X and the City: Modeling Aspects of Urban Life

by Adam, John A.

  While the change from driving at 70 mph (as many do) to 60 mph (about 14%) may not be considered “small,” we’ll use it anyway, and conclude that there is a nearly 30% drop in fuel consumption. This makes a lot of sense!

  X = d0: TRAFFIC SIGNALS IN THE CITY

  What thoughts typically run through your mind as you approach a traffic signal? Here are some likely ones: will it stay green long enough for me to continue through? Will it turn red in enough time for me to stop? What if it turns yellow and someone is close behind me—should I try to stop or go through? And perhaps related to the latter thought, “Is there a police car in the vicinity?”

  Obviously we assume that the car is being driven at or below the legal speed. If the light turns yellow as you approach the signal you have a choice to make: to brake hard enough to stop before the intersection, or to accelerate (or coast) and continue through the intersection legally before the light turns red. Unfortunately, many accidents are caused by drivers misjudging the latter (or going too fast) and running a red light.

  The mathematics involved in describing the limits of legal maneuvers is straightforward: integration of Newton’s second law of motion. Suppose that the width of the intersection is s ft and that at the start of the deceleration (or acceleration), time t = 0, and the vehicle is a distance d0 from the intersection and traveling at speed v0. If the duration of the yellow light is T seconds, and the maximum acceleration and deceleration are denoted by a+ and −a− respectively, then we have all the initial information we need to find expressions for the two situations above: to stop or continue through. A suitable form of Newton’s second law relates displacement x and acceleration a as

  from which follow the speed and displacement equations

  We have chosen x(0) = 0. In order to stop before the intersection in at most T seconds, it follows from the second of these equations that

  On the other hand, to continue through, we note that the vehicle must travel a distance d0 + s in less than T, so that requiring x(T) ≥ d0 + s in the third equation above yields the inequality

  In each inequality, we have assumed that the maximum deceleration and acceleration are applied accordingly. Before treating the inequalities graphically, we write them in dimensionless form. This will reduce the number of parameters needed from five to four. To illustrate this, if we divide the distance d0 by s, we obtain a dimensionless measure of distance in units of s, namely δ = d0/s. Similarly, we define dimensionless speed by σ = v0/(s/T), time by τ = t/T, and acceleration by α± = a±/(s/T2). Notice that τ is a measure of time in units of the yellow light duration. In these new units, the combined inequalities show that a driver may successfully choose either legal alternative provided that

  Now we are in a position to discuss reasonable ranges on these parameters, starting with the physical data. Typical ranges have been taken from the literature [17]. If we adopt the range for the speed approaching the intersection as 10 mph ≤ v0 ≤ 70 mph (or approximately 15 fps ≤ v0 ≤ 100 fps), with the bounds 20 ft ≤ d0 ≤ 600 ft, 30 ft ≤ s ≤ 100 ft, 2 s ≤ T ≤ 6 s, and 3 ft/s2 ≤ | a± | ≤ 10 ft/s2, then we find that 0.3 ≤ σ ≤ 16.7, 0.2 ≤ δ ≤ 20, and 0.12 ≤ α± ≤ 8.3. Since we are measuring time in units of T, the results are relative, even though T itself can and does vary. In Figure 8.1 we plot a generic graph for the bounds on δ(σ) (based on equation (8.6)). There are four regions to consider. The region OAB corresponds to a domain with relatively low speed and small distance, and represents a portion of (σ, δ) space for which either option—stop or continue—is viable. Continuing, the region above ABC is a (theoretically infinite!) region with relatively low speed and large distance, so it is easy to stop before reaching the intersection. BCD is a region with relatively high speed and large distance, and represents a domain in which a violation (or accident) is likely. Finally, below OBD the region corresponds to relatively high speed and small distance, implying it is possible to continue through the intersection before the light turns red.

