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Delays caused by a queue at a freeway exit ramp
Transportation Research Part B 33 (1999) 337±350
Delays caused by a queue at a freeway exit ramp G.F. Newell* Institute of Transportation Studies, 109 McLaughlin Hall, University of California, Berkeley, CA 94720, USA Received 29 January 1998; received in revised form 11 August 1998; accepted 17 August 1998
Abstract We consider here what happens to trac on a freeway when a queue from an exit ramp backs onto the freeway causing a partial blockage of the right lane. Exiting vehicles are con®ned to the right lane but through vehicles can travel in any lane. The two vehicle types interact but their queues must be treated separately. This illustrates a special case of a model of ``freeways with special lanes'' formulated by Daganzo (1997). Whereas Daganzo presented a numerical scheme of calculating ¯ows, the emphasis here is on graphical evaluation of the complete evolution of the queues. The graphical solution more clearly illustrates the practical issues, at least in this special situation. # 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction Our goal is to describe what happens to trac on a freeway if a queue from an exit ramp should back onto the freeway causing a partial blockage of the right lane (or lanes if trac can exit from more than one freeway lane). The geometry is shown schematically in Fig. 1. We consider two types of vehicles. Through vehicles will be designated as type 1 vehicles and those which will exit the freeway as type 2 vehicles. Type 1 vehicles can travel in any freeway lane but type 2 are assumed to stay in the right lane throughout the length of freeway section under consideration. This study was motivated by papers of Daganzo (1997) and Daganzo et al. (1997) which analyzed freeway trac with ``special lanes''. This example of an exit ramp was used by Daganzo as an illustration of their general method of simulating trac ¯ow with two vehicle types one of which was excluded from using certain freeway lanes (the 2-vehicles cannot use the left lanes). Whereas Daganzo's paper dealt mostly with computational methods appropriate for computer simulation of a general class of ¯ow patterns, the present paper deals mainly with graphical * Tel.: +1-510-642-3585; fax: +1-510-643-8919. 0191-2615/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0191-2615(98)00039-3
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Fig. 1. A freeway o-ramp.
solution methods and interpretation of issues for this particular example. Some of the methods described here, however, could be extended to other situations with two or more vehicle types. In Section 2, we describe how the two vehicle types react when a queue backs onto the freeway from the exit ramp, and some of the issues. In Sections 3 and 4 we show how one can determine delays for the two vehicle types from postulates about capacity constraints alone, without specifying anything about vehicle dynamics. To determine where the vehicles are stored and the location of shocks, however, one must introduce some model of kinematic waves (as in Daganzo's work). This is done in Section 5 using graphical techniques similar to those described by Newell (1993). 2. Description Prior to the time that the exit queue backs onto the freeway, we assume that some 1-vehicles were traveling in the right lane and that the velocities of the vehicles are equal in all lanes. Either all vehicles are traveling at the free-¯ow velocity or there is some bottleneck downstream of the exit which constrains the ¯ow of the 1-vehicles and, therefore, also the velocity of both the 1- and 2-vehicles upstream of the exit. When the queue of 2-vehicles backs onto the right lane of the freeway (illustrated in Fig. 1 by the single hatched area), the velocity of the vehicles drops as they enter the queued section. If there were any 1-vehicles caught in this queue, they would see that the velocity is higher in the left lanes and they would squeeze into the left lanes. Near the back of this queue of 2-vehicles there might be some short section of the highway in which any 1-vehicles, which had been traveling in the right lane upstream of the queue, merge into the left lanes to avoid the queue. The width of
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this merge section should be (nearly) independent of the length of the queued section. We might idealize this by imagining that this merge section has a negligible width, essentially zero (a shock). If the rate at which 1-vehicles wish to pass this queue section is less than the capacity of the left lanes, the 1 -vehicles will suer a negligible delay due to the queue of 2-vehicles. They will just bypass the queue. The 1-vehicle ¯ow past the exit will be dictated either by the arriving ¯ow from upstream if the 1-vehicles are traveling at the free-¯ow velocity, or by the capacity of a downstream bottleneck if this constrained ¯ow is less than the capacity of the left lanes. To determine delays to 2-vehicles in this case one can disregard the 1-vehicles and treat this as a conventional queuing problem for just the 2-vehicles, constrained by some speci®ed rate at which they can exit the freeway. If, however, the arrival rate of 1-vehicles to this freeway section exceeds the capacity of the left lanes, a queue of 1-vehicles will develop (illustrated in Fig. 1 by the crosshatched region) even if there is sucient excess capacity in the right lane to accommodate any 1-vehicles that cannot be carried in the left lanes. Note that the restriction on the 2-vehicles is assumed to be somewhere on the exit ramp, not on the freeway itself, so the right lane could accommodate some 1-vehicles if they wished to use it. There will still be a section of freeway containing 2-vehicles in the right lane waiting in a queue to exit the freeway. If there were any 1-vehicles in this right lane section and they saw that the velocity was higher in the left lanes, they would still squeeze into the left lanes but at the expense of vehicles already in the left lanes, because the ¯ow on the left lanes cannot exceed the capacity of these lanes. If some vehicles did squeeze in, this would cause the density in the left lanes to be higher than that for maximum ¯ow. But this disturbance would have a negative wave velocity and force the ¯ow in the left lanes to decrease upsteam of the entering vehicles to make room. So we conclude that there will be no 1-vehicles in the 2-vehicle queue in the right lane, but the left lanes adjacent to this 2vehicle queue will operate at capacity and at a velocity probably nearly equal to the free ¯ow velocity. This is a clear example of the dierence between a ``user optimal'' strategy and a ``system optimal'' strategy. A system optimal strategy would assign some 1-vehicles to the right lane to use any excess capacity in the right lane even though the travel time is longer than in the left lanes. A person in a 1-vehicle, however, when he reaches the back end of the 2-vehicle queue will choose the fastest lane from this point (even though he may have waited in a queue to reach this point), at the expense of others who may be waiting in a queue. The queue of 1-vehicles which develops upstream of the 2-vehicle queue will also contain some 2-vehicles, but we still refer to it as a 1-vehicle queue because the 2-vehicles in this queue are not constrained by the o-ramp ¯ow. They are simply being delayed in reaching their own queue where they will wait some more. 3. Evaluation of delays From the rules of behavior described above one can evaluate the delays caused by the exit blockage without specifying any actual equations of motion or determining where the queues are located, provided that the queues caused by the blockage do not back up past another on-ramp or o-ramp so as to distort the mix of trac entering the freeway upstream and provided that there is no interaction with possible bottlenecks downstream.
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We treat trac as a continous deterministic ¯uid (disregarding the integer nature of vehicle counts and stochastic ¯uctuations) as in Newell (1993). Let D1 t; D2 t = cumulative number of 1-vehicles, 2-vehicles which actually pass the exit ramp by time t starting from the time t 0 when the exit queue ®rst backs onto the freeway. A1 t; A2 t = cumulative number of 1-vehicles, 2-vehicles which would have passed the exit ramp by time t if there were no exit queue. The A1 t; A2 t might be determined from the arrivals upstream or from what could pass some downstream bottlenecks, but the A1 t; A2 t are presumed to be speci®ed for all t. If one could determine the curves D1 t and D2 t then one would know the delays caused by the exit queue. The 1-vehicles are assumed to be served in a ®rst come ®rst served order among themselves so the delay to some nth 1-vehicles (caused by the exit queue) would be the horizontal distance at height n between the A1 t and D1 t curves. Similarly for the 2-vehicles. One would also know the hypothetical ``vertical'' queues at time t, the dierences A1 t ÿ D1 t, A2 t ÿ D2 t. The blockage is assumed to be caused by some bottleneck on the exit ramp, an accident or a trac signal at the downstream end of ramp. Whatever it may be, it will presumably dictate the maximum number of vehicles which can leave the freeway. In particular it will determine the D2 t from time 0 until the exit queue vanishes on the freeway. We do not know yet, however, when the 2-vehicle queue vanishes or what happens thereafter, because, if a 1-vehicle queue forms, this will also delay the arrival of 2-vehicles to the ramp. If dA1 t=dt < Q1 , the capacity of the left lanes, then there is no 1-vehicle queue, D1 t A1 t. The 1-vehicles have no eect on the 2-vehicle queue and the 2-vehicle queue vanishes when D2 t A2 t. If, however, dA1 t=dt > Q1 at t 0, then D1 t Q1 t at least until either the 2vehicle queue or the 1-vehicle queue vanishes. In Fig. 2 , the ``given'' data, A1 t and A2 t for all t and the D1 t; D2 t until some queue vanishes are illustrated by the thick lines. In typical application these curves would be smooth but with time-dependent slopes. For purposes of illustration, however, we represent them in Fig. 2 by some (arbitrary) piecewise linear curves. Actually, in Fig. 2, the arrival rates are nearly contant, but the D2 t behaves as if there were some obstruction on the exit ramp which appears as t 0 and is later suddenly removed. The following graphical constructions, however, apply to any arbitrary shapes of these given curves. To determine when the 2-vehicle queue vanishes if there is a 1-vehicle queue, we must determine how much the 2-vehicles are delayed by the 1-vehicle queue. In Fig. 2, the delay to any nth 1-vehicle is equal to the horizontal distance between the curves A1 t and D1 t at height n, the time displacement between points 1 and 3. Suppose that a 1vehicle suers negligible delay (compared with the queuing delay) in traversing the highway section adjacent to the 2-vehicle queue (see Fig. 1), i.e. the velocity at capacity ¯ow is nearly equal to the free-¯ow velocity. The (known) delay to the 1-vehicles occurs (almost) entirely upstream of the 2-vehicle queue. In this 1-vehicle queuing section upstream of the 2-vehicle queue, the 1-vehicles will be distributed over all lanes of the freeway so that the velocity is the same in all lanes (otherwise the 1-vehicles would shift to the faster lane). The 2-vehicles in this section will also travel at the same
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velocity as the 1-vehicles. Thus an n0 th 2-vehicle which enters the 1-vehicle queue at the same time as the nth 1-vehicle will have experienced the same delay as the nth 1-vehicle by the time it reaches the rear of the 2-vehicle queue. In the terminology of queuing theory, the 1-vehicle queue section acts like a ®rst-in-®rst-out (FIFO) server which serves 1-vehicles at a ®xed rate Q1 (independent of the mix of 1- and 2-vehicles).
