Analysis of kinematic waves arising in diverging traffic flow models
aa r X i v : . [ m a t h . D S ] S e p Analysis of kinematic waves arising in diverging trafficflow models
Wen-Long Jin ∗ October 23, 2018
Abstract
Diverging junctions are important network bottlenecks, and a better understanding of di-verging traffic dynamics has both theoretical and practical implications. In this paper, we firstintroduce a continuous multi-commodity kinematic wave model of diverging traffic and thenpresent a new framework for constructing kinematic wave solutions to its Riemann problemwith jump initial conditions. In supply-demand space, the solutions on a link consist of an inte-rior state and a stationary state, subject to admissible conditions such that there are no positiveand negative kinematic waves on the upstream and downstream links respectively. In addi-tion, the solutions have to satisfy entropy conditions consistent with various discrete divergemodels. In the proposed analytical framework, kinematic waves on each link can be uniquelydetermined by the stationary and initial conditions, and we prove that the stationary states andboundary fluxes exist and are unique for the Riemann problem of diverge models when all orpartial of vehicles have predefined routes. We show that the two diverge models by Lebacque ∗ Department of Civil and Environmental Engineering, California Institute for Telecommunications and InformationTechnology, Institute of Transportation Studies, 4000 Anteater Instruction and Research Bldg, University of California,Irvine, CA 92697-3600. Tel: 949-824-1672. Fax: 949-824-8385. Email: [email protected]. Corresponding author nd Daganzo are asymptotically equivalent. We also prove that the supply-proportional andpriority-based diverge models are locally optimal evacuation strategies. With numerical exam-ples, we demonstrate the validity of the analytical solutions of interior states, stationary states,and corresponding kinematic waves. This study presents a unified framework for analyzingtraffic dynamics arising in diverging traffic and could be helpful for developing emergencyevacuation strategies. Key words : Kinematic wave models, diverging traffic, Riemann problem, supply-demand space,stationary states, interior states, boundary fluxes, turning proportions, First-In-First-Out, evacua-tion strategies
Essential to effective and efficient transportation control, management, and planning is a betterunderstanding of the evolution of traffic dynamics on a road network, i.e., the formation, propa-gation, and dissipation of traffic congestion. The seminal work by (Lighthill and Whitham, 1955;Richards, 1956) (LWR) describes traffic dynamics with kinematic waves, including shock and rar-efaction waves, in density ( r ), speed ( v ), and flux ( q ). Based on a continuous version of trafficconservation, ¶r¶ t + ¶ q ¶ x =
0, and an assumption of a speed-density relationship, v = V ( r ) , the LWRmodel can be written as ¶¶ t r + ¶¶ x r V ( r ) = , (1)which is for a homogeneous road link with time and location independent traffic characteristics,such as free flow speed, jam density, capacity, and so on. In general, V ( r ) is a non-increasingfunction, and v f = V ( ) is the free flow speed. In addition, q = Q ( r ) ≡ r V ( r ) is unimodal withcapacity C = Q ( r c ) , where r c is the critical density. Traffic states with density higher than r c arecongested or over-critical, and those with lower density are free flowing or under-critical. Here wedenote the jam density by r j , and r ∈ [ , r j ] . 2n a road network, however, more important and interesting are the formation, propagation, anddissipation of traffic queues caused by network bottlenecks, including merges, diverges, and othernetwork junctions (Daganzo et al., 1999). But compared with numerous studies on the LWR modeland higher-order models of traffic flow on a road link (Federal Highway Administration, 2004),studies on traffic dynamics at merging, diverging, and other junctions are scarce. In (Fazio et al.,1990), behavioral models were proposed to capture individual vehicles’ diverging maneuvers. In(Papageorgiou, 1990), diverging flows of vehicles on a path are determined by pre-defined splittingrates. In (Daganzo, 1995), the First-In-First-Out (FIFO) principle was explicitly introduced so thatdiverging flows are proportional to turning proportions, which can be time-dependent. But it wasnoted that the FIFO principle could be violated when one downstream branch is heavily congested.In (Liu et al., 1996; Ngoduy et al., 2006), diverging traffic was considered in a so-called frictionterm of a higher-order model, where diverging flow to an off-ramp is determined by expecteddiverging flow and the congestion level of the off-ramp. In (Mu˜noz and Daganzo, 2002), it wasshown that First-In-First-Out (FIFO) blockage caused by one congested downstream branch couldsignificantly reduce the discharging flow-rate of the whole diverge, and vehicles may not follow theFIFO principle strictly. Diverging traffic with two or more vehicles have been studied in (Daganzo,1997; Daganzo et al., 1997; Newell, 1999). In (Cassidy, 2003), metering strategies were discussedfor diverging junctions. As pointed out in (Daganzo, 1999), different network bottlenecks caninduce different traffic behavior; at diverging junctions, which are different from merging andother junctions, not only capacities of all branches but also the combinations of diverging vehicleson the upstream branch could determine the formation and dissipation of queues. In addition, abetter understanding of diverging traffic flow could also lead to more efficient evacuation strategies(Sheffi et al., 1982). In this study, we are interested in traffic dynamics arising from divergingjunctions for one type of vehicles within the framework of the LWR model.Considering the analytical power and simplicity of the LWR model, many researchers haveattempted to study traffic dynamics arising in general transportation networks in the framework of3inematic wave models. In one line, Daganzo (1995) and Lebacque (1996) extended the Godunovdiscrete form of the LWR model for computing traffic flows through diverging, diverging, andgeneral junctions. Hereafter we call such models as Cell Transmission Models (CTM). In CTM,so-called traffic demand and supply functions are introduced, and boundary fluxes through vari-ous types of junctions can be written as functions of upstream demands and downstream supplies.In CTM, various physically meaningful rules can be used to compute boundary fluxes, such asthe First-In-First-Out diverging principle (Papageorgiou, 1990; Daganzo, 1995) and the fair merg-ing principle (Jin and Zhang, 2003b). CTM are discrete in nature and only suitable for numericalsimulations. Thus they do not provide any analytical insights on traffic dynamics at a networkintersection as the LWR model. In another line, Holden and Risebro (1995) and Coclite et al.(2005) attempted to solve a Riemann problem of an intersection with m upstream links and n downstream links. In both of the analytical studies, all links are homogeneous and have the samespeed-density relations, and traffic dynamics on each link are described by the LWR model. In(Holden and Risebro, 1995), the Riemann problem with jump initial conditions is solved by in-troducing an entropy condition that maximizes an objective function of all boundary fluxes. In(Coclite et al., 2005), the Riemann problem is solved to maximize total flux with turning propor-tions. Both studies were able to describe basic waves arising from a network intersection butalso subject to significant shortcomings: (i) All links are assumed to have the same fundamen-tal diagram in both studies; (ii) In (Holden and Risebro, 1995), vehicles can travel to an arbitrarydownstream link, and the entropy conditions used are pragmatic and lack of physical interpreta-tions; and (iii) In (Coclite et al., 2005), results are only valid for restricted turning proportions andjunctions with no fewer downstream links; i.e., n ≥ m . In addition, neither of these studies presenta unified continuous model of network vehicular traffic.As in (Holden and Risebro, 1995; Coclite et al., 2005), in this study we attempt to analyticallyobtain kinematic wave solutions of traffic dynamics arising at a diverging junction. However, ourstudy does not bear the same limitations as in these studies: all links can be mainline freeways or4ff-ramps with the same or different characteristics, and our solutions are physically meaningfuland consistent with the discrete supply-demand models of diverging traffic, e.g. those proposedin (Daganzo, 1995; Lebacque, 1996). We first present a continuous kinematic wave model ofmulti-commodity diverging traffic flow based on the conservation of commodity traffic. Followingthe new framework used to solve Riemann problems for inhomogeneous LWR model at a linearjunction (Jin et al., 2009) and for merging traffic flow (Jin, 2010), we present a new frameworkfor solving the Riemann problem for diverge models. In the Riemann solutions, there can be astationary state and an interior state for each branch. Here stationary states are the self-similarstates at the boundary. That is, in the Riemann solutions, stationary states prevail all links aftera long time. In contrast, interior states do not take any space in the continuous