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dynamic traffic assignment ( dta ) is the descriptive modeling of time - varying flows on traffic networks consistent with traffic flow theory and travel choice principles . dta models describe and predict departure rates , departure times and route choices of travelers over a given time horizon . it seeks to describe the dynamic evolution of traffic in networks in a fashion consistent with the fundamental notions of traffic flow and travel demand ; see @xcite for some review on dta models and recent developments . _ dynamic user equilibrium _ ( due ) of the open - loop type , which is one type of dta , remains a major modern perspective on traffic modeling that enjoys wide scholarly support . it captures two aspects of driving behavior quite well : departure time choice and route choice @xcite . within the due model , travel cost for the same trip purpose is identical for all utilized route - and - departure - time choices . the relevant notion of travel cost is _ effective travel delay _ , which is a weighted sum of actual travel time and arrival penalties . in the last two decades there have been many efforts to develop a theoretically sound formulation of dynamic network user equilibrium that is also a canonical form acceptable to scholars and practitioners alike . there are two essential components within the due models : ( 1 ) the mathematical expression of nash - like equilibrium conditions ; and ( 2 ) a network performance model , which is , in effect , an embedded _ dynamic network loading _ ( dnl ) problem . the dnl model captures the relationships among link entry flow , link exit flow , link delay and path delay for any given set of path departure rates . the dnl gives rise to the notion of _ path delay operator _ , which is viewed as a mapping from the set of feasible path departure rates to the set of path travel times or , more generally , path travel costs . properties of the path delay operator are of fundamental importance to due models . in particular , continuity of the delay operators plays a key role in the existence and computation of due models . the existence of dues is typically established by converting the problem to an equivalent mathematical form and applying some version of brouwer s fixed - point existence theorem ; examples include @xcite and @xcite . all of these existence theories rely on the continuity of the path delay operator . on the computational side of analytical due models , every established algorithm requires the continuity of the delay operator to ensure convergence ; an incomplete list of such algorithms include the fixed - point algorithm @xcite , the route - swapping algorithm @xcite , the descent method @xcite , the projection method @xcite , and the proximal point method @xcite    it has been difficult historically to show continuity of the delay operator for general network topologies and traffic flow models . over the past decade , only a few continuity results were established for some specific traffic flow models . @xcite use the link delay model @xcite to show the continuity of the path delay operator . their result relies on the _ a priori _ boundedness of the path departure rates , and is later improved by a continuity result that is free of such an assumption @xcite . in @xcite , continuity of the delay operator is shown for networks whose dynamics are described by the lwr - lax model @xcite , which is a variation of the lwr model that does not capture vehicle spillback . their result also relies on the _ a priori _ boundedness of path departure rates . @xcite consider vickrey s point queue model @xcite and show the continuity of the corresponding path delay operator for general networks without invoking the boundedness on the path departure rates . all of these aforementioned results are established for network performance models that do not capture vehicle spillback . to the best of our knowledge , there has not been any rigorous proof of the continuity result for dnl models that allow queue spillback to be explicitly captured . on the contrary , some existing studies even show that the path travel times may depend discontinuously on the path departure rates , when physical queue models are used . for example , @xcite uses the cell transmission model and signal control to show that the path travel time may depend on the path departure rates in a discontinuous way . such a finding suggests that the continuity of the delay operator could very well fail when spillback is present . this has been the major hurdle in showing the continuity or identifying relevant conditions under which the continuity is guaranteed . this paper bridges this gap by articulating these conditions and providing accompanying proof of continuity . this paper presents , for the first time , a rigorous continuity result for the path delay operator based on the lwr network model , which explicitly captures physical queues and vehicle spillback . in showing the desired continuity , we propose a systematic approach for analyzing the well - posedness of two specific junction models : a merge and a diverge model , both originally presented by @xcite . the underpinning analytical framework employs the _ wave - front tracking _ methodology @xcite and the technique of _ generalized tangent vectors _ @xcite . a major portion of our proof involves the analysis of the interactions between kinematic waves and the junctions , which is frequently invoked for the study of well - posedness of junction models ; see @xcite for more details . such analysis is further complicated by the fact that vehicle turning ratios at a diverge junction are determined endogenously by drivers route choices within the dnl procedure . as a result , special tools are developed in this paper to handle this unique situation . as we shall later see , a crucial step of the process above is to estimate and bound from below the minimum network supply , which is defined in terms of local vehicle densities in the same way as in @xcite . in fact , if the supply somewhere tends to zero ( that is , when traffic approaches the jam density ) , the well - posedness of the diverge junction may fail , as we demonstrate in section [ subsubsecexample ] . this has also been confirmed by the earlier study of @xcite , where a wave of jam density is triggered by a signal red light and causes spillback at the upstream junction , leading to a jump in the path travel times . remarkably , in this paper we are able to show that ( 1 ) if the supply is bounded away from zero ( that is , traffic is bounded away from the jam density ) , then the diverge junction model is well posed ; and ( 2 ) the desired boundedness of the supply is a natural consequence of the dynamic network loading procedure that involves only the merge and diverge junction models we study here . this is a highly non - trivial result because it not only plays a role in the continuity proof , but also implies that gridlock can never occur in the network loading procedure in any finite time horizon . the final continuity result is presented in section [ subseccontinuity ] , following a number of preliminary results set out in previous sections . although our continuity result is established only for networks consisting of simple merge and diverge nodes , it can be extended to networks with more complex topologies using the procedure of decomposing any junction into several simple merge and diverge nodes @xcite . moreover , the analytical framework employed by this paper can be invoked to treat other and more general junction topologies and/or merging and diverging rules , and the techniques employed to analyze wave interactions will remain valid . the main contributions made in this paper include :    * formulation of the lwr - based dynamic network loading ( dnl ) model with spillback as a system of _ partial differential algebraic equations _ ( pdaes ) ; * a continuity result for the path delay operator based on the aforementioned dnl model ; * a novel method for estimating the network supply , which shows that gridlock can never occur within a finite time horizon . the rest of this paper is organized as follows . section [ seclwrdnl ] recaps some essential knowledge and notions regarding the lwr network model and the dnl procedure . section [ secpdae ] articulates the mathematical contents of the dnl model by formulating it as a pdae system . section [ subsecwellposed ] introduces the merge and diverge models and establishes their well - posedness . section [ seccontinuityproof ] provides a final proof of continuity and some discussions . finally , we offer some concluding remarks in section [ secconclusion ] . throughout this paper , the time interval of