Fig. 2. Cumulative arrival and departure curves.
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Regardless of whether the A1 t; A2 t curves were determined from arrivals upstream or, under congested conditions, by some bottleneck downstream, it is assumed that, in the absence of the exit queue, the 1-vehicles and 2-vehicles would travel at the same velocity everywhere within this section of freeway. Thus, in Fig. 2, if the nth arriving 1-vehicle is identi®ed as point 1 on the curve A1 t, the n0 th 2-vehicle is identi®ed as point 2 on the curve A2 t at the same time. The nth 1-vehicle reaches the rear of the 2-vehicle queue at the time corresponding to point 3 on the D1 t curve. Actually this is a hypothetical 2-vehicle queue which occupies no space or, more precisely, one for which the free ¯ow trip time over the 2-vehicle queue is zero. The n0 th 2-vehicle reaches this point at the same time at point 4. It reaches the exit at point 5. The horizontal distance between points 2 and 4 is the delay experienced by the nth 2-vehicle in the 1-vehicle queue and the horizontal distance between points 4 and 5 is the delay experienced in the 2-vehicle queue. As one moves point 1 along the curve A1 t, the locus of point 4 generates a curve A2 t shown by the broken line
A2 t = cumulative number of 2-vehicles to reach the end of the 2-vehicle queue by time t after passing through the 1-vehicle queue (if the free ¯ow trip time over the 2vehicle queue is disregarded). Actually, this construction is not always valid. It is possible that the broken line curve of Fig. 2 evaluated in this manner will lie below D2 t even immediately after t 0. But this cannot be correct. What happens in this case is that the ¯ow of 1-vehicles past the exit drops suddenly when the queue of 2-vehicles starts to back onto the freeway. But this, in turn, also causes a sudden drop in the ¯ow of 2-vehicles approaching the exit. If the reduced ¯ow of 2-vehicles is less than the given rate at which 2-vehicles can leave the exit, the 2-vehicle queue will disappear and the ¯ow of 1-vehicles will go back up. What must ®nally evolve is that a short queue of 2-vehicles forms on the freeway. Some of the 1-vehicles which were traveling in the right lane may no longer care to switch lanes for only a short distance. The queue of 2-vehicles will stabilize at such a length that the ¯ow of 1-vehicles past the exit will generate a ¯ow of 2-vehicles equal to the exit rate. The modi®cation in Fig. 2 is that the curve A2 t must now (nearly) follow the curve D2 t, i.e. points 4 and 5 coincide, but the trip times of the 1- and 2-vehicles in the 1-vehicle queue are still equal. The displacement 1±3 is equal to that of 2±4, but this now determines the locus of points 3, i.e. a curve D1 t with slope larger than Q1 . The condition that a 2-vehicle queue forms at t 0 and continues to grow depends on the ¯ows immediately after t 0. If we let Qe = exit ¯ow at t 0 dD2 t=dt at t 0 q2 = arrival rate of 2-vehicles at t 0 dA2 =dt at t 0 q1 = arrival rate of 1-vehicle at t 0 and denotes the time of points 1 and 2 of Fig. 2 soon after t 0, then the time of point 5 is q2 =Qe and the time of point 3 is q1 =Q1 . The condition that point 5 is later than point 3 (a 2vehicle queue forms) is that 1 < q1 =Q1 < q2 =Qe0 :
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If, however, 1 < q2 =Qe < q1 =Q1 , the time coordinate of point 3 is equal to that of point 5, which means that the ¯ow of 1-vehicles past the exit is q1 = q2 =Qe q1 =q2 Qe ; (which is longer than Q1 but less than q1 ). If there is a 2-vehicle queue, it will eventually vanish when the curve D2 t meets the curve A2 t. If the 1-vehicle queue vanishes ®rst, the A2 t curve intersects and joins the curve A2 t before the D2 t meets it, but suppose that there is a 1-vehicle queue when the 2-vehicle queue vanishes, as illustrated in Fig. 2. When the 2-vehicle queue vanishes, the total ¯ow past the exit can potentially jump to the combined capacity of the right and left lanes with the 1-vehicles using any residual capacity of the right lanes not used by the 2-vehicles. To analyze the subsequent evolution one should also draw curves for A2 t A2 t and D1 t D2 t by adding the known component curves until the time the 2-vehicle queue vanishes at point 40 , 50 of Fig. 2. If the total ¯ow at point 70 on the curve D1 t D2 t can increase to the capacity of all lanes, we will know the new slope of this curve beyond point 70 . We can then construct the rest of this curve until the combined queue vanishes when this curve meets the curve A1 t A2 t at point 8. From the curves for the combined ¯ows, one can now construct the curves for the component ¯ows because the trip time through the 1-vehicle queue is the same for either the 1- or 2-vehicles. At the time the 2-vehicle queue vanished, the horizontal distance between point 20 and 40 ,50 was equal to that between points 10 and 30 , but also between points 60 and 70 on the combined curves. This condition remains true. At a later time, the distance between points 100 and 200 or 300 and 400 are equal to the known distance from 600 to 700 . The locus of points 200 and 400 de®ne the curves D1 t and D2 t as shown by the broken lines until all queues disappear. There is a possible modi®cation of this solution analogues to that which can occur at t 0 as described previously. When the obstruction is removed there will be a sudden increase in the ¯ow past the exit for both the 1- and 2-vehicles. The slope of the line 40 ±400 , however, cannot exceed the slope of D2 t at point 40 because that would imply that the 2-vehicle queue backs onto the freeway again. If this happens a small queue must form so as to restrict the combined ¯ow just enough that the 2-vehicle ¯ow matches the exit rate. In this case the graphical construction is similar to that shown in Fig. 2 except that the curve D2 t after point 40 remains (for a while) as the given cumulative ¯ow which can exit the freeway, i.e. the 2-vehicle queue is kept at essentially zero. The horizontal distance 200 ±400 in Fig. 2 is, therefore, known and this determines the distances 100 ±300 and 600 ±700 and thus the locus of points 200 and 700 until such time that the resulting slope of D1 t D2 t reaches the combined capacity of all lanes or the queue vanishes. If the former happens ®rst, then the subsequent graphical construction is determined from the constraint on the slope of D1 t D2 t as in Fig. 2. 4. Some complications In view of the complexity of Daganzo's analysis of all possible boundary conditions for ¯ow patterns with special lanes one certainly would not expect the simple graphical solution of Fig. 2 to tell the complete story even as applied to this illustration of a queue backing onto the freeway.
