X and the City: Modeling Aspects of Urban Life

by Adam, John A.

  Question: jerk is defined as the time derivative of the acceleration, that is, j(t) = da/dt. Do you think there is any usefulness to defining a quantity similar acceleration noise (as in equation (8.10), namely jerk noise?

  Figure 8.7. Headway geometry for a two-lane highway.

  X = Th: OVERTAKING IN THE CITY

  We conclude this chapter with a simple mathematical model of overtaking and passing on a straight section of road. In Figure 8.7 a vehicle in the right lane moves into the left lane at speed V1 > V0 to overtake and pass a slower-moving vehicle. The equations, if not the diagram, are independent of which side of the Atlantic the vehicle resides! The headway TH is the time gap between adjacent oncoming vehicles moving at speed V2.

  Clearly,

  Suppose that a reasonable time in which to complete a passing maneuver by a car traveling at 55 mph is 8 seconds, the headway required for an oncoming vehicle traveling at 65 mph in the opposing lane is TH ≈ 15 seconds. For interstates or other multilane highways, the needed gaps are smaller; we just change the sign of V2 (obviously we require V1 < V2). In this case for the same speeds TH ≈ 1.2 seconds!

  Chapter 9

  PROBABILITY IN THE CITY

  There are many topics under the umbrella “road traffic” that are amenable to mathematical investigation. Examples include traffic flow on the open road and at intersections, parking problems, accident rates, design of road systems for new towns and expanding cities, traffic lights and other control systems, transportation and scheduling problems, to name but a few. And as in so many aspects of mathematical modeling, there are two basic choices: to define the problem in a deterministic or a probabilistic context. The former can subdivide farther into continuum or discrete (or equivalently, macroscopic or microscopic) approaches, each with its own advantages and disadvantages. In the continuum approach the flow of traffic is treated as a fluid; properties of individual vehicles (“fluid particles”) are not considered. Such models are often referred to as being kinematic—the motion (as opposed to the “forces” behind that motion) is of prime consideration. By contrast, car-following models are dynamic in the sense that accelerations and decelerations (and by implication, the forces) are implicit in their description. Although it is outside the scope of this book, it is perhaps worth mentioning that the discrete approach is often used to model flocking in starlings, shoaling in fish, and other swarming behavior (in ant, bee, and locust populations, for example).

  One very interesting feature common to both natural and man-made phenomena is the topic of stability vs. instability. Will those waves on the lake grow or die out? Will that crack in the windshield continue to grow or stop? (Almost certainly the former!) Will that traffic bottleneck disappear by the time I get there, or will it get worse? A typical traffic “instability” arises from time lags in the response of a driver to the accelerations and decelerations of the vehicle in front. A small lag can grow as it passes, in a wavelike manner from car to car. Indeed, it is accurate to say that terms like “shock wave” and “expanding wave” are quite appropriate to describe some traffic patterns. Typically the existence of such waves in traffic flow can be deduced from both kinematic (continuous fluid-like) models and discrete (“particle”) car-following models. Most are too detailed for inclusion here, but we can gain some insight into these phenomena by examining some simple “steady-state” models. In this context, the steady state represents an equilibrium or perhaps a neutrally stable solution, much like a ball on a flat table—if disturbed, it will not move away indefinitely (instability) or return to its starting point (stability) but will remain where it is placed. This is neutral stability. But we start with an introduction to the probabilistic approach to traffic flow. And as pedestrians, we should be particularly interested in the gaps between the vehicles! We shall think about the gaps in what follows.

  X = Pr: PROBABILITY IN THE CITY

  Probabilistic (or stochastic) models incorporate, by definition, an element of randomness. This word is not to be understood in the common sense as haphazard; a more precise definition is given below. In this context it can mean that there is a probability distribution for, say, the size of gaps in a line of traffic. We can view traffic or the gaps in traffic as a distribution in space or time. In space, a length of single lane road (for simplicity) will have a distribution of vehicles along it at any given moment in time—a snapshot view. Alternatively we may identify a fixed location on the road with vehicles passing this position as time goes on. The first situation is a distribution of intervals in space, and the second is a distribution in time. Such distributions (or series of events) are termed random provided that [19]

  (i) each event (e.g., vehicle arrival time) is independent of any other, and

  (ii) equal intervals of time (or space) are equally likely to contain equal numbers of events (e.g., vehicles).

  There are several important distributions of interest in traffic flow studies; we shall briefly examine two of them—the Poisson and (displaced) negative exponential distributions. The latter is a simple generalization of a negative exponential distribution. The former gives the probability of a specified number of vehicles along a section of road at a given time, or passing a given point in a certain time interval. The latter provides the probability of a time or distance “gap” of a specified length in a specified time or distance. More precisely, it describes the time between events in a Poisson process, that is, a process in which events occur continuously and independently at a constant average rate. A derivation of the Poisson distribution is given in Appendix 4.