analysis is a single commuting period expressed as @xmath0 $ ] for some @xmath1 . we let @xmath2 be the set of all paths employed by travelers . for each path @xmath3 we define the path departure rate which is a function of departure time @xmath4 $ ] : @xmath5~\rightarrow~\mathbb{r}_+\ ] ] where @xmath6 denotes the set of non - negative real numbers . each path departure rate @xmath7 is interpreted as a time - varying path flow measured at the entrance of the first arc of the relevant path , and the unit for @xmath7 is _ vehicles per unit time_. we next define @xmath8 to be a vector of departure rates . therefore , @xmath9 can be viewed as a vector - valued function of @xmath10 , the departure time . instead of @xmath11 to denote path flow vectors . ] the single most crucial ingredient t is the path delay operator , which maps a given vector of departure rates @xmath12 to a vector of path travel times . more specifically , we let @xmath13,\quad   \forall p\in \mathcal{p}\ ] ] be the path travel time of a driver departing at time @xmath10 and following path @xmath14 , given the departure rates @xmath12 associated with all the paths in the network . we then define the path delay operator @xmath15 by letting @xmath16 , which is a vector consisting of time - dependent path travel times @xmath17 . we recap the network extension of the lwr model @xcite , which captures the formation , propagation , and dissipation of spatial queues and vehicle spillback . discussion provided below relies on general assumptions on the fundamental diagram and the junction model , and involves no _ ad hoc _ treatment of flow propagation , flow conservation , link delay , or other critical model features . we consider a road link expressed as a spatial interval @xmath18\subset \mathbb{r}$ ] . the _ partial differential equation _ ( pde ) representation of the lwr model is the following scalar conservation law @xmath19\times[a,\,b]\ ] ] with appropriate initial and boundary conditions , which will be discussed in detail later . here , @xmath20 denotes vehicle density at a given point in the space - time domain . the fundamental diagram @xmath21\rightarrow [ 0,\ , c]$ ] expresses vehicle flow at @xmath22 as a function of @xmath20 , where @xmath23 denotes the jam density , and @xmath24 denotes the flow capacity . throughout this paper , we impose the following mild assumption on @xmath25 :    * ( f ) . * the fundamental diagram @xmath25 is continuous , concave and vanishes at @xmath26 and @xmath27 . an essential component of the network extension of the lwr model is the specification of boundary conditions at a road junction . derivation of the boundary conditions should not only obey physical realism , such as that enforced by entropy conditions @xcite , but also reflect certain behavioral and operational considerations , such as vehicle turning preferences @xcite , driving priorities @xcite , and signal controls @xcite . articulation of a junction model is facilitated by the notion of _ riemann problem _ , which is an initial value problem at the junction with constant initial condition on each incident link . there exist a number of junction models that yield different solutions of the same riemann problem . in one line of research , an entropy condition is defined based on a minimization / maximization problem @xcite . in another line of research , the boundary conditions are defined using link _ demand _ and _ supply _ @xcite , which represent the link s sending and receiving capacities . models following this approach include @xcite and @xcite . the solution of a riemann problem is given by the _ riemann solver _ ( rs ) , to be defined below . we consider a general road junction @xmath28 with @xmath29 incoming roads and @xmath30 outgoing roads , as shown in figure [ chapdnl : figlwrjunction ] . incoming links and @xmath30 outgoing links . ] we denote by @xmath31 the incoming links and by @xmath32 the outgoing links . in addition , for every @xmath33 , the dynamic on @xmath34 is governed by the lwr model @xmath35\times [ a_i,\,b_i]\ ] ] where link @xmath34 is expressed as the spatial interval @xmath36 , and we always use the subscript ` @xmath37 ' to indicate dependence on the link @xmath34 . the initial condition for this conservation law is @xmath38\ ] ]    notice that the above @xmath39 initial value problems are coupled together via the boundary conditions to be specified at the junction . such a system of coupling equations is commonly analyzed using the riemann problem . [ chapdnl : rpdef]*(riemann problem ) * the riemann problem at the junction @xmath28 is defined to be an initial value problem on a network consisting of the single junction @xmath28 with @xmath29 incoming links and @xmath30 outgoing links , all extending to infinity , such that the initial densities are constants on each link : @xmath40,\qquad \qquad i\in\{1,\,\ldots,\,m\ } \\ \rho_j(0,\,x)~\equiv~\hat\rho_j\qquad & x\in [ a_j,\,+\infty ) ,   \qquad\qquad j\in\{m+1,\,\ldots,\,m+n\ } \end{cases}\ ] ] where @xmath41 $ ] are constants , @xmath42 .    a riemann solver for the junction @xmath28 is a mapping that , given any @xmath39-tuple of riemann initial conditions @xmath43 , provides a unique @xmath39-tuple of boundary conditions @xmath44 such that one can solve the initial - boundary value problem for each link , and the resulting solutions constitute a weak entropy solution of the riemann problem at the junction . a precise definition of the riemann solver is given as follows . [ chapdnl : lwrrsdef ] * ( riemann solver ) * a riemann solver for the junction @xmath28 with @xmath29 incoming links and @xmath30 outgoing links is a mapping @xmath45 & ~\longrightarrow~ \prod_{k=1}^{m+n}\left[0,\,\rho^{jam}_k\right ] \\ \big(\hat\rho_{1},\,\ldots,\ , \hat\rho_{m+n}\big)&~\mapsto~\big(\overline\rho_1,\,\ldots,\,\overline\rho_{m+n}\big)\end{aligned}\ ] ] which relates riemann initial data @xmath46 to boundary conditions @xmath47 , such that the following hold .    1 . the solution of the riemann problem restricted on each link @xmath48 is given by the solution of the initial - boundary value problem with initial condition @xmath49 and boundary condition @xmath50 , @xmath42 . the rankine - hugoniot condition ( flow conservation ) holds : @xmath51 3 . the _ consistency condition _ holds : @xmath52\big]~=~rs[\hat\rho]\ ] ]    three conditions must be satisfied by the riemann solver ( rs ) . item @xmath53 above requires that the boundary condition on each link must be properly given so that the initial value problems not only have well - defined solutions , but these solutions must also be compatible and form a sensible solution at the junction . simply stipulates the conservation of flow across the junction . is a desirable property and is sometimes referred to as the _ invariance property _ @xcite .    for the same riemann problem , there exist many riemann solvers that satisfy conditions ( i)-(iii ) above . despite their varying forms , most existing riemann solvers rely on a flow maximization problem at the relevant junction subject to constraints related to turning ratio , right - of - way , or signal controls ; see @xcite and @xcite .      for each link @xmath34 , we let @xmath54 be the critical density at which the flow is maximized . the demand @xmath55 for incoming links @xmath34 and the supply @xmath56 for outgoing links @xmath57 are defined in terms of the density near the exit and entrance of the link , respectively @xcite : @xmath58 in prose , the demand represents the maximum flow at which cars can be discharged from the incoming link ; and the supply represents the maximum flow at which cars can enter the outgoing link . notice that the demand and supply are both expressed as functions of density , and they are always greater than or equal to the fundamental diagram @xmath59 or @xmath60 ; see figure [ figds ] for an illustration . in our subsequent presentation , without causing confusion we will use notations @xmath55 and @xmath61 interchangeably where the former indicates the demand as a time - varying quantity , and the latter emphasizes demand as a function of density . the same applies to the supply . the aim of this section is to formulate the lwr - based dynamic network loading ( dnl ) problem as a system of partial differential algebraic equations ( pdaes ) . the proposed pdae system uses vehicle density and queues as the primary unknown variables , and computes link dynamics , flow propagation , and path delay for any given vector of departure rates . the pdae system captures vehicle spillback explicitly , and accommodates a wide range of junction types and riemann solvers . we consider a network @xmath62 expressed as a directed graph with @xmath63 being the set of links and @xmath64 being the set of nodes . let @xmath2 be the set of paths employed by travelers , and @xmath65 be the set of origin - destination pairs . each path @xmath3 is expressed as an ordered set of links it traverses : @xmath66 where @xmath67 is the number of links in this path . there are several crucial components of a complete network loading model , each of which is elaborated in a subsection below . throughout the rest of this paper , for each node @xmath68 , we denote by @xmath69 the set of incoming links and @xmath70 the set of outgoing links .      