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We discuss here a few possible complications but we do not attempt to give a complete analysis of everything that could happen. 4.1. Congested trac The implication in Section 3 is that this graphical construction is valid if the freeway was initially uncongested and the A1 t; A2 t were determined from inputs to the freeway upstream, or the freeway was initially congested due to some bottleneck downstream. In the latter case, the construction in Fig. 2 may be valid up to a certain point but the interpretation of the results may be quite dierent. Suppose that the downstream bottleneck is immediately downstream of the exit, i.e. the prior bottleneck was on the through lanes at the exit ramp possibly due to a lane drop. If the exit queue backs onto the freeway this will certainly (further) delay the 2-vehicles and, if a 1-vehicle queue forms, it will also delay the 1-vehicles at least temporarily. The issue is what happens after the 2vehicle queue vanishes (at point 40 , 50 of Fig. 2). The constraint after point 40 , 50 may not be the capacity of all lanes as illustrated in Fig. 2, but the capacity of the through lanes immediately downstream due to the prior bottleneck. If this capacity is Q1 (with Q1 > Q1 ), the slope of D1 t after point 30 will be Q1 and this determines the curve D1 t after point 30 . The time interval 200 to 400 is still equal to the known time between points 100 and 300 and this will determine the locus of point 400 , i.e. the D2 t after point 40 , 50 (provided that the resulting slope of the D2 t does not exceed the given slope at point 40 , 50 ). If, however, the curve A1 t was dictated by the downstream bottleneck of capacity Q1 , then the slope of A1 t is also Q1 and the curves A1 t; D1 t become parallel. The eect of the exit queue persists until the end of the period of congestion when there is no longer a queue caused by the downstream bottleneck. What is happening in this case is that the temporary bottleneck caused by the exit queue chokes o the ¯ow to the downstream bottleneck. There is a loss of utilization of the downstream bottleneck which cannot be recovered. Suppose, on the other hand, that the downstream bottleneck is far downstream and that the D1 t; D2 t proceed as in Fig. 2 at least for a while after point 40 , 50 . When the 1-vehicle queue formed at time 0, the freeway is assumed to be congested between the exit ramp and the downstream bottleneck, but if the downstream bottleneck has a capacity Q1 > Q1 , the new bottleneck will choke o the ¯ow downstream and the queue between the two bottlenecks will start to dissipate. The 1-vehicles passing the exit ramp will travel at the free ¯ow velocity until they overtake and join the end of the downstream queue. When the exit queue vanishes at point 30 of Fig. 2, however, the 1-vehicle ¯ow rises to a value above Q1 which will eventually cause the downstream queue to back up to the exit ramp again. The critical issue here is whether or not the initial downstream queue is suciently large to keep the downstream bottleneck busy during this period of reduced ¯ow. If so, then the graphical solution of Fig. 2 is valid, but there is no net delay to the 1-vehicles caused by the exit queue. Fig. 2 describes the delay to the 1-vehicles caused by the exit queue prior to the passage of the exit ramp. However, they recover these losses before they pass the downstream bottleneck, because the cumulative departure curve at the downstream bottleneck is unaected by the exit queue. (There may, however, be some complications if there are other on or o ramps between these two locations.)
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If the initial downstream queue is not large enough to keep the downstream bottleneck busy during this period of reduced ¯ow then the bottleneck ¯ow will be interrupted when the down stream queue vanishes. This in turn will cause the slope of the D1 t curve to drop to Q1 before the D1 t curve can meet the A1 t curve. To analyze this case one must also specify how many vehicles are stored between the exit ramp and the downstream bottleneck at t 0. In the illustration above with the downstream bottleneck immediately downstream of the exit ramp, there was no storage. 4.2. Multiple queuing sections Suppose the freeway was uncongested prior to the formation of the queues so that the A1 t; A2 t were determined from arrivals upstream. At t 0 certainly q2 > Qe (otherwise there is no queue) and q2 < Q2 , the capacity of the right lanes. If a 1-vehicle queue also forms then q1 > Q1 and, therefore, q1 =q2 > Q1 =Q2 . This means that initially some of 1-vehicles were traveling in the right lanes in the 1-vehicle queue, because, if the velocities are equal in the right and left lanes, the actual ¯ows in these lanes must be in the ratio Q1 =Q2 and the 2-vehicles are all in the right lanes. At some later time, however, the arriving ¯ow q1 t could drop and/or the q2 t could increase so that q1 t=q2 t < Q1 =Q2 . The ratio of these ¯ows within the 1-vehicle queue, however, must be a least Q1 =Q2 . What happens now is that a 2-vehicle queue also forms upstream of the 1-vehicle queue with the ¯ow of 2-vehicles into the 1-vehicle queue restricted to q1 tQ2 =Q1 . One can easily modify Fig. 2 to adjust for this, if necessary. One simply draws another cumulative curve of (known) maximum slope q1 tQ2 =Q1 for the 2-vehicles which can enter the 1-vehicle queues as if this were the arriving 2-vehicle ¯ow. The dierence between this curve and the actual A2 t determines the new 2-vehicle queue. If at some later time the 1-vehicle queue should vanish, the 2-vehicle queues upstream and downstream of the 1-vehicle queue coalesce into a single queue. The above eect is not very likely to happen and is of little practical consequence, but it does add complications to the theory. One can also imagine situations in which the q1 t=q2 t could oscillate above and below Q1 =Q2 so as to create a sequence of 1- and 2-vehicle queuing sections. One can construct (rather arti®cial) arbitrarily complex patterns. 4.3. Some reservations The model described here seems to be qualitatively consistent with what one can observe when a queue from an exit ramp backs onto a freeway, but perhaps it raises more questions than it answers. It was assumed here that the exiting vehicles would stay in the right lane, but for how far upstream? It is certainly not obvious how to model what happens if the queue from one exit ramp backs up to an on ramp or another o ramp. In the former case, entering vehicles would have to weave across the congested right lanes and in the latter case it seems likely that the queue would spill over into other lanes. One might speculate here as to what would happen if the 2-vehicle queue becomes so long that some of the 2-vehicles, seeing that the velocity is higher in the left lanes, would try to by-pass part
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of their queue by traveling in the left lanes and then squeezing back into the right lanes closer to the exit. This would change the ``priority rule'' for the 2-vehicles; they would not be served FIFO, but it would not aect the given curve D2 t and would, therefore, have no immediate eect on the total delay to the 2-vehicles. If, however, there is a 1-vehicle queue constrained by the capacity of the left lanes and some 2vehicles were to travel in the left lanes, the combined ¯ow of 1-vehicles plus 2-vehicles would be constrained by the capacity of the left lanes. The 1-vehicle ¯ow would be reduced by the amount of 2-vehicle ¯ow at the point where the 2-vehicle ¯ow is largest in the left lanes. Where the 2vehicles leave the left lanes, the ¯ow in the left lanes will be below capacity. In particular it will be less than capacity at the exit ramp. The immediate eort of this is that the delay to the I-vehicles increases, but a secondary eect is that the 2-vehicles are also delayed in reaching the 2-vehicle queue section which may cause the 2vehicle queuing section to end sooner. The later eect, however, cannot compensate for the former. 5. Location of the shocks In the previous sections it was shown how one could determine the delays to the 1- and 2vehicles from given curves for the cumulative arrivals of 1- and 2-vehicles, the cumulative maximum exit ¯ow of 2-vehicles, and the capacities of the right and left lanes. In doing so we made no postulates about the equations of motion of vehicles, except that the velocity of 1-vehicles at capacity ¯ow was assumed to be (nearly) equal to the free-¯ow velocity, and that the velocities in various lanes would be equal if they had equal ¯ows per lane. From these postulates we cannot also predict the location of the shocks de®ning the ends of the queues, nor can we evaluate the delays if the velocity of the 1-vehicles at capacity ¯ow is suciently below that of the free-¯ow velocity that one need consider the delay caused by this dierence in velocities (because this delay will depend on how far the 1-vehicles travel at the reduced velocity which, in turn, depends on the physical length of the 2-vehicle queue section). To determine the location of shocks, one must, in essence, evaluate the complete solution N01 x; t0 ; N02 x; t0 =cumulative number of 1-vehicles, 2-vehicles to pass location x by time t0 , starting from the passage of some hypothetical reference vehicle moving at the free-¯ow velocity, and with t0 measured at each x from the passage of the reference vehicle (moving coordinates), for all x and t0 , from some dynamical equations of motion. We will consider here only the simple kinematic wave model with a triangular ¯ow-density relation (Newell, 1993) which in a moving coordinate system is equivalent to having an in®nite free-¯ow velocity and a ¯ow-density relation as shown in Fig. 3 for each lane. The q0 here represents the ¯ow considered as a function of the moving coordinates x and t0 , and the ``density'', k0 is the number of vehicles per unit distance but along a path traveling at the free¯ow velocity. Q is the capacity and kj is the ``jam density'' per lane. This model is quite realistic and is much simpler to analyze than other types of ¯ow-density relations. One simplifying feature is that the velocity at capacity ¯ow is indeed equal to the free¯ow velocity (in®nite in the moving coordinates), so we can make use of the results of the previous sections. Another feature is that all disturbances travel at the wave velocity ÿQ=kj for k0 > 0 (in the moving coordinates).
