Transpn. Res.-B. Vol 27B, No.4, pp 289-303,1993 Printed In Great Bntam.

0191-2615193 $600 + .00 ib 1993 Pergamon Press Ltd

A SIMPLIFIED THEORY OF KINEMATIC WAVES IN HIGHWAY TRAFFIC,

PART II: QUEUEING AT FREEWAY BOTTLENECKS

G. F. NEWELL Institute of Transportation Studies, University of California,

Berkeley, CA 94720, U.S.A.

(Received 9 March 1992)

Abstract-For a freeway having various entrance and exit ramps, the methods described in Part I are used to relate the cumulative flow curve at any junction to the net cumulative entrance flow at this junction, and the cumulative flow curves for the freeway at the next upstream junction and/ or the next downstream junction. If the type of flow-density relations typical of freeway traffic are idealized by a triangular shaped curve with only two wave speeds, one for free-flowing traffic (positive) and the other for congested traffic (negative), then the relationship is easy to evaluate. The cumulative flow curve at the junction is simply the lower envelope of a translation of the cumulative curve from upstream and a different translation of the cumulative curve from down­stream. This relationship is the basic building block for a freeway flow prediction model described in Part III.

1. INTRODUCTION

In Part I, Newell (1993), it was shown that if one solves the kinematic wave equations of Lighthill, Whitham and Richards (L-W-R) for the cumulative flows, the recipe of taking the lower envelope of all formal solutions for A(x, t) automatically incorporates any shock conditions. Having simplified the mathematical analysis to the point that the solu­tion of L-R-W equations is a straightforward exercise, we can now turn our attention to where and how to apply the theory for the analysis of highway traffic flow patterns.

The key hypothesis in the L-W-R theory is that there is some functional relation between the density k(x, t) at location x and time t and the flow q(x, t), (1-1), (1-2).1 For highway traffic, this relation, for any given location x, is assumed to have a form as illustrated in Figs. la and b. The flow q*(k, x) as a function of k vanishes at k == 0 and also at the "jam density" kix). It has a maximum Q(x), the "capacity" at x. The inverse relation k*(q, x) is a two-valued function of q for 0 < q < Q(x). The lower branch is associated with "free flow" traffic, the upper branch with congested traffic.

For any value of k, the "wave velocity" is the slope of the curve q*. For any value of q, the "pace" of the wave w(q, x), (1-3), is the slope of the curve k*. It is positive for the lower branch and negative for the upper branch.

The L-W-R theory is known to have serious deficiencies, resulting from the fact that the relation between q and k is not really valid under time-dependent conditions. Despite this, however, the theory does give reasonable estimates of some things of interest to the analysis of traffic flow, particularly the delays caused by queues and their dependence upon entrance and exit flows. The reason is that queueing delays depend mostly on the capacities Q(x) and the input flows, but are insensitive to the details of the relation between q and k.

The details of the L-W-R theory are certainly not realistic for light or even moder­ately heavy ("free-flowing") traffic. At very low densities each driver chooses a "desired speed" nearly independent of the average spacing and nearly independent of any other driver. When a fast vehicle overtakes a slow vehicle, the fast vehicle may be slowed down and lose time because it cannot pass the slow vehicle immediately. The result of this is that the average speed of all vehicles (particularly the fast ones) decreases with increasing density or flow, because the passing rate increases. This, however, does not imply that the average relation between v and k or q and k, etc., determined under stationary

)Equations from Part I will be labeled (1-#).

289

290

* rr ;Q(xl o

LL

G. F. NEWELL

k j (xl

Density, k Flow, q

(a) (b)

Fig. 1. A typical relation between flow and density.

conditions, can be used to predict the time-dependent behavior of q as described in the L-W-R theory. The mechanism of time-dependence is quite different (and quite compli­cated, because a nonuniform flow or density will result in a space and time-dependent distribution of the desired speeds of drivers).

For most freeways (and perhaps also for arterials with synchronized signals) the velocity v is not very sensitive to the flow until q is close to Q. The L-W-R theory predicts, under such conditions, that the wave velocity and the vehicle velocity are nearly equal. An increase or decrease in flow propagates forward with the vehicles themselves and the horizontal distance between the cumulative curves A(x" t) and A (Xi+ I> t) at two locations x, and X'+l is nearly UO(Xi+l - x,), independent of t, with Uo the free-flow vehicle pace. This would be approximately true also for a more precise theory for which the time lost in passing is relatively small, but the spread in the velocity distribution will tend to disperse changes in flow (there would certainly be no shocks). The average trip time of vehicles, however, would still be determined by the average pace Uo.

As q approaches the capacity Q, passing may be restricted somewhat and the average speed will decrease. The details of how this happens are not well understood. Delays caused by heavy but undersaturated flows are typically quite small as compared with the queueing delays at oversaturated locations. In many cases one could neglect these com­pletely by postulating that v is completely independent of q for q < Q (on the lower branch of the k*-curve). If one does wish to estimate these delays, it would typically suffice to obtain just crude estimates of the increased trip times caused by a pace U > Uo averaged over many vehicles.

