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70. A Nearly Uniform Traffic Flow Example In this section another type of traffic problem involving a nearly uniform traffic density will be solved. Suppose that the initial traffic density is constant for the semi-infinite expressway illustrated in Fig. 70-1. How many cars per hour would have to continually enter in order for the traffic flow to remainuniform ? The traffic flow at the entrance must be p0u(po), the flow corresponding to the uniform density p0. To prove this statement (th
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  70. A Nearly Uniform Traffic Flow Example In this section another type of traffic problem involving a nearly uniform traffic density will be solved. Suppose that the initial traffic density is constant for the semi-infinite expressway illustrated in Fig. 70-1. How many cars per hour would have to continually enter in order for the traffic flow to remainuniform ? The traffic flow at the entrance must be  p 0  u(po), the flow corresponding to the uniform density  p 0  . To prove this statement (though to many of you a mathematical proof of this should not be necessary), consider the interval of roadway between the entrance and the point  x = a. Using the integral conservation of cars, Since the traffic density is prescribed to be constant, the left hand side is zero. Thus the flow at  x —  a must be the same as the flow at the entrance q(a, t) = g(0, t). But the flow at  x = a is  p 0  u(po). Thus q(Q, t) = /> 0 w(/> 0 )-In other words, the flow in must equal the flow out, as the number of cars in between stays the same assuming constant density. However, suppose that the flow in of cars is slightly different (and varies in time) from that flow necessary for a uniform density, with #,(/) known. What is the resulting traffic density ? The partial differential equation is the same as before: being derived from The traffic is assumed initially to be uniform, so that the initial condition is (This could be generalized to also include initial densities that vary slightly from the uniform case.) Note that the initial condition is only valid for  x > 0 (rather than in the previous sections in which —  oo <  x < oo). The initial condition must be supplemented by the flow condition, equation 70.1, called a boundary condition since it occurs at the boundary of the roadway, the entrance to the expressway at  x = 0. The general solution to the partial differential equation has already been Obtained or equivalently Let us use the concepts of characteristics assuming light traffic, i.e., c > 0 (heavy traffic is discussed in the exercises). The characteristics are the lines  x —  ct = constant, sketched in Fig. 70-2. The density  p { is constant along Figure 70-2 Characteristics along which the density is constant. these lines. Hence, in the shaded region in Fig. 70-2, the density  p l = 0 or the total density  p = p 0  , since  p = p 0 at t = 0. The unshaded region is where on the highway it is noticed that cars are entering at a nonuniform rate. In this region the traffic density only differs slightly from a uniform density, equation 70.3. What is the density of cars if the density remains the same moving at speed cl From the diagram in Fig. 70-2, the traffic density at (x, t) is the same as the traffic density at the entrance at a time  x/c earlier,  xfc is the time it takes a wave to move a distance  x at speed c. Thus the density at the entrance at time t —  (x/c) yields the density  x miles along the roadway at time t. The traffic density at the entrance can be determined since the traffic flow is prescribed there (use equation 70.1 assuming  p is near />„)ã  The traffic flow, q(p) = q(p 0 + €g), may be expressed using Taylor series methods, The traffic flow is approximated by since c = q'(po). Thus the perturbed traffic flow is simply c times the perturbed  density. Since the perturbed traffic flow is known at the entrance, q v  (t), then and thus by letting z = —  ct Consequently the total car density is given by equation 70.3 as In summary This solution clearly indicates that information (that the traffic is entering at  x = 0) is propagated at a velocity c, and hence at position  x the information has taken time  x/c to travel. 71. Nonuniform Traffic  —  The Method of Characteristics The nonlinear first-order partial differential equation derived from conservation of cars and the Fundamental Diagram of Road Traffic is In the previous sections we considered approximate solutions to this equation in cases in which the density is nearly uniform. The traffic was shown to vary via density waves. We will find the techniques of nearly uniform traffic density to be of great assistance. Again consider an observer moving in some prescribed fashion  x(f). The density of traffic at the observer changes in time as the observer moves about, By comparing equation 71.1 to equation 71.2, it is seen that the density will remain constant from the observer's viewpoint, or  p is a constant, if For this to occur the observer must move at the velocity q'(p), the velocity at which nearly uniform traffic density waves propagate. Since this velocity depends on the density (which may dramatically vary from one section of roadway to another), this velocity is called the local wave velocity. If the observer moves at the local wave velocity, then the traffic density will appear constant to that observer. Thus there exist certain motions for which an observer will measure a constant traffic density, as shown in Fig. 71-1. Since equations 71.3 and 71.4 are ordinary differential equations, these curves are again called characteristics.  