On the Variational Theory of Traffic Flow: Well-Posedness, Duality and Applications

: This paper describes some simplifications allowed by the variational theory of traffic flow (VT). It presents general conditions guaranteeing that the solution of a VT problem with bottlenecks exists, is unique and makes physical sense ABSTRACT This paper describes some simplifications allowed by the variational theory of traffic flow (VT). It presents general conditions guaranteeing that the solution of a VT problem with bottlenecks exists, is unique and makes physical senseB i.e., that the problem is well-posed. The requirements for well-posedness are mild and met by practical applications. They are consistent with narrower results available for kinematic wave or Hamilton-Jacobi theories. The paper also describes some duality ideas relevant to all these theories. Duality and VT are used to establish the equivalence of eight traffic models. Most of these are not new but VT-duality considerations offer a new insight into their relationship.


INTRODUCTION
Consider an infinite one-directional road on which vehicles cannot pass and move in the direction of increasing distance, I. If at some location I I 0 we assign consecutive integers to the vehicles we observe as time increases from -! to K!, then the space-time trajectories of all the vehicles are completely defined as proposed in Moskowitz (1NO5), and further elaborated in Makigami et al (1N71), by the integer contours of a surface. This surface is characterized by a continuous function N(t, I) I n. The floor " n # is the number of the last vehicle to have advanced beyond I by time t. Since passing is not allowed the ordering of the vehicles is preserved everywhere. Therefore we can assume without loss of generality that 2 N(t, I) is non-increasing in t for every I. Moreover, since vehicles move in the direction of increasing I; we can also assume that N is non-decreasing in I for every t8 The simplest model of traffic flow further assumes that that N is differentiable almost everywhere (except possibly along some curves that would form ridges in the surface defined by N) and that the first partial derivatives of N are related by a functionB i.e.: This is a Hamilton-Jacobi (HJ) equation with O as the Hamiltonian. Note that $NY$t (abbreviated +) is the flow and %$NY$I (abbreviated B) is the density, and that meaningful solutions require flow and density to be non-negative.
The function O is called the "fundamental diagram" (FD) by traffic engineers. We assume in rough agreement with experiments that O is piecewise differentiable and concave in its first argument and returns non-negative values (for every t and I) if the first argument is in an interval \0, '(t, I)] such that O(0, t, I) I O(', t, I) I 0, with '(t, I) ^ !. The parameter ' is called the "jam density." The maximum of O, + maI , is called the "capacity"B see Fig. 1(a). Note that if (1) is differentiated with respect to I and expressed in terms of density it reduces to the conservation law: $BY$t K ($OY$B)($BY$I) I %$OR$I. This is the classical kinematic wave (`W) formulation of aighthill and Whitham (1N55), and Richards (1N5O). A simplified solution of some `W problems in terms of the Moskowitz function was later proposed in Newellds seminal trilogy (Newell,1NN3). Newellds results have been recently formalized and extended in Daganzo (2003Daganzo ( , 2003aDaganzo ( and 2005, which proposes a variational theory (VT) able to capture bottlenecks of all types. (