for each @xmath71 , the density dynamic is governed by the scalar conservation law @xmath72~=~0\qquad ( t,\,x)\in[0,\,t]\times [ a_i,\,b_i]\ ] ] subject to initial condition and boundary conditions to be determined in section [ subsecpdaebdycond ] . the fundamental diagram @xmath73 satisfies condition * ( f ) * stated at the beginning of section [ secpreliminaries ] . in order to explicitly incorporate drivers route choices , for every @xmath3 such that @xmath74 we introduce the function @xmath75 , @xmath76\times [ a_i,\,b_i]$ ] , which represents , in every unit of flow @xmath77 , the fraction associated with path @xmath14 . we call these variables _ path disaggregation variables _ ( pdv ) . for each car moving along the link @xmath34 , its surrounding traffic can be distinguished by path ( e.g. 20% following path @xmath78 , 30% following path @xmath79 , 50% following path @xmath80 ) . as this car moves , such a composition will not change since its surrounding traffic all move at the same speed under the first - in - first - out ( fifo ) principle ( i.e. no overtaking is allowed ) . in mathematical terms , this means that the path disaggregation variables , @xmath81 , are constants along the trajectories of cars @xmath82 in the space - time diagram , where @xmath83 is the trajectory of a moving car on link @xmath34 . that is , @xmath84 which , according to the chain rule , becomes @xmath85 which further leads to another set of partial differential equations on link @xmath34 : @xmath86 here , @xmath87 is the solution of . the following obvious identity holds @xmath88 where @xmath89 means path @xmath14 contains ( or traverses ) link @xmath34 " , and the summation appearing in is with respect to all such @xmath14 . by convention , if @xmath90 , then @xmath91 for all @xmath89 . for reason that will become clear later , we introduce the concept of an _ ordinary node_. an ordinary node is neither the origin nor the destination of any trip . we use the notation @xmath92 to represent the set of ordinary nodes in the network .    as mentioned earlier , the partial differential equations on links incident to @xmath28 are all coupled together through a given junction model , i.e. , a riemann solver . a common prerequisite for applying the riemann solver is the determination of the flow distribution ( turning ratio ) matrix @xcite , which relies on knowledge of the pdvs @xmath93 for all @xmath94 . we define the time - dependent flow distribution matrix associated with @xmath28 : @xmath95^{|\mathcal{i}^j| +   |\mathcal{o}^j|}\ ] ] where by convention , we use subscript @xmath37 to indicate incoming links , and @xmath96 to indicate outgoing links . each element @xmath97 represents the turning ratios of cars discharged from @xmath34 that enter downstream link @xmath57 . then , for all @xmath14 that traverses @xmath28 , the following holds . @xmath98 it can be easily verified that @xmath99 $ ] and @xmath100 according to . we are now ready to express the boundary conditions for the ordinary junction @xmath101 . let @xmath102 \rightarrow \prod_{k=1}^{|\mathcal{i}^j|+|\mathcal{o}^j|}[0,\,\rho^{jam}_k]\ ] ] be a given riemann solver . notice that the dependence of the riemann solver on @xmath103 has been indicated with a superscript . the boundary conditions for pdes read @xmath104\qquad   \forall i_k \in \mathcal{i}^j \\ \label{pdaersdef2 } \rho_{l}(t,\,a_l)&~=~rs^{a^j}_l\left[\big(\rho_{i}(t,\ , b_i-)\big)_{i_i\in\mathcal{i}^j}~,~\big(\rho_{j}(t,\,a_j+)\big)_{i_j\in\mathcal{o}^j}\right]\qquad \forall i_l\in \mathcal{o}^j\end{aligned}\ ] ] where @xmath105 $ ] denotes the @xmath106-th component of the mapping , @xmath107 .    intuitively , - mean that , given the current traffic states @xmath108 and @xmath109 adjacent to the junction @xmath28 , the riemann solver @xmath110 specifies , for each incident link @xmath48 or @xmath111 , the corresponding boundary conditions @xmath112 or @xmath113 . in prose , at each time instance the riemann solver inspects the traffic conditions near the junction , and proposes the discharging ( receiving ) flows of its incoming ( outgoing ) links . such a process is based on the flow distribution matrix @xmath114 and often reflects traffic control measures at junctions . furthermore , the riemann solver operates with knowledge of every link incident to the junction , thus the boundary condition of any relevant link is determined jointly by all the links connected to the same junction . therefore , the lwr equations on all the links are coupled together through this mechanism . for this reason the lwr - based dnl models are highly challenging , both theoretically and computationally . the upstream boundary conditions associated with pdes are : @xmath115 where the numerator @xmath116 expresses the exit flow on link @xmath34 associated with path @xmath14 , which , by flow conservation , is equal to the entering flow of link @xmath57 associated with the same path @xmath14 ; the denominator represents the total entering flow of link @xmath57 . unlike the density - based pde , the pdv - based pde does not have any downstream boundary condition due to the fact that the traveling speeds of the pdvs are the same as the car speeds ( they can be interpreted as lagrangian labels that travel with the cars ) ; thus information regarding the pdvs does not propagate backwards or spills over to upstream links . we consider a node @xmath68 that is either the origin or the destination of some path @xmath14 . one immediate observation is that the flow conservation constraint no longer holds at such a node since vehicles either are ` generated ' ( if @xmath117 is an origin ) or ` vanish ' ( if @xmath117 is a destination ) . a simple and effective way to circumvent this issue is to introduce a _ virtual link_. a virtual link is an imaginary road with certain length and fundamental diagram , and serves as a buffer between an ordinary node and an origin / destination ; see figure [ chapdnl : figvirtuallink ] for an illustration . by introducing virtual links to the original network , we obtain an augmented network @xmath118 in which all road junctions are ordinary , and hence fall within the scope of the previous section .    . right : a virtual link connecting a destination ( t ) to an ordinary junction @xmath28 . ] let us denote by @xmath119 the set of origins in the augmented network @xmath118 . for any @xmath120 , we denote by @xmath121 the set of paths that originate from @xmath122 , and by @xmath123 the virtual link incident to this origin . for each @xmath124 we denote by @xmath7 the departure rate ( path flow ) along @xmath14 . it is expected that a buffer ( point ) queue may form at @xmath122 in case the receiving capacity of the downstream @xmath123 is insufficient to accommodate all the departure rates @xmath125 . for this buffer queue , denoted @xmath126 , we employ a vickrey - type dynamic @xcite ; that is , @xmath127 where @xmath128 denotes the supply of the virtual link @xmath123 . the only difference between and vickrey s model is the time - varying downstream receiving capacity provided by the virtual link . it remains to determine the dynamics for the path disaggregation variables ( pdv ) . more precisely , we need to determine @xmath129 for the virtual link @xmath123 where @xmath124 , and @xmath130 is the upstream boundary of @xmath123 . this will be achieved using the vickrey - type dynamic and the fifo principle . specifically , we define the queue exit time function @xmath131 where @xmath10 denotes the time at which drivers depart and join the point queue , if any ; @xmath131 expresses the time at which the same group of drivers exit the point queue . clearly , fifo dictates that @xmath132 where the two integrands on the left and right hand sides of the equation are flow entering the queue and flow leaving the queue , respectively . we may determine the path disaggregation variables as : @xmath133 notice that , if @xmath134 , then the flow leaving the point queue at time @xmath131 is also zero ; thus there is no need to determine the path disaggregation variables . therefore , the identity is well defined and meaningful .      