The fact that the L-W-R theory predicts the formation of shocks due to the focusing of even forward moving waves is somewhat of a nuisance. (One can avoid this if one assumes that the v is independent of q.) In any graphical or numerical solution one would certainly evaluate the A-curves only at certain discrete times or for certain discrete cumulative counts, and then interpolate either graphically or numerically to obtain inter­mediate values. As explained in Part I, if one should compute multiple values of the A for the same x and t, one should adopt the smallest count. This would signal the existence of a shock somewhere, but it is not necessary that one determines the exact passage time of the shock or even worry about how the shock might affect the interpolation between discrete time points. One is primarily interested in evaluating the counts correctly (at selected times). One is not particularly interested in knowing the "instantaneous" values of the flows.

The L-W-R theory also has deficiencies in describing the time-dependent behavior of traffic within a queue. It is known that, for a given speed, drivers will choose a different spacing if they are accelerating than if they are decelerating and that there may be some "instability" in long queues resulting in "stop and go traffic." The capacities of road sections, however, seem to be reasonably well-defined, although there will be random fluctuations in the counts over finite time intervals due to random variations in the

Theory of kinematic waves in highway traffic, Part II 291

headways between individual vehicles. It is certainly true that the queue which backs up behind a bottleneck occupies space. Despite the possible instability, it is also true that a moving queue (with a positive flow q) will (on the average) have a lower density than a stationary queue (with a "jam density").

Since the detailed form of the upper branch of the k*-curve is not known (because the density depends on things other than just the q), it would suffice to take any form for this function which is consistent with the above qualitative properties. We do not really need to know exactly where the end of the queue is at any time except for the purpose of determining if or when a queue from some bottleneck at X,+ 1 backs up past some entrance at X,; we only need to know how many vehicles can be stored in any highway section, as a function of the flow.

The fact that the L-W-R theory predicts that backward moving waves may focus and cause minor shocks if the upper branch of the k*-curve is nonlinear, is again somewhat of a mathematical nuisance not necessarily in agreement with what actually happens. Of course, we do expect a "shock" where the backward waves meet the forward waves to define the back of the queue.

To avoid some unpleasant mathematical complications, it might be reasonable to assume that the upper branch of the k*-curve is linear so that all backward waves have the same pace. This would not only avoid the formation of minor shocks but also greatly simplify the graphical constructions. The assumption that the upper portion of the k*­curve is linear is equivalent to assuming that the car velocity v is a linear function of the spacing (for spacings less than that for maximum flow), a not too unreasonable postulate.

2. FURTHER SIMPLIFICATIONS

If we postulate that the q-k relation is piecewise linear as illustrated in Fig. 2, the L-W-R theory is really simple. For a homogeneous section of highway, there are only two possible values for the wave pace, one positive and the other negative (independent of q). Since cause and effect relations must proceed forward in time, the value of A (x, t) for any X and t must either be determined from initial or boundary conditions upstream at earlier times along a wave of positive pace, or by initial or boundary conditions downstream at earlier times along a wave of negative pace.

The surface A (x, t) generated by the positive wave is a single-valued ruled surface with only one direction, i.e., a cylindrical surface. For the q-k relation of Fig. 2, the positive wave pace w is equal to the vehicle pace Uo = kl q and along this wave, according to (1-8)

dAldx = (-k + qw) = (-k + quo) = o.

This is simply a confirmation of the obvious fact that the number on the vehicle A(x, t) must be constant along a vehicle trajectory which, in this case, is also the trajectory of the wave.

Another geometrical interpretation of this is that the curve of A(x, t) vs. t at some

k· J

* ~ >. ...... (f)

c Q)

0

Flow,

Fig. 2. An idealized q-k relation. TR(B) 27:4-0

292

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LL Q)

>

o :J

E :J U

G. F. NEWELL

Time, t

Fig. 3. Propagation of the forward waves.

fixed location x is simply a horizontal translation of the curve A(xo> t) vs. t by a time displacement uo(x - xo) for x > Xo as illustrated in Fig. 3, i.e. the trip time of any vehicle from Xo to x is uo(x - xo), independent of t.

The surface A(x, t) generated by the negative wave pace is also a single-valued ruled surface with only one direction, another cylindrical surface. For the q-k relation of figure 2 with slope - Wo

dA/dx = -k - qwo = -kJ'

The geometrical interpretation of this analogous to Fig. 3 is illustrated in Fig. 4. The curve A(x, t) vs. t for fixed x is simply a translation of the curve A(xl> t) for Xl > x horizontally by a time WO(xl - x) and vertically by an amount kj(x1 - x), independent of t.

Because cylinders are single-valued ruled surfaces, boundary or initial conditions could provide at most two values of A (x, t) at some x and t, one from conditions upstream as in Fig. 3, the other from conditions downstream as in Fig. 4. If two cylinders intersect, the path of intersection is the shock and the correct value of A(x, t) is the smaller. We will give specific examples later, but we will first simplify the construction even more.

3. MOVING COORDINATES

For the purpose of evaluating "delays," it is convenient to subtract the free-speed trip time from the actual travel time of any vehicle. Equivalently, one can imagine that an observer at location x measures not only the cumulative curve A(x, t) from the passage

3: o

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> .... o :J

E :J U

Time, t

Fig. 4. Propagation of the backward waves.

