X and the City: Modeling Aspects of Urban Life
Page 10
u, whereas if u decreases with k, the speed of the “wave” c < u. This gives some basic insight into traffic jams: although u is non-negative, a sufficiently negative value of du/dk can render c < 0; that is, changes in the traffic conditions propagate backward at speed | c |.
We may also infer some qualitative traffic behavior under the very reasonable assumption that the flow is in general a decreasing function of traffic density k. Now suppose that k decreases gradually in the forward direction, that is, k′(x) < 0. Then the front region of the flow moves out faster than the regions behind, the decrease in density is smeared out, and provided the range of k-values are less than km (when q = qmax), everything propagates in the forward direction. If this is not the case some of the traffic flow conditions may propagate backward.
By contrast, if there is a sharp decrease in k, such as at a traffic signal red light, the road behind the light will have a high density while that ahead will be empty of vehicles for some distance. The density behind is k = kj, the jam density, while that ahead is k = 0. Consequently there is an abrupt change when the light changes to green. We can think of this discontinuity as being a highly compressed collection of all possible k-values, which starts to “decompress” as traffic begins to flow. Each value of the density then propagates away at its own speed; those corresponding to q > qmax travel backward, those for which q < qmax travel forward, and since c = q′(km) = 0 at q = qmax, the latter do not move at all. Referring to Figure 10.1 we see that the fastest forward speed corresponds to the tangent slope at k = 0 (a clear road ahead) and the fastest backward speed to that at k = kj. This is the rear boundary along which the traffic jam resolves.
At the traffic light, k = km corresponding to capacity flow, since q′(km) = 0. This means that a traffic signal is useful for determining the capacity flow by allowing jam conditions to build up before turning to green. If R is the duration of the red phase, G that of the green, and vehicles arrive at the light at a rate qa and leave at a rate ql, the total number of cars arriving in a complete cycle is qa(R + G). The condition for them all to pass through the junction during the succeeding green phase is qa(R + G) < qlG. Therefore the capacity of the junction is given by the upper bound Gql/(R + G).
If the density k increases with forward distance there will be a “pile-up” problem, because the flow is greater for lower concentrations. This results—hydrodynamically speaking—in a shock wave: a rapid transition from light traffic in the rear to heavy traffic in the front. No doubt we have all experienced this. Suppose that the set {q>, k>} describes the conditions just ahead of the shock front, and {q<, k<} behind it, and the shock moves at speed U. The rate at which vehicles emerge from the front of the shock wave is q> − Uk>; the rate at which they enter from the rear is q< − Uk<. The number of vehicles is conserved, so these rates are equal, so
This is the slope of the chord on the q-k diagram for a given x-value joining the shock wave entry and exit points. Note that in the limit as these differences tend to zero, the chord becomes the tangent line, and for that value of x
In summary, we have seen that according to the kinematic theory of traffic flow, the front of heavy traffic concentration tends to smooth out, whereas the rear steepens and forms a jam. A vehicle approaching from the rear encounters the heavy traffic suddenly, but exits it gradually. But we all know that from personal experience! And below we put a little more mathematical meat on those most annoying events, namely:
X = d: TRAFFIC SIGNAL DELAYS
Let’s start this subsection off with a bang:
So there! What on earth does this formula mean? It is based on a model (Webster 1958) for the average delay per vehicle (d) at an intersection controlled by a traffic signal, and unfortunately it is too complicated to derive here. It was originally formulated for traffic in the UK and has been described as one of the most influential and useful results in this field of traffic control, known now as the Webster delay formula, so we had better pay it some attention! The various parameters on the right-hand side of equation (10.7) are listed below:
c = cycle length, i.e., length of one complete sequence of phases (seconds);
m = “g/c”, i.e., the proportion of the cycle that is “effectively green”; *
g = “green time” (seconds)
q = flow, i.e., average number of vehicles/s (v/s);
s = saturation flow, i.e., maximum capacity of road in vehicles/s;
x = q/λs, the degree of saturation; 0 < x < 1; if the light were continually green, λ = 1, q = s, and therefore x = 1.
It should be pointed out that the light sequence in the UK is red, red and amber (i.e., orange) together, green, amber, red, etc. The “effective green time” is (green + amber − 2) seconds, the 2 seconds being an allowance for delay in starting once the signal is green. The effective green time is easily adapted to the U.S. system in which the sequence is red, green, yellow, red, etc.