with all preceding discussions , we may finally express the path travel times , which are the outputs of a complete dnl model . the path travel time consists of link travel times plus possible queuing time at the origin . mathematically , the link exit time function @xmath135 for any @xmath34 is defined , in a way similar to , as @xmath136    for a path expressed as @xmath137 , the time to traverse it is calculated as @xmath138 where @xmath139 means the composition of two functions . this is due to the assumption that cars leaving the previous link ( or queue ) immediately enter the next link without any delay . we are now ready to present a generic pdae system for the dynamic network loading procedure . let us begin by summarizing some key notations . *   the original network with link set @xmath63 and node set @xmath64 ; *   the set of virtual links ; *   the augmented network including virtual links ; *   the set of origins in @xmath140 ; *   the set of paths originating from @xmath120 ; *   the set of ordinary junctions in @xmath140 ; *   the set of incoming links of a junction @xmath101 ; *   the set of outgoing links of a junction @xmath101 ; *   the flow distribution matrix associated with junction @xmath28 ; *   the riemann solver for junction @xmath28 , which depends on @xmath103 .    we also list some key variables of the pdae system below .    *   the path departure rate along @xmath3 ; *   the vehicle density on link @xmath141 ; *   the proportion of flow on link @xmath34 associated with path @xmath14 ( path disaggregation variable ) ; *   the point queue at the origin @xmath120 ; *   the point queue exit time function at origin @xmath120 .    given any vector of path departure rates @xmath142 , the proposed pdae system for calculating path travel times is summarized as follows . @xmath143=0\qquad & ( t,\,x)\in[0,\,t]\times [ a_i,\,b_i ] \\ \label{chapdnl : lwrpdaeeqn3 } \partial_t\mu_i^p\big(t,\,x\big)+v_i\big(\rho_i(t,\,x)\big)\cdot \partial_x\mu_i^p\big(t,\,x\big)=0\qquad   & ( t,\,x)\in[0,\,t]\times [ a_i,\,b_i ] \\ \label{chapdnl : lwrpdaeeqn65 } \mu_s^p\big(\lambda_s(t),\,a_s\big)~=~ { h_p(t)\over \sum_{q\in\mathcal{p}^s}h_q(t ) } \qquad & \forall s\in\mathcal{s},\ , p\in\mathcal{p}_s \\ \label{chapdnl : lwrpdaeeqn66 } \mu_{j}^{p}(t,\ , a_j)~=~{f_{i}\big(\rho_{i}(t,\ , b_i)\big)\cdot \mu_{i}^p(t,\ , b_i ) \over f_{j}\big(\rho_{j}(t,\,a_j)\big ) }   \qquad & \forall p \supset \{i_i,\,i_j\ } \\ \label{chapdnl : lwrpdaeeqn4 } a^j(t)=\big\{\alpha^j_{ij}(t)\big\},\quad    \alpha^j_{ij}(t)=\sum_{p\ni i_i,\,i_j } \mu_{i}^p(t,\,b_i ) \qquad & \forall i_i\in\mathcal{i}^j,\ , i_j\in\mathcal{o}^j \\ \label{chapdnl : lwrpdaeeqn5 } \rho_{k}(t,\,b_k)~=~rs^{a^j}_k\left[\big(\rho_{i}(t,\ , b_i-)\big)_{i_i\in\mathcal{i}^j}~,~\big(\rho_{j}(t,\,a_j+)\big)_{i_j\in\mathcal{o}^j}\right]\qquad   & \forall i_k \in \mathcal{i}^j \\ \label{chapdnl : lwrpdaeeqn6 } \rho_{l}(t,\,a_l)~=~rs^{a^j}_l\left[\big(\rho_{i}(t,\ , b_i-)\big)_{i_i\in\mathcal{i}^j}~,~\big(\rho_{j}(t,\,a_j+)\big)_{i_j\in\mathcal{o}^j}\right]\qquad   & \forall i_l\in \mathcal{o}^j \\ \label{chapdnl : lwrpdaeeqn7 } d_p(t,\,h)~=~\lambda_s \circ \lambda_1 \circ \lambda_2 \ldots \circ\lambda_{m(p ) } ( t ) \qquad & \forall p\in\mathcal{p},\quad \forall t\in[0,\,t]\end{aligned}\ ] ]    eqn describes the ( potential ) queuing process at each origin . eqns and express the queue exit time function for a point queue , and the link exit time function for a link , respectively . eqns - express the link dynamics in terms of car density and pdv ; eqns - specifies the upstream boundary conditions for the pdv as these variables can only propagate forward in space . eqns - determine the boundary conditions at junctions . finally , eqn determines the path travel times . the above pdae system involves partial differential operators @xmath144 and @xmath145 . solving such a system requires solution techniques from the theory of numerical _ partial differential equations _ ( pde ) such as finite difference methods @xcite and finite element methods @xcite . in mathematical modeling , the term _ well - posedness _ refers to the property of having a unique solution , and the behavior of that solution hardly changes when there is a slight change in the initial / boundary conditions . examples of well - posed problems include the initial value problem for scalar conservation laws @xcite , and the initial value problem for the hamilton - jacobi equations @xcite . in the context of traffic network modeling , well - posedness is a desirable property of network performance models capable of supporting analyses and computations of dta models . it is also closely related to the continuity of the path delay operator , which is the main focus of this paper . this section investigates the well - posedness ( i.e. continuous dependence on the initial / boundary conditions ) of two specific junction models . these two junctions are depicted in figure [ figtwojunc ] , and the corresponding merge and diverge rules are proposed initially by @xcite in a discrete - time setting with fixed vehicle turning ratios and driving priority parameters . we first consider the diverge junction shown on the left part of figure [ figtwojunc ] , with one incoming link @xmath146 and two outgoing links @xmath147 and @xmath148 . the demand @xmath149 and the supplies , @xmath150 and @xmath151 , are defined by and respectively . the riemann solver for this junction relies on the following two conditions . * cars leaving @xmath146 advance to @xmath147 and @xmath148 according to some turning ratio which is determined by the pdv @xmath152 in the dnl model . * subject to ( a1 ) , the flow through the junction is maximized .    in the original diverge model @xcite , the vehicle turning ratios , denoted @xmath153 and @xmath154 with obvious meaning of notations , are constants known _ a priori_. this is not the case in a dynamic network loading model since they are determined endogenously by drivers route choices , as expressed mathematically by eqn . the diverge junction model , described by ( a1 ) and ( a2 ) , can be more explicitly written as : @xmath155 where @xmath156 denotes the exit flow of link @xmath146 , and @xmath157 denotes the entering flow on link @xmath57 , @xmath158 . we now turn to the merge junction in figure [ figtwojunc ] , with two incoming links @xmath159 and @xmath160 and one outgoing link @xmath161 . in view of this merge junction , assumption ( a1 ) becomes irrelevant as there is only one outgoing link ; and assumption ( a2 ) can not determine a unique solution . to address this issue , we consider a _ right - of - way _ parameter @xmath162 and the following priority rule : + ( r1 ) the actual link exit flows satisfy @xmath163 . +    ) are the optimal solution sets of . left : rule ( r1 ) is compatible with ( a2 ) ; and there exists a unique point @xmath164 satisfying both ( a2 ) and ( r1 ) . right : ( r1 ) is incompatible with ( a2 ) ; in this case , the model selects point @xmath164 within the set @xmath165 that is closest to the ray ( @xmath166 ) from the origin with slope @xmath167 . ] notice that ( r1 ) may be incompatible with assumption ( a2 ) ; and we refer the reader to figure [ figmerge ] for an illustration . whenever there is a conflict between ( r1 ) and ( a2 ) , we always respect ( a2 ) and relax ( r1 ) so that the solution is chosen to be the point that is closest to the line @xmath168 among all the points yielding the maximum flow . clearly , such a point is unique . mathematically , we let @xmath165 be the set of points @xmath169 that solves the following maximization problem : @xmath170    moreover , we define the ray @xmath171 . then the solution of the merge model is defined to be the projection of @xmath166 onto @xmath165 ; that is , @xmath172\ ] ] where @xmath173 $ ] denotes the euclidean distance between the point @xmath174 and the ray @xmath166 : @xmath175~=~\min_{(x,\,y)\in r}\left\| ( f^{out}_{4},\,f^{out}_{5 } ) -   ( x,\,y)\right\|_2\ ] ]      in this section , we investigate the well - posedness of the diverge junction model . unlike previous studies @xcite , a major challenge in this case is to incorporate drivers route choices , expressed by the _ path disaggregation variable _ @xmath176 , into the model and the analysis . in effect , we need to establish the continuous dependence of the model on the initial / boundary conditions in terms of both @xmath177 and @xmath176 . as we shall see in section [ subsubsecexample ] below , such a continuity does not hold in general . following this , section [ subsubsecsufficient ] provides sufficient conditions for the continuity to hold . these sufficient conditions are crucial for the desired continuity of the delay operator .      