Theory of kinematic waves in highway traffic, Part II 293

of some reference vehicle traveling at pace uo, but he also measures time relative to the passage time of this vehicle, i.e. he observes a "time"

t' (x, t) = t - to - uo(x - xo), (1)

with t equal to the "absolute time" and to the time at which the reference vehicle passes some point Xo. Of course to depends upon the origin of the time coordinate and Xo depends upon the origin of the space coordinate, both of which are arbitrary. One could measure x relative to some arbitrary point so that Xo = 0 and measure t relative to the time the reference vehicle passes Xo = 0 so that to = 0, i.e.

t' (x, t) = t - UoX. (2)

We can now define a new cumulative count

A' (x, t') = A (x, t' + uoX), (3)

as the count measured relative to the time t'. In this "moving coordinate system," a vehicle traveling at pace Uo takes zero "time" to travel any distance; it has "pace" zero in the new coordinates or infinite speed. A vehicle traveling at pace u in the real coordinates has pace u - Uo in the new coordinates, and its "trip time" is actually its "delay."

This is particularly convenient if the q-k relation is as shown in Fig. 2. The solution of the equations for the forward moving waves is now

A' (x, t') = A' (xo, t')

independent of x. The cylindrical A ' -surface now has an axis along the x-axis. Whereas in the x, t coordinates

aA (x, t) k(x, t) = - and q(x, t)

ax

aA (x,t)

at

(4)

it is logical that in the moving coordinates, one should define a new "density" and "flow" as

k' (x, t) -aA'(x,t')

ax =

aA (x, t' + uoX)

ax

= k(x, t' + uoX) - uoq(x, t' + uoX),

The (spacial) rate at which A' decreases along a path moving with pace uo, and

, ( ) aA' (x, t') aA (x, t' + uoX) (' q x t == = = q x t + uoX), , at' at' ,

which is simply the flow measured in the new coordinates.

(5)

(6)

The assumption that for a homogeneous highway, k(x, t) is a function k*(q(x, t), x) = k*(q(x, t» implies also that k' (x, t') is a function, k' *, of q' (x, t'); namely

k'(x,t') = k*(q'(x,t'» - uoq'(x,t'). (7)

In particular, if the k* relation is as in Fig. 2, then the relation k' * is as in Fig. 5 with the lower branch being k' *(q) = 0 for 0 < q < Q, and the upper branch having a "pace" -(wo + uo) = -k/Q.

The new form of the equations are, in effect, "special cases" of the original form. In Fig. 3 the A' -solution for the forward waves is equivalent to setting Uo = O. For the backward waves the A ' -solution corresponds to replacing Wo by Wo + Uo in Fig. 4, but

294 G. F. NEWELL

Flow. ql

Fig. 5. The q-k relation in a moving coordinate system.

now the slope of the translation in Fig. 4 becomes kixl - x)/(wo + UO)(xl - x) = Q. The only parameters in Fig. 5 are Q and kj ; the Uo has been completely eliminated.

We have been assuming here that the highway is homogeneous, but the consequences of going to a moving coordinate system resulted from having the wave pace independent of q and therefore equal to the vehicle pace (at all locations). If the vehicle pace depends on x (but not q), one could simply measure the time (' relative to the variable pace trajectory of the reference vehicle. The k' *-relation in Fig. 5 would still apply but with kj

and Q dependent on x. The A ' -solution for the forward waves would still be independent of x.

If kj(x) and Q(x) depend on x but the relation k' * is linear as in Fig. 5 for all x, so that the wave pace kix)/Q(x) is independent of q, a construction analogous to Fig. 4 still applies. The A '(x, (')-curve is simply a translation of the curve A '(Xl' ('), but the horizontal and vertical displacements are

rXI kj(z) d d rx

, k ( )d J -- z an Jx

j Z Z, x Q(z)

the time it takes a wave to travel (upstream) from Xl to X and the jam density storage of the highway between X and Xl' respectively.

We have not yet specified any units of time or length. For a completely homogeneous highway with a single value of Q and of kj we can measure time in units of lIQ and distance in units 11k)' One would draw Fig. 5 as k'*lkj vs. qlQ, a line of slope (wave speed) - 1 and intercepts 1. The cumulative curves would be drawn vs. the "dimension­less" time Qt, the maximum number of vehicles which could pass in time t, so that they would have slope 1 for q = Q. The scale of the graph would likely be in thousands of vehicles, but at least the vertical and horizontal scales would be the same.

For an inhomogeneous highway one could measure time in units of lIQ(xo) for any specified choice of Xo but it would not be advantageous to use different units of time at different locations. There is more flexibility in choosing the scale of length because the units need not be the same everywhere. For example, one could measure "distance" from some point Xo as the (dimensionless) number of vehicles that could be stored between Xo

and x

location of X = rx kj(z)dz, to which would, in effect, make the jam density equal to 1 everywhere, but the wave pace would likely vary with x. Since, however, an increase in road width or the number of lanes would likely cause kj{x) and Q(x) to increase in the same proportion, the wave pace kj(x)/Q(x) might be nearly independent of X if distance is measured on any uniform scale of distance such as the average spacing between vehicles for a single lane.

Theory of kinematic waves in highway traffic, Part II 295

Fig. 6. A freeway merge point.