In equation (10.7) the first term represents the delay to the vehicles assuming a uniform arrival rate. The second term, a correction to the first, is the additional delay due to the randomness of vehicle arrivals. It is related to the probability that sudden surges in vehicle arrivals may cause temporary “oversaturation” of the signal operation. The third term, a subtractive one, is an empirical correction factor to correct the delay estimates consistent with observational data. It amounts to about 10% of the sum of the other terms. It is not surprising therefore that the following simplification to equation (10.7) is commonly used, namely,
Webster tabulated the delays (according to equation (10.7) for s using increments Δs = 300 in the range [900, 3600]; for c (Δc = 5) in the range [30, 60], and (Δc = 10) in the range [60, 120]; for q (Δq = 25) in the range [50, 1200] and for g (Δg = 5) in the range [10, 100]. Let us calculate d from equation (10.8) for the values
Then
Webster also derived two expressions for the average length L of a traffic line (or queue) at the beginning of a green phase (usually the maximum length in the cycle). The first and least accurate one is
where the new parameter r is the “red time.” This underestimates the length by 5–10% because it is based on the assumption that vehicles do not join the line until they have reached the stop line. A more accurate formula incorporating three new parameters is
that is, each term is increased by the factor , where y is the average spacing between vehicles in the line, a is the number of lanes and v is the “free running speed” of the traffic.
X = oh no!: TUNNEL TRAFFIC IN THE CITY
How does the fundamental diagram “square” with known traffic configurations in tunnels (where the likelihood of traffic jams is quite high, especially during rush hour)? The Lincoln Tunnel (under the Hudson River) in New York City has a maximum traffic flow of about 1600 vph (vehicles per hour) at a density of about 82 vehicles per mile moving at a mean speed of 19 mph (see Table 10.1 and the scatter plot in Figure 10.2). As noted, if the density is almost zero, the traffic flows at the maximum speed, and for a range of small densities remains nearly constant, so the flow varies almost linearly with density in this range, q ≈ kumax. This is consistent with measurements in both the Lincoln Tunnel and other NYC tunnels such as the Holland and Queens Midtown Tunnels. As the density increases, with a corresponding increase in flow, these same measurements show differences in the q-k curves: both the capacity and the corresponding speed are lower in the older tunnels (Haberman 1977). This should not be too surprising given that newer tunnels typically have greater lane width and improved lighting that permit higher speeds at the same traffic concentrations. Furthermore, the capacity of a given tunnel may vary along the road—regions with lower capacity than elsewhere are called bottlenecks, and in tunnels they typically occur on the “upward and outward” part of the tunnel (why do you think this is?).
TABLE 10.1
Figure 10.2. Scatter plot of data from Table 10.1.
Ideally, traffic should be forced somehow to maintain maximum flow by manipulating the density an
d speed appropriately. In my (limited) experience this works well on the M25 “orbital” motorway around London, which has variable speed limits (and speed cameras) during rush hour. In the Holland Tunnel, momentarily stopping traffic resulted in increased flow! This is because a traffic signal at the entrance to the tunnel was suitably timed to permit traffic to flow in intervals with density corresponding to the maximum flow.
Chapter 11
CAR FOLLOWING IN THE CITY—I
Don’t some cars inevitably follow others, and not just in the city? They certainly do, but the phrase as used here means that we model traffic by identifying each car as a separate object, not just part of the flow of a fluid called “traffic.” We’ll start by setting up a particular type of differential equation for this (now) discrete system.
X = xn (t): A STEADY-STATE CAR-FOLLOWING MODEL
Suppose that the position of the nth car on the road is xn(t). If we disallow passing in this model we can assume that the motion of any car depends only on that of the car ahead. A simple approach is to set the car’s acceleration proportional to the relative speed between it and the car in front; thus we have
This means that if the nth car is traveling faster than the one in front, it must decelerate to avoid a collision (generally a good idea). Conversely, if it is not traveling as fast as the one in front, the driver will (in this model) accelerate accordingly. Note that the model, simplistic as it is, does not give the driver the choice of maintaining a constant speed unless the right-hand side of equation (11.1) is zero. Additionally, the equation takes no account of the time lag due to the reaction time (T) of the driver in responding to the changing conditions ahead of her.
(At this point the reader may throw up his hands in disgust and say—“These mathematicians! Nothing is ever realistic—when do such conditions ever occur?” He has my sympathies, but this is the way modeling is usually done: take the simplest nontrivial situation and see what the implications are for the real-world problem, and modify, tweak, and improve as necessary . . . trial and error are important in constructing models.)