it has been shown by @xcite that the diverge model ( section [ subsecdiverge ] ) with constant turning ratios , @xmath153 and @xmath154 , is well - posed . however , the assumption of fixed and exogenous turning ratios does not hold in dnl models ; see . as a result the well - posedness is no longer true , which is demonstrated by the following counterexample . we consider the diverge junction ( figure [ figtwojunc ] ) and assume the same fundamental diagram @xmath25 for all the links for simplicity . we consider a series of constant initial data parameterized by @xmath178 on the three links @xmath179 , and @xmath148 : @xmath180,\qquad\qquad \rho_3(0,\,x)~\equiv~\hat\rho_3^{\,\varepsilon}\in ( 0,\,\rho^c)\ ] ] where @xmath181 and @xmath23 are the critical density and the jam density , respectively . @xmath182 , @xmath183 , and @xmath184 satisfy : @xmath185 where @xmath186 is a parameter . this configuration of initial data implies that link @xmath146 and link @xmath147 are both in the congested phase , while link 3 is in the uncongested ( free - flow ) phase , see figure [ figcounterex ] for an illustration . two paths exist in this example : @xmath187 and @xmath188 . the initial conditions for @xmath189 and @xmath190 , which arise from travelers route choices , are as follows . @xmath191 notice that these initial conditions are part of the initial value problem at the junction . the solution of this initial value problem depends on the value of @xmath178 ; in particular , we have the following two cases . * when @xmath192 , we claim that the initial conditions @xmath193 and @xmath194 satisfying - constitute a constant solution at the junction . to see this , we follow the junction model and the definitions of demand and supply - to get @xmath195 where @xmath24 denotes the flow capacity . thus @xmath196 , @xmath197 and @xmath194 form a constant solution at the junction . * when @xmath198 , the turning ratios satisfy @xmath199 , @xmath200 . effectively , link @xmath146 is only connected to link @xmath148 . we easily deduce that @xmath201 as a result , the solution on link @xmath146 is given by a backward - propagating rarefaction wave ( or expansion wave , fan wave ) with @xmath196 and @xmath181 on the two sides . on link @xmath148 , a forward - propagating rarefaction wave with @xmath181 and @xmath202 ( that is , the limit of @xmath184 as @xmath203 ) on the two sides is created . link @xmath147 remains in a completely jam state with full density @xmath23 . see figure [ figsolution ] for an illustration of the solutions .    , @xmath147 and @xmath148 for the two cases : @xmath192 and @xmath198 . the dashed lines represent the kinematic waves ( characteristics ) , and darker color indicates higher density . notice that jumps in the boundary conditions across these two cases exist on @xmath146 and @xmath148 . ] the two sets of solutions corresponding to @xmath192 and @xmath198 are shown in figure [ figsolution ] . we first notice that in these two cases , the boundary flows are very different , despite the infinitesimal difference in the pdv , namely from @xmath192 to @xmath198 . in particular , both @xmath156 and @xmath204 jump from @xmath205 and @xmath206 ( when @xmath192 ) to @xmath24 ( when @xmath198 ) . this is a clear indication of the discontinuous dependence of the diverge junction model on its initial conditions . let us zoom in on the mechanism that triggers such a discontinuity . according to , the expression for @xmath156 when @xmath192 is @xmath207 as long as @xmath178 is positive , the fraction @xmath208 . however , when @xmath198 we have an expression of ` @xmath209 ' , which should be equal to @xmath210 since link @xmath147 has effectively no influence on the junction , and we have @xmath211 . the above argument amounts to the following statement : @xmath212 which explains the jump in the solutions when @xmath178 tends to zero . the fact that @xmath213 tends to @xmath23 as @xmath214 plays a key role in this example . as we shall see later in theorem [ divwpthm ] , bounding @xmath213 away from the jam density is essential for the well - posedness . as we have previously demonstrated , the diverge model with time - varying vehicle turning ratios may not depend continuously on its initial conditions , which are defined in terms of the two - tuple @xmath215 . in this section , we propose additional conditions that guarantee the continuous dependence with respect to the initial data at the diverge junction . our analysis relies on the method of wave - front tracking @xcite and the technique of _ generalized tangent vectors _ @xcite . in order to be self - contained while keeping our presentation concise , we move some general background on these subjects and essential mathematical details to [ secappemb ] .    * ( well - posedness of the diverge model)*[divwpthm ] consider the diverge junction ( figure [ figtwojunc ] ) and assume that    1 . there exists some @xmath216 such that the supplies @xmath217 , @xmath218 for all @xmath10 ; 2 . the path disaggregation variable @xmath219 has bounded total variation in @xmath10 . then the solution of the diverge junction depends continuously on the initial and boundary conditions in terms of @xmath177 and @xmath176 . the proof is long and technical , and thus will be presented in [ appsecthm1 ] following necessary mathematical preliminaries in [ secappemb ] .      for the merge junction depicted in figure [ figtwojunc ] , the path disaggregation variables @xmath176 become irrelevant since there is only one downstream link . according to theorem 5.3 of @xcite , the solution at the merge junction , in terms of density @xmath177 , depends continuously on the initial and boundary conditions . moreover , according to , the propagation speed of @xmath176 is the same as the vehicle speed @xmath220 , we thus conclude that @xmath176 also depends continuously on the initial and boundary conditions . this shows the well - posedness of the merge junction . the proof of the continuity of the path delay operator is outlined as follows . we first show that the first hypothesis of theorem [ divwpthm ] holds for networks that consists of only the merge and diverge junctions discussed in section [ secmergediverge ] ; this will be done in section [ secsupplymin ] . then , we propose some mild conditions on the fundamental diagram and the departure rates in section [ secpdv ] to ensure that the second hypothesis of theorem [ divwpthm ] holds . it then follows that both the merge and diverge models are well posed . finally , in section [ sec : dep ] we show the well - posedness of the model in terms of @xmath177 and @xmath176 at each origin , where a point queue may be present . put altogether , these results lead to the desired continuity for the delay operator . this section provides a lower bound on the supply , which is a function of density , on any link in the network during the entire time horizon @xmath0 $ ] . our finding is that the jam density can never be reached within any finite time horizon @xmath221 , and the supply anywhere in the network is bounded away from zero . as a consequence , grid lock will never occur in the dynamic network . we denote by @xmath222 the set of destinations in the augmented network with virtual links ( see section [ subsecvirtuallink ] ) ; that is , every destination @xmath223 is incident to a virtual link that connects @xmath224 to the rest of the network ; see figure [ figflowintod ] . for each @xmath223 , we introduce its supply , denoted @xmath225 , to be the maximum rate at which cars can be discharged from the virtual link connected to @xmath224 . effectively , there exists a bottleneck between the virtual link and the destination ; and the supply of the destination is equal to the flow capacity of this bottleneck ; see figure [ figflowintod ] . notice that in some literature such a bottleneck is completely ignored and the destination is simply treated as a sink with infinite receiving capacity . this is of course a special case of ours once we set the supply @xmath225 to be infinity . however , such a supply may be finite and even quite limited under some circumstances due to , for example , ramp metering , limited parking spaces , or the fact that the destination is an aggregated subnetwork that is congested .    . ]    we introduce below a few more concepts and notations . @xmath226 : :    : the minimum link length in the network , including virtual links . @xmath227 : :    : the minimum link flow capacity in the network , including virtual    links . @xmath228 : :    : the maximum backward wave speed in the network . @xmath229 : :    , where @xmath230 is the priority parameter for the merge    junction @xmath28 ( see section [ subsecmergemodel ] ) ,    @xmath231 denotes the set of merge junctions in the    network . @xmath232 : :    : the minimum supply among all the destination nodes . @xmath233}\inf_{x\in[a_i,\,b_i]}s_i\left(\rho_i(t,\,x)\right)$ ] : :    : the minimum supply at any location in the network during time    interval    @xmath234 ,    where @xmath235    [ thmdelta ] consider a network consisting of only merge and diverge junctions as shown in figure [ figtwojunc ] . given any vector of path departure rates , the dynamic network loading procedure described by the pdae system yields the following property in the solution : @xmath236    we postpone the proof until [ subsecappthmdelta ] for a more compact presentation . theorem [ thmdelta ] guarantees that the supply values anywhere in the network during the period @xmath237 $ ] are uniformly bounded below by some constant that depends only on @xmath106 , although such a constant may decay exponentially as @xmath106 increases . given that any dynamic network loading problem is conceived in a finite time horizon , we immediately obtain a lower bound on the supplies , as shown in the corollary below . [ cordelta ] under the same setup as theorem [ thmdelta ] , we have @xmath238}s_i\big(\rho_i(t,\,x)\big)~\geq~\bar p^{\lfloor { t\over l/\lambda}\rfloor } \left\{\delta^{\mathcal{d}}~,~\bar p c^{min}\right\}\qquad\forall t\in[0,\,t]\ ] ] and @xmath238}\inf_{t\in[0,\,t]}s_i\big(\rho_i(t,\,x)\big)~\geq~\bar p^{\lfloor { t\lambda\over l}\rfloor}\min\left\{\delta^{\mathcal{d}}~,~\bar p c^{min}\right\}\ ] ] where the operator @xmath239 rounds its argument to the nearest integer from below . both inequalities are immediately consequences of . corollary [ cordelta ] shows that , for any network consisting of the merge and diverge junctions , the supply values are uniformly bounded from below at any point in the spatial - temporal domain . in particular , a jam density can never occur . moreover , such a network is free of complete gridlock . is formed somewhere in the network and the static queues do not dissipate in finite time . ]      in this section we establish some properties of the path disaggregation variable @xmath176 , which serve to validate the second hypothesis of theorem [ divwpthm ] . [ lemma : ben ] assume that there exists @xmath240 and @xmath241 so that the following hold :    1 .   for all links @xmath34 , the fundamental diagrams @xmath59 are uniformly linear near the zero density ; more precisely , @xmath242 is constant for @xmath243 $ ] for all @xmath141 . each @xmath59 has non - vanishing derivative , i.e. @xmath244 $ ] . the departure rates @xmath245 are uniformly bounded , have bounded total variation ( tv ) , and bounded away from zero when they are non - zero ; i.e. @xmath246 and @xmath247 $ ] for all @xmath3 and almost every @xmath4 $ ] .    then the path disaggregation variables @xmath248 are either zero , or uniformly bounded away from zero . moreover , they have bounded variation . the proof is moved to [ subsecapplemma : ben ] . notice that assumptions 1 and 2 of lemma [ lemma : ben ] are satisfied by the newell - daganzo ( triangular ) fundamental diagrams , where both the free - flow and congested branches of the fd are linear . moreover , given an arbitrary fundamental diagram , one can always make minimum modifications at @xmath26 and @xmath249 to comply with conditions 1 and 2 . assumption 3 of lemma [ lemma : ben ] is satisfied by any departure rate that is a result of a finite number of cars entering the network . and , again , any departure rate can be adjusted to satisfy this condition with minor modifications .      in this section we discuss the continuous dependence of the queues @xmath126 and the solutions @xmath250 and @xmath251 with respect to the path departure rates @xmath7 , @xmath3 . the analysis below relies on knowledge of the wave - front tracking algorithm and generalized tangent vector , for which an introduction is provided in [ secappemb ] . following @xcite , we introduce a generalized tangent vector @xmath252 for the triplet @xmath253 where @xmath254 is a scalar shift of the queue @xmath255 , i.e. the shifted queue is @xmath256 , while the tangent vectors of @xmath250 and @xmath251 are defined in the same way as in [ secgtv ] . the tangent vector norm of @xmath257 is simply its absolute value @xmath258 . [ lemmabenqueue ] assume that the departure rates @xmath259 are piecewise constant , and let @xmath260 be a tangent vector defined via shifting the jumps in @xmath261 . then the tangent vector @xmath252 is well defined , and its norm is equal to that of @xmath260 and bounded for all times .    the proof is postponed until [ subsecapplemmabenqueue ] .      at the end of this paper , we are able to prove the continuity result for the delay operator based on a series of preliminary results presented so far .    * ( continuity of the delay operator ) * consider a network consisting of merge and diverge junctions described in section [ secmergediverge ] , under the same assumptions stated in lemma [ lemma : ben ] , the path delay operator , as a result of the dynamic network loading model - , is continuous . we have shown that at each node ( origin , diverge node , or merge node ) , the solution depends continuously on the initial and boundary values . in addition , between any two distinct nodes , the propagation speeds of either @xmath177-waves or @xmath176-waves are uniformly bounded . thus such well - posedness continues to hold on the network level . consequently , the vehicle travel speed @xmath262 for any @xmath34 depends also continuously on the departure rates . we thus conclude that the path travel times depend continuously on the departure rates . the assumption that the network consists of only merge and diverge nodes is not restrictive since junctions with general topology can be decomposed into a set of elementary junctions of the merge and diverge type @xcite . in addition , junctions that are also origins / destinations can be treated in a similar way by introducing virtual links .    here , we would like to comment on the assumptions made in the proof of the continuity . in a recent paper @xcite , a counterexample of uniqueness and continuous dependence of solutions is provided under certain conditions . more precisely , the authors construct the counterexample by assuming that the path disaggregation variables @xmath251 have infinite total variation . this shows the necessity of the second assumption of theorem [ divwpthm ] . moreover , again in @xcite , the authors prove that for density oscillating near zero the solution may not be unique ( even in the case of bounded total variation ) , which shows the necessity of the third assumption in lemma [ lemma : ben ] . @xcite provides an example of discontinuous dependence of the path travel times on the path departure rates using the cell transmission model representation of a signal - controlled network . in particular , the author showed that when a queue generated by the red signal spills back into the upstream junction , the experienced path travel time jumps from one value to another . this , however , does not contradict our result presented here for the following reason : the jam density caused by the red signal in @xcite does not exist in our network , which has only merge and diverge junctions ( without any signal controls ) . indeed , as shown in corollary [ cordelta ] , the supply functions at any location in the network are uniformly bounded below by a positive constant , and thus the jam density never occurs in a finite time horizon . the reader is reminded of the example presented in section [ subsubsecexample ] , where the ill - posedness of the diverge model is caused precisely by the presence of a jam density . the counterexample from @xcite is constructed essentially in the same way as our example , by using signal controls that create the jam density . we offer some further insights here on the extension of the continuity result to second - order traffic flow models . to the best of our knowledge , second - order models on traffic networks are mainly based on the aw - rascle - zhang model @xcite and the phase - transition model @xcite . solutions of the aw - rascle - zhang system may present the vacuum state , which , in general , may prevent continuous dependence @xcite . on the other hand , the phase transition model on networks behaves in a way similar to the lwr scalar conservation law model . however , continuous dependence may be violated for density close to maximal ; see @xcite . therefore , extensions of the continuity result to second - order models appear possible only for the phase - transition type models with appropriate assumptions on the maximal density achievable on the network , but this is beyond the scope of this paper . this paper presents , for the first time , a rigorous continuity result