4. A SINGLE BOTTLENECK

The most elementary application of this theory is to the queueing at a single bottle­neck. Suppose one has a uniform and arbitrarily long freeway with a single entrance ramp feeding additional vehicles to some through traffic as illustrated schematically in Fig. 6. The bottleneck is at the point X, just downstream of the merge, having a capac­ity Q.

In the conventional "deterministic queueing" analysis of this, one would first con­struct a curve A(x" I) equal to the cumulative number of vehicles which would arrive at X,

by time I if there were no queue. This is the combined number of vehicles from both the ramp A,(/) and the freeway A (X,- , I) with x,- a point just upstream of the ramp, measured relative to some reference vehicle numbered ° passing X, at time 0,

A(x"t) =A(x,-,/) + A,(t). (8)

If the slope of the curve A(x" I) should exceed Q, or the slope of A vs. QI should exceed 1 as illustrated in Fig. 7, a "departure curve" D(x

" I) is constructed as the maxi­

mum curve subject to the constraints that D(x" I) ~ A(x" I) and dD(x" 1)/ dl ~ Q. For a typical "rush hour," the D-curve is obtained by pushing a line of slope Q tangent to the curve A(x" I) (slope 1 on the scale of Fig. 7). The D(x" I) represents the cumulative number of vehicles which actually pass X, by time I.

The present purpose of this construction is simply to establish the "boundary condi­tions" for the flows upstream or downstream of X" If there are no further constraints downstream, the D(x" I)-curve is the input to the downstream section and in the moving coordinate system,

3 o

LL

<ll >

o :J

E :J

U

Time,Ot

Fig. 7. Construction of cumulative curves at the bottleneck.

296 G. F. NEWEll

A' (x, I') = D(x" I') for all x > Xl; (9)

if the t' = I at X = x,. How many through vehicles pass x, by time I depends on the priorities at the merge,

but suppose that the A,(t) is the cumulative number of ramp vehicles which aClually pass x, by time I. The cumulative number of through vehicles which can pass x, by time I is then

D(x,-,I) =D(x"t) -A,(t), (10)

(which will certainly have a flow no larger than Q). In deterministic queueing theory the vertical distance

Q(x" I) = A (x" I) - D(x" t)

between the curves D(x" I) and A(x" I) at time I is called the "queue." This is not the number of vehicles in the "physical queue," the number of vehicles between the shock wave and the bottleneck; it is simply the difference between the number of vehicles which would like to pass x, by time I and the number which actually pass. In the present situation it is also the "queue" for the through traffic

Q(x"t) = A(x,-,t) - D(x,-,I),

since the ramp vehicles are assumed to pass x, with no delay. If there are no entrances or exits upstream of x" the physical queue backs up some

finite distance behind x,. In the moving coordinate system the curve A '(x, I') for x upstream of the physical queue is

A'(x,I') = A(x,-,I'), (11)

independent of x (since I = I' at the bottleneck). Indeed the A(x" I') is determined by observing or specifying the A ' (x, I) at some point x upstream of the queue, since the A (x, -, I) was defined as the number of vehicles which would pass Xi by time I if there were no queue.

If vehicles do not pass each other, the value of I' such that A ' (x, I') = j represents the "time" I' when the jth through vehicle actually passes x if x is upstream of the queue. This is independent of x for x upstream of the queue, since the vehicle is traveling with the moving coordinates. This is also the time I when A(x,-, I) = j, the time when the jth vehicle would reach x, if there were no queue. The time at which this vehicle actually passes x" however, is the value of I such that D(x,-, I) = j. The "delay" to thejth vehicle is the horizontal distance from the A(x" I)-curve to the D(x,-, I)-curve at height j as illustrated in Fig. 7.

An important point to observe here is that this delay depends on the "given" arrival curves A(x" t) and A,(t), and Q, but does not depend on where the physical queue may be. One does not need to solve the L-W-R equations to evaluate this. Whether or not the L-W-R theory is correct is irrelevant. If one knows when a vehicle would like to pass the bottleneck and when it actually passes, it makes no difference where or how the vehicle is delayed. The reason one might like to know how far upstream the physical queue extends is that there may be other entrance or exit ramps somewhere upstream and the flow pattern will depend on whether or not the queue backs up past another ramp. If it does not, one probably does not care where it is.

To determine if or when the physical queue backs up to some point x (or any set of locations), one simply translates the curve D(x,-, I) to the right by (k/Q) (x, -x) and upward by kj(x, -x) as in Fig. 4 in a direction with slope Q. The actual curve of A '(x, I') which one might now relabel as D' (x, I') is the minimum of the original A' (x, I') curve and this translated departure curve. The physical queue will back up to x if and when the latter is the smaller.

== o [L:

Q)

> ..... o ::l

E ::l U

Theory of kinematic waves in highway traffic, Part II

tl

Time, Qt'

Fig. 8. Propagation of the queue.

297

/

This is illustrated in Fig. 8. There is only one "undelayed" A '(x, I')-curve; namely, the A (Xi- , I) but a succession of D'(x, I')-curves for various x values, three of which are shown by the curves labeled 0, 1,2. The D'-curve labeled 0 is the result of a hypothetical flow pattern initiated by a large but steady ramp flow from 10 to 11 and therefore a low flow for the through traffic at x,. This causes a queue to form on the freeway starting at time 10, At time 11 the ramp flow drops to a low value and the through traffic flow increases nearly to Q. The queue disappears at time 12 when the D(x,-, I)-curve meets the A ' (x, I' )-curve.