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Fig. 3. Flow density relation in moving time coordinates.
The simplest way to construct the solution is to integrate along the characteristic curves of slope ÿQ=kj in the t0 ; x plane as illustrated in Fig. 4(a). Note that the direction of the characteristics is the same in each lane separately or in any set of lanes collectively. In Fig. 4(a), if we measure time t0 in units of 1/Q (the minimum average headway per lane) and distances x in units of 1/kj (the minimum average spacing per lane) i.e. plot curves as a function of Qt0 and kj x, then the slope of the characteristics will be ÿ1, as illustrated by the broken line 1±2±4 in Fig. 4(a). If we measure t0 from the passage of some hypothetical reference vehicle which passes the exit ramp, x 0, at t 0, then t0 t at x 0. Along the line x 0 in Fig. 4(a), we know N02 0; t0 D2 t from the construction in Fig. 2, also reproduced in Fig. 4(b). Thus, starting from time 0 where N02 0; 0 0, we can follow along the line x 0 to some arbitrary point 1 where N02 0; t0 is known. From there we can follow along the characteristic curve through point 1. Along this line the ¯ow, density and velocity of the 2-vehicles are constant and N02 x; t0 will increase linearly at a rate Q per lane relative to the t0 -coordinate (rate 1 relative to Qt0 ) and kj relative to the x-coordinate (rate 1 relative to kj x). Thus we will know N2 x; t0 everywhere along this line until we reach the shock de®ning the rear of the 2-vehicle queue. From the construction of Fig. 2, the A2 t0 , in the moving coordinates, is precisely the cumulative number of 2-vehicles to reach the end of the 2-vehicles queue by ``time'' t0 . Thus
N02 x; t0 A2 t0 along the shock path (shown in Fig. 4(a) as the solid line 0, 2, 3) de®ning the rear of the 2-vehicle queue. The curve A2 t0 is reproduced in Fig. 4(b) as the broken line 0, 2, 3.
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Fig. 4. Construction of shock paths.
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In Fig. 4(b) we can draw a graph of N02 x; t0 along the path 0, 1, 2 of Fig. 4(a) from any point 1. This is D2 t0 from point 0 to 1 and then a line segment 1, 2 of slope 1 in Fig. 4(b) from point 1 (for a single 2-vehicle lane). Where this curve meets the curve A2 t0 , point 2 in Fig. 4(b), de®nes the time at which a wave starting at point 1 meets the shock, thus the time and spatial coordinates of point 2 in Fig. 4(a). The locus of point 2 generated from various locations of point 1 in Fig. 4(a) de®nes the shock path. One can also read the location of point 2 directly from Fig. 4(b). The spatial coordinate of point 2 is simply Q=kj times the horizontal (time) component of the line 1±2 of Fig. 4(b). If we measured time as Qt0 and distance as kj x, the horizontal component of the line 1±2 is the distance kj x. If we interpret point 2 as some n0 th 2-vehicle which passes the rear of the 2-vehicle queue at the time coordinate of point 2, the horizontal component of the line 1±2 de®nes where the vehicle is at that time. Note that within the 2-vehicle queue, N02 x; t0 for any x; t0 depends only on the D2 t00 for 00 t < t0 , the given cumulative exit ¯ow at earlier times. The N01 x; t0 N01 0; t0 D1 t0 is also known everywhere within this section. It is only the location of the shock path at any time that depends on the previous arrivals of 1- and 2-vehicles. Having determined the N01 x; t0 and N02 x; t0 along the shock path of the 2-vehicle queue, we can now proceed to determine the solution in the 1-vehicle queue section. In this section the characteristic has the same velocity as in the 2-vehicle queue, so we can just continue the straight line 1±2 of Fig. 4(a). We cannot yet predict how N01 x; t0 or N02 x; t0 themselves increase along this characteristic because we do not yet know the mix of trac in the right lane, but we do know that N01 x; t0 N02 x; t0 increases linearly along the characteristic at rate nQ if they are n freeway lanes. Thus we can evaluate N01 x; t0 N02 x; t0 along the characteristic curve through point 2 of Fig. 4(a). In Fig. 4(b), from the known curves D1 t0 and A2 t0 obtained from Fig. 2, we can also draw a curve D1 t0 A2 t0 , which represents the cumulative count of 1- plus 2-vehicles along the 2vehicle shock. For any point 2 in Fig. 4(a) along the shock the count N01 x; t N02 x; t0 is identi®ed in Fig. 4(b) by point 20 . From here we can draw a line 20 ±40 of slope nQ [in Fig. 4(b), n 2] which determines the count along the characteristic curve 2±4 of Fig. 4(a). If the freeway had initially been uncongested and the A1 t0 A2 t0 were determined by conditions upstream, then the intersections of the line 20 ±40 with the curve A1 t0 A2 t0 de®nes the rear of the 1-vehicle queue. This, in turn, determines the location of point 4 in Fig. 4(a). The locus of point 4 generated from various locations of point 1 (or 2) de®nes the shock path 0, 4, 5, 6 in Fig. 4(a). In this particular illustration, the exit ¯ow increased after point 1. The exit queue vanishes at point 3 and the velocity of vehicles after point 3 returns to the free-¯ow velocity (in®nite in the moving coordinates). When the wave from point 1 reaches the end of the 2-vehicle queue at point 2 in Fig. 4(a), the 2-vehicle shock moves forward and when this wave hits the rear of the 1-vehicle queue at point 4, the 1-vehicle shock changes velocity. When the 2-vehicle shock vanishes at point 3, a wave from point 3 reaches the 1-vehicle shock at point 5, after which all vehicles are traveling at the free-¯ow velocity (in®nite), including the shock path 5±6. To complete the solution, one may wish to determine the N02 x; t0 in the 1-vehicle queue section [which will, in turn, also determine the N01 x; t0 , since we already know N01 x; t0 N02 x; t0 ]. We can evaluate this by exploiting the fact that the velocity of a vehicle anywhere in the 1-vehicle queuing section is independent of its type.
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If we take any point a on the characteristic 2±4 of Fig. 4(a), it is identi®ed as the vehicle at point a on the line 20 ±40 of Fig. 4(b) with the same time coordinate. If we draw a horizontal line through point a in Fig. 4(b), point b represents the time this vehicle passed the 1-vehicle shock and point c represents the time it entered the 2-vehicle queue. In Fig. 4(a), the points b, a, c lie on the vehicle trajectory for this vehicle. If this vehicle were a 2-vehicle, it would be identi®ed in Fig. 4(b) with the line segment b0 ; c0 with the same time coordinates as b, c. But also the time this vehicle passes point a of Fig. 4(a) is the same for all vehicles, i.e. point a0 must be on the curve N02 x; t0 along the characteristic 2±4. The locus of points a0 evaluated in this way de®nes the N02 x; t0 everywhere along the characteristic 2±4, and by varying the points 1 or 2 and a one determines N02 x; t0 everywhere in the 1-vehicle queue. The above construction is valid if the freeway was initially uncongested and the A1 t0 ; A2 t0 were determined from conditions upstream. If the freeway were initially congested with A1 t0 ; A2 t0 determined from conditions downstream, the construction is slightly dierent. The construction of the 2-vehicle shock path 0, 2, 3 in Fig. 4(a) is still valid but there is no shock path for the 1-vehicle queue. The integration of N01 x; t0 N02 x; t0 along the characteristic 2, 4 or 3, 5 of Fig. 4(a) continues inde®nitely (or until the wave reaches some uncongested region at an unspeci®ed location far upstream). To evaluate N02 x; t0 along the characteristic in Fig. 4(b) one did not really need to know the location of point b. It suces to know just the location of a and c to determine the a0 , c0 . After point 3, however, the ¯ow past the exit increases. To follow the subsequent developments one must look downstream of the exit to see when the queue from downstream backs up past the exit, so that one can construct the D2 t0 after point 3. 6. Conclusion We have analyzed here in some detail the consequences of having a queue from an exit ramp back onto a freeway. This is a special case of a general theory of ``freeways with special lanes'' proposed by Daganzo. The technique of analysis here, however, is quite dierent. The graphical solution more clearly illustrates the issues than the numerical procedures of Daganzo, but may he dicult to implement in more complex situations. Acknowledgement The author was goaded into this analysis by Carlos Daganzo, who did not think it could be done. References Daganzo, C.F., 1997. A continuum theory of trac dynamics for freeways with special lanes. Transportation Research B 31, 83±102. Daganzo, C.F., Lin, W.-H., del Castillo, J.M., 1997. A simple physical principle for the simulation of freeways with special lanes and priority vehicles. Transportation Research B 31, 103±125. Newell, G.F., 1993. A simpli®ed theory of kinematic waves in highway trac. Transportation Research B 27, 281±314.