Variational theory
Variational theory also assumes that (1) holds where N is differentiable and that O is concave, but treats the problem as a capacity-constrained optimization problem. An intuitive explanation is as follows. We know that the flow at any point in space-time is bounded from above by + maI., the capacity. A similar capacity constraint should also hold if the road is viewed from a rigid frame of reference that moves with speed I'. In this case the capacity relative to the frame (the "relative capacity") is the maximum rate at which traffic can pass an observer attached to the frame. Since an observer that moves with speed I' next to a traffic stream with density B and flow + is passed by traffic at a rate + f BI', the FD for the moving frame is O(B, t, I) % B I' and its relative capacity is: Figure 1(a) shows these relations geometricallyB note that the relative capacity P is the intercept of the tangent to curve O with slope I'. Figure 1(b) shows the relative capacity function (also called the "cost function" in variational theory) with I' as the argument. Note that %P is the aegendre-Fenchel transform of O, and that as a result, as shown by Fig. 1(b), P is convex and strictly decreasing in the range of "valid" slopes where O is non-negativeB i.e., for I' , -\w(', t, I), w(0, t, I)]. Note as well that P ! 0 and P(u) I 0, since curve O touches the origin. Thus, in traffic flow the aegendre-Fenchel transform has an intuitive physical interpretation, which makes its application fairly intuitive as we shall now see.
Clearly, an observer travelling with a valid speed IQ(t) -\w('), w(0)] along a "valid" space-time path ! from point D to point P cannot see a change in vehicle number greater than the integral with respect to time of its relativistic capacitiesB i.e., an upper bound to change is: where t D and t P are the times associated with the path endpoints. Therefore, an upper bound to the vehicle number N P observed at a point P can be written by considering the set P of all valid observer paths to P from the points of a boundary D where the vehicle numbers are known. In other words, if D(!)-D is the beginning of a valid path !, and N D(!) is the known vehicle number at D(!), then it must be true that N P , must satisfy: kquation (4) is the capacity constraint mentioned at the outset.
In variational theory the solution domain S is the set of points P such that all infinitely long valid paths ending at P intersect the boundary. For example, the solution domain for the initial value problem (IVP) is the half plane, t ! 0. Variational theory assumes that capacity constraint (4) is bindingB i.e., that the actual value of N P for P -S is the largest possible allowed by (4): This is a calculus of variations problem. It is well known that under some regularity conditions (5) characterizes both the viscosity solution of the HJ-IVP and also the entropy solution of the `W-IVP. 2 A key advantage of VT over the HJ and `W theories is its natural framework for expressing the relatively complicated problems arising in traffic flow applications (including bottlenecks and finite roads), and the convenient way in which the "well-posedness" of such problems can be assessedB see below.

Homogeneous problems with point bottlenecks: solution existence and uniqueness.
In traffic flow theory it is often necessary to consider "point bottlenecks". These are usually slower vehicles or fixed obstructions that reduce the maximum rate at which traffic can flow past them. A point bottleneck is defined by its space-time trajectory I A (t), assumed to be a valid path, and by its relative capacity (maximum passing rate) r A (t). The bottleneck imposes the condition dN(t, I A (t))Ydt " r A (t) in HJ theory. This type of constraint seems not to have received much attention in the mathematics literature. The constraint is even more complicated when expressed in terms of `W theory. But the complication disappears in VT.
In VT a bottleneck reduces the original relative capacity of the road along the bottleneck trajectory. This is recognized by using r A (t) instead of P as the integrand in (3) for the portion of any path that overlaps I A (t). 3 Nothing else needs to be changed: (4) and (5) continue to apply. Hence, in VT, point bottlenecks are just shortcuts through space-time, which preserve the shortest-path character of the problem without increasing its complexity. The solution should be equally easy to find. The question is whether the solution with bottlenecks is continuous and varies with t and with I at allowable rates.
We look for solutions that satisfy the following aipschitz-continuity conditions: A solution satisfying (O) is obviously continuousB thus, vehicles have continuous trajectories. Furthermore, if (O) holds, vehicles can neither reverse direction nor overtake an object moving with speed uB i.e. their average speed is always bounded in a physically meaningful way. 4 Therefore, solutions satisfying (O) will be called "valid". A VT problem whose solution is valid will be said to be "well-posed". We examine below whether these conditions are satisfied for "homogeneous" problems in which O and P are time-independent and space-independent. Therefore, they will be expressed from now on as functions of one argument, O(B) and P(I')B the parameters', + maI , etc. will become constants. It will be useful to keep in mind that for homogeneous problem without bottlenecks straight lines turn out to be optimum paths and the RHS of (3) . In this special case, thus, the calculus of variations problem (5) reduces to an ordinary minimization for the point on the boundary (D-D) that produces the minimum cost. We can now state the following.   for problems with complex boundaries: namely, that the least cost of reaching a boundary point with a path from the boundary be no less than the cost specified for that pointB i.e., that N(t, 0) I n(t, 0) and N(t, I o ) I n(t, I o ) for t 1 0. In our case, this means that the boundary lines I I 0 and I[ I o must be optimum paths.
THkORkM 2: A FHP with bottlenecBs is well posed if: (i) the boundary data n(t, I) satisfies (O)B (ii) the bottlenecBs satisfy r A (t) 1 0; dI A (t)Rdt 1 0 and I A (t) 1 0B and (iii) the consistency condition is satisfied: N(t; 0) I n(t; 0) and N(t, I o ) I n(t, I o ) for t " 0.
In applications, Theorem 2 can be interpreted in terms of a competition between "upstream demand" and "downstream capacity". aet U(t) be a real function satisfying (Ob), giving the number of cumulative desired entrances at I I 0B and let N D (t, 0) be the available capacity at I [ 0 from conditions downstream. We define N D (t, 0) as the infimum of the costs of reaching point (t, 0) from the boundary with valid paths in the solution domain starting at an earlier timeB i.e., from the set of points and 6(t) can be chosen in any way consistent with (Ob). For example, if there is a highway with bottlenecks and different O and P for I ^ 0, we can define U(t) for the downstream problem as the demand at I I 0 arising from the upstream problem: U(t) # N U (t, 0)B and choose 6(t) for the upstream problem as the available capacity arising from the downstream problem at I [ 0: N D (t, 0). To stitch together the two solutions we stipulate n(t; 0) I minhN U (t, 0), N D (t, 0)i for both problems. This ensures that both problems are well-posed, and is a natural way to treat inhomogeneous highways.
The Theorems and Corollary are consistent with results available for `W (or aWR) theory for the case without bottlenecks. The results also generalize the demand vs. capacity metaphor of the cell-transmission model (Daganzo,1NN3,1NN4) and the related formulations in Daganzo (1NN3a), aebacque (1NN3). 5 The results can also be applied to time-dependent problems. Well-posedness can be checked in this case by slicing the solution space into successive time-independent problems and verifying that each time-independent slice satisfies the conditions of one of the above theorems. Unfortunately, well-posedness cannot always be tested a priori (before solving the problem) as in the time-independent case because for the initial data of a slice to be valid (and the theorems to hold) the solution obtained at the end of the previous slice must satisfy (Oa) with the jam density ' specified a priori for the current slice.