  The various δi(σ) functions chosen here (with i = 1, 2, 3, 4, also based on equation (8.6)) are defined as follows:

  As in Figure 8.1, Figure 8.2 shows dimensionless position/speed graphs identifying regions of safety. Obviously this is a simplistic analysis of the stop-light problem; an experienced and careful driver will have developed some measure of intuition (and caution) concerning whether a successful “continue through” is possible. We have not considered the possibility of skids; they are likely to occur if the deceleration is too large, and poor road conditions (wet, icy, etc.) will greatly affect the required stopping distance. In addressing this problem, Seifert (1962) has suggested posting signs along the roadside indicating a speed at which it is safe to continue through or stop from that location. It’s a thought!

  Figure 8.1. Dimensionless distance-speed graphs indicating regions of legal/illegal options based on equation (8.6); δ1(σ) = σ + 3.15 and δ2(σ) = σ2/10.

  Figure 8.2. Additional dimensionless distance-speed graphs based on equation (8.6) (see Figure 8.1). Here δ1 is as in Figure 8.1, and δ3(σ) = σ − 0.94; δ4(σ) = σ2/16.6; and δ5(σ) = σ2/0.24.

  X = β: AVOIDING ACCIDENTS IN THE CITY

  Should the driver of a car try to stop or turn in order to avoid a collision? We shall examine this question for several different situations, the first being when a car approaches a T-intersection with a brick wall directly ahead across the intersection. We shall assume that the junction is free of other vehicles, so the only possibility of collision involves the car hitting the wall. Furthermore, we shall assume that there is no skidding, in which case the coefficient of friction in a turn may be considered to be the same as that in the forward direction. (Skidding would involve the coefficient of sliding friction, in general different from that for rolling friction.)

  We can examine three possible choices [18] as illustrated in Figure 8.3: (i) to steer straight ahead and apply the brakes for maximum deceleration; (ii) to turn in a circular arc without braking (using all the available force for centripetal acceleration); or (iii) to choose some combination of (i) or (ii), such as turning first and then steering straight (or vice versa), or even steering in a spiral path. In fact, option (ii) can be ruled out immediately by means of a simple (but nontrivial) argument as follows.

  Suppose that the distance of the car from the wall is l and that its speed at that point is v0. The force required to turn the vehicle (of mass m) in a circular arc of radius l is , but the force required to bring the vehicle to stop in a distance l is . This means that if the car can be turned without hitting the wall, it can be brought to a stop halfway to the wall. Regarding option (iii), a rather more subtle argument [18] shows that the appropriate choice is still to stop in the direction of motion. Apart from a brief discussion below, this will not be elaborated on here; instead we shall examine some other potential driving hazards. It should be no surprise that the worst highway accidents are those involving head-on collisions.

  A related problem is this: suppose that we are driving along a road in the right-hand lane for which, at the present speed, our stopping distance is D.

  Figure 8.3. The three maneuver options open to the driver.

  There is an obstacle ahead—it might be a repair crew, a stalled truck, or even a vehicle moving more slowly than we are (in the latter case we must adjust our speed in the calculations to that relative to the vehicle). What is the maximum obstacle width W that can just be avoided by turning left in a circular arc (if traffic in the adjacent lane permits this maneuver)?

  We already know that the turning radius is twice the stopping distance. From Figure 8.4 we see that the angle β = α/2. Here β is the angle subtended by the obstacle and the vehicle before the maneuver begins, and α = arcsin (1/2) = 30°, so β = 15°. The width of the obstacle at this distance is therefore D tan β ≈ 0.268 D. A wider obstacle cannot be avoided except by stopping.

  Exercise: From the figure show that tan 15° = 2 − .