solution and onlyshow up in one cell in the numerical solutions as observed in (van Leer, 1984). We introduce aso-called supply-demand diagram and discuss the problem in supply-demand space, rather thanin r − q space as in (Holden and Risebro, 1995; Coclite et al., 2005). After deriving admissiblesolutions for upstream and downstream stationary and interior states, we introduce an entropycondition based on various diverge models. We then prove that stationary states and boundaryfluxes are unique for given upstream demand and downstream supplies (but interior states maynot). Then, kinematic waves on a link are determined by the corresponding LWR model with thestationary state and the initial state. In a sense, kinematic waves of the Riemann problem can beconsidered as continuous solutions of the discrete Cell Transmission Model with various divergingrules in (Daganzo, 1995; Lebacque, 1996).Different from (Holden and Risebro, 1995; Coclite et al., 2005), where the Riemann solutionsonly comprise of initial and stationary states, here we have additional interior states. Interior stateswere observed when the inhomogeneous LWR model was used to simulate traffic dynamics on aring road (Jin and Zhang, 2003a; Jin et al., 2009). Although interior states are not directly relatedto kinematic waves on all links, they are used in the entropy condition and therefore essential topicking out unique physical solutions. As we can see later, interior states are essential to construct5inematic wave solutions for different diverge models.The rest of the paper is organized as follows. In Section 2, we introduce a continuous multi-commodity kinematic wave model of diverging traffic. In Section 3, we introduce a new frameworkfor solving the kinematic waves of the Riemann problem with jump initial conditions in supply-demand space. In particular, we derive traffic conservation conditions, admissible conditions ofstationary and interior states, and additional entropy conditions based on various discrete divergemodels. In Section 4, we solve stationary states and boundary fluxes for diverge models whenvehicles have predefined routes. In Section 5, we discuss diverge models in various evacuationstrategies. In Section 6, we demonstrate the validity of the proposed analytical framework withnumerical examples. In Section 7, we summarize our findings and present some discussions. n x p = 0 −∞ ∞ Figure 1: An illustration of a diverge networkWe consider a diverge network with m ≥ m + m paths. We differentiate all vehicles into P = m commodities according to their paths. We denote the link-commodity incidence variable by d p , a , which equals 1 if commodity p ( p = , · · · , m ) uses link a ( a = , · · · , m +
1) and 0 otherwise.6hus P a = (cid:229) Pp = d p , a is the number of commodities on link a : P a = a = , · · · , m , and P a = m for a =
0. On a link a , the location is denoted by link coordinate x a ∈ [ X a , X a + L a ] , where L a is thelength of link a , and x a = X a and X a + L a are the upstream and downstream boundaries respectively.On the path of a commodity p , the location is denoted by commodity coordinate x p ∈ [ X p , X p + L p ] ,where L p = (cid:229) a d p , a L a and we assume that there is no loop on a path. is the length of path p , and x p = X p and X p + L p are at the origin and destination respectively. If d p , a =
1, we denote L p , a as thedistance from the origin of path p to the upstream boundary of link a , and x a and x p follows a one-to-one relation: if x p ∈ [ X p + L p , a , X p + L p , a + L a ] , then x p is on link a and x a = x p − X p − L p , a + X a .That is, d p , a ( x a − X a − x p + L p , a + X p ) = a = , · · · , m + p = , · · · , m For commodity p , we denote density, speed, and flux by r p ( x p , t ) , v p ( x p , t ) , and q p ( x p , t ) = r p ( x p , t ) v p ( x p , t ) , respectively. From traffic conservation of commodity p , we can have the follow-ing continuous conservation equation ¶r p ¶ t + ¶ q p ¶ x p = , (2)whose derivation is the same as that for single commodity (e.g. Haberman, 1977; Newell, 1993).For link a , we denote density, speed, and flux by r a ( x a , t ) , v a ( x a , t ) , and q a ( x a , t ) = r a ( x a , t ) v a ( x a , t ) ,respectively. Then we have that r a ( x a , t ) = (cid:229) p d p , a r p ( x a , t ) and q a ( x a , t ) = (cid:229) p d p , a q p ( x a , t ) . Notethat, r p ( x a , t ) exists only when link a is on path p and r p ( x a , t ) = r p ( x p , t ) with x a = x p − L p , a − X p + X a . It is the same for v p ( x a , t ) and q p ( x a , t ) . We assume that traffic streams ofdifferent commodities on link a are homogeneous and share the same speed at the same loca-tion and time. That is, we have the following speed-density relationships (Greenshields, 1935;Del Castillo and Benitez, 1995) v p ( x a , t ) = v a ( x a , t ) = V ( x a , r a ( x a , t )) . (3)Generally, V a ( x a , r a ) is non-increasing in r a , and Q ( x a , r a ) ≡ r a V ( x a , r a ) is unimodal in r a withits maximum as capacity at x a . We can see that conservation laws of multi-commodity flows in (2)7ead to the following LWR model ¶¶ t r a ( x a , t ) + ¶¶ x a r a ( x a , t ) V ( x a , r a ( x a , t )) = , (4)which can work for inhomogeneous roads. Correspondingly, we can have the following trafficconservation equation for commodity p and x p ∈ [ X p + L p , a , X p + L p , a + L a ] ¶¶ t r p ( x p , t ) + ¶¶ x p r p ( x a , t ) V ( r a ( x a , t )) = , p = , · · · , m (5)where x a = x p − X p − L p , a + X a . For commodity p , the traffic stream evolves on the correspondingpath, and we obtain a one-dimensional hyperbolic conservation law. However, all traffic streamsinteract with each other on the network, and we have a system of network hyperbolic conservationlaws. We hereafter call (5) as a multi-commodity kinematic wave (MCKW) model of divergingtraffic.We can see that traffic flow on a road network cannot be modeled by either one-dimensionalor two-dimensional conservation laws, since vehicles of different commodities interact with eachother on their shared links. In particular, for a diverge network with m downstream links, trafficstreams of m commodities interact with each other on the upstream link. Traffic dynamics insideeach link can be studied by the LWR models in (1) or (4), and the remained task is to study trafficdynamics at the diverging junction. Here we consider the Riemann problem for the MCKW modelof diverging traffic in (5) with jump initial conditions. Without loss of generality, we assumethat all links are homogeneous and infinitely long. For link a = , · · · , m +
1, we assume that itsflow-density relation is q a = Q a ( r a ) , critical density r c , a , and its capacity C a . For the network inFigure 1, we set X p = − ¥ , X p + L p = ¥ , and x p = p = , · · · , m ; X a = X a + L a = ¥ for a = , · · · , m ; and X = − ¥ and X m + + L m + =
0. Therefore, L p , = L p , p = ¥ , and d p , a ( x a − x p ) = a = p = , · · · , m .For commodity p = , · · · , m , we have the following jump initial conditions: r p ( x p , ) = r p , L , x p ∈ ( − ¥ , ] r p , R , x p ∈ ( , + ¥ ) . (6)8hen upstream link 0 and downstream link i = , · · · , m have constant initial conditions: r ( x , ) = r ≡ (cid:229) p r p , L , x ∈ ( − ¥ , ) , (7) r i ( x i , ) = r i ≡ r i , R , x i ∈ ( , + ¥ ) , i = , · · · , m (8) For link a = , · · · , m , we define the following demand and supply functions with all subscript a suppressed (Engquist and Osher, 1980; Daganzo, 1995; Lebacque, 1996) D ( r ) = Q ( min { r , r c } ) = Q ( r ) , if r ≤ r c C , if r ≥ r c , = Z r c ( s ) Q ′ ( s ) ds = Z r max { Q ′ ( s ) , } ds (9) S ( r ) = Q ( max { r , r c } ) = Q ( r ) , if r ≥ r c C , if r ≤ r c , = C + Z r ( − c ( s )) Q ′ ( s ) ds = C + Z r min { Q ′ ( s ) , } ds , (10)where c ( r ) equals 1 iff Q ′ ( r ) ≥ U = ( D , S ) . This is different frommany existing studies, in which traffic states are considered in r - q space. For the demand andsupply functions in (9) and (10), we can see that D is non-decreasing with r and S non-increasing.Thus D ≤ C , S ≤ C , max { D , S } = C , and flow-rate q ( U ) = min { D , S } . In addition, D = S = C ifftraffic is critical; D < S = C iff traffic is strictly under-critical (SUC); S < D = C iff traffic is strictlyover-critical (SOC). Therefore, state U = ( D , S ) is under-critical (UC), iff S = C , or equivalently D ≤ S ; State U = ( D , S ) is over-critical (OC), iff D = C , or equivalently S ≤ D .In Figure 2(b), we draw a supply-demand diagram for the two fundamental diagrams in Figure2(a). On the dashed branch of the supply-demand diagram, traffic is UC and U = ( D , C ) with9 ≤ C ; on the solid branch, traffic is OC and U = ( C , S ) with S ≤ C . Compared with the fun-damental diagram of a road section, the supply-demand diagram only considers its capacity C and criticality, but not other detailed characteristics such as critical density, jam density, or theshape of the fundamental diagram. That is, different fundamental diagrams can have the samedemand-supply diagram, as long as they have the same capacity and are unimodal, and their criti-cal densities, jam densities, or shapes are not relevant. However, given a demand-supply diagramand its corresponding fundamental diagram, the points are one-to-one mapped. ρqC DSC C (a) (b)