Along a characteristic,  p is constant; the density is the same density as it is at the position at which the characteristic intersects the initial data. In the case of nearly uniform flow, and thus all the curves (characteristics) were parallel straight lines. In nonuniform traffic flow, the observer moves at the local wave velocity. For each observer, the traffic density remains the same, and therefore the local wave velocity for this observer remains the same! The velocity at which each observer moves is constant! Each observer moves at a constant velocity, but different observers may move at different constant velocities, since they may start with different initial traffic densities. Each moves at its own local wave velocity. Each characteristic is thus a straight line as in the case of nearly uniform flow. However, the slopes (related to the velocities) of different characteristics may be different. The characteristics may not be parallel straight lines. Consider the characteristic which is initially at the position  x = a on the  highway, as shown in Fig. 71-2. Along the curve dx/dt = q'(p), dp\dt = 0 or  p is constant. Initially  p equals the value at  x = a (i.e., at t = 0). Thus along this one characteristic, which is a known constant. The local wave velocity which determines the characteristic is a constant, dx/dt = q '(/>«). Consequently, this characteristic is a straight line, where k, the x-intercept of this characteristic, equals a since at / = 0,  x = a. Thus the equation for this one characteristic is  Along this straight line, the traffic density  p is a constant, Similarly, for the characteristic initially emanating from  x = ft, also a straight line characteristic, but with a different slope (and corresponding different velocity) if #'(/>«) =£ q'(pp)- Thus, for example, we have Fig. 71-3. Figure 71 -3 Possibly nonparallel straight line characteristics. In this manner the density of cars at a future time can be predicted. To determine the density at some later time t = t* at a particular place  x = x#, the characteristic that goes through that space-time point must be obtained (see Fig. 71-4). If we are able to determine such a characteristic, then since the density is constant along the characteristic, the density of the desired point is given by the density at the appropriate jc-intercept, This technique is called the method of characteristics. Figure 71-4 Using characteristics to determine the future traffic density. The density wave velocity, dqjdp, is extremely important. At this velocity the traffic density stays the same. Let us describe some properties of this density wave velocity. We have assumed dqfdp decreases as  p increases (see Fig. 63-3); the density wave velocity decreases as the traffic becomes denser. Furthermore, we will now show a relationship between the two velocities, density wave velocity and car velocity. To do so the characteristic velocity is conveniently expressed in terms of the traffic velocity and density. Since we know q = pu(p), du/dp < 0 by the srcinal hypothesis that cars slow down as the traffic density increases, see Fig. 71-5. (Equality above is valid only in very light traffic when speed limits, rather than the interaction with other cars, control an auto's velocity.) Consequently, dqfdp < u, that is the density of automobiles (or density wave) always moves at a slower velocity than the cars themselves! 72. After a Traffic Light Turns Green In the past sections the intent has been to develop in each reader a sufficient understanding of the assumptions under which we have formulated a mathematical model of traffic. The time has come to solve some problems and explain what kinds of qualitative and quantitative information the model yields. In this section we will formulate and solve one such interesting problem. Suppose that traffic is lined up behind a red traffic light (or behind a railroad crossing, with a train stopping traffic). We call the position of the traffic light  x = 0. Since the cars are bumper to bumper behind the traffic light,  p = /? max for  x < 0. Assume that the cars are lined up indefinitely and, of course, are not moving. (In reality the line is finite, but could be very long.  Our analysis is limited then to times and places at which the effects of a thinning of the waiting line can be ignored.) If the light stops traffic long enough, then we may also assume that there is no traffic ahead of the light,  p = 0 for  x > 0. Thus the initial traffic density distribution is as sketched in Fig. 72-1 Suppose that at / = 0, the traffic light turns from red to green. What is the density of cars for all later times? The partial differential equation describing conservation of cars, must be solved with the initial condition Note the initial condition is a discontinuous function. Before solving this problem, can we guess what happens from our own observations of this type of traffic situation ? We know that as soon as the light turns green, the traffic starts to thin out, but sufficiently far behind the light, traffic hasn't started to move even after the light changes. Thus we expect the density to be as illustrated in Fig. 72-2. Traffic is less dense further ahead on the road; the density is becoming thinner or rarefied and the corresponding solution will be called a rarefactive wave. Figure 72-2 Traffic density: expected qualitative behavior after red light turns green. We will show the solution of our mathematical model yields this type of result. Partial differential equation 72.1 may be solved by the method of characteristics as discussed in Sec. 71. As a brief review, note that ifdx/dt =  p(x, t) is constant along the characteristics, which are given by dq/dp, then dpjdt = (dp/dt) + (dx/df)(dp/dx) = 0. Thus the traffic density  p(x, t) is constant along the characteristics, which are given by The density propagates at the velocity dqjdp. Since  p remains constant, the density moves at a constant velocity. The characteristics are straight lines. In the  x-t plane where each characteristic may have a different integration constant k. Let us analyze all characteristics that intersect the initial data at  x > 0. There  p(x, 0) = 0. Thus  p = 0 along all lines such that where this velocity has been evaluated using equation 72.2. The characteristic velocity for zero density is always w max , the car velocity for zero density. The characteristic curves which intersect the *-axis for  x > 0 are all straight lines with velocity t/ max . Hence the characteristic which emanates from  x = x  0 (x  0 > 0) at t = 0 is given by Various of these characteristics are sketched in Fig. 72-3. The first characteristic in this region starts at  x = 0 and hence  x = « max f. Thus below in the lined region (x > w max O, the density is zero; that is no cars have reached that region. At a fixed time if one is sufficiently far from the traffic light, then no cars have yet arrived and hence the density is zero. In fact imagine you are in the first car. As soon as the light changes you observe zero density ahead of you, and therefore in this model you accelerate instantaneously to the speed Wmax- You would not reach the point  x until t —   Jt/w max , and thus there would be no cars at  x for t < #/w max . Figure 72-3 Characteristics corresponding to no traffic. Now we analyze the characteristics that intersect the initial data for  x < 0, where the cars are standing still being at maximum density,  p = /> m «. P = />ma* along these characteristics determined from equation 72.2 where we have used the fact that u(p mtK   ) = 0. This velocity is negative since
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