  Let’s try to estimate so
me typical stopping distances for a range of speeds. This distance depends on the coefficient of friction (μ) between the tires and the road, and the driver’s reaction time. The minimum such distance Dm can be found by ignoring the latter, as long as one adds the “reaction time × speed” distance Dr afterward. The frictional force must reduce the kinetic energy of the car to zero over the distance Dm. Provided the wheels do not lock during the deceleration (no sliding or skidding occurs), we use the coefficient of static friction. If the wheels are locked, the braking force is due to sliding friction, which is in general different, as noted earlier. In the case of static friction, for a car of mass m, the equation to be satisfied is therefore

  Figure 8.4. Geometry for turning to avoid a slower vehicle.

  from which it follows that

  Examining this result, we note that it is independent of the mass (or weight) of the car. It is also proportional to the square of the initial speed; thus doubling the speed quadruples the minimum stopping distance. The value of the coefficient μ depends on the quality of the tires and the prevailing road conditions; probably the best realistic value is μ = 0.8, but for more worn tires, or wet roads, a somewhat lower value 0.7 or 0.6 is probably appropriate (or even lower for tires in poor condition). Here are some minimal (rounded) stopping distances for various speeds, taking μ = 0.65. Also included, for illustrative purposes, is Dr, the distance covered in a nominal (and somewhat slow) reaction time of one second. The fourth column is the approximate total distance (DT) required to stop at these speeds, given the above assumption.

  For simplicity we now consider a minimal stopping distance of 100 ft, corresponding to a speed of about 44 mph (71 km/hr). If the vehicle is able to pass the obstacle without braking at all, this maneuver will begin after the reaction time. This means that the vehicle can pass a large obstacle of width nearly 27 feet if traffic in the adjacent lane(s) permits. But then the direction of the car will be at an angle of 30° to the original direction, a dangerous predicament to be sure!

  To improve the safety of this maneuver, consider the following modification: we require that once abreast of the obstacle, the car should be moving parallel to the road in the new lane. For this case, the geometry changes a little (see Figure 8.5). Now the car’s “trajectory” will be a sigmoidal-type shape composed of two smoothly joined circular arcs as shown. As before β = α/2, but now α = arcsin(1/4) ≈ 14.48°, so β ≈ 7.24°, meaning that the width of the obstacle should not exceed D tan β ≈ 0.127 D for the maneuver to be executable. For a value of D = 100 ft, this is just less than 13 ft, which allows for a few feet of clearance around a large truck.

  Figure 8.5. Modified geometry for turning to avoid a slower vehicle.

  Next we consider two cars approaching an intersection perpendicularly at the same speed, as shown in Figure 8.6. Suppose that each driver instinctively tries to swerve to the side by at least 45°; we will take this as a lower bound, for then they end up moving parallel to each other (if road conditions permit, of course). The angles of the truncated triangle are each 45° and therefore by symmetry the line joining each vertex to the corner of the junction makes an angle exactly half this with the hypotenuse. Recall that the minimum distance required for a “straight stop” is D and that for a circular arc is 2D. Now the radius of the arc shown in Figure 8.6 is r = D cot(π/8) ≈ 2.41D. From equation (8.8) we can compare the corresponding speeds for the circular arc (1) and the 45° swerve (2):

  This implies that v0(2)/v0(1) ≈ 1.1, that is, the speed can be about 10% greater in the 45° swerve.

  Figure 8.6. Two vehicles approaching on an initially perpendicular path.

  A compromise of sorts between the straight stop of length D and the circular turn of radius 2D is the “spiral stop.” This is accomplished by turning and braking simultaneously, and the vehicle will trace out a spiral path. The straight stop is achieved by applying (in the simple case) a constant force at 180° to the direction of motion. The circular path arises when that force is applied at 90° to the direction of motion. When this force is applied at some other constant angle γ to the direction of motion, the result is an equiangular spiral trajectory. In polar coordinates the equation of the path takes the form r1(θ) = r0e(2cotγ)θ, where r0 is a constant (see Appendix 6 for details).

  X = σ : ACCELERATION “NOISE” IN THE CITY

  What factors are important in studying what might be called “traffic engineering”? Clearly, weather and road conditions and the driver’s response to a changing traffic environment are all significant and indeed, interrelated. A twisty two-lane road poses different problems from a six-lane Interstate highway or “main drag” in a city. The effects of increased traffic volume, road repairs, or adjacent building projects on congestion are generally difficult to answer quantitatively, but the concept of acceleration noise—the root-mean-square of the acceleration of a vehicle—is a useful one in determining some answers to questions of this type.