Delays caused by a queue at a freeway exit ramp G.F. Newell* Institute of Transportation Studies, 109 McLaughlin Hall, University of California, Berkeley, CA 94720, USA Received 29 January 1998; received in revised form 11 August 1998; accepted 17 August 1998
Abstract We consider here what happens to trac on a freeway when a queue from an exit ramp backs onto the freeway causing a partial blockage of the right lane. Exiting vehicles are con®ned to the right lane but through vehicles can travel in any lane. The two vehicle types interact but their queues must be treated separately. This illustrates a special case of a model of ``freeways with special lanes'' formulated by Daganzo (1997). Whereas Daganzo presented a numerical scheme of calculating ¯ows, the emphasis here is on graphical evaluation of the complete evolution of the queues. The graphical solution more clearly illustrates the practical issues, at least in this special situation. # 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction Our goal is to describe what happens to trac on a freeway if a queue from an exit ramp should back onto the freeway causing a partial blockage of the right lane (or lanes if trac can exit from more than one freeway lane). The geometry is shown schematically in Fig. 1. We consider two types of vehicles. Through vehicles will be designated as type 1 vehicles and those which will exit the freeway as type 2 vehicles. Type 1 vehicles can travel in any freeway lane but type 2 are assumed to stay in the right lane throughout the length of freeway section under consideration. This study was motivated by papers of Daganzo (1997) and Daganzo et al. (1997) which analyzed freeway trac with ``special lanes''. This example of an exit ramp was used by Daganzo as an illustration of their general method of simulating trac ¯ow with two vehicle types one of which was excluded from using certain freeway lanes (the 2-vehicles cannot use the left lanes). Whereas Daganzo's paper dealt mostly with computational methods appropriate for computer simulation of a general class of ¯ow patterns, the present paper deals mainly with graphical * Tel.: +1-510-642-3585; fax: +1-510-643-8919. 0191-2615/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0191-2615(98)00039-3
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Fig. 1. A freeway o-ramp.
solution methods and interpretation of issues for this particular example. Some of the methods described here, however, could be extended to other situations with two or more vehicle types. In Section 2, we describe how the two vehicle types react when a queue backs onto the freeway from the exit ramp, and some of the issues. In Sections 3 and 4 we show how one can determine delays for the two vehicle types from postulates about capacity constraints alone, without specifying anything about vehicle dynamics. To determine where the vehicles are stored and the location of shocks, however, one must introduce some model of kinematic waves (as in Daganzo's work). This is done in Section 5 using graphical techniques similar to those described by Newell (1993). 2. Description Prior to the time that the exit queue backs onto the freeway, we assume that some 1-vehicles were traveling in the right lane and that the velocities of the vehicles are equal in all lanes. Either all vehicles are traveling at the free-¯ow velocity or there is some bottleneck downstream of the exit which constrains the ¯ow of the 1-vehicles and, therefore, also the velocity of both the 1- and 2-vehicles upstream of the exit. When the queue of 2-vehicles backs onto the right lane of the freeway (illustrated in Fig. 1 by the single hatched area), the velocity of the vehicles drops as they enter the queued section. If there were any 1-vehicles caught in this queue, they would see that the velocity is higher in the left lanes and they would squeeze into the left lanes. Near the back of this queue of 2-vehicles there might be some short section of the highway in which any 1-vehicles, which had been traveling in the right lane upstream of the queue, merge into the left lanes to avoid the queue. The width of
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this merge section should be (nearly) independent of the length of the queued section. We might idealize this by imagining that this merge section has a negligible width, essentially zero (a shock). If the rate at which 1-vehicles wish to pass this queue section is less than the capacity of the left lanes, the 1 -vehicles will suer a negligible delay due to the queue of 2-vehicles. They will just bypass the queue. The 1-vehicle ¯ow past the exit will be dictated either by the arriving ¯ow from upstream if the 1-vehicles are traveling at the free-¯ow velocity, or by the capacity of a downstream bottleneck if this constrained ¯ow is less than the capacity of the left lanes. To determine delays to 2-vehicles in this case one can disregard the 1-vehicles and treat this as a conventional queuing problem for just the 2-vehicles, constrained by some speci®ed rate at which they can exit the freeway. If, however, the arrival rate of 1-vehicles to this freeway section exceeds the capacity of the left lanes, a queue of 1-vehicles will develop (illustrated in Fig. 1 by the crosshatched region) even if there is sucient excess capacity in the right lane to accommodate any 1-vehicles that cannot be carried in the left lanes. Note that the restriction on the 2-vehicles is assumed to be somewhere on the exit ramp, not on the freeway itself, so the right lane could accommodate some 1-vehicles if they wished to use it. There will still be a section of freeway containing 2-vehicles in the right lane waiting in a queue to exit the freeway. If there were any 1-vehicles in this right lane section and they saw that the velocity was higher in the left lanes, they would still squeeze into the left lanes but at the expense of vehicles already in the left lanes, because the ¯ow on the left lanes cannot exceed the capacity of these lanes. If some vehicles did squeeze in, this would cause the density in the left lanes to be higher than that for maximum ¯ow. But this disturbance would have a negative wave velocity and force the ¯ow in the left lanes to decrease upsteam of the entering vehicles to make room. So we conclude that there will be no 1-vehicles in the 2-vehicle queue in the right lane, but the left lanes adjacent to this 2vehicle queue will operate at capacity and at a velocity probably nearly equal to the free ¯ow velocity. This is a clear example of the dierence between a ``user optimal'' strategy and a ``system optimal'' strategy. A system optimal strategy would assign some 1-vehicles to the right lane to use any excess capacity in the right lane even though the travel time is longer than in the left lanes. A person in a 1-vehicle, however, when he reaches the back end of the 2-vehicle queue will choose the fastest lane from this point (even though he may have waited in a queue to reach this point), at the expense of others who may be waiting in a queue. The queue of 1-vehicles which develops upstream of the 2-vehicle queue will also contain some 2-vehicles, but we still refer to it as a 1-vehicle queue because the 2-vehicles in this queue are not constrained by the o-ramp ¯ow. They are simply being delayed in reaching their own queue where they will wait some more. 3. Evaluation of delays From the rules of behavior described above one can evaluate the delays caused by the exit blockage without specifying any actual equations of motion or determining where the queues are located, provided that the queues caused by the blockage do not back up past another on-ramp or o-ramp so as to distort the mix of trac entering the freeway upstream and provided that there is no interaction with possible bottlenecks downstream.
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We treat trac as a continous deterministic ¯uid (disregarding the integer nature of vehicle counts and stochastic ¯uctuations) as in Newell (1993). Let D1 t; D2 t = cumulative number of 1-vehicles, 2-vehicles which actually pass the exit ramp by time t starting from the time t 0 when the exit queue ®rst backs onto the freeway. A1 t; A2 t = cumulative number of 1-vehicles, 2-vehicles which would have passed the exit ramp by time t if there were no exit queue. The A1 t; A2 t might be determined from the arrivals upstream or from what could pass some downstream bottlenecks, but the A1 t; A2 t are presumed to be speci®ed for all t. If one could determine the curves D1 t and D2 t then one would know the delays caused by the exit queue. The 1-vehicles are assumed to be served in a ®rst come ®rst served order among themselves so the delay to some nth 1-vehicles (caused by the exit queue) would be the horizontal distance at height n between the A1 t and D1 t curves. Similarly for the 2-vehicles. One would also know the hypothetical ``vertical'' queues at time t, the dierences A1 t ÿ D1 t, A2 t ÿ D2 t. The blockage is assumed to be caused by some bottleneck on the exit ramp, an accident or a trac signal at the downstream end of ramp. Whatever it may be, it will presumably dictate the maximum number of vehicles which can leave the freeway. In particular it will determine the D2 t from time 0 until the exit queue vanishes on the freeway. We do not know yet, however, when the 2-vehicle queue vanishes or what happens thereafter, because, if a 1-vehicle queue forms, this will also delay the arrival of 2-vehicles to the ramp. If dA1 t=dt < Q1 , the capacity of the left lanes, then there is no 1-vehicle queue, D1 t A1 t. The 1-vehicles have no eect on the 2-vehicle queue and the 2-vehicle queue vanishes when D2 t A2 t. If, however, dA1 t=dt > Q1 at t 0, then D1 t Q1 t at least until either the 2vehicle queue or the 1-vehicle queue vanishes. In Fig. 2 , the ``given'' data, A1 t and A2 t for all t and the D1 t; D2 t until some queue vanishes are illustrated by the thick lines. In typical application these curves would be smooth but with time-dependent slopes. For purposes of illustration, however, we represent them in Fig. 2 by some (arbitrary) piecewise linear curves. Actually, in Fig. 2, the arrival rates are nearly contant, but the D2 t behaves as if there were some obstruction on the exit ramp which appears as t 0 and is later suddenly removed. The following graphical constructions, however, apply to any arbitrary shapes of these given curves. To determine when the 2-vehicle queue vanishes if there is a 1-vehicle queue, we must determine how much the 2-vehicles are delayed by the 1-vehicle queue. In Fig. 2, the delay to any nth 1-vehicle is equal to the horizontal distance between the curves A1 t and D1 t at height n, the time displacement between points 1 and 3. Suppose that a 1vehicle suers negligible delay (compared with the queuing delay) in traversing the highway section adjacent to the 2-vehicle queue (see Fig. 1), i.e. the velocity at capacity ¯ow is nearly equal to the free-¯ow velocity. The (known) delay to the 1-vehicles occurs (almost) entirely upstream of the 2-vehicle queue. In this 1-vehicle queuing section upstream of the 2-vehicle queue, the 1-vehicles will be distributed over all lanes of the freeway so that the velocity is the same in all lanes (otherwise the 1-vehicles would shift to the faster lane). The 2-vehicles in this section will also travel at the same
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velocity as the 1-vehicles. Thus an n0 th 2-vehicle which enters the 1-vehicle queue at the same time as the nth 1-vehicle will have experienced the same delay as the nth 1-vehicle by the time it reaches the rear of the 2-vehicle queue. In the terminology of queuing theory, the 1-vehicle queue section acts like a ®rst-in-®rst-out (FIFO) server which serves 1-vehicles at a ®xed rate Q1 (independent of the mix of 1- and 2-vehicles).