Linear cost functions
In this subsection O is triangular in B. Now, the problem simplifies even more because the cost function (3) is linear (Daganzo, 2003a). If we use u , w(0) and fw , w(') for be the slopes of the rising and dropping branches of O (in traffic flow lingo u is the "free-flow speed" and w the "backward wave speed"), then (2) becomes: Note that P(I') decreases. We shall abbreviate its maximum: P(%w) I (1_wRu)+ max , by the symbol r. This parameter (the maximum relative capacity) will be of importance later. kxperiments show that r is about 15q greater than + maI .
This case is so simple because when P is linear the cost of a path (4) is a linear function of the pathds duration and distanceB i.e., if ! goes from D to P: Hence, if as is often the case in traffic flow the boundary data (i.e., the coordinates t D and I D and the values N D for all points D-D) are given as piecewise linear functions of a parameter, t D I t(2), I D I I(2)) and N D I n(2), then (5) becomes: which is just the minimization of a piecewise linear function. Obviously, we can find its minimum by inspecting the corners of the objective function.
The solution can also be found with network algorithmsB see e.g., Daganzo (2003a). These methods are advantageous when the solution is sought at many points in the solution domain. The networks in question are digraphs with nodes L embedded in space-time, with directed arcs LLQ. Arcs are defined only for node pairs that can be connected by a valid path. We call these "valid node pairs." kach arc is assigned a cost, c LL' , equal to that of an optimum continuum path between its end nodesB e.g., as given by (8) when O is triangular. Of interest are networks whose shortest "walks" (network paths) between all valid node pairs are shortest continuum paths. These networks are said to be "sufficient" because by solving the shortest path problem on the network one solves the continuum problem exactly for all its valid node pairs. This is useful because if one puts nodes of a sufficient network on the corners of a piecewise linear boundary, then the network solution identifies the exact N at every node. The solution can be found with the usual dynamic programming recursion: where F(LQ) is the set of "from" nodes of LQ.
For problems with linear P sparse sufficient networks with as few as 2 links per node can be constructedB thus (10) can be computed fast. The rest of this paper does not consider bottlenecks and uses sufficient networks of the "lopsided" type defined in Daganzo (2003a). O A lopsided network (see Fig. 2) is a network with the following properties: (i) its nodes are on a rectangular lattice with space separation 4 and time step 5, (ii) the set of links pointing to O aopsided neworks can only be used if there are no bottlenecksB otherwise they need to be modified. This is done by overlying a discrete shortcut with appropriately reduced link costs over the networkB see Menendez and Daganzo (2005). any node is translationally symmetric, (iii) two of the links in this set have slopes u and %w, and (iv) none of the links spans a distance greater than 4. Note: since the nodes are on a rectangular lattice, 4Yu and 4Y(%w) must be integer multiples of 5, assuming u, w 6 0. These networks will help us compare different ways of finding N8 But before this is done we introduce some duality ideas, which will allow us to double the number of models covered under the same umbrella.