The resulting curve D(x,-, I), also labeled as 0 in Fig. 8, is now translated along the direction of the broken lines (slope 1 in the scale of Fig. 8) to obtain a curve 1 at one location upstream of x, and a curve 2 at a second location further upstream. The vertical component of this translation, kix, - x), is the distance upstream from the bottleneck measured in units of the jam spacing 11 kr The actual departure curve at the first location is the minimum of the curve 1 and the curve A '(x, I'). The shock wave passes this location at the time I' when curve 1 first meets the A '(x, I' )-curve and then again on the way back when the A' (x, I' )-curve later drops below curve 1. The curve 2 is for the location corresponding to the maximum physical length of the queue. Curve 2 is tangent to but never below the A '(x, I' )-curve.

The family of curves for all values of x generated by the curves A ' (x, I') and D(x,- , I) describes everylhing; the shock path, the trajectories of all vehicles, flows, densities, etc. For example, if one draws a horizontal line at heightj representing vehicle numbered j, the time when this horizontal line meets A '(x, I') is the time I' (in the moving coordi­nates) when the jth vehicle would pass any of the x locations if it were not delayed (which is independent of x). The horizontal distance from the A' (x, I' )-curve to the D(x,-, I) curve at height j is the total delay for the jth vehicle, as was already shown in Fig. 7. The distance from the A' (x, I' )-curve to curve 1 is the delay suffered by the time the jth vehicle reaches location 1 and the horizontal distance from there to the D(x, -, I)-curve is the delay suffered between location 1 and Xi' Note that this construction determines the trip times of vehicles between various locations without an explicit evaluation of the velocity or pace, which may be highly time-dependent.

5. INHOMOGENEOUS HIGHWAY, POINT BOTTLENECKS

In the previous section we considered a homogeneous highway which, however, had a single entrance ramp. The bottleneck was caused by the merging flow plus the through traffic exceeding the capacity of the freeway, in particular the downstream section.

298 G. F. NEWElL

33

7 Fig. 9. A freeway section.

If one has an inhomogeneous highway section with capacity and jam density varying with location, we have also seen that the cumulative flow past some point x can still be obtained as the smaller of a cumulative curve from upstream or a suitable horizontal and vertical translation of a cumulative curve from a downstream bottleneck. The horizontal translation was by the trip time of a wave from the bottleneck to x and the vertical translation was by the jam density storage between the bottleneck and x. Both of these quantities could be measured directly in the field. One does not need to know the values of Q(x) and kj(x) at all locations x and, indeed, the details are irrelevant except insofar as they affect the total wave trip time or the jam density storage.

The fact that the flow past any point x cannot exceed the capacity Q(x) at x does, however, lead to some potential complications if Q(x) varies with x (and/or with time) because "bottlenecks" may occur at multiple locations within a freeway section.

Suppose, for example, one has a freeway section as illustrated in Fig. 9 between a junction at XI and another junction at Xs. There are possible restrictions at XI and Xs and also at some intermediate points X2' x3, and X4 but nowhere else. The locations x2 , x3, X4

may be at a curve, a hill, a sight restriction or whatever. We assume that Q(x3) < Q(X2) < Q(xl ) and Q(X3) < Q(x4) < Q(xs) i.e. the most restrictive bottleneck is at x3.

Suppose that the flow past XI should increase with time and exceed the capacities at both X2 and x3• If we disregard the possible constraint at x2, the cumulative departure curve at X3 (in the moving coordinate system) is obtained by drawing a line of slope Q(X3) tangent to the arrival curve A '(Xio t') as illustrated in Fig. 10. The corresponding depar­ture curve past X2 is obtained by translating the curve D'(x3, t') by the appropriate horizontal and vertical displacements and taking the smaller of this and the curve A ' (XI' t'). Since Q(x2) > Q(x3), this translated curve will certainly have a slope less than Q(X2)'

3 .2 LL Q)

> +-C :J

E :J

U

Time, t'

Fig. 10. Queueing at two locations.

Theory of kinematic waves in highway traffic, Part II 299

but it is possible that the slope of the A I -curve could exceed Q(x2) before the queue from X3 backs-up to X2• If so, the D ' (X2, (')-curve is obtained by drawing a line of slope Q(X2) tangent to the A I (Xl! (' )-curve, indicating that a queue starts to form at X2 before the queue from X3 backs-up to X2 and further restricts the flow.

If a queue does form at X2, the departure curve past a point upstream of X2 is obtained by an appropriate translation of the departure curve past x2• Thus, if the queue backs up to Xl' the flow past Xl might drop to Q(X2) for a time before it drops to Q(X3) or it could be further restricted by the capacity Q(xl) before the queue from either X2 or X3 can reach Xl'

If there are no constraints downstream of X3, i.e. every point downstream of X3 can pass a flow Q(x3), and also every point upstream of X3 can pass a flow of at least Q(x3), the curve D I (x3, (I) as drawn in Fig. 10 is valid independent of the actual values of the Q(x), X * x3• Also, the cumulative curves downstream of X3 will be the same as at x3, i.e. min Q(x), Xl < X < Xs is a critical parameter of the freeway section.