Fig. 2. Cumulative arrival and departure curves.
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Regardless of whether the A1 t; A2 t curves were determined from arrivals upstream or, under congested conditions, by some bottleneck downstream, it is assumed that, in the absence of the exit queue, the 1-vehicles and 2-vehicles would travel at the same velocity everywhere within this section of freeway. Thus, in Fig. 2, if the nth arriving 1-vehicle is identi®ed as point 1 on the curve A1 t, the n0 th 2-vehicle is identi®ed as point 2 on the curve A2 t at the same time. The nth 1-vehicle reaches the rear of the 2-vehicle queue at the time corresponding to point 3 on the D1 t curve. Actually this is a hypothetical 2-vehicle queue which occupies no space or, more precisely, one for which the free ¯ow trip time over the 2-vehicle queue is zero. The n0 th 2-vehicle reaches this point at the same time at point 4. It reaches the exit at point 5. The horizontal distance between points 2 and 4 is the delay experienced by the nth 2-vehicle in the 1-vehicle queue and the horizontal distance between points 4 and 5 is the delay experienced in the 2-vehicle queue. As one moves point 1 along the curve A1 t, the locus of point 4 generates a curve A2 t shown by the broken line
A2 t = cumulative number of 2-vehicles to reach the end of the 2-vehicle queue by time t after passing through the 1-vehicle queue (if the free ¯ow trip time over the 2vehicle queue is disregarded). Actually, this construction is not always valid. It is possible that the broken line curve of Fig. 2 evaluated in this manner will lie below D2 t even immediately after t 0. But this cannot be correct. What happens in this case is that the ¯ow of 1-vehicles past the exit drops suddenly when the queue of 2-vehicles starts to back onto the freeway. But this, in turn, also causes a sudden drop in the ¯ow of 2-vehicles approaching the exit. If the reduced ¯ow of 2-vehicles is less than the given rate at which 2-vehicles can leave the exit, the 2-vehicle queue will disappear and the ¯ow of 1-vehicles will go back up. What must ®nally evolve is that a short queue of 2-vehicles forms on the freeway. Some of the 1-vehicles which were traveling in the right lane may no longer care to switch lanes for only a short distance. The queue of 2-vehicles will stabilize at such a length that the ¯ow of 1-vehicles past the exit will generate a ¯ow of 2-vehicles equal to the exit rate. The modi®cation in Fig. 2 is that the curve A2 t must now (nearly) follow the curve D2 t, i.e. points 4 and 5 coincide, but the trip times of the 1- and 2-vehicles in the 1-vehicle queue are still equal. The displacement 1±3 is equal to that of 2±4, but this now determines the locus of points 3, i.e. a curve D1 t with slope larger than Q1 . The condition that a 2-vehicle queue forms at t 0 and continues to grow depends on the ¯ows immediately after t 0. If we let Qe = exit ¯ow at t 0 dD2 t=dt at t 0 q2 = arrival rate of 2-vehicles at t 0 dA2 =dt at t 0 q1 = arrival rate of 1-vehicle at t 0 and denotes the time of points 1 and 2 of Fig. 2 soon after t 0, then the time of point 5 is q2 =Qe and the time of point 3 is q1 =Q1 . The condition that point 5 is later than point 3 (a 2vehicle queue forms) is that 1 < q1 =Q1 < q2 =Qe0 :
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If, however, 1 < q2 =Qe < q1 =Q1 , the time coordinate of point 3 is equal to that of point 5, which means that the ¯ow of 1-vehicles past the exit is q1 = q2 =Qe q1 =q2 Qe ; (which is longer than Q1 but less than q1 ). If there is a 2-vehicle queue, it will eventually vanish when the curve D2 t meets the curve A2 t. If the 1-vehicle queue vanishes ®rst, the A2 t curve intersects and joins the curve A2 t before the D2 t meets it, but suppose that there is a 1-vehicle queue when the 2-vehicle queue vanishes, as illustrated in Fig. 2. When the 2-vehicle queue vanishes, the total ¯ow past the exit can potentially jump to the combined capacity of the right and left lanes with the 1-vehicles using any residual capacity of the right lanes not used by the 2-vehicles. To analyze the subsequent evolution one should also draw curves for A2 t A2 t and D1 t D2 t by adding the known component curves until the time the 2-vehicle queue vanishes at point 40 , 50 of Fig. 2. If the total ¯ow at point 70 on the curve D1 t D2 t can increase to the capacity of all lanes, we will know the new slope of this curve beyond point 70 . We can then construct the rest of this curve until the combined queue vanishes when this curve meets the curve A1 t A2 t at point 8. From the curves for the combined ¯ows, one can now construct the curves for the component ¯ows because the trip time through the 1-vehicle queue is the same for either the 1- or 2-vehicles. At the time the 2-vehicle queue vanished, the horizontal distance between point 20 and 40 ,50 was equal to that between points 10 and 30 , but also between points 60 and 70 on the combined curves. This condition remains true. At a later time, the distance between points 100 and 200 or 300 and 400 are equal to the known distance from 600 to 700 . The locus of points 200 and 400 de®ne the curves D1 t and D2 t as shown by the broken lines until all queues disappear. There is a possible modi®cation of this solution analogues to that which can occur at t 0 as described previously. When the obstruction is removed there will be a sudden increase in the ¯ow past the exit for both the 1- and 2-vehicles. The slope of the line 40 ±400 , however, cannot exceed the slope of D2 t at point 40 because that would imply that the 2-vehicle queue backs onto the freeway again. If this happens a small queue must form so as to restrict the combined ¯ow just enough that the 2-vehicle ¯ow matches the exit rate. In this case the graphical construction is similar to that shown in Fig. 2 except that the curve D2 t after point 40 remains (for a while) as the given cumulative ¯ow which can exit the freeway, i.e. the 2-vehicle queue is kept at essentially zero. The horizontal distance 200 ±400 in Fig. 2 is, therefore, known and this determines the distances 100 ±300 and 600 ±700 and thus the locus of points 200 and 700 until such time that the resulting slope of D1 t D2 t reaches the combined capacity of all lanes or the queue vanishes. If the former happens ®rst, then the subsequent graphical construction is determined from the constraint on the slope of D1 t D2 t as in Fig. 2. 4. Some complications In view of the complexity of Daganzo's analysis of all possible boundary conditions for ¯ow patterns with special lanes one certainly would not expect the simple graphical solution of Fig. 2 to tell the complete story even as applied to this illustration of a queue backing onto the freeway.