  If v(t) and a(t) are respectively the speed and acceleration of a vehicle at time t, assumed continuous, then the average acceleration over a trip lasting time T is

  It is interesting to note that if the initial and final speeds are the same, then ā = 0. This will certainly be the case if the vehicle starts from rest and stops at the end of the trip!

  The acceleration noise σ is the RMS (root-mean-square) of a(t) − ā, that is,

  Exercise: Show that

  Clearly, when ā = 0, then

  Exercise: Calculate this quantity for several simple analytic choices of a(t).

  Why might this concept be a useful one for traffic engineering? If we think about a car that is driven fairly smoothly (i.e., with no violent acceleration or braking), we would expect the quantity σ to be small (in a sense to be discussed later). If the vehicle is driven with such accelerations and decelerations, σ will be large. Recall that the slope of a speed-time graph at a given point is the acceleration at that point. In effect, the acceleration noise is a measure of the smoothness of the speed-time graph—the smaller σ, the smoother the journey. A narrow, crowded road with sharp turns will give rise, other things being equal, to a higher value of σ. Those reckless drivers (never ourselves, of course) we see so frequently weaving in and out of traffic will engender high σ-values. Instead of wishing to shake a fist at them, or inwardly raging at them, perhaps we should just content ourselves with this fact. At this point, an amusing image comes to mind: after passengers alight from a car, they each in turn raise a card above their heads with an estimate of the σ-value for the just-completed trip!

  In a very interesting article by Jones and Potts (1962), several possible answers to this question are provided, based on experimental data. After extensive investigations in Adelaide and its environs, they were able to draw several conclusions, some of them perhaps not surprising:

  1. Given, say, a two-lane road and a four-lane road through hilly countryside, σ is much greater for the former than the latter.

  2. For roads in hilly countryside, σ is smaller for an uphill journey than for a downhill one.

  3. If two drivers drive at different speeds below the “design speed” of a highway, σ is about the same.

  4. If one or both drivers exceeds the design speed of the highway, σ is higher for the faster driver.

  5. An increase in the volume of traffic increases σ.

  6. Similarly, an increase in traffic congestion resulting from parked cars, frequently stopping buses, cross traffic, pedestrians, etc., increases σ.

  7. A suitably calibrated value of σ may provide a better measure of traffic congestion than travel or stopped times.

  8. High values of σ indicate a potentially dangerous situation; they may be a measure of higher accident rates.

  Naturally we must ask, what are typically high and low values for σ? The authors found that σ = 1.5 ft/s2 is a high value, and σ = 0.7 ft/s2 is a low one.

  Another factor influenced by the size of σ is an economic one: fuel efficiency. This is not just im
portant for trucking companies or individual truckers, of course: it is increasingly important for the average driver as well. Trucks are fitted with tachographs to record the driving behavior of the truckers, and presumably have the effect of providing an incentive to (in effect) lower the value of σ.

  Generally, then, the smoothness of a journey can be measured by the acceleration noise—the standard deviation of the accelerations; furthermore, it is known that the acceleration distribution is essentially normal. It could well be a useful measure of the danger posed by erratic drivers, for whom σ is high. It also increases in tunnels, the reasons probably being narrow lanes, artificial lighting, and confined conditions. However, it is not always sensitive to changing traffic conditions, especially in city centers or major highways leading into them, where traffic congestion is common and the average speeds are low. It is also the case that a given value of σ may correspond to more than one traffic situation, for example, journeys at low speeds in dense traffic or faster journeys interspersed with traffic lights. One possibility to avoid this ambiguity is to use a modification of σ, namely , where is the average speed. may be interpreted as a measure of the mean change in speed per unit distance of the journey.