If Q(x) > Q(x2) for X2 < X < X3, then any flow which can pass X2 can also pass points between X2 and X3. The actual values of the Q(x), x2 < X < X3 are, in this case, irrelevant except insofar as they may affect the wave trip time between X2 and X3• The only places (such as X2) where a queue can form are places where Q(x) > Q(X2) for all x,

Xl < X < x2. The formation of queues at multiple locations occurs only in rather special circum­

stances and, even when it does occur, it is often of little practical consequence. If one has a curved section of road, for example, there will be some point X3 in the curve where Q(x) is least. At other neighboring points X2 upstream of x3, Q(x2) may be slightly larger. In order for a queue to form at X2 before the queue from X3 backs up to X2, it would be necessary that the flow approaching X2 increase from Q(x3) to more than Q(X2) before the queue from X3 backs up to X2' If X2 is sufficiently close to X3 (a hundred feet or so), this would affect the D I (x2, {' )-curve at most for only a short time (a minute or so) if it happened at all.

Actually one is not particularly interested in knowing the cumulative curves at all locations. One is primarily interested in how constraints at points X2, X3, etc., might affect the flows past the junction Xl' When one translates the curve D I (X2' (I) to the location Xl' only that portion of the translated curve which is below A I (Xl! (') is relevant. The initial phases of the D I (X2, (' )-curve, before the queue backs-up to Xl' are irrelevant. In order for a queue at X2 to affect the cumulative curve at Xl' it is necessary that the queue from X2 itself back-up to Xl before the queue from X3 would back-up to Xl'

The situation in which constraints at two or more locations X2' X3 are most likely to affect the flow at Xl is if X2 is "close" to Xl but X3 is not. If X2 is essentially at Xl, for example, X2 is at the end of the merge of an on-ramp, a surge of traffic from the on-ramp might create a flow approaching X2 exceeding the capacity Q(X2), causing a queue to propagate upstream of Xl' A flow between the values Q(X3) and Q(x2) will, of course, also cause a queue to form at X3 "earlier" (in the moving coordinates), but it is quite possible if X3 is several miles downstream, that the queue from X3 would not back-up to X2 before the flow approaching X2 exceeded Q(x2). This could be of particular importance if Q(x2) is nearly equal to Q(x3) as for a nearly homogeneous freeway section from Xl to X3. A queue may form at X2 at nearly the same "time" as at X3, but the effect of a queue at X3 would not be known at X2 until at least a wave trip time after the queue forms at X3•

Although it is possible to deal with constraints at many possible locations, we expect that, in most practical situations, the only capacity constraints which affect the flow at Xl are the capacities at or near Xl and the capacity Q(X3) = min Q(x).

There is an analogous argument regarding the effect of an inhomogeneous Q(x) on the flow past xs. Suppose that a queue has backed-up past X3 from xs, possibly caused by a heavy on-ramp flow at Xs or at points downstream of x s, but then the constraint is relaxed permitting a flow past Xs exceeding Q(X3)' In the absence of any constraint up­stream of xs, the cumulative departure curves at any X,X < xs, would be obtained by an appropriate translation of the curve at Xs. The slope of this curve, however, cannot exceed Q(x) at any x.

300

3 o

LL Q)

> ..-o ::l

E ::l U

G. F. NEWELL

Time, t l

Fig. 11. Queueing at two locations.

Suppose, as illustrated in Fig. 11, the flow past Xs should increase so as to exceed Q(X3). A wave trip time later, the flow past X3 would try to exceed Q(X3), but it cannot. The actual D'(x3, (')-curve is obtained by drawing a line of slope Q(x3) tangent to this translation of the D' (xs, (' )-curve. This curve now becomes an "arrival curve" for all points downstream of X3. Since the D' (xs, (' )-curve cannot exceed this arrival curve, this curve of slope Q(x) becomes the D' (xs, (' )-curve once it drops below the original D' (xs, (')-curve; the Xs location is being "starved" by the upstream restriction.

The effect on the through traffic of a time-dependent ramp flow at Xs or at down­stream ramps is similar to having a time-dependent "capacity" at Xs. If this "capacity" had been less than Q(x) at earlier times, it was the "bottleneck." But when this constraint is relaxed, the bottleneck shifts to x3• The effect of the bottleneck at X3 will not be felt at xs, however, until the queue of vehicles that existed between X3 and Xs has passed Xs.

Analogous to the effects illustrated in Fig. 10, it is also possible to have constraints at multiple locations between X3 and Xs if the flow past Xs should try to increase too rapidly. Suppose that there were some point X4 between X3 and Xs at which Q(x4) > Q(x3) and the flow past Xs should increase so as to exceed Q(x3) and then also to exceed Q(x4).

In the absence of other constraints, the departure curve past X4 is obtained by an appropriate translation of the curve from Xs. A wave trip time after the flow exceeds Q(x4) at xs, the flow will try to exceed Q(x4) at x4, but it cannot. One must draw a line of slope Q(x4) tangent to this translated curve. This will have no effect on the departure curve at X), which will already have achieved a lower slope before a wave of flow Q(x4)

can reach x3• However, the departure curve at X4 cannot exceed that at X3. The capacity at X4 will have no effect at all if the flow at X4 is constrained already by

the departure curve at X3 by the time a wave of flow Q(x4) can reach X4 (the queue between X3 and X4 is already gone). But even if the constraint at X4 does affect the flow past x4 , we are primarily interested in what effect this has on the flow past Xs. It will have no effect unless the departure curve at X4 of slope Q(x4) drops below the original curve at Xs before the departure curve at X3 constrains the curves at both X4 and xs, i.e. the queue between X4 and Xs dissipates before the queue between X3 and X4 clears out.