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We discuss here a few possible complications but we do not attempt to give a complete analysis of everything that could happen. 4.1. Congested trac The implication in Section 3 is that this graphical construction is valid if the freeway was initially uncongested and the A1 t; A2 t were determined from inputs to the freeway upstream, or the freeway was initially congested due to some bottleneck downstream. In the latter case, the construction in Fig. 2 may be valid up to a certain point but the interpretation of the results may be quite dierent. Suppose that the downstream bottleneck is immediately downstream of the exit, i.e. the prior bottleneck was on the through lanes at the exit ramp possibly due to a lane drop. If the exit queue backs onto the freeway this will certainly (further) delay the 2-vehicles and, if a 1-vehicle queue forms, it will also delay the 1-vehicles at least temporarily. The issue is what happens after the 2vehicle queue vanishes (at point 40 , 50 of Fig. 2). The constraint after point 40 , 50 may not be the capacity of all lanes as illustrated in Fig. 2, but the capacity of the through lanes immediately downstream due to the prior bottleneck. If this capacity is Q1 (with Q1 > Q1 ), the slope of D1 t after point 30 will be Q1 and this determines the curve D1 t after point 30 . The time interval 200 to 400 is still equal to the known time between points 100 and 300 and this will determine the locus of point 400 , i.e. the D2 t after point 40 , 50 (provided that the resulting slope of the D2 t does not exceed the given slope at point 40 , 50 ). If, however, the curve A1 t was dictated by the downstream bottleneck of capacity Q1 , then the slope of A1 t is also Q1 and the curves A1 t; D1 t become parallel. The eect of the exit queue persists until the end of the period of congestion when there is no longer a queue caused by the downstream bottleneck. What is happening in this case is that the temporary bottleneck caused by the exit queue chokes o the ¯ow to the downstream bottleneck. There is a loss of utilization of the downstream bottleneck which cannot be recovered. Suppose, on the other hand, that the downstream bottleneck is far downstream and that the D1 t; D2 t proceed as in Fig. 2 at least for a while after point 40 , 50 . When the 1-vehicle queue formed at time 0, the freeway is assumed to be congested between the exit ramp and the downstream bottleneck, but if the downstream bottleneck has a capacity Q1 > Q1 , the new bottleneck will choke o the ¯ow downstream and the queue between the two bottlenecks will start to dissipate. The 1-vehicles passing the exit ramp will travel at the free ¯ow velocity until they overtake and join the end of the downstream queue. When the exit queue vanishes at point 30 of Fig. 2, however, the 1-vehicle ¯ow rises to a value above Q1 which will eventually cause the downstream queue to back up to the exit ramp again. The critical issue here is whether or not the initial downstream queue is suciently large to keep the downstream bottleneck busy during this period of reduced ¯ow. If so, then the graphical solution of Fig. 2 is valid, but there is no net delay to the 1-vehicles caused by the exit queue. Fig. 2 describes the delay to the 1-vehicles caused by the exit queue prior to the passage of the exit ramp. However, they recover these losses before they pass the downstream bottleneck, because the cumulative departure curve at the downstream bottleneck is unaected by the exit queue. (There may, however, be some complications if there are other on or o ramps between these two locations.)
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If the initial downstream queue is not large enough to keep the downstream bottleneck busy during this period of reduced ¯ow then the bottleneck ¯ow will be interrupted when the down stream queue vanishes. This in turn will cause the slope of the D1 t curve to drop to Q1 before the D1 t curve can meet the A1 t curve. To analyze this case one must also specify how many vehicles are stored between the exit ramp and the downstream bottleneck at t 0. In the illustration above with the downstream bottleneck immediately downstream of the exit ramp, there was no storage. 4.2. Multiple queuing sections Suppose the freeway was uncongested prior to the formation of the queues so that the A1 t; A2 t were determined from arrivals upstream. At t 0 certainly q2 > Qe (otherwise there is no queue) and q2 < Q2 , the capacity of the right lanes. If a 1-vehicle queue also forms then q1 > Q1 and, therefore, q1 =q2 > Q1 =Q2 . This means that initially some of 1-vehicles were traveling in the right lanes in the 1-vehicle queue, because, if the velocities are equal in the right and left lanes, the actual ¯ows in these lanes must be in the ratio Q1 =Q2 and the 2-vehicles are all in the right lanes. At some later time, however, the arriving ¯ow q1 t could drop and/or the q2 t could increase so that q1 t=q2 t < Q1 =Q2 . The ratio of these ¯ows within the 1-vehicle queue, however, must be a least Q1 =Q2 . What happens now is that a 2-vehicle queue also forms upstream of the 1-vehicle queue with the ¯ow of 2-vehicles into the 1-vehicle queue restricted to q1 tQ2 =Q1 . One can easily modify Fig. 2 to adjust for this, if necessary. One simply draws another cumulative curve of (known) maximum slope q1 tQ2 =Q1 for the 2-vehicles which can enter the 1-vehicle queues as if this were the arriving 2-vehicle ¯ow. The dierence between this curve and the actual A2 t determines the new 2-vehicle queue. If at some later time the 1-vehicle queue should vanish, the 2-vehicle queues upstream and downstream of the 1-vehicle queue coalesce into a single queue. The above eect is not very likely to happen and is of little practical consequence, but it does add complications to the theory. One can also imagine situations in which the q1 t=q2 t could oscillate above and below Q1 =Q2 so as to create a sequence of 1- and 2-vehicle queuing sections. One can construct (rather arti®cial) arbitrarily complex patterns. 4.3. Some reservations The model described here seems to be qualitatively consistent with what one can observe when a queue from an exit ramp backs onto a freeway, but perhaps it raises more questions than it answers. It was assumed here that the exiting vehicles would stay in the right lane, but for how far upstream? It is certainly not obvious how to model what happens if the queue from one exit ramp backs up to an on ramp or another o ramp. In the former case, entering vehicles would have to weave across the congested right lanes and in the latter case it seems likely that the queue would spill over into other lanes. One might speculate here as to what would happen if the 2-vehicle queue becomes so long that some of the 2-vehicles, seeing that the velocity is higher in the left lanes, would try to by-pass part
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of their queue by traveling in the left lanes and then squeezing back into the right lanes closer to the exit. This would change the ``priority rule'' for the 2-vehicles; they would not be served FIFO, but it would not aect the given curve D2 t and would, therefore, have no immediate eect on the total delay to the 2-vehicles. If, however, there is a 1-vehicle queue constrained by the capacity of the left lanes and some 2vehicles were to travel in the left lanes, the combined ¯ow of 1-vehicles plus 2-vehicles would be constrained by the capacity of the left lanes. The 1-vehicle ¯ow would be reduced by the amount of 2-vehicle ¯ow at the point where the 2-vehicle ¯ow is largest in the left lanes. Where the 2vehicles leave the left lanes, the ¯ow in the left lanes will be below capacity. In particular it will be less than capacity at the exit ramp. The immediate eort of this is that the delay to the I-vehicles increases, but a secondary eect is that the 2-vehicles are also delayed in reaching the 2-vehicle queue section which may cause the 2vehicle queuing section to end sooner. The later eect, however, cannot compensate for the former. 5. Location of the shocks In the previous sections it was shown how one could determine the delays to the 1- and 2vehicles from given curves for the cumulative arrivals of 1- and 2-vehicles, the cumulative maximum exit ¯ow of 2-vehicles, and the capacities of the right and left lanes. In doing so we made no postulates about the equations of motion of vehicles, except that the velocity of 1-vehicles at capacity ¯ow was assumed to be (nearly) equal to the free-¯ow velocity, and that the velocities in various lanes would be equal if they had equal ¯ows per lane. From these postulates we cannot also predict the location of the shocks de®ning the ends of the queues, nor can we evaluate the delays if the velocity of the 1-vehicles at capacity ¯ow is suciently below that of the free-¯ow velocity that one need consider the delay caused by this dierence in velocities (because this delay will depend on how far the 1-vehicles travel at the reduced velocity which, in turn, depends on the physical length of the 2-vehicle queue section). To determine the location of shocks, one must, in essence, evaluate the complete solution N01 x; t0 ; N02 x; t0 =cumulative number of 1-vehicles, 2-vehicles to pass location x by time t0 , starting from the passage of some hypothetical reference vehicle moving at the free-¯ow velocity, and with t0 measured at each x from the passage of the reference vehicle (moving coordinates), for all x and t0 , from some dynamical equations of motion. We will consider here only the simple kinematic wave model with a triangular ¯ow-density relation (Newell, 1993) which in a moving coordinate system is equivalent to having an in®nite free-¯ow velocity and a ¯ow-density relation as shown in Fig. 3 for each lane. The q0 here represents the ¯ow considered as a function of the moving coordinates x and t0 , and the ``density'', k0 is the number of vehicles per unit distance but along a path traveling at the free¯ow velocity. Q is the capacity and kj is the ``jam density'' per lane. This model is quite realistic and is much simpler to analyze than other types of ¯ow-density relations. One simplifying feature is that the velocity at capacity ¯ow is indeed equal to the free¯ow velocity (in®nite in the moving coordinates), so we can make use of the results of the previous sections. Another feature is that all disturbances travel at the wave velocity ÿQ=kj for k0 > 0 (in the moving coordinates).
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Fig. 3. Flow density relation in moving time coordinates.