DUALITY
In this section O(B) is general-not necessarily triangular. The results apply to problems where N(t, I) strictly decreases with I for every t in the relevant solution domain. aemma 5b of Appendix A shows that an IVP with strictly decreasing n(0, I) satisfies this condition if it has no bottlenecks with zero relative capacityB and also if there are bottlenecks with zero relative capacity but the solution is only sought upstream of them. 7 Since N is continuous and declines with I, the relation N(t, I) I n defines an implicit function for I in terms of t and n, I I `(t, n). This function gives the position of vehicle n at time t. It is also continuous and declines with n. Both functions describe the same Moskowitz surface. The two functions are connected by the relation: which merely expresses that the position at time t of the vehicle that was at I at time t must be I. Conversely, we can also write: since the vehicle number found at time t at the position of vehicle n at time t is n. Note that (11b) is obtained from (11a), and vice versa, by interchanging (I, `) with (n, N). Since the (primal) results of the previous section were derived with N as the unknown, this suggests that The mathematical e+uivalence of eight traffic flow models N similar (dual) results can be derived with ` as the unknown after swapping the variables and functions for position and vehicle number. Differentiation of (11a) with respect to t and x yields the following relation among the partial derivatives of the primal and dual functions: The same expressions with (I; `) and (n, N) interchanged are obtained if one differentiates (11b). The quantity v [ $`Y$t is the vehicle speed, and the quantity s I %$`Y$n the reciprocal of densityB i.e., the continuum version of vehicular spacing. If we now insert (12) into (1) we find: where U is related to O by the following transformation: 8 kquation (13), like (1), is an HJ equation. Since we have not reversed the direction of time the solution of (13), which is obtained by transforming with (11) the stable (viscosity) solution of (1), is also a stable solution. Thus, for any given set of boundary conditions \t D I t(2), I D I I(2)) and N D I n(2)] the stable solutions of (13) and (1) describe the same Moskowitz surface. Since U, like O, is concave in the relevant range of its argument, s -\1Y', !), `, like N, can be found with VT. Thus, the Moskowitz surface can be found by solving with the same methods either a primal problem (1) or a dual problem (13). N The dual cost function P d is given by (2) with U substituted for O8 We find that P d is the inverse of P, with the roles of speed I' and passing rate n' reversed, and that it still is convex and decreasing in the relevant range of passing rates. In the triangular case the dual cost function is the inverse of (7), and is still linear: 8 Note that the transformation O 7 U is an involution, which should not be surprising since the swap of I and n is a reflection. N Note too that one can define by differentiating (13) with respect to n a conservation law, $sY$t K ($U(s)Y$s) $sY$n I 0, which is the dual of $BY$t K ($O(B)Y$B)$BY$I I 0. The analyses and methods relevant to the primal conservation law also apply to the dual. Therefore, the sufficient lopsided networks that one could use with (10) now have slopes equal to the cost rates of the primal (0, and r) and cost rates equal to the slopes of the primal (u and %w, respectively). Variational theory in its primal and dual forms is used in the next section to examine the connection between eight different traffic models.