The arguments here are a "mirror image" of the arguments relative to multiple constraints between Xl and x3, except that vehicles are replaced by "holes" or deficiencies of vehicles. Again we can argue that a capacity constraint at a point X4 is not likely to affect the flow at Xs unless X4 is "close" to Xs but X3 is not. The most likely location of a constraint would be immediately upstream of a freeway off-ramp. If some constraint downstream of an off-ramp were suddenly removed, the capacity of the freeway at the

Theory of kinematic waves in highway traffic, Part II 301

off-ramp may restrict the flow temporarily, until it is further restricted by a more severe bottleneck upstream. This could be of particular importance if Q(x4) is nearly equal to Q(X3)' For example, the bottleneck at X3 may be due to a lane drop, but the highway is nearly homogeneous from X3 to xs. If the flow tries to increase at x s, it would likely to be restricted by the Q(x4) before it is further restricted by the Q(X3)'

In summary, in order to relate the cumulative flow at one end of an inhomogeneous freeway section to the cumulative flow at the other end (XI to Xs or vice versa), it should typically suffice to know

1. The wave trip time from Xs to XI and the jam density storage between XI and x s, so that one can relate the cumulative curve at XI to that at Xs if a queue propagates upstream fromxs·

2. The capacity min Q(x) so that one can check to see if a queue forms within the section (at X3 in this example). If so, one should also know the wave trip time and jam density storage from the bottleneck X3 to XI and from Xs to X3 (the former if X3 restricts the flow upstream, the latter if it restricts the flow downstream).

3. The capacities at or near XI and Xs if these capacities might constrain the flows entering or leaving the freeway section.

Any other features of the freeway section are likely to be irrelevant. To deal with possible "incidents" (accidents) within a freeway section, one should

also know the maximum (time-dependent) flow which can pass the location of the inci­dent (assumed to be sometimes less than Q(x3»; also the wave trip time and jam density storage between the incident location and other points XI. X3• and Xs.

6. MULTIPLE FREEWAY SECTIONS

We consider now a long stretch of freeway with many entrance and exit ramps as illustrated schematically in Fig. 12. The points X, now designate either entrance ramps or exit ramps or both (if exit and entrance ramps are so close together that one can neglect the separation between them).

Let A' (x, I') represent the cumulative number of vehicles to pass X by "time" I' counted from some hypothetical reference vehicle 0 which travels at its free speed in each section (not necessarily the same speed at all locations), entering at Xo but never exiting, and relative to the time I' measured at each X from the passage time of this reference vehicle. We assume that the cumulative input at xo, A' (xo, I') is given and that, if queues form anywhere on the freeway, they do not back-up past xo•

Let A,{/') be the net cumulative number of vehicles to enter the freeway by time I' at x" i.e. the number that enter minus those that leave. The All') and/or dA,(/')/dl' could be negative (at an exit). For now we will treat the A,(/') as "known" even though one may not be able to predict the number of vehicles exiting the freeway without knowing how long the vehicles have been delayed. (We will address this question in Part III.) There are certain restrictions which must apply to the Ai(/') so as to guarantee that the flow on the freeway is never negative and never exceeds the capacity in any section, but we assume that the given A,(/') are consistent with any such restrictions.

The problem now is to evaluate the A ' (x, I') for all X and I', but particularly at the junctions x, from the given values of A ' (xo• I') and A,(/'), and the relevant properties of the freeway sections.

The A,(t') are considered to be the net number of vehicles which actually enter at x,

~I~ x· I

Fig. 12. A sequence of on-ramps and/or off-ramps.

302 O. F. NEWELL

by time I'. We are not concerned here with how these may depend on possible control strategies. The A,(/') cause a discontinuity in the A' (x, I') at x, such that

A'(x,+, I') = A'(x,-, I') + A,(t'). (12)

as in eqn (8). No queues will form anywhere until such time I' ° when the flow past some location

x tries to exceed the capacity Q(x). Until then

A'(x,_I+, I') = A'(x, I') = A'(xi-, I') (13)

independent of x for X,_I < x < x, and I' < 1'0' Thus, starting from the given A , (xo, I'), we can successively evaluate A'(xl -, I') = A'(xo, I') from eqn (13), A'(xl+, I') from eqn (12), then A ' (x2-, I'), etc. After queues form in various sections, the relation between A ' (Xi -I +, I') and A ' (Xi -, t') at the two ends of section i is as described in the previous section.

In the last section we argued that if the freeway between Xi-I and Xi were inhomoge­neous, the flow pattern at the junctions would likely be affected only by the capacities near the ends of the section and/or by min Q(x), X'_I < x < Xi' A bottleneck within a section, however, can be treated as if it were a special case, a "junction" with zero entrance or exit flow and equal capacities immediately upstream and downstream of the "junction." To simplify the notation it is convenient to include in the set of points x, not only any entrance or exit ramps but also each point between successive ramps where Q(x) is a minimum between ramps (if this point is not already near a ramp). Hereafter the {x,} will be the ordered set of all such junctions with X,_I < x,. We need not worry anymore about the capacities at points between junctions.