Figure 2: Fundamental diagrams and their corresponding supply-demand diagramsIn supply-demand space, initial conditions in (7) and (8) are equivalent to (Here i = , · · · , m ifnot otherwise mentioned) U ( x , ) = ( D , S ) , x ∈ ( − ¥ , ) , (11) U i ( x i , ) = ( D i , S i ) , x i ∈ ( , + ¥ ) . (12)(13)In the solutions of the Riemann problem for (5) with initial conditions (11-12), a shock wave or ararefaction wave could initiate on a link from the diverging junction at x =
0, and traffic states onall links become stationary after a long time. We hereafter refer to these states as stationary states.At the boundary, there can also exist interior states (van Leer, 1984; Bultelle et al., 1998), which10ake infinitesimal space and only exist in one cell in numerical solutions. We denote the stationarystates on upstream link 0 and downstream link i by U − and U + i , respectively. We denote theinterior states on links 0 and i by U ( − , t ) and U i ( + , t ) , respectively. The structure of Riemannsolutions on upstream and downstream links are shown in Figure 3, where arrows illustrate thedirections of possible kinematic waves. Then the kinematic wave on upstream link 0 is the solutionof the corresponding LWR model with initial left and right conditions of U and U − , respectively.Similarly, the kinematic wave on downstream link i is the solution of the corresponding LWRmodel with initial left and right conditions of U + i and U i , respectively.Since vehicles’ proportions travel forward along vehicles (Lebacque, 1996), traffic dynamicson the upstream link follow the First-In-First-Out (FIFO) principle (Papageorgiou, 1990). If thecommodity proportions x i are predefined and constant, as a result of the global FIFO principle, inthe Riemann solutions we have q i = x i q , (14)which serves as the First-In-First-Out principle (Papageorgiou, 1990). Also we have that, in thestationary state U − , vehicles’ proportions are the same as predefined ones. However, we couldhave different proportions in the interior state U ( − , t ) and denote the corresponding proportionof commodity i by x i ( − , t ) . U (0 − , t ) U − U − x U i (0 + , t ) U + i U i x (a) (b) Figure 3: The structure of Riemann solutions: (a) Upstream link 0; (b) Downstream link i We denote q → i as the flux from link 0 to link i for t >
0. The fluxes are determined by thestationary states: the out-flux of link 0 is q = q ( U − ) , and the in-flux of link i is q i = q ( U + i ) .11urthermore, from traffic conservation at a diverging junction, we have at stationary states q → i = q i = q ( U + i ) , q = q ( U − ) = m (cid:229) i = q ( U + i ) . (15) As observed in (Holden and Risebro, 1995; Coclite et al., 2005), the speed of a kinematic wave onan upstream link cannot be positive, and that on a downstream link cannot be negative. We havethe following admissible conditions on stationary states.
Theorem 3.1 (Admissible stationary states)
For initial conditions in (11) and (12), stationarystates are admissible if and only ifU − = ( D , C ) or ( C , S − ) , (16) where S − < D , and U + i = ( C i , S i ) or ( D + i , C i ) , (17) where D + i < S i . The proof is quite straightforward and omitted here. The regions of admissible upstream stationarystates in both supply-demand and fundamental diagrams are shown in Figure 4, and the regionsof admissible downstream stationary states are shown in Figure 5. From the figures, we can alsodetermine the types and traveling directions of waves with given stationary and initial states onall links. In particular, the types of kinematic waves and the signs of the wave speeds can bedetermined in the supply-demand diagram, but the absolute values of the wave speeds have to bedetermined in the fundamental diagram.
Remark 1. U − = U and U + i = U i are always admissible. In this case, the stationary states arethe same as the corresponding initial states, and there are no waves.12 DSC C x U = U − ( C , D ) U − DSC C x U U − (a) (b) Figure 4: Admissible stationary states for upstream link 0: marked by black dots DSC i C i x U i U + i DSC i C i x U i = U + i ( S i , C i ) U + i (a) (b) Figure 5: Admissible stationary states for downstream link i : marked by black dots Remark 2.
Out-flux q = min { D − , S − } ≤ D and in-flux q i = min { D + i , S + i } ≤ S i . That is, D isthe maximum sending flow and S i is the maximum receiving flow in the sense of (Daganzo, 1994,1995). Remark 3.
In (Lebacque and Khoshyaran, 2005), a so-called “invariance principle” is proposedas follows: if D − = C , then q ( U − ) < D ; if S + i = C i , then q ( U + i ) < S . We can see that Theorem3.1 is consistent with the “invariance principle”. Corollary 3.2
For the upstream link , q ≤ D ; q < D if and only if U − = ( C , q ) , and q = D f and only if U − = ( D , C ) . For the downstream link i, q i ≤ S i ; q i < S i if and only if U + i = ( q i , C i ) ,and q i = S i if and only if U + i = ( C i , S i ) . That is, given out-fluxes and in-fluxes, the stationary statescan be uniquely determined. For interior states, the waves of the Riemann problem on link 0 with left and right initialconditions of U − and U ( − , t ) cannot have negative speeds. Similarly, the waves of the Riemannproblem on link i with left and right initial conditions of U i ( + , t ) and U + i cannot have positivespeeds. Therefore, interior states U ( − , t ) and U i ( + , t ) should satisfy the following admissibleconditions. Theorem 3.3 (Admissible interior states)
For asymptotic stationary states U − and U + i , interiorstates U ( − , t ) and U i ( + , t ) in (20) are admissible if and only ifU ( − , t ) = ( C , S − ) = U − , when U − is SOC; i.e., S − < D − = C ( D ( − , t ) , S ( − , t )) , when U − is UC; i.e., D − ≤ S − = C (18) where S ( − , t ) ≥ D − , andU i ( + , t ) = ( D + i , C i ) = U + i , when U + i is SUC; i.e., D + i < S + i = C i ( D i ( + , t ) , S i ( + , t )) , when U + i is OC; i.e., S + i ≤ D + i = C i (19) where D i ( + , t ) ≥ S + i . The proof is quite straightforward and omitted here. The regions of admissible upstream interiorstates in both supply-demand and fundamental diagrams are shown in (6), and the regions of ad-missible downstream interior states are shown in (7). From the figures, we can also determinethe types and traveling directions of waves with given stationary and interior states on all links,but these waves are suppressed and cannot be observed, and we are only able to observe possibleinterior states in numerical solutions.
Remark 1.
Note that U ( − , t ) = U − and U i ( + , t ) = U + i are always admissible. In this case,the interior states are the same as the stationary states.14 DSC C x U − ( C , D − ) U (0 − , t ) 0 DSC C x U − = U (0 − , t ) (a) (b) Figure 6: Admissible interior states for upstream link 0: marked by black dots DSC i C i x U + i = U i (0 + , t ) 0 DSC i C i x U + i ( S + i , C i ) U i (0 + , t ) (a) (b) Figure 7: Admissible interior states for downstream link i : marked by black dots Corollary 3.4
For upstream link , q ≤ D ; q < D if and only if U ( − , t ) = U − = ( C , q ) , andq = D if and only if U − = ( D , C ) , and U ( − , t ) = ( D ( − , t ) , S ( − , t )) with S ( − , t ) ≥ D .For the downstream link i, q i ≤ S i ; q i < S i if and only if U i ( + , t ) = U + i = ( q i , C i ) , and q i = S i ifand only if U + i = ( C i , S i ) , and U i ( + , t ) = ( D i ( + , t ) , S i ( + , t )) with D i ( + , t ) ≥ S i . .2 Entropy conditions consistent with discrete diverge models In order to uniquely determine the solutions of stationary states, we introduce a so-called entropycondition in interior states as follows: q i = F i ( U ( − , t ) , U ( + , t ) , · · · , U m ( + , t ) , x ( − , t ) , · · · , x m ( − , t )) . (20)That is, the entropy condition uses “local” information in the sense that it determines boundaryfluxes from interior states. In the discrete version of (5), the entropy condition is used to determineboundary fluxes from cells contingent to the diverging junction. Thus, F i ( U ( − , t ) , U ( + , t ) , · · · , U m ( + , t ) , x ( − , t ) , · · · , x m ( − , t )) in (20) can be considered as local, discrete flux functions.In (Daganzo, 1995), F ( U ( − , t ) , U ( + , t ) , · · · , U m ( + , t ) , x ( − , t ) , · · · , x m ( − , t )) was proposed to solve the following local optimization problemmax U − , U + i , U ( − , t ) , U ( + , t ) , ··· , U m ( + , t ) , x ( − , t ) , ··· , x m ( − , t ) { q } (21)subject to q ≤ D ( − , t ) , q i ≤ S i ( + , t ) , x i ( − , t ) = the proportion of vehicles choosing path i . Thus, we obtain the total flux as F ( U ( − , t ) , U ( + , t ) , · · · , U m ( + , t ) , x ( − , t ) , · · · , x m ( − , t )) = x i ( − , t ) m min i = { D ( − , t ) , S i ( + , t ) x i ( − , t ) } . In the literature, a number of other diverge models have been proposed. In (Lebacque, 1996),the upstream demand is split into commodity demands according to predefined turning proportions,16nd the in-flux of each downstream link is the minimum of its supply and commodity demand.In (Jin and Zhang, 2003b), turning proportions were proposed to be determined by downstreamsupplies when vehicles have no predefined routes. In (Sheffi et al., 1982), turning proportions wereproposed to be determined by downstream speeds in a myopic evacuation scheme. All these local,discrete diverge models can be considered as entropy conditions, so that we have correspondingcontinuous diverge models (5).