for the path delay operator based on the lwr network model with spillback explicitly captured . this continuity result is crucial to many dynamic traffic assignment models in terms of solution existence and computation . similar continuity results have been established in a number of studies , all of which are concerned with non - physical queue models . as we show in section [ subsubsecexample ] , the well - posedness of a diverge model may not hold when spillback occurs . this observation , along with others made in previous studies @xcite , have been the major source of difficulty in showing continuity of the delay operator . in this paper , we bridge this gap through rigorous mathematical analysis involving the wave front tracking method and the generalized tangent vectors . in particular , by virtue of the finite propagation speeds of @xmath177-waves and @xmath176-waves , the continuity of the delay operator boils down to the well - posedness of nodal models , including models for the origins , diverge nodes , and merge nodes . minor assumptions are made on the fundamental diagram and the path departure rates in order to provide an upper bound on the total variations of the density @xmath177 and the path disaggregation variables @xmath176 , which subsequently leads to the desired well - posedness of the nodal models and eventually the continuity of the operator . a crucial step of the above process is to estimate and bound from below the minimum network supply , which is defined in terms of local vehicle densities . in fact , if the supply of some link tends to zero , the well - posedness of the diverge junction may fail as we demonstrate in section [ subsubsecexample ] . this has also been confirmed by an earlier study @xcite , where a wave of jam density is triggered by a signal red light and causes spillback at the upstream junction , leading to a jump in the path travel times . remarkably , in this paper we are able to show that ( 1 ) if the supply is bounded away from zero , then the diverge junction model is well posed ; and ( 2 ) the desired boundedness of the supply is a natural consequence of the dynamic network loading procedure that involves only the simple merge and diverge junction models . this is a highly non trivial result because it not only plays a role in the continuity proof , but also implies that gridlock can never occur in the network loading procedure in a finite time horizon . however , we note that in numerical computations gridlock may very well occur due to finite approximations and numerical errors , while our no - gridlock result is conceived in a continuous - time and analytical ( i.e. , non - numerical ) framework . it is true that , despite the correctness of our result regarding the lower bound on the supply , zero supply ( or gridlock ) can indeed occur in real - life traffic networks as a result of signal controls . in fact , this is how @xcite constructs the counter example of discontinuity . nevertheless , signal control is an undesirable feature of the dynamic network loading model for far more obvious reasons : the on - and - off signal control creates a lot of jump continuities in the travel time functions , making the existence of dynamic user equilibria ( due ) almost impossible . given that the main purpose of this paper is to address fundamental modeling problems pertinent to dues , it is quite reasonable for us to avoid junction models that explicitly involve the on - and - off signal controls . nevertheless , there exist a number of ways in which the on - and - off signal control can be approximated using continuum ( homogenization ) approaches . some examples are given in @xcite and @xcite . regarding the exponential decay of the minimum supply , we point out that , as long as the jam density ( gridlock ) does not occur , our result is still applicable . from a practical point of view most networks reach congestion during day time but are almost empty during night time . in other words , the asymptotic gridlock condition is never reached in reality . therefore , we think that our result is a reasonable approximation of reality for due problems . we would also like to comment on the continuity result for other types of junction models . prove that lipschitz continuous dependence on the initial conditions may fail in the presence of junctions with at least two incoming and two outgoing roads . notably , the counterexample that they use to disprove the lipschitz continuity is valid even when the turning ratio matrix @xmath103 is constant and not dependent on the path disaggregation variables @xmath251 . it is interesting to note that , in the same book , continuity results for general networks are provided , but only for riemann solvers at junctions that allow re - directing traveling entities  a feature not allowed in the dynamic network loading model but instead is intended for the modeling of data networks . the findings made in this paper has the following important impacts on dta modeling and computation . ( 1 ) the analytical framework proposed for proving or disproving the well - posedness of junction models and the continuity property can be applied to other junction types and riemann solvers , leading to a set of continuity / discontinuity results of the delay operators for a variety of networks . ( 2 ) the established continuity result not only guarantees the existence and computation of continuous - time dues based on the lwr model , but also sheds light on discrete - time due models and provides useful insights on the numerical computations of dues . in particular , the various findings made in this paper provide valuable guidance on numerical procedures for the dnl problem based on the cell transmission model @xcite , the link transmission model @xcite , or the kinematic wave model @xcite , such that the resulting delay operator is continuous on finite - dimensional spaces . as a result , the existence and computation of discrete - time dues will also benefit from this research . the wave - front tracking ( wft ) method was originally proposed by @xcite as an approximation scheme for the following initial value problem @xmath263 where the initial condition @xmath264 is assumed to have _ bounded variation _ ( bv ) @xcite . the wft approximates the initial condition @xmath264 using _ piecewise constant _ ( pwc ) functions , and approximates @xmath25 using _ piecewise affine _ ( pwa ) functions . it is an event - based algorithm that resolves a series of wave interactions , each expressed as a riemann problem ( rp ) . the wft method is primarily used for showing existence of weak solutions of conservation laws by successive refinement of the initial data and the fundamental diagram ; see @xcite and @xcite . @xcite extend the wft to treat the network case and show the existence of the weak solution on a network . we provide a brief description of this procedure below . fix a riemann solver ( rs ) for each road junction in the network . consider a family of piecewise constant approximations @xmath265 of the initial condition @xmath266 on each link and a family of piecewise affine approximation of the fundamental diagrams @xmath267 , where @xmath178 is a parameter such that @xmath268 and @xmath269 as @xmath214 . a wft approximate solution on the network is constructed as follows .    1 .   within each link , solve a riemann problem at each discontinuity of the pwc initial data . at each junction , solve a riemann problem with the given rs . 2 .   construct the solution by gluing together the solutions of individual rps up to the first time when two traveling waves interact , or when a wave interacts with a junction . 3 .   for each interaction , solve a new rp and prolong the solution up to the next time of any interaction . 4 .   repeat the processes 2 - 3 .    to show that the procedure described above indeed produces a well - defined approximate solution on the network , one needs to ensure that the following three quantities are bounded : ( 1 ) the total number of waves ; ( 2 ) the total number of interactions ( including wave - wave and wave - junction interactions ) ; and ( 3 ) the _ total variation _ ( tv ) of the piecewise constant solution at any point in time . these quantities are known to be bounded in the single conservation law case ; in fact , they all decrease in time @xcite . however , for the network case , one needs to proceed carefully in estimating these quantities as they may increase as a result of a wave interacting with a junction , which may produce new waves in all other links incident to the same junction . the reader is referred to @xcite for more elaborated discussion on these interactions . for a sequence of approximate wft solutions @xmath270 , @xmath192 , if one can show that the total variation of @xmath270 is uniformly bounded , then as @xmath214 a weak entropy solution on the network is obtained . the generalized tangent vector is a technique proposed by @xcite to show the well - posedness of conservation laws , and is used later by @xcite to show the well - posedness of junction models in connection with the lwr model . its mathematical contents are briefly recapped here .    