  X = P(t): TRAFFIC GAPS IN THE CITY?

  This is an important question that has direct relevance to a topic mentioned above: pedestrians crossing roads. When I walk to work I have several roads to cross, and not all of them have crosswalks. There have also been quite a few occasions when I am in the middle of a crosswalk and cars go right by me (once it was a police cruiser that nearly knocked me down). There is a flavor of probability theory in this chapter, but not to worry, the applications we’ll be making are very straightforward.

  We’ll call P(t) the probability that no vehicle passes a certain point in a time interval t. Suppose that, over another period of time T a large number N of cars pass that same point. What do we mean by large in this context? Let’s take N > 100. The average number of cars in a time interval t is then n = Nt/T. For events that are equally likely to occur at any time, the distribution of times between the events is well described exponential distribution. For example, it is often used for modeling the behavior of items with a constant failure rate. It also has the advantage of taking a simple mathematical form. We define the exponential distribution

  P(t) also describes the probability that there is a gap of at least t seconds between the passage of any two consecutive vehicles. Therefore for N cars the average number of gaps ≥ t is given by

  An old survey [19] (see Gerlough 1955) of traffic gaps in the Pasadena (Arroyo Seco) Freeway over a period of 1753 seconds (!) in one lane found that 214 cars passed the observation point. By fitting a curve to the experimental data, that is, a plot of the number of gaps of length L or greater vs. the length of the gaps, the traffic researchers found quite good agreement between the data and the curve predicted by equation (9.2), namely

  X = N(W): GAPS AT SCHOOL CROSSINGS

  Similar principles apply here, though a little preliminary work is necessary. What is the average walking speed of a school-aged child? Let’s take the arithmetic mean of 1 mph (for the very young, or dawdlers!) and 3 mph for older children. Now

  by the way; a result used in Chapter 6. If the width of the street in feet is W, then the time to cross the street is tc = W/3 seconds. Of course, school crossings generally have guards who halt the traffic when the build-up of waiting children is sufficiently large. We shall dispense with the guards here and rely on natural gaps in the traffic to permit an opportunity to cross, say on average once every minute. We’ll assume that the traffic flow past the c
rossing is governed by a distribution with a flow rate of N vehicles per hour. We wish to find the maximum flow rate, Nmax, which would permit the above crossing opportunity. If N > Nmax then we’ll reinstate the crossing guards!

  The expected (or average) number of vehicles in the crossing tc is Ntc/3600, so as above, the probability that no vehicle will pass in that interval is

  For k successive intervals of this length, on average kP of these will have no vehicular traffic, and for there to be one of these, k = P−1. These k intervals correspond to a time . Therefore

  and so

  that is, if = 60s (corresponding to at least one crossing opportunity per minute),

  A graph of this function is shown in Figure 9.1, drawn for a minimum width W = 20 ft. Notice how rapidly the maximum permissible flow rate decreases with the width of the street, particularly in the 20–40 ft range. It makes much sense then, for crossing guards to be in place especially in the vicinity of schools located near large highways.

  We return to the question: how might the vehicles be distributed (in time) along the roadway? One possibility is the above-mentioned Poisson distribution (Chapter 3; see also Appendix 4), from which the probability of n arrivals in unit time is

  Another possible model is the second one described above: the displaced exponential distribution with density function

  From this, the probability of encountering a gap exceeding a given time interval T is

  Figure 9.1. Maximum flow rate vs. width of road.

  The constant a is the minimum time gap (≈ 1 second for example). For a traffic flow rate of 600 vehicles per hour we can take λ = 1/6 vehicle/second. Suppose that a (fairly nimble) pedestrian requires a gap of at least T = 6 seconds to successfully cross the road; then with a = 1 the corresponding probability of being able to do so is e−5/6 ≈ 0.43. Note also that

  Increasing volume of traffic has obvious consequences. It generally causes a reduction in the mean speed of vehicles and also can affect the mean spacing between them. If this mean spacing is measured in terms of time rather than distance, as introduced at the end of Chapter 8, it is called a headway.