To solve the Riemann problem for (5) with the initial conditions in (11)-(12), we will first findstationary and interior states that satisfy the aforementioned entropy condition, admissible condi-tions, and traffic conservation equations. Then the kinematic wave on each link will be determinedby the Riemann problem of the corresponding LWR model with initial and stationary states asinitial conditions. Here we will only focus on solving the stationary states on all links, since thekinematic waves of the LWR model have been well studied in the literature. From all the condi-tions, we can see that the feasible domains of stationary and interior states are independent of theupstream supply, S i , and the downstream demand, D m + . That is, the same upstream demand anddownstream supply will yield the same solutions of stationary and interior states. However, theupstream and downstream wave types and speeds on each can be related to S i as shown in Figure4(d) and D m + as shown in Figure 5(d). In this paper, we solve the Riemann problem for a diverging junctions with two downstream links;i.e., m =
2. In this section, we consider two entropy conditions, i.e., two diverge model. Here ve-hicles have predefined routes; i.e., x i are predefined constants, determined by vehicle route choicebehaviors. We attempt to find the relationships between the boundary fluxes and the initial condi-17ions. q i = ˆ F i ( U , U , U ) . (22)In contrast to local, discrete flux functions F i ( U ( − , t ) , U ( − , t ) , U ( + , t )) , ˆ F i ( U , U , U ) can beconsidered as global, continuous. With the global, continuous fluxes, we can find stationary statesfrom Corollary 3.2. With the solution framework in the preceding section, we can then find thekinematic waves of the Riemann problem of (5) with initial conditions ( U , U , U ) . In (Daganzo, 1995), a FIFO diverge model was proposed based on (21) q = min { D ( − , t ) , S ( + , t ) x ( − , t ) , S ( + , t ) x ( − , t ) } , (23)and a local FIFO principle q i = x i ( − , t ) q . (24)Comparing (24) and (14), we obviously have x i ( − , t ) = x i . That is, the commodity proportions inthe stationary state are the same as predefined. Thus in Riemann solutions, stationary and interiorstates have to satisfy (23), traffic conservation, and the corresponding admissible conditions. Theorem 4.1
For the Riemann problem of the MCKW model of merging traffic in (5) with ini-tial conditions in (11) and (12), boundary fluxes satisfying the entropy condition in (23), trafficconservation equations, and the corresponding admissible conditions are:q = min { D , S x , S x } , (25) and q i = x i q . The corresponding stationary and interior states are in the following:1. If D > q , the stationary state of the upstream link is SOC, and U ( − , t ) = U − = ( C , q ) ; ifD = q , the stationary state of the upstream link is UC, U − = ( D , C ) , and U ( − , t ) = U − or U ( − , t ) = ( D ( − , t ) , S ( − , t )) with D ( − , t ) > D and S ( − , t ) ≥ D . . If S i x i > q , the stationary state of downstream link i (i = , ) is SUC, and U i ( + , t ) = U + i =( q i , C i ) ; if S i x i = q , the stationary state of downstream link i is OC, U + i = ( C i , S i ) , andU i ( + , t ) = U + i or U i ( + , t ) = ( D i ( + , t ) , S i ( + , t )) with D i ( + , t ) ≥ S i and S i ( + , t ) > S i . The proof of the theorem is given in Appendix A. The solutions of fluxes are illustrated in Figure8, in which the starting point of an arrow represents the initial condition ( S , S ) , and the endingpoint represents the solution ( q , q ) . That is, in region I, D < min i S i x i , and we have q i = x i D ; inregion II, C x < min { D , C x } , and we have q = S and q = S x x ; in region III, C x < min { D , C x } ,and we have q = S and q = S x x ; on the boundary line between regions I and II, or the boundaryline between regions I and III, q i = x i D ; on the boundary line between regions II and III, q i = S i .We can see that, in region I, D < S + S , and q = min { D , S + S } . In regions II and III, q < min { D , S + S } . That is, due to vehicles’ route choice behaviors, the capacity of the diverge,min { D , S + S } , is generally under-utilized. S , q S , q D D ξ D ξ D III III C C C C Figure 8: The solutions of fluxes for a FIFO diverging junction19omparing (23) and (25), we can see that the global, continuous fluxes have the same func-tional form as the local, discrete fluxes. In this sense, the FIFO diverge model (23) is “invariant”.Hereafter, we consider a model invariant if and only if the global, continuous fluxes have the samefunctional form as the local, discrete fluxes.
In (Lebacque, 1996), the following diverge model was proposed q i = min { x i ( − , t ) D ( − , t ) , S i ( + , t ) } , (26)and q = q + q . Compared with Daganzo’s model (23), this model is locally non-FIFO, and itssolutions are the following. Theorem 4.2
For the Riemann problem of the MCKW model of merging traffic in (5) with ini-tial conditions in (11) and (12), boundary fluxes satisfying the entropy condition in (26), trafficconservation equations, and the corresponding admissible conditions are the same as in (25); i.e.,q = min { D , S x , S x } , and q i = x i q . The corresponding stationary and interior states are in the following:1. If D > q , the stationary state of the upstream link is SOC, and U ( − , t ) = U − = ( C , q ) ; ifD = q , the stationary state of the upstream link is UC, U − = ( D , C ) , and U ( − , t ) = U − or U ( − , t ) = ( D ( − , t ) , S ( − , t )) with D ( − , t ) > D and S ( − , t ) ≥ D .2. If S i x i > q , the stationary state of downstream link i (i = , ) is SUC, and U i ( + , t ) = U + i =( q i , C i ) ; if S i x i = q , the stationary state of downstream link i is OC, U + i = ( C i , S i ) , andU i ( + , t ) = U + i or U i ( + , t ) = ( D i ( + , t ) , S i ( + , t )) with D i ( + , t ) ≥ S i and S i ( + , t ) > S i .3. The interior turning proportions x i ( − , t ) can be determined by interior states and stationarystates. We consider the a diverging rule proposed in (Jin and Zhang, 2003b), in which q i = min { , D ( − , t ) S ( + , t ) + S ( + , t ) } S i ( + , t ) , i = , . (27)In this diverging rule, vehicles do not have predefined routes and belong to the same commodity.This diverge model was applied for emergency evacuation situations in a road network (Qiu and Jin,2008). In this model, we have q = min { D ( − , t ) , S ( + , t ) + S ( + , t ) } , and the turning proportions are time-dependent x i = S i ( + , t ) S ( + , t ) + S ( + , t ) , i = , . (28)Thus in the Riemann solutions, stationary and interior states have to satisfy (27), traffic conser-vation, and the corresponding admissible conditions.21 heorem 5.1 For the Riemann problem of the MCKW model of diverging traffic in (5) with initialconditions in (11) and (12), stationary and interior states satisfying the entropy condition in (27),traffic conservation equations, and the corresponding admissible conditions are the following:1. When S + S < D , U + i = U i ( + , t ) = ( C i , S i ) (i = , ) and U − = U ( − , t ) = ( C , S + S ) ;2. When S + S = D , U + i = U i ( + , t ) = ( C i , S i ) (i = , ), U − = ( D , C ) , U ( − , t ) = ( D , C ) or ( D ( − , t ) , S ( − , t )) with D ( − , t ) ≥ D and S ( − , t ) > D when D < C ;3. When S i > C i C + C D (i = , ), U − = U ( − , t ) = ( D , C ) , and U − i = U i ( + , t ) = ( C i C + C D , C i ) .4. When S + S > D and S i ≤ C i C + C D (i , j = or 2 and i = j), U − = U ( − , t ) = ( D , C ) ,U + i = ( C i , S i ) , U i ( + , t ) = ( C i , C j D − S i S i ) , and U + j = U j ( + , t ) = ( D − S i , C j ) . The proof of the theorem is given in Appendix C.