Q/ Qi-

ri ni

We now claim that the only potentially relevant properties of the freeway are

= = = =

Q(x,+) = capacity near the downstream side of x,; Q(x, -) = capacity near the upstream side of x" (Q, - = Q, + if x, is not at a ramp); wave propagation time from x, to X'-l (in the moving coordinates) jam density storage between X,_I and x,.

A computer algorithm for evaluation of the A' (x±, I'), would proceed in a quite different matter than a graphical hand calculation. A computer algorithm would evaluate the A' (x, \ I') only at discrete times (for example, at integer multiples of some lattice time r) but would be designed to deal with any possible surges in the arrivals or departures at any time or any location.

One possible scheme would proceed iteratively in space and time. If one has already evaluated the A'(x,-, I') for all i at previous times I' = k'r, k' = 0, 1, ... , k - 1 and also A' (X,_I -, kr), one would next evaluate A' (x,-, kr).

The A ' (Xi - , kr) is the smallest of:

(a) A '(X,_I +, kr) = A '(X,_I-, kr) + Ai_l(kr) the maximum number of arrivals from upstream, or

(b) A , (Xi- ,(k - l)r) + Q,-, r, if the flow is constrained by the capacity Q-" or

(c) A' (x,- ,(k - l)r) + A,«k - l)r) - A,(kr) + Q,+ r, if the flow past Xi + , is constrained by the capacity q, + , or

(d) A '(XI+I-' kr - rl+l) + nl+ l - A,(kr) if the flow is constrained by a queue from downstream.

If x, is not at a ramp, Ai(t) = 0 and Q,- = Q,+, then conditions (b) and (c) are the same. In condition (d), the r,+ I is not generally an integer multiple of r. If r,+ I < r we

Theory of kinematic waves in highway traffic, Part II 303

would treat the points x/ and X/+ I as a single junction neglecting the separation. Otherwise the A' -function in (d) can be evaluated by interpolation between values from earlier times.

In most practical applications the on-ramp flows have a single surge typical of a "rush hour." A queue is most likely to form at only one point, the "main bottleneck," but at most at only a few points. Queues could, however, back-up past several junctions. Instead of following the above scheme, a manual graphical procedure would take advan­tage of what one already knows. One would not need to keep checking for possible constraints that one knows will not apply. A manual procedure could proceed as follows.

One might start by constructing the unconstrained A ' (x/ ±, t') curves only at those locations where queues are likely to form. By comparing the slopes of the A '(x± , t') with the capacities at x" one would identify the first time and place when a queue would form anywhere. Temporarily one will disregard any other capacity constraints and evaluate the departure curves that would exist at this location x/o if this were the only constraint. One would draw a line of slope Q(xio ) tangent to the arrival curve as in Fig. 7 until it intersects the arrival curve again. This may not necessarily give the final solution at Xio but it does give an upper bound, since any constraints due to capacities at other locations (either upstream or downstream) can only cause the departure curve at x/o to decrease even further at later times.

The effect of a constraint at Xio is to decrease the cumulative curve at x/o at time t' . The departure curve at x/o' however, determines the arrival curves downstream; so, for any downward displacement of the departure curve at x io ' one must apply an equal displacement to the curves at time t' for all Xl downstream of x/o•

One can also follow the propagation of the queue upstream. Each time the queue passes a junction, one subtracts off the known entrance and/or exit flows and continues upstream. Having determined the flow pattern constrained by the single constraint at x/o'

one can inspect the resulting cumulative curves to see if the flow past any other location exceeds the capacity. If so, pick the first such event at location X'I and time t l •

One can again construct a constrained departure curve at Xi, as in Fig. 7 relative to the previously evaluated curve. If X'I < x/o' the constraint at X'I will affect the arrival curve at x/o but only for t' > t; > to. This will have no effect on the departure curve at x/o until the displaced arrival curve at x/o drops below the departure curve, at which time one simply replaces the departure curve by the new arrival curve. Regardless of whether or not the queue from x/o may have backed up to x/" one can determine the propagation of the queue from X'I upstream as in Figs. 7 and 8.

If X'I > x/o the flow downstream of X'I will be further constrained by the departures from Xi, and a queue will start to propagate upstream of X'I. At some later time this queue might reach xio ' i.e. the departure curve at x/o determined from the constraint at X'I may drop below that determined previously from the constraint at x/o' at which time the latter is replaced by the former. One can now test to see if the solution constrained at both x/o and X'I violates any other capacity constraints at later times t' > t;. One can obviously continue this until there are no new constraints.

This graphical procedure exploits the fact that it is very easy to construct a line of given slope tangent to a curve, and it is also fairly easy to make a uniform translation of a curve and to take the lower envelope of two curves. The addition and subtraction of the A,(t) may, however, be a bit tedious. For a computer, it is better to have a simple program even if it is not computationally very "efficient." The above algorithm would run so fast that one need not worry about efficiency.

REFERENCE

Newell G. F. (1993) A simplified theory of kinematic waves in highway traffic, Part I: General theory. Transpn. Res., 27D, 281-287.