  It’s time for a little integration practice; let’s combine it with some useful definitions. This will enable us to derive in a straightforward manner some results of interest in studies of pedestrian delays and minor-road delays to vehicles [19]. As is standard practice in probability theory, we define the expectation E(x) for a continuous random variable taking on values in (b, c) with probability density function (p.d.f.) f(x); it is

  Note that by definition of a p.d.f., the integral , so that for future reference we can write the expectation value as

  The expectation can be thought of as a mean value of the random variable. We shall consider the above displaced exponential distribution, which gives the distribution of lengths of intervals ≥ a between vehicles (headways). We can therefore calculate the mean headway time for all vehicles as

  Obviously this reduces to λ−1 when a = 0, as it should (recall that λ is just the mean number of vehicles arriving in unit time). There are several other related properties of this distribution that are of interest to traffic engineers. The proportion of intervals in the interval (a, t) is

  The proportion of intervals > t seconds is therefore e−λ(t−a). The proportion of time occupied by intervals ≤ t seconds is the weighted average

  Not surprisingly, the proportion of time occupied by intervals > t seconds is

  Finally, the mean headway time for all intervals ≤ t seconds is given by the expression

  The corresponding result for all intervals greater than t (> a) seconds is

  which is a result (not surprisingly) independent of a.

  Exercise: Practice your integration by verifying equations (9.9)–(9.11).

  Negative exponential distributions are sometimes considered even more important than the Poisson distribution in traffic flow, since they provide information about headways. Realistically, the hypothesis of random traffic distributions best describes situations where the traffic flow is light and vehicles can pass freely. Then the vehicles can be considered to be approximately randomly distributed along a road. In practice, however, as drivers well know, passing can be partially restricted by other vehicles in the passing lane(s), bends in the road, brows of hills, and so on. Some drivers who catch up to cars moving slightly less fast are content to stay behind them (though I find it a little frustrating), so bunching of vehicles is very common. Under these circumstances other distributions are more relevant to understanding traffic flow: for example the intervals between the “endpoints” of such bunches may follow a negative exponential or other distribution.

  Chapter 10

  TRAFFIC IN THE CITY

  I hooked up my accelerator pedal in my car to my brake lights. I hit the gas, people behind me stop, and I’m gone.

  —Steven Wright

  X = q: KINEMATICS IN THE CITY

  There is a fundamental relationship between the flow of traffic q in vehicles per unit time, the concentration k in vehicles per unit distance, and the speed u of the traffic. It is q = ku. In general each of these quantities is a function of distance (x) and time (t), but the form q(k) = ku(k) may also be valuable. Another useful quantity is the spacing per vehicle, s = k−1. If s0 is the minimum possible spacing, that is, when the vehicles are stationary (or almost so), then is referred to as the jam concentration; but it has nothing to do with preservatives! Associated with the above “fundamental relationship” q = ku is, not surprisingly, a “fundamental diagram.” The overall features can be inferred as follows. When the concentration is zero, the flow must be zero, so q(0) = 0. Furthermore, the flow is zero when k = kj, so q(kj) = 0. Since q ≥ 0, ruling out the trivial case q ≡ 0 there must be an absolute maximum q = qmax somewhere in the interval (0, kj). This is obviously of interest to traffic engineers (and indirectly, to those in traffic).

  There may be more than one relative maximum of course, but the case we’ll examine will have a single maximum—the “capacity” of the road. Figure 10.1 shows a typical q-k diagram for this situation.

  Suppose we take two measurements of the flow q(x, t) a short distance Δx apart, at points A and B, respectively (traffic moving from A to B). The flow is defined to be the number of vehicles passing a given location in time Δt; hence the change in q between the points A and B is given by

  Within Δx the change in the traffic density (or concentration) k is

  To see this, suppose without loss of generality that NA > NB, meaning that there is a build-up of cars between A and B, so the density in that spatial interval increases, that is, Δk > 0, as indicated above in this case. It is assumed of course that there is no creation or loss of cars from within the interval (no white holes, sinkholes, or UFO abductions)—the total number of cars is constant. From these two equations for ΔN we see that

  or

  Now we invoke the continuum hypothesis (some shortcomings of which are discussed in Chapter 15 and Appendix 8); we assume the quotients above possess well-defined limits as the discrete increments tend to zero, thus obtaining the limiting equation

  Figure 10.1. Flow-concentration diagram. The maximum is at the point of tangency (km, qmax).

  This is the equation of continuity for the kinematic model. Note that it can be adapted to include the effects of entrances, exits and intersections, etc. by adding a term, g(x, t) say, to the right hand side. We will examine some simple consequences of equation (10.1); suppose that q = q(k); assuming the differentiability of q we have that

  In the simplest possible case dq/dk = c, a constant, so the resulting equation is

  This has the general solution

  as is readily confirmed using the chain rule. In equation (10.4), h is a differentiable but otherwise arbitrary function. Its form depends on the so-called “initial conditions” at time t = 0 (say).

  The solution (10.4) implies that k (and therefore q) travels to the right with “shape” h and speed c. If we rec
all that the mean speed of vehicles at a point is u = q/k then it follows that

  This is directly analogous to the relationship between the speed of individual waves in a medium and the speed of a group of them (a “wave packet”) in fluid dynamics. If u increases with traffic density (unlikely), then c >