Corollary 5.2
For the Riemann problem of the MCKW model of diverging traffic in (5) with ini-tial conditions in (11) and (12), boundary fluxes satisfying the entropy condition in (27), trafficconservation equations, and the corresponding admissible conditions are the following:1. When S + S ≤ D , q i = S i (i = , ) and q = S + S ;2. When S i > C i C + C D (i = , ), q i = C i C + C D and q = D ;3. When S + S > D and S i ≤ C i C + C D (i , j = or 2 and i = j), q i = S i , q j = D − S i , andq = D .That is, for i , j = or 2 and i = j, q = min { D , S + S } .q i = min { S i , max { D − S j , D C + C C i }} . (29)22he solutions of fluxes in four different regions are shown in Figure 9, in which the starting pointsof arrows represent the initial conditions in ( D , D ) , and the ending points represent the solutionsof fluxes ( q , q ) . In the figure, we can see four regimes: In regime I, both downstream linkshave OC stationary states; In regime II and IV, one downstream link has SUC and the other OCstationary states; In regime III, both downstream links have SUC stationary states. Comparing(29) and (27), we can see that the evacuation diverge model is not “invariant”. Compared with thediverge model in the preceding section, this model is optimal, since q = min { D , S + S } . S , q S , q D D C C + C D C C + C D I IIIII IV C C C C Figure 9: Solutions of fluxes for a supply-proportional emergency evacuation diverge model
Corollary 5.3
If U i (i = , ) and U satisfy min { D i , S i } = min { S i , max { D − S j , D C + C C i }} , min { D , S } = min { S + S , D } , then the unique stationary states are the same as the initial states, and traffic dynamics at thediverging junction are stationary. .2 A priority-based evacuation strategy Inspired by (29), we propose a priority-based evacuation strategy ( i , j = , i = j ) q i = min { S i ( + , t ) , max { D ( − , t ) − S j ( + , t ) , a i D ( − , t ) }} , (30)where a i ∈ [ , ] and a + a = Theorem 5.4
For the Riemann problem of the MCKW model of diverging traffic in (5) with ini-tial conditions in (11) and (12), boundary fluxes satisfying the entropy condition in (30), trafficconservation equations, and the corresponding admissible conditions are given byq i = min { S i , max { D − S j , a i D }} , (31) and q = min { D , S + S } . The proof of the theorem is given in Appendix D. The solutions of fluxes ( q , q ) from ( S , S , D ) are illustrated in Figure 10. Clearly we can see that fluxes in (29) can be considered as a specialcase when x i = C i / ( C + C ) , and the priority-based evacuation diverge model is invariant. Anextreme case is to give one downstream link an absolute priority for evacuation, e.g., a = a =
0. This can happen when link 1 is shorter or less congestion prone. In this case the fluxes in(31) become q = min { S , D } , q = min { S , max { D − S }} . By a partial evacuation scenario, we mean that some vehicles have predefined routes and others donot. For example, x ∈ [ , ] and x ∈ [ , ] are the predefined portions of vehicles choosing link 124 S , q S , q D D α D α D I IIIII IV C C C C Figure 10: Solutions of fluxes for a priority-based evacuation diverge modeland 2, respectively, but x + x may be smaller than 1. That is, the remaining portion 1 − x − x can take either route. For this scenario, we propose the following evacuation strategy ( i , j = , i = j ) q i = min { S i ( + , t ) , x j S j ( + , t ) − S j ( + , t ) , max { D ( − , t ) − S j ( + , t ) , a i D ( − , t ) }} , (32)where a i ∈ [ x i , − x j ] and a + a = Theorem 5.5
For the Riemann problem of the MCKW model of diverging traffic in (5) with ini-tial conditions in (11) and (12), boundary fluxes satisfying the entropy condition in (32), trafficconservation equations, and the corresponding admissible conditions are given byq i = min { S i , x j S j − S j , max { D − S j , a i D }} , (33) and q = q + q . The proof of the theorem is omitted here. The solutions of ( q , q ) are illustrated in Figure 11, inwhich there are six regimes. Furthermore, we can show that (1) q i ≥ x i q ; (2) When x + x = x = x = S , q S , q D D I IVII VIIII V α D α D ξ D (1 − ξ ) D (1 − ξ ) D ξ D C C C C Figure 11: Solutions of fluxes for a generalized diverge model
In this section, we numerically solve various diverge model and demonstrate the validity of ouranalytical results. Here, both links 0 and 1 are two-lane mainline freeways with a correspondingnormalized maximum sensitivity fundamental diagram (Del Castillo and Benitez, 1995) is ( r ∈ [ , ] ) Q ( r ) = r (cid:26) − exp (cid:20) − exp (cid:18) ( r − ) (cid:19)(cid:21)(cid:27) . r ∈ [ , ] ) Q ( r ) = r (cid:26) − exp (cid:20) − exp (cid:18) ( r − ) (cid:19)(cid:21)(cid:27) . Note that here the free flow speed on the off-ramp is half of that on the mainline freeway, which is1. Thus we have the capacities C = C = C = . r c = r c = r c = . L =
10, and the simulationtime duration is T = M cells and divide the simulation timeduration T into N steps. The time step D t = T / N and the cell size D x = L / M , with D t = . D x ,satisfy the CFL condition (Courant et al., 1928) v f D t D x = D t D x = . ≤ . Then we use the following finite difference equation for link i = , , r n + i , m = r ni , m + D t D x ( q ni , m − / − q ni , m + / ) , where r ni , m is the average density in cell m of link i at time step n , and the boundary fluxes q ni , m − / are determined by supply-demand methods. For example, for downstream links i = q ni , m + / = min { D ni , m , S ni , m + } , m = , · · · , M , where D ni , m is the demand of cell m on link i , S ni , m + is the supply of cell m , and S ni , M + is the supplyof commodity i . For link 0, the in-fluxes are q n , m − / = min { D n , m − , S n , m } , m = , · · · , M , where D n , is the demand at the origin. Then the in-fluxes of the downstream links and the out-fluxof the downstream link are determined by diverge models, which are discrete versions of (20): q ni , / = F i ( D n , M , S n , , S n , ) , n , M + / = q n , / + q n , / . We also track the commodity proportions in cell m of link 0, x ni , m , as follows (Jin and Zhang, 2004) x n + i , m = r n , m r n + , m x ni , m + D t D x q n , m − / x ni , m − − q , m + / x ni , m r n + , m . Note that x i is the predefined proportion of commodity i .In our numerical studies, we only consider Lebacque’s diverge model (26) and its invariantcounterpart (23). For Lebacque’s diverge model, we have q ni , / = min { D n , M x ni , M , S ni , } , q n , M + / = q n , / + q n , / . In the invariant Daganzo’s diverge model, we have ( i , j = , i = j ) q n , M + / = min { D n , M , S n , x n , M , S n , x n , M } , q ni , / = x ni , M q n , M + / . In this subsection, we study numerical solutions of Lebacque’s diverge model in (26). Initially,links 0 and 1 carry OC flows with r = r =
1, and 30% of the vehicles on link 0 diverge to link 2starting at t =
0; i.e., x = .
7, and x = .
3. The initial density on link 2 is r = .
1. That is, the ini-tial conditions in supply-demand space is U = U = ( . , . ) and U = ( . , . ) .Here we use the Neumann boundary condition in supply and demand (Colella and Puckett, 2004): D n , = D n , , S n , M + = S n , M , and S n , M + = S n , M . Therefore, we have a Riemann problem here.In this case, S x < D < S x . Thus according to Theorem 4.2, we should have the follow-ing stationary and interior states U − = U ( − , t ) = ( C , S x ) , U + = U ( + , t ) = ( x x S , C ) , and28 + = U ( + , t ) = ( C , S ) = ( C , C ) . From the LWR model, there should be a back-travelingrarefaction wave on link 0 connecting U to U − , since S < S x ; a forward-traveling shock wave onlink 1 connecting U + to U , since x x S < S ; and a forward-traveling rarefaction wave on link 2connecting U + to U . Furthermore, from (35), we should have that x ( − , t ) = . x t (a) Contour plot of r −8 −6 −4 −2050100150200250300350 0.80.820.840.860.880.90.920.940.960.98 x t (b) Contour plot of r t (c) Contour plot of r Figure 12: Solutions of Lebacque’s diverge model (26): M = N = r , r , and r are demonstrated with M =
160 and N = t = T the approximate asymptotic values: U − = U ( − , t ) = ( . , . ) , and r − = r ( − , t ) = . U + = U ( + , t ) = ( . , . ) , and r + = r ( + , t ) = . U + = U ( + , t ) = ( . , . ) , and r + = r ( + , t ) = . ≈ r c . These numbers are all veryclose to the theoretical values and get closer if we reduce D x or increase T . That is, the results areconsistent with theoretical results asymptotically.In Figure 13, we demonstrate the evolution of the in-flux of link 1 and the proportion of com-29 x ( [ − D x , ],t ) (b) Proportion of commodity 1 in the last cell of link 00 5 10 15 20 250.1950.20.2050.210.2150.220.2250.230.2350.24 t q ( ,t ) (a) Out−flux of link 1 D x=1 D x=1/4 D x=1/16 D x=1 D x=1/4 D x=1/16 Figure 13: Evolution of the out-flux and the density in the downstream cell of link 2 for Lebacque’sdiverge model (26)modity 1 vehicles in the last cell of link 0 for three different cell sizes. From Figure 13(a) wecan see that, initially, the out-flux of link 2 is min { x D , S } = . x x S = . In this subsection, we compare the numerical solutions of Lebacque’s diverge model (26) withits invariant counterpart, Daganzo’s diverge model (23). Initially, links 0 and 1 carry OC flows30ith r = r =
1, and 30% of the vehicles on link 0 diverge to link 2 starting at t =
0; i.e., x = .