APPLICATION: EIGHT MODELS OF A TRAFFIC STREAM
In this section the highway is homogeneous and the FD is triangular. We classify traffic models into 4 categories distinguished by the number of variables that are treated discretely: 0-models treat all variables continuously, as in the discussion up to this pointB 1-models treat one independent variable (I or n) discretelyB 2-models treat both independent variables (t and IB or t and n) discretelyB and 3-models treat all variables (t, I and n) discretely. Here we shall present primal and dual VT models for each category (eight models in total) and see how they relate to existing ones.

0-models:
These are fluid models. Our primal 0-model is (3, 5). We have already seen that it has the following dual 0-model, where ! is a dual path n(t) from (t D , n D ) to (t P ,n P ): The last equality follows from (15). Note the similarity of .( !) in (1O) and (8).

1-models:
These are queuing and car-following models. An example of a primal 1model is Newellds queuing formula (Newell,1NN3) The reader can verify that (17) is the result of applying (N) to our boundary data. 10 We now apply (1O) to a "lead vehicle problem". This is a dual problem with boundary conditions: `(t, 0) I I 0 (t) for t 1 0 and `(G, n) I I n (0) for n 1 0. Assume the I n (0) is linear in n (vehicles are uniformly spaced) and that dI 0 (t)Ydt " u. Then, an optimum path to reach point (t; n) for some integer n must begin at one of the two extreme points of the relevant part of the boundary for point (t, n): either point (0, n) or point (t% nYr, 0). The result is: `(t, n) I minh I n (0) K ut, I 0 (t% nYr) % ns j i, 10 Since N U and N D cannot increase at a rate that exceeds + maI , an optimum path to point "P" must emanate from a point on the (upstream or downstream) boundary with the highest possible t8 Only two such points generate valid paths. They correspond to the two arguments of (13).
The mathematical e+uivalence of eight traffic flow models 11 which is the trajectory of vehicle n8 The parameter 1Yr (comparable with a second) has the interpretation of a reaction time and we denote it by 8. In practice we are usually interested in the 1-model that seeks the values of ` for all integer n. A recursive expression is obtained by setting n I 1 in the above and applying the same recipe to all consecutive vehicle pairs. The result is the following car-following law (Newell, 2002):

2-models:
These are numerical versions of fluid models. For the primal we use (10) with a lopsided network with two links per node. We choose 4 I u58 Therefore, the links with slope u (and zero cost) span one time step. The links with slope %w (and cost rate r) span 9 I uYw time steps. Hence their cost is c [ r95 [r4Yw. Note: since 9 must be integer we are assuming that uYw is an integer-this ratio is comparable with O in practice. If we now use sub-indices l and m to identify the time and distance steps, i.e., so that N lm , N(l5, m4), (10) becomes: kquation (1N) expresses the ACT (asynchronous cell transmission) model for cells of size 4B see Appendix B. 11 A dual 2-model is obtained by applying (10) to a lopsided network on the (t, n) plane as described above with arc slopes (0, r) and arc cost rates (u, fw). We choose the step for variable n to be 1 and the time step, 5 I 1Yr [ 8. This achieves a rectangular lattice, since 5r [ 1. The link costs become as a result: u8 and %w8 I %s j . Therefore, with the convention: `l m , `(l8, m), recursion (10) reduces to: `l m I minh`l %1,m K u8, `l %1,m%1 % s j i.