The simplest way to construct the solution is to integrate along the characteristic curves of slope ÿQ=kj in the t0 ; x plane as illustrated in Fig. 4(a). Note that the direction of the characteristics is the same in each lane separately or in any set of lanes collectively. In Fig. 4(a), if we measure time t0 in units of 1/Q (the minimum average headway per lane) and distances x in units of 1/kj (the minimum average spacing per lane) i.e. plot curves as a function of Qt0 and kj x, then the slope of the characteristics will be ÿ1, as illustrated by the broken line 1±2±4 in Fig. 4(a). If we measure t0 from the passage of some hypothetical reference vehicle which passes the exit ramp, x 0, at t 0, then t0 t at x 0. Along the line x 0 in Fig. 4(a), we know N02 0; t0 D2 t from the construction in Fig. 2, also reproduced in Fig. 4(b). Thus, starting from time 0 where N02 0; 0 0, we can follow along the line x 0 to some arbitrary point 1 where N02 0; t0 is known. From there we can follow along the characteristic curve through point 1. Along this line the ¯ow, density and velocity of the 2-vehicles are constant and N02 x; t0 will increase linearly at a rate Q per lane relative to the t0 -coordinate (rate 1 relative to Qt0 ) and kj relative to the x-coordinate (rate 1 relative to kj x). Thus we will know N2 x; t0 everywhere along this line until we reach the shock de®ning the rear of the 2-vehicle queue. From the construction of Fig. 2, the A2 t0 , in the moving coordinates, is precisely the cumulative number of 2-vehicles to reach the end of the 2-vehicles queue by ``time'' t0 . Thus
N02 x; t0 A2 t0 along the shock path (shown in Fig. 4(a) as the solid line 0, 2, 3) de®ning the rear of the 2-vehicle queue. The curve A2 t0 is reproduced in Fig. 4(b) as the broken line 0, 2, 3.
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Fig. 4. Construction of shock paths.
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In Fig. 4(b) we can draw a graph of N02 x; t0 along the path 0, 1, 2 of Fig. 4(a) from any point 1. This is D2 t0 from point 0 to 1 and then a line segment 1, 2 of slope 1 in Fig. 4(b) from point 1 (for a single 2-vehicle lane). Where this curve meets the curve A2 t0 , point 2 in Fig. 4(b), de®nes the time at which a wave starting at point 1 meets the shock, thus the time and spatial coordinates of point 2 in Fig. 4(a). The locus of point 2 generated from various locations of point 1 in Fig. 4(a) de®nes the shock path. One can also read the location of point 2 directly from Fig. 4(b). The spatial coordinate of point 2 is simply Q=kj times the horizontal (time) component of the line 1±2 of Fig. 4(b). If we measured time as Qt0 and distance as kj x, the horizontal component of the line 1±2 is the distance kj x. If we interpret point 2 as some n0 th 2-vehicle which passes the rear of the 2-vehicle queue at the time coordinate of point 2, the horizontal component of the line 1±2 de®nes where the vehicle is at that time. Note that within the 2-vehicle queue, N02 x; t0 for any x; t0 depends only on the D2 t00 for 00 t < t0 , the given cumulative exit ¯ow at earlier times. The N01 x; t0 N01 0; t0 D1 t0 is also known everywhere within this section. It is only the location of the shock path at any time that depends on the previous arrivals of 1- and 2-vehicles. Having determined the N01 x; t0 and N02 x; t0 along the shock path of the 2-vehicle queue, we can now proceed to determine the solution in the 1-vehicle queue section. In this section the characteristic has the same velocity as in the 2-vehicle queue, so we can just continue the straight line 1±2 of Fig. 4(a). We cannot yet predict how N01 x; t0 or N02 x; t0 themselves increase along this characteristic because we do not yet know the mix of trac in the right lane, but we do know that N01 x; t0 N02 x; t0 increases linearly along the characteristic at rate nQ if they are n freeway lanes. Thus we can evaluate N01 x; t0 N02 x; t0 along the characteristic curve through point 2 of Fig. 4(a). In Fig. 4(b), from the known curves D1 t0 and A2 t0 obtained from Fig. 2, we can also draw a curve D1 t0 A2 t0 , which represents the cumulative count of 1- plus 2-vehicles along the 2vehicle shock. For any point 2 in Fig. 4(a) along the shock the count N01 x; t N02 x; t0 is identi®ed in Fig. 4(b) by point 20 . From here we can draw a line 20 ±40 of slope nQ [in Fig. 4(b), n 2] which determines the count along the characteristic curve 2±4 of Fig. 4(a). If the freeway had initially been uncongested and the A1 t0 A2 t0 were determined by conditions upstream, then the intersections of the line 20 ±40 with the curve A1 t0 A2 t0 de®nes the rear of the 1-vehicle queue. This, in turn, determines the location of point 4 in Fig. 4(a). The locus of point 4 generated from various locations of point 1 (or 2) de®nes the shock path 0, 4, 5, 6 in Fig. 4(a). In this particular illustration, the exit ¯ow increased after point 1. The exit queue vanishes at point 3 and the velocity of vehicles after point 3 returns to the free-¯ow velocity (in®nite in the moving coordinates). When the wave from point 1 reaches the end of the 2-vehicle queue at point 2 in Fig. 4(a), the 2-vehicle shock moves forward and when this wave hits the rear of the 1-vehicle queue at point 4, the 1-vehicle shock changes velocity. When the 2-vehicle shock vanishes at point 3, a wave from point 3 reaches the 1-vehicle shock at point 5, after which all vehicles are traveling at the free-¯ow velocity (in®nite), including the shock path 5±6. To complete the solution, one may wish to determine the N02 x; t0 in the 1-vehicle queue section [which will, in turn, also determine the N01 x; t0 , since we already know N01 x; t0 N02 x; t0 ]. We can evaluate this by exploiting the fact that the velocity of a vehicle anywhere in the 1-vehicle queuing section is independent of its type.
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If we take any point a on the characteristic 2±4 of Fig. 4(a), it is identi®ed as the vehicle at point a on the line 20 ±40 of Fig. 4(b) with the same time coordinate. If we draw a horizontal line through point a in Fig. 4(b), point b represents the time this vehicle passed the 1-vehicle shock and point c represents the time it entered the 2-vehicle queue. In Fig. 4(a), the points b, a, c lie on the vehicle trajectory for this vehicle. If this vehicle were a 2-vehicle, it would be identi®ed in Fig. 4(b) with the line segment b0 ; c0 with the same time coordinates as b, c. But also the time this vehicle passes point a of Fig. 4(a) is the same for all vehicles, i.e. point a0 must be on the curve N02 x; t0 along the characteristic 2±4. The locus of points a0 evaluated in this way de®nes the N02 x; t0 everywhere along the characteristic 2±4, and by varying the points 1 or 2 and a one determines N02 x; t0 everywhere in the 1-vehicle queue. The above construction is valid if the freeway was initially uncongested and the A1 t0 ; A2 t0 were determined from conditions upstream. If the freeway were initially congested with A1 t0 ; A2 t0 determined from conditions downstream, the construction is slightly dierent. The construction of the 2-vehicle shock path 0, 2, 3 in Fig. 4(a) is still valid but there is no shock path for the 1-vehicle queue. The integration of N01 x; t0 N02 x; t0 along the characteristic 2, 4 or 3, 5 of Fig. 4(a) continues inde®nitely (or until the wave reaches some uncongested region at an unspeci®ed location far upstream). To evaluate N02 x; t0 along the characteristic in Fig. 4(b) one did not really need to know the location of point b. It suces to know just the location of a and c to determine the a0 , c0 . After point 3, however, the ¯ow past the exit increases. To follow the subsequent developments one must look downstream of the exit to see when the queue from downstream backs up past the exit, so that one can construct the D2 t0 after point 3. 6. Conclusion We have analyzed here in some detail the consequences of having a queue from an exit ramp back onto a freeway. This is a special case of a general theory of ``freeways with special lanes'' proposed by Daganzo. The technique of analysis here, however, is quite dierent. The graphical solution more clearly illustrates the issues than the numerical procedures of Daganzo, but may he dicult to implement in more complex situations. Acknowledgement The author was goaded into this analysis by Carlos Daganzo, who did not think it could be done. References Daganzo, C.F., 1997. A continuum theory of trac dynamics for freeways with special lanes. Transportation Research B 31, 83±102. Daganzo, C.F., Lin, W.-H., del Castillo, J.M., 1997. A simple physical principle for the simulation of freeways with special lanes and priority vehicles. Transportation Research B 31, 103±125. Newell, G.F., 1993. A simpli®ed theory of kinematic waves in highway trac. Transportation Research B 27, 281±314.