' ...
2. Is "Delta t" a kind of observation period? For example, if we want to know the number of arrivals during the red phase, "Delta t" will be the red duration. Is it correct?
3. I can't follow well your derivation. I don't quite understand why the occupany (or the smeared occupancy) could be equivalent to the expected number of actuations. By definition (or even in your derivation), occupancies (Expressions 4 and 5 in your note), mean the proportion of time (or which can also be regarded as a 'probability) that a detector being occupied by a vehicle. It's supposed to be a value between 0 and 1. I don't understand how it can be a "expected number of vehicles".
4. I fully understand and agree with your Expression 4, but I don't understand the physical meaning of your Expression (5). Given the spacings (I assume h is spacing) remain the same, I would expect the measured occupany would be the same (instead of being smeared), no matter what the speed of the vehicles are. What do you think?
5. Assume that your derivation is correct, let's go to your final result which is Expression 12:
a. The dimension is correct here as both sides have the unit of 'number of vehicles';
b. If we look into Expression 12, the first term actually gives you 'q*Delta_t' since 'pho*v = q' where q is the flow. Do you agree?
c. If point (b) above is correct, then we will come up with the SAME detector model, except that you have an additonal second term. Do you agree?
d. Now, let's try to understand the second term 'pho*l'. Again, does it have any physcial meaning? It's also a bit strange to me since it's independent of the 'Delta t'.
Furthermore, I noted that if we consider an extreme case (zero speed; jam density): Assume a jam density = 250veh/mile;
and l (typical car length)
= 7 ft = 7*5280 mile;
we are going to have:
lamda = l*pho = 7*250/5280 = 0.33 instead of a value closed to '1' as you expect...
e. If we compare the magnitudes of the two terms in Expression 12, the first term seems to be able to dominate the second term (especially for a long Delta t), taking 'lamda = flow rate' doesn't seem to be a very bad assumption... What do you think?
Please advise. Please let me know if I misunderstand anything.
All best,
Andy
A small typo:
In point 5d:
l = 7/5280 (mile)
instead of '7*5280' miles ...
Andy
1. Yes, h is the distance headway.
2. lambda is the expected number of observations during the time period Delta t. There is probably some restriction on Delta t, perhaps that it should be no greater than the simulation dt (since v and h are only valid for that long)
3. Perhaps "occupancy" in "smeared occupancy" is not the right term. lambda is as if you took an overhead photograph of the traffic with a shutter speed of Delta t. The vehicles appear smeared in the picture. Lambda>=1 corresponds to a picture where you can see no pavement (because Delta t is long). Now, if we randomly place a point detector somewhere on the road, what is the probability that it will register a vehicle? That is the same as the probability that we randomly place it on a vehicle in the smeared photograph.
4. This confussion is again due to the term "occupancy". We should think of something better.
5b. Yes.
5c. Yes.
5d. l*rho is the expected number of actuations in total jam density (v=0). This should be less but near 1, assuming rho=1/(l+h). It is also expected that it is independent of Delta t, since the traffic is stopped.
On the numbers, the average length of a car in the US is 16'4''. 250 veh/mile is 21 ft/veh, which means the jam headway h is around 5 ft and lambda(v=0)=0.77. So 77% of the jammed freeway is covered by cars, and you therefore have .77 probability of detecting a vehicle with a point detector.
5e. It dominates only if v is large. It fails when v is small.
Gabriel, I think I have now understood the difference between the detector models of yours and mine. In your model, you take explicitly the physical dimension of the vehicles (which is the parameter 'l') into account. This gives you a residual term 'rho*l' in your expression.
I looked at vehicles as 'particles' (I think that's an underlying point of view in q-k-v macroscopic model as well.) and so I don't have that term in my expression.
Apart from this, I think essentially we are deriving the same thing.
As we discussed, there will be a maximum difference in the estimation of ~0.7 in the expected number of actuations between the models of yours and mine. This is about +/- 1 actuation, which to me, it's quite insignificant..
Gabriel's response to the above:
I agree that the difference is not large generally, but it is
significant in congestion and become more significant as speed
decreases. This "small" difference could lead to very large differences once we consider the control algorithm. For example, consider an algorithm that only provides green if it receives an actuation on a given phase. If the speed on that phase becomes zero (i.e. the queue goes over the detector), then under the "flow model" detector that phase
will lock and remain red for all time. UNder the "smeared occupancy" detector the controller will behave as desired.
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