Incorporating the reaction time T results in the modification
According to one source, T is approximately 1.5 seconds for half of all drivers, and in the range 1–2.2 sec for all drivers, though over two seconds seems rather high to me. Equation (11.2) is actually a system of delay-differential equations (n = 1, 2, 3, . . .) and in general these are notoriously difficult to solve. It can be integrated directly however, to yield
where cn is a constant of integration. This equation defines the speed of the nth car in terms of the separation from the car in front at an earlier time. Let’s examine the special case of a steady state (or time-independent) situation in which all the cars are spaced equidistantly, and hence moving at the same speed. Then, since
equation (11.3) may be reformulated as
We may now ask how the spacing xn(t) − xn−1 (t) = −d might depend on the traffic concentration k. In the situation described by equation (11.4), consider all vehicles to have length l, so that the number of cars per mile (or km) will be the constant value k = (l + d)−1. From equation (11.4) with cn = c
Imposing the reasonable requirement that at the maximum possible density kmax (bumper-to-bumper traffic), u = 0, we can solve for the constant c to obtain the simple result
There is a problem, however; this equation predicts that u → ∞ as k → 0(!). But it is easily resolved, because we know that for small enough densities, 0 < k < kc say, u ≈ umax. By requiring u(k) to be continuous it is necessary to choose
so that the traffic flow is
A typical q-k graph is shown in Figure 11.1. It is a piecewise linear approximation to the concave-down curve discussed in the previous model. One unfortunate feature is that the maximum flow occurs at k = kc, which is unlikely to be the case. The corresponding u-k graph is shown as a dotted line.
The above model is rather limited in its scope, and can be improved somewhat by modifying the proportionality parameter b. This is likely to depend on the distance between the car and the one it is following; it seems reasonable to conclude that the closer it follows, the larger will be the accelerative or decelerative response. This quantity is in effect a sensitivity term. To this end, we choose
Figure 11.1. q(k) and u(k) profiles based on equations (11.6) and (11.5), respectively.
so that instead of the linear equation (11.2) we have the following nonlinear version:
As above, this can be integrated, yielding
Again, we choose a steady-state traffic flow for which this reduces to
Since
we simplify this to the case when the traffic density is much less than the bumper-to-bumper density kmax = l/l (for which u = 0), that is, k l/l. Then (11.9) reduces to
upon which, as before, we impose the low density condition u = umax to avoid singular behavior as k → 0. The flow is therefore given by
For u to be continuous the choice of
must be made, but in this model the coefficient B has a more interesting interpretation. From equation (11.10) the maximum flow occurs when
and this occurs when k = kmax/e. At this value of k, the speed of traffic is, from (11.10) simply u(kmax/e) = B. For consistency with the continuity requirement, kc must be such that
or more explicitly,
Figure 11.2. q(k) and u(k) profiles based on equations (11.11) and (11.10), respectively.
A generic sketch of both u(k) (dotted line) and q(k) is shown in Figure 11.2.
Chapter 12
CAR FOLLOWING IN THE CITY—II
I love to watch clouds; their changing forms are indicative of the different kinds of hydrodynamical process that are present in the upper atmosphere, such as convection, shear flow, and turbulence. Unfortunately, I am rather prone to do this while driving. Probably the worst example of this occurred many years ago when my wife and I were on our way to the local hospital (she was in labor with our third child). I won’t elaborate here, except to say that she rightly urged me to concentrate on the road. Distractions such as cloud-watching while driving increase the reaction time for avoiding traffic hazards (and therefore should not be engaged in!). This next set of models incorporate reaction times in a simple and rather natural manner.
X = Q: ALTERNATIVE CAR-FOLLOWING MODELS
We now consider a driver traveling at speed u who tries to maintain a constant distance between his car and the one ahead of him. Since he will wish to be able to stop suddenly if the vehicle ahead does, and to do so without hitting it, the spacing s can be written (in particular) as a quadratic function of speed. (Why is this so?) Consider the expression
From equation (12.1) the constants s0, a, and b must have dimensions of distance, time, and (acceleration)−1, respectively. Therefore s0 might be, for instance, the minimum spacing from the back of car n to the front of the following car (n − 1) or front to front. The constant a could be the reaction time to a sudden braking of the car ahead, and b could be the maximum deceleration, which would modify the speed u in order to keep s constant. We can solve equation (12.1) for u:
where α = a/s0 and β = b/s0. The positive solution (+ root) can be recast directly into a form related to q = q(k) by defining some new parameters:
from which we obtain the expression
In view of the above discussion about the shape of the q(k) (and now the q(K)) graph, we seek the location of the maximum from dq/dK = 0, that is, where
Simplifying, we obtain the following quadratic equation in K:
This equation has the roots
On substituting these into equation (12.3) it transpires that only the – root gives q > 0. We can write this value as
Finally, writing Q = q/qmax we have an expression for Q(K), that is,
A typical graph of Q(K) is shown in Figure 12.1. The maximum occurs at
Figure 12.1. Modified q-k diagram based on equation (12.6).
The position of the maximum changes relatively little with γ; in the interval
1 < γ < ∞, for example, 0.25 < Km < 0.5.
Note also that
This explains the shape of the graph near the origin; as K → 0+, Q′(K) → ∞ (as does q′(k) of course). As the concentration goes to zero, any interaction between vehicles will become negligible, and the flow would become an average “free speed,” which renders this car-following model inapplicable. In practice, the real flow at any concentration is probably less than this, so the slope of the graph near the origin will not be so steep (see Figure 10.1).