7, and x = .
3. The initial density on link 2 is r = .
1. Different from the example in thepreceding subsection, here we use the following boundary conditions: D n , = D n , , S n , M + = S n , M ,and S n , M + = . + .
03 sin ( n p D t / ) . Thus we have a periodic supply on link 2. D t e ( n D t ) D x=1 D x=1/4 D x=1/16 Figure 14: Difference in the solutions between Lebacque’s diverge model (26) and its invariantcounterpart (23)We use r ni , m for the discrete density from Lebacque’s diverge model (26) and ¯ r ni , m from itsinvariant counterpart (23). Then we denote the difference between the two solutions by e ( n D t ) = (cid:229) i = M (cid:229) m = | r ni , m − ¯ r ni , m | D x . (34)In Figure 14, we can see that the difference decreases if we decreases the cell size. This clearlydemonstrates that Lebacque’s diverge model (26) converges to its invariant counterpart (23).31 Conclusion
In this paper, we first introduced a continuous multi-commodity kinematic wave model for a di-verge network and defined its Riemann problem. Then, we introduced the supply-demand diagramof traffic flow and proposed a solution framework for the Riemann problem. In the Riemann solu-tions, each link has two new states: an interior state and a stationary state; and the kinematic waveson a link are determined by the initial state and the stationary state. We then derived admissibleconditions for interior and stationary states and introduced entropy conditions consistent with var-ious discrete diverge models. In the analytical framework we proved that the stationary states andboundary fluxes exist and are unique for the Riemann problem for normal diverge, in which ve-hicles have predefined routes, and evacuation models, in which vehicles may not have predefinedroutes. With numerical examples, we demonstrated the validity of the solution framework devel-oped here and that Lebacque’s diverge model converges to its invariant counterpart, Daganzo’sdiverge model, when we decrease the cell size.An important observation is that, for both (26) and (27), fluxes computed by discrete supply-demand methods are different from the continuous fluxes. For example, the local fluxes fromLebacque’s diverge model, (26), are q i = min { x i D , S i } , i = , . When x i D > S i ; i.e., when the upstream demand is very heavy, we have q i = S i . In this case, q i is not proportional to the turning proportion. Thus Lebacque’s diverge model violates the FIFOprinciple. However, from the analysis in Section 4.2 and the numerical example in Section 6.2, wefind that Lebacque’s diverge model has the same continuous flux solutions as Daganzo’s model,which observes the FIFO principle. Therefore, we conclude that Lebacque’s diverge model is notstrictly non-FIFO. As another example, for the supply-proportional evacuation model, at t = q i = min { , D S + S } S i , i = , , which are different from (29) when only one downstream is SUC; i.e., when S + S > D and S i ≤ C i C + C D . However, the analytical results here suggest that the discrete fluxes converge to thecontinuous ones after a sufficient amount of time or at a given time but with decreasing period ofa time interval.Comparing kinematic wave solutions of Daganzo’s and Lebacque’s diverge models, we findthat, given the same initial conditions, they have the same stationary states and kinematic wavesolutions, but different interior states. In this sense, interior states are essential to distinguishdifferent diverge models. Numerical simulations in Section 6.2 also demonstrate the existenceof interior states. Therefore, interior states are essential in understanding diverging traffic flow.This is different from the LWR model for a homogeneous link, in which interior states couldexist (Jin and Zhang, 2003a; Jin et al., 2009) but are not essential to constructing kinematic wavesolutions.Here we showed that both supply-proportional and priority-based diverge models can be con-sidered locally optimal evacuation strategies. But how to analyze kinematic waves arising in aspeed-dependent evacuation model (Sheffi et al., 1982) is subject to further investigations. In ad-dition to theoretical implications, this study, by improving our understanding of the formation andpropagation of traffic congestion caused by diverging bottlenecks, could be helpful for developing,calibrating, and validating diverge models and associated emergency evacuation strategies in thefuture. For example, with different a and a , (30) is a priority-based invariant diverge model,which can be used to evacuate vehicles to shorter or less congestion prone links without wastingthe capacity of a diverging junction. In the future, we will also be interested in studying kinematicwave solutions of general junctions with multiple upstream and downstream junctions.33 cknowledgements The author would like to thank two anonymous reviewers for their helpful comments. The viewsand results contained herein are the author’s alone.
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SIAM Journal on Scientific and Statistical Computing , 5(1):1–20.
Appendix A: Proof of Theorem 4.1
Proof . From traffic conservation equations in (15), admissible conditions of stationary states,and the global FIFO principle (14), we have q i = x i q ≤ S i and q ≤ D . Thus, we have q ≤ min { D , S x , S x } . 37ote that (23) is equivalent to q = min { D ( − , t ) , S ( + , t ) x , S ( + , t ) x } . We first prove (25). Otherwise, q < min { D , S x , S x } . (i) Since q < D , from (16) and (18), link 0is SOC, and U ( − , t ) = U − = ( C , q ) . (ii) Since q i = x i q < S i , from (17) and (19), link i is SUC,and U i ( + , t ) = U + i = ( x i q , C i ) . Hence from (23) we have q = min { C , C x , C x } < D ≤ C . Thus q = min { C x , C x } . From the FIFO principle (14) we have q i = x i q = x i ( − , t ) min { C x , C x } ≥ C i ,and q = q i / x i ≥ C i x i , which contradicts q ≤ min { D , S x , S x } .We consider the following cases.(1) When only one of the three terms on the right hand-side of (25) equals q , we have (i) D = q < min { S x , S x } , (ii) S x = q < min { D , S x } , or (iii) S x = q < min { D , S x } . Here we onlyshow the solutions of stationary and interior states for (i), and solutions for (ii) and (iii) can beobtained in a similar fashion. When min i S i x i > D = q , from (25) we have q = D , and q i = x i q < S i . From (16) and (18), we have U − = ( D , C ) , and U ( − , t ) = ( D ( − , t ) , S ( − , t )) with S ( − , t ) ≥ D . From (17) and (19), we have U i ( + , t ) = U + i = ( x i q , C i ) . Then from(23) we have q = min i { D ( − , t ) , C i x i } . Since min i C i x i ≥ min i S i x i > D = q , we have D ( − , t ) = q = D , and U ( − , t ) = U − =( D , C ) . That is, the upstream and downstream interior states are the same as the corre-sponding stationary states.(2) When two of the three terms on the right hand-side of (25) equals q , we have (i) D = S x = q < S x , (ii) D = S x = q < S x , or (iii) S x = S x = q < D . Here we only show the solutionsfor (i), and solutions for (ii) and (iii) can be obtained in a similar fashion. When D = S x < S x , from Corollary 3.4, we have U − = ( D , C ) , and U ( − , t ) = ( D ( − , t ) , S ( − , t )) with38 ( − , t ) ≥ D ; U + = ( C , S ) , and U ( + , t ) = ( D ( + , t ) , S ( + , t )) with D ( + , t ) ≥ S ;and U ( + , t ) = U + = ( q , C ) . Then from (23) we have q = min { D ( − , t ) , S ( + , t ) x , C x } , which leads to D ( − , t ) = D or S ( + , t ) = x D = S . In this case, we can have the follow-ing interior states for links 1 and 2: (a) U ( − , t ) = U − , and U ( + , t ) = ( D ( + , t ) , S ( + , t )) with D ( + , t ) ≥ S and S ( + , t ) > S ; (b) U ( + , t ) = U + , and U ( − , t ) = ( D ( − , t ) , S ( − , t )) with D ( − , t ) > D and S ( − , t ) ≥ D .(3) When all the three terms on the right hand-side of (25) equals q , we have D = S x = S x = q .From Corollary 3.4, we have U − = ( D , C ) , and U ( − , t ) = ( D ( − , t ) , S ( − , t )) with S ( − , t ) ≥ D ; U + = ( C , S ) , and U ( + , t ) = ( D ( + , t ) , S ( + , t )) with D ( + , t ) ≥ S ;and U + = ( C , S ) , and U ( + , t ) = ( D ( + , t ) , S ( + , t )) with D ( + , t ) ≥ S . In this case,at least one of U ( − , t ) = U − , U ( + , t ) = U + , and U ( + , t ) = U + should be satisfied, andthe other can be U ( − , t ) = ( D ( − , t ) , S ( − , t )) with D ( − , t ) > D and S ( − , t ) ≥ D , U ( + , t ) = ( D ( + , t ) , S ( + , t )) with D ( + , t ) ≥ S and S ( + , t ) > S , or U ( + , t ) =( D ( + , t ) , S ( + , t )) with D ( + , t ) ≥ S and S ( + , t ) > S . (cid:4) Appendix B: Proof of Theorem 4.2