3-models:
kxamples of 3-models are cellular automata (CA) models, where cars are assumed to jump on a lattice. Most CA models are described in dual space, but as we now show primal models can also be derived. Simply, use 4 I s j , wRr in (1N), which yields c [ 1, and therefore: This expression returns an integer if the input vehicle numbers are integer. Therefore it is a CA model. The expression indicates that the vehicle count at a point in space increases by one if and only if vehicle number at the downstream lattice point had reached the target number 9 time steps agoB i.e., if the previous vehicle had jumped from m and left it vacant for at least 9 time steps. This is the CA(M) rule described in Daganzo (200O). Consider now the dual formula (20) and express it in dimensionless distance, c [ `Ys j .. It becomes: 12 c lm I minhc l%1,m K u8Rs j , c l%1,m%1 %1i I minhc l%1,m K 9 , c l%1,m%1 %1 i (22) We see that if vehicles are initially on the lattice (the cds are integer) and if 9 is an integer, then (22) keeps vehicles on the latticeB i.e., it is a CA model. kquation (22) is the unbounded acceleration model of Nagel and Schreckenberg (1NN2), called the CA(a) model in Daganzo (200O).
This concludes our review. Duality and variational theory provided a framework that clearly established the equivalence of our models. The best model for any given application depends on the form of the data and the requirements of the output.

COMPOSITION INTO NETWORKS AND DISCUSSION
Primal analysis looks for the flow of vehicles from the perspective of the roadB and dual analysis the "flow of road" from the perspective of the vehicles. 12 Fixed bottlenecks such as merges and lane-drops are understood by scientists in primal space, from the perspective of the road, since this is the form in which data are available. Moving bottlenecksB e.g., those caused by slow-moving obstructions are understood from the perspective of the moving bottleneck, since data from this perspective is available. The moving-bottleneck effects of lane-changing are most easily expressed in dual spaceB those of fixed bottlenecks in primal space. The ideas in this paper allow us to combine the effects of fixed and moving bottlenecks, including lane-changing, consistently in whatever framework is most useful (primal or dual) for a practical application.
Since lane changes to a faster stream act as moving bottlenecks on the target lane, and lane changes to a slower stream act as moving bottlenecks on the source lane, the ability to treat moving bottlenecks allows us to compose the very basic component described in this paper-a single lane of traffic-into complex multi-lane streams quite realisticallyB this approach was explained, proposed and tested with encouraging results in aaval and . It has proven to be parsimonious and surprisingly accurate for lane drops, moving bottlenecks and mergesB see also aaval et al, (2005). A variant of it has also been applied to HOV lanes, with considerable success (Menendez and Daganzo,200O).
Composition of links into networks is possible along traditional lines, e.g., as in the CTM (Daganzo, 2004). But the ideas of this paper allow us to treat turning movements as lane-changes, and junctions as complex multi-lane links. Therefore, they allow us to compose multi-lane links into networks in quite a bit of detail without introducing extra parameters. 12 A possible interpretation of dual VT and its constraints is as follows. Imagine uniformly spaced parked (dual) vehicles by the side of the road. Then, dual VT describes the flow of these vehicles from the perspective of a flexible frame of reference attached to the moving (primal) vehiclesB i.e., where (dual) distance increases by a unit with each (primal) moving vehicle. From this frame of reference, the (dual) flow is the rate at which dual vehicles (i.e., units of primal distance) flow past fixed positions in the dual frame (i.e., moving-primal vehicles). Thus, dual flow I primal speed. Conversely, the rate at which a dual vehicle overcomes dual distance (i.e., moving vehicles) is both the dual speed and the primal flow. And the number of parked vehicles between two consecutive moving vehicles is both the dual density and, the primal spacing. Thus, dual-VT can also be interpreted in terms of flows and densities, and its constraints described in terms of relative capacities, but all from the perspective of the flexible frame of reference. Thus, the dual relative capacity is the maximum flow of parked vehicles that can be seen by an observer jumping from primal vehicle to primal vehicle with a fixed jump frequency.
The mathematical e+uivalence of eight traffic flow models 13 The composition rules, however, require additional dataB most notably, the destinations of the vehicles making up the stream. This make-up strongly affects the discharge rates of diverge bottlenecks (see Munoz and Daganzo, 2002) and the performance of intersections controlled by traffic signals. Unfortunately, as a network grows in size, the number of possible destinations grows and the availability of the required input data diminishes. Thus, the practical limit to composition is not theoretical (we could model relatively well almost anything if we knew where vehicles were going) but informational.
We believe that the results in this paper can be of use for the design of small networks such as complex interchanges, but other approaches should be sought for very large networks. See Daganzo (200Oa) for some ideas in this direction