Proof . From traffic conservation equations in (15), admissible conditions of stationary states,and the global FIFO principle (14), we have q i = x i q ≤ S i and q ≤ D . Thus, we have q ≤ min { D , S x , S x } .We first prove that q is given by (25). Otherwise, q < min { D , S x , S x } , which leads to q < D and q i = x i q < S i . From Corollary 3.4, we have U ( − , t ) = U − = ( C , q ) and U i ( + , t ) = U + i = q i , C i ) . Then from (26) we have q i = min { x i ( − , t ) C , C i } < S i ≤ S i . Thus q i = x i ( − , t ) C , and q = q + q = C , which contradicts q < D ≤ C .We consider the following cases.(1) When only one of the three terms on the right hand-side of (25) equals q , we have (i) D = q < min { S x , S x } , (ii) S x = q < min { D , S x } , or (iii) S x = q < min { D , S x } . Here weonly show the solutions of stationary and interior states for (i) and (ii), and solutions for (iii)can be obtained in a similar fashion.(i) When min i S i x i > D = q , from (25) we have q = D , and q i = x i q < S i . From Corol-lary 3.4, we have U − = ( D , C ) , U ( − , t ) = ( D ( − , t ) , S ( − , t )) with S ( − , t ) ≥ D ,and U i ( + , t ) = U + i = ( x i q , C i ) . Then from (26) we have q i = min { x i ( − , t ) D ( − , t ) , C i } = x i ( − , t ) D ( − , t ) < C i . Then we have q = D ( − , t ) = D , and U ( − , t ) = U − = ( D , C ) . Further we have x i ( − , t ) = x i . In this case, the upstream and downstream interior states are the same asthe corresponding stationary states.(ii) When S x < min { D , S x } , from (25) we have q = S x < D , q = S , q = x x S < S .From Corollary 3.4, we have U ( − , t ) = U − = ( C , S x ) , U + = ( C , S ) , U ( + , t ) =( D ( + , t ) , S ( + , t )) with D ( + , t ) ≥ S , and U ( + , t ) = U + = ( x x S , C ) . Then from(26) we have q = min { x ( − , t ) C , S ( + , t ) } , q = min { x ( − , t ) C , C } = x ( − , t ) C < C . Thus we have x ( − , t ) = x S x C , (35)40nd q = min { C , S ( + , t ) + x ( − , t ) C } = S ( + , t ) + x ( − , t ) C < C . Thus, q = S ( + , t ) = S , and U ( + , t ) = U + = ( C , S ) .(2) When two of the three terms on the right hand-side of (25) equals q , we have (i) D = S x = q < S x , (ii) D = S x = q < S x , or (iii) S x = S x = q < D . Here we only show the solutionsfor (i), and solutions for (ii) and (iii) can be obtained in a similar fashion. When D = S x < S x , from Corollary 3.4, we have U − = ( D , C ) , and U ( − , t ) = ( D ( − , t ) , S ( − , t )) with S ( − , t ) ≥ D ; U + = ( C , S ) , and U ( + , t ) = ( D ( + , t ) , S ( + , t )) with D ( + , t ) ≥ S ;and U ( + , t ) = U + = ( q , C ) . Then from (25) we have q = min { x ( − , t ) D ( − , t ) , S ( + , t ) } , q = min { x ( − , t ) D ( − , t ) , C } = x ( − , t ) D ( − , t ) < C . Thus D ( − , t ) = D or S ( + , t ) = x D = S . In this case, we can have the follow-ing interior states for links 1 and 2: (a) U ( − , t ) = U − , x i ( − , t ) = x i , and U ( + , t ) =( D ( + , t ) , S ( + , t )) with D ( + , t ) ≥ S and S ( + , t ) > S ; (b) U ( + , t ) = U + , x ( − , t ) = x D / D ( − , t ) , and U ( − , t ) = ( D ( − , t ) , S ( − , t )) with D ( − , t ) > D and S ( − , t ) ≥ D .(3) When all the three terms on the right hand-side of (25) equals q , we have D = S x = S x = q .From Corollary 3.4, we have U − = ( D , C ) , and U ( − , t ) = ( D ( − , t ) , S ( − , t )) with S ( − , t ) ≥ D ; U + = ( C , S ) , and U ( + , t ) = ( D ( + , t ) , S ( + , t )) with D ( + , t ) ≥ S ;and U + = ( C , S ) , and U ( + , t ) = ( D ( + , t ) , S ( + , t )) with D ( + , t ) ≥ S . In this case,at least one of U ( − , t ) = U − , U ( + , t ) = U + , and U ( + , t ) = U + should be satisfied, andthe other can be U ( − , t ) = ( D ( − , t ) , S ( − , t )) with D ( − , t ) > D and S ( − , t ) ≥ D , U ( + , t ) = ( D ( + , t ) , S ( + , t )) with D ( + , t ) ≥ S and S ( + , t ) > S , or U ( + , t ) = D ( + , t ) , S ( + , t )) with D ( + , t ) ≥ S and S ( + , t ) > S . Here x i ( − , t ) can be deter-mined once the interior states are determined. (cid:4) Appendix C: Proof of Theorem 5.1
Proof . From traffic conservation equations in (15) and admissible conditions of stationary states,we can see that q ≤ min { S + S , D } . We first demonstrate that it is not possible that q < min { S + S , D } ≤ min { C + C , C } . Other-wise, from (17) and (19) we have U ( − , t ) = U − = ( C , q ) with q < D ; Since q ( U + ) + q ( U + ) = q < S + S , then we have q ( U + i ) < S i for at least one downstream link, e.g., q < S . From (16)and (18) we have U ( + , t ) = U + = ( q , C ) . Then from the entropy condition in (27) we have q = min { C + S ( + , t ) , C } , q = min { , C C + S ( + , t ) } C . Since q < C , from the first equation we have q = C + S ( + , t ) < C , and from the secondequation we have q = C , which contradicts q < S . Therefore, q + q = q = min { S + S , D } . That is, the diverge model (27) yields the optimal fluxes for any initial conditions.(1) When S + S < D , we have q = S + S < D . We have U ( − , t ) = U − = ( S + S , C ) .Since q + q = S + S and q i ≤ S i , we have q i = S i , and U + i = ( C i , S i ) . From (19) we have U i ( + , t ) = ( D i ( + , t ) , S i ( + , t )) with D i ( + , t ) ≥ S + i = S i . From (27) we have q = min { S ( + , t ) + S ( + , t ) , C } = S + S < D ≤ C , q i = min { , C S ( + , t ) + S ( + , t ) } S i ( + , t ) = S i . S i ( + , t ) = S i ≤ S i ( + , t ) . Then U i ( + , t ) = U + i = ( C i , S i ) . In this case, there are nointerior states on all links.(2) When S + S = D , we have q = D , and q i = S i . We have U − = ( D , C ) and U ( − , t ) =( D ( − , t ) , D ( − , t )) with S ( − , t ) ≥ D − = D , and U + i = ( C i , S i ) and U i ( + , t ) = ( S i ( + , t ) , S i ( + , t )) with D i ( + , t ) ≥ S + i = S i . From (27) we have q = min { S ( + , t ) + S ( + , t ) , D ( − , t ) } = S + S = D , q i = min { , D ( − , t ) S ( + , t ) + S ( + , t ) } S i ( + , t ) = S i . We can have the following two scenarios.(2-i) If S ( + , t ) + S ( + , t ) ≥ D ( − , t ) = S + S = D ≤ S ( − , t ) , then U ( − , t ) = U − =( D , C ) and there is no interior state on link 0. Moreover, we have S + S S ( + , t ) + S ( + , t ) S i ( + , t ) = S i , which leads to S i ( + , t ) ≤ S i . From the assumption that S ( + , t ) + S ( + , t ) ≥ S + S ,we have S i ( + , t ) = S i . Further we have U i ( + , t ) = U + i = ( C i , S i ) , and there are nointerior states on links 1 or 2.(2-ii) If D ( − , t ) > S ( + , t ) + S ( + , t ) = S + S = D , S i ( + , t ) = S i . Thus U i ( + , t ) = U + i = ( C i , S i ) , and there are no interior states on links 1 or 2. Moreover, U ( − , t ) satisfies S ( − , t ) > D and D ( − , t ) ≥ D . Thus there can be multiple interior stateson link 0 when D < C .(3,4) When S + S > D , then q = q + q = D . We have U − = ( D , C ) , and U ( − , t ) =( D ( − , t ) , D ( − , t )) with S ( − , t ) ≥ D − = D . For downstream links, at least one of thestationary states is SUC. Otherwise, from (17) we have U + i = ( C i , S i ) , and q + q = S + S > D , which is impossible. From (27) we have q = min { S ( + , t ) + S ( + , t ) , D ( − , t ) } = D < S + S , i = min { , D ( − , t ) S ( + , t ) + S ( + , t ) } S i ( + , t ) . If S ( + , t ) + S ( + , t ) ≤ D ( − , t ) , then S ( + , t ) + S ( + , t ) = D < S + S and q i = S i ( + , t ) . This is not possible for the SUC stationary state U + i = U i ( + , t ) = ( q i , C i ) with q i < S i ≤ C i . Thus D ( − , t ) < S ( + , t ) + S ( + , t ) , D ( − , t ) = D < S + S , and U ( − , t ) = U − = ( D , C ) . Hence for both downstream links q i = D S ( + , t ) + S ( + , t ) S i ( + , t ) . (3) When S i > C i C + C D ( i = , U + i = U i ( + , t ) = ( q i , C i ) with q i < S i . Otherwise, we assume that link 1 is SUC with U ( + , t ) = U + = ( q , C ) and link 2 is OC with U + = ( S , C ) . Then S = D C + S ( + , t ) S ( + , t ) ≤ D C + C C < S , which is impossible. From (27), we have q i = D C + C C i , and U i ( + , t ) = U + i = ( q i , C i ) .(4) When S + S > S and S i ≤ C i C + C D ( i , j = i = j ), we can show thatstationary states on links j and i are SUC and OC respectively with U + j = U j ( + , t ) =( q j , C j ) with q j < S j , U + i = ( C i , S i ) , and S i ( + , t ) ≥ S i . Otherwise, U i ( + , t ) = U + i =( q i , C i ) with q i < S i , and q i = D C i + S j ( + , t ) C i ≥ C i C + C D ≥ S i , which is impossible. Since at least one of the downstream links has SUC stationarystate, the stationary states on links i and j are OC and SUC respectively. From (27), wehave a unique interior state on link i , U i ( + , t ) = ( C i , S i D − S i C j ) , and q j = D − S i .For the four cases, it is straightforward to show that (29) always holds. (cid:4) ppendix D: Proof of Theorem 5.4 Proof .First (30) implies that q = min { S ( + , t ) + S ( + , t ) , D ( − , t ) } , which can be shown for three cases: (i) S ( + , t ) + S ( + , t ) < D ( − , t ) , (ii) S i ( + , t ) ≥ a i S ( − , t ) ,and (iii) S ( + , t ) + S ( + , t ) ≥ D ( − , t ) and S i ( + , t ) ≤ a i S ( − , t ) .(1) When S + S < D , q = q + q ≤ S + S < D ≤ C . Thus the downstream stationarystate is SOC with U − = U ( − , t ) = ( C , q ) . In the following, we prove that q i = S i , whichis consistent with (31).(i) Assuming that q i < S i ≤ C i , then the stationary state on link i is SUC with U + i = U i ( + , t ) = ( q i , C i ) . From (30), we have q i = min { C i , max { C − S j ( + , t ) , a i C }} = max { C − S j ( + , t ) , a i C } < C i , q j = min { S j ( + , t ) , max { C − C i , a j C }} . We show that the two equations have no solutions for either a j C ≤ S j ( + , t ) or a j C > S j ( + , t ) . Thus q i = S i .(a) When a j C ≤ S j ( + , t ) , we have a i C ≥ C − S j ( + , t ) . From the first equationwe have q i = a i C . From the second equation we have q j = S j ( + , t ) ≥ a j C or q j = max { C − C i , a j C } ≥ a j C . Thus q i + q j ≥ C ≥ D , which contradicts q < D .(b) When a j C > S j ( + , t ) , we have a i C < C − S j ( + , t ) . From the first equation wehave q i = C − S j ( + , t ) . From the second equation we have q j = S j ( + , t ) . Thus q i + q j = C , which contradicts q < D .452) When S i ≥ a i D , D − S j ≤ a i D . In the following we show that q = D and q i = a i D ,which is consistent with (31).(i) If q < D , then the stationary state on link 0 is SOC with U − = U ( − , t ) = ( C , q ) .Also at least one of the downstream stationary states is SUC, since, otherwise, q + q = S + S ≥ D . Here we assume U + i = U i ( + , t ) = ( q i , C i ) . From (30) we have q i = min { C i , max { C − S j ( + , t ) , a i C }} = max { C − S j ( + , t ) , a i C } , q j = min { S j ( + , t ) , max { C − C i , a j C }} . We show that the two equations have no solutions for either C − S j ( + , t ) ≥ a i C or C − S j ( + , t ) < a i C . Thus q = D .(a) If C − S j ( + , t ) ≥ a i C , S j ( + , t ) ≤ a j C . From the first equation we have q i = C − S j ( + , t ) . From the second equation we have q j = S j ( + , t ) . Thus q i + q j = C , which contradicts q < D ≤ C .(b) If C − S j ( + , t ) < a i C , S j ( + , t ) > a j C . From the first equation we have q i = a i C . From the second equation we have q j = S j ( + , t ) > a j C or q j = max { C − C i , a j C } ≥ a j C . Thus q i + q j ≥ C , which contradicts q < D ≤ C .(ii) If q i < a i D ≤ S i ≤ C i for any i = ,
2, then U + i = U i ( + , t ) = ( q i , C i ) . From (30) wehave q i = max { D ( − , t ) − S j ( + , t ) , a i D ( − , t ) } < C i , q j = min { S j ( + , t ) , max { D ( − , t ) − C i , a j D ( − , t ) }} . The first equation implies that a i D ( − , t ) < a i D ; i.e., D ( − , t ) < D . In addition, D ( − , t ) − S j ( + , t ) < a i D . Thus, D ( − , t ) − C i < D − C i < D − a i D = a j D ,and max { D ( − , t ) − C i , a j D ( − , t ) } < a j D . From the second equation we have q j < a j D . Thus q i + q j < D , which contradicts q i + q j = D . Thus q i ≥ a i D for i = ,
2. Since q i + q j = D , q i = a i D .463) When S i + S j ≥ D and S i ≤ a i D for i , j = i = j . In the following we show that q = D and q i = S i , which is consistent with (31).(i) If q < D , then the stationary state on link 0 is SOC with U − = U ( − , t ) = ( C , q ) .We first prove that at least one downstream stationary state is SUC and then that noneof the downstream stationary states can be SUC. Therefore, q = D .(a) If none of the downstream stationary states are SUC, then q + q = S + S ≥ D ,which contradicts q < D . Thus, at least one of the downstream stationary statesis SUC.(b) Assuming that q i < S i , then U + i = U i ( + , t ) = ( q i , C i ) . From (30) we have q i = min { C i , max { C − S j ( + , t ) , a i C }} = max { C − S j ( + , t ) , a i C } < S i , which is not possible, since S i ≤ a i D . Thus q i = S i .(c) Assuming that q j < S j , then U + j = U j ( + , t ) = ( q j , C j ) . Since q = min { S i ( + , t ) + C j , C } < D , we have q = S i ( + , t ) + C j < D . From (30) we have S i = q i ≤ S i ( + , t ) . Thus S i + S j ≤ S i ( + , t ) + C j < D , which contradicts S i + S j ≥ D .(ii) If q i < S i , then U + i = U i ( + , t ) = ( q i , C i ) . From (30), we have q i = max { D ( − , t ) − S j ( + , t ) , a i D ( − , t ) } < S i ≤ a i D , q j = min { S j ( + , t ) , max { D ( − , t ) − C i , a j D ( − , t ) }} . From the first equation we have that D ( − , t ) < D . We show that the two equa-tions have no solutions for either D ( − , t ) − S j ( + , t ) ≥ a i D ( − , t ) or D ( − , t ) − S j ( + , t ) < a i D ( − , t ) . Therefore q i = S i .(a) When D ( − , t ) − S j ( + , t ) ≥ a i D ( − , t ) , we have S j ( + , t ) ≤ a j D ( − , t ) . Thus q i = D ( − , t ) − S j ( + , t ) and q j = S j ( + , t ) . Then q i + q j = D ( − , t ) < D ,which contradicts q i + q j = D .47b) When D ( − , t ) − S j ( + , t ) < a i D ( − , t ) , we have q i = a i D ( − , t ) and D ( − , t ) − C i < D ( − , t ) − q i = a j D ( − , t ) . Thus q j ≤ a j D ( − , t ) , and q i + q j ≤ D ( − , t ) < D , which contradicts q = D . (cid:4)48