given a piecewise constant function @xmath271\to \mathbb{r}$ ] , a _ tangent vector _ is defined in terms of the shifts of the discontinuities of @xmath272 . more precisely , let us indicate by @xmath273 the discontinuities of @xmath274 where @xmath275 , and by @xmath276 the values of @xmath274 on @xmath277 . a tangent vector of @xmath272 is a vector @xmath278 such that for each @xmath241 , one may define the corresponding perturbation of @xmath272 , denoted by @xmath279 and given by @xmath280 for @xmath281 , where we set @xmath282 . the norm of the tangent vector @xmath283 is defined as @xmath284 in other words , the norm of the tangent vector is the sum of each @xmath285 multiplied by the magnitude of the shifted jumps . the procedure of showing the well - posedness of a junction model using the tangent vectors is as follows . given piecewise constant initial / boundary conditions on each link incident to the junction , one considers their tangent vectors . by showing that the norms of their tangent vectors are uniformly bounded in time among all approximate wave - front tracking solutions , it is guaranteed that the @xmath286 distance of any two solutions is bounded , up to a multiplicative constant , by the @xmath286 distance of their respective initial / boundary conditions . more precisely , we have the following theorem    if the norm of a tangent vector at any time is bounded by the product of its initial norm and a positive constant , among all approximate wave - front tracking solutions , then the junction model is well posed and has a unique solution . see @xcite .    throughout this paper , we employ the notation @xmath287 to represent a wave interacting with the junction from road @xmath34 , where @xmath288 is the density value in front of the wave ( in the same direction as the traveling wave ) and @xmath250 is the density behind the wave . after the interaction , a new wave may be created on some road @xmath57 ( it is possible that @xmath289 ) , and we use @xmath290 and @xmath291 to denote the density at @xmath28 on road @xmath57 before and after the interaction , respectively . [ lemma : tangent - vectors - diverge ] consider the diverge model in section [ subsecdiverge ] . if a wave @xmath287 on @xmath34 interacts with @xmath28 then the shift @xmath292 produced on any @xmath57 , as a result of the shift @xmath293 on @xmath34 , satisfies @xmath294 where @xmath295 are the jumps in flow across these two waves .    to fix the ideas , we assume that @xmath34 is the incoming road @xmath146 ( the other cases can be treated in a similar way ) . let @xmath296 be the interacting wave , then it must be that @xmath297 .    if no wave is reflected on road @xmath146 after the interaction ( i.e. no new backward wave @xmath298 is created on @xmath146 as a result of the interaction ) , then we can conclude the lemma by following the same estimate as shown in @xcite . if , on the other hand , a wave @xmath299 is reflected then @xmath300 , @xmath296 is a rarefaction wave , and @xmath301 . in other words , part of the change in the flow caused by the interaction passes through the junction , and part of it is reflected back onto @xmath146 . we may then apply the same argument as in @xcite to the part that passes through the junction ( @xmath302 ) and conclude the lemma . if a @xmath177-wave from @xmath34 interacts with the junction , producing a @xmath177-wave on link @xmath57 , we call @xmath34 and @xmath57 the source and recipient of this interaction , respectively . such an event is denoted @xmath303 . we observe that @xmath304 is precisely the @xmath286-distance of two initial conditions on @xmath34 with and without the shift @xmath293 at the discontinuity @xmath287 . similarly , @xmath305 is the @xmath286-distance of the two solutions on @xmath57 as a result of having or not having the initial shift @xmath293 on @xmath34 , respectively . furthermore , if @xmath293 is the only shift in the initial condition on @xmath34 , then @xmath306 is the norm of the tangent vector for @xmath34 ( see definition ) before the interaction , and @xmath307 is the norm of the tangent vector for @xmath57 after the interaction . from , we have @xmath308 therefore , to bound the norm of the tangent vectors one has to check that the multiplication factors @xmath309 remain uniformly bounded , regardless of the number of interactions that may occur at this junction . let us consider the diverge junction , and assume that a wave interacts with @xmath28 from road @xmath34 and produces a wave on road @xmath57 , where @xmath310 . as in the proof of lemma [ lemma : tangent - vectors - diverge ] , we can restrict to the flow passing through the junction ( indeed the reflected part has the same flow variation and smaller shift since the reflected wave is a slow big shock ) . according to , @xmath311 and @xmath312 . consequently , we have the following matrix of multiplication factors @xmath313 : @xmath314 we denote this matrix by @xmath315 . according to @xcite , in order to estimate the tangent vector norm , it suffices to keep tract of just one single shift and show the corresponding tangent vector norm is bounded regardless of the number of wave interactions . to this end , we consider the only meaningful sequence of wave interaction , which is of the form @xmath303 , @xmath316 ,  . we observe that , if @xmath153 and @xmath154 are nonzero constants , then @xmath317 for any @xmath318 . this means that no matter how many interactions occur , the multiplication factor is always an element of the matrix @xmath319 and is uniformly bounded . thus the diverge model with fixed turning ratios is well posed . however , in the dnl model where @xmath153 and @xmath154 are time - varying , this is no longer true as we have shown in the counterexample in section [ subsubsecexample ] . the well - posedness requires some additional conditions to hold , and the corresponding proof needs more elaborated arguments that take into account the @xmath176-waves . these will be done in theorem [ divwpthm ] . this proof is completed in several steps . + * [ step 1 ] . * we invoke the wave - front tracking framework and the generalized tangent vector to show the desired continuous dependence . as discussed in [ secgtv ] , it boils down to showing that the increase in the tangent vector norm , as a result of arbitrary number of wave interactions ( including both @xmath177-wave and @xmath176-wave ) , remains bounded . + * [ step 2 ] . * first of all , the end of [ secgtv ] shows that , for constant values of @xmath176 ( that is , with fixed vehicle turning ratios ) , the tangent vector norm is uniformly bounded regardless of the interactions between the @xmath177-waves and the junction . next , one needs to consider the case where @xmath176 changes from one value to another , i.e. when a @xmath176-wave interacts with the junction . in general , we consider two consecutive interaction times of the @xmath176-waves : @xmath320 and @xmath321 , and let @xmath322 be the multiplication factor for the tangent vector norm of @xmath177 relative to the times @xmath320 and @xmath321 . more precisely , we have @xmath323 , where @xmath324 is the tangent vector norm of @xmath177 at time @xmath10 . clearly , @xmath322 can only take value in the following matrix where @xmath325 and @xmath326 are given by the constant @xmath176 value during @xmath327 ; see . @xmath328,\qquad k=0,\,1,\,2,\,\ldots\ ] ]    similar arguments can be carried out to all other intervals @xmath329 $ ] , @xmath330 $ ] and so forth . in the rest of the proof , our goal is to estimate the multiplication factor for the tangent norm of @xmath177 under the wft framework , assuming arbitrary interaction patterns of the @xmath177-waves and @xmath176-waves . + * [ step 3 ] . * we define @xmath331 where @xmath332 is stated in the hypothesis of this theorem and @xmath333 is the flow capacity of link @xmath146 . without loss of generality , we let @xmath334 . throughout * step 3 * , we assume that @xmath335 and @xmath336 ( the other cases will be treated in * step 4 * ) .    as demonstrated at the end of [ secgtv ] , in order to estimate the increase in the tangent vector norms it suffices to consider the following type of sequence in which @xmath177-waves interacts with the junction : @xmath303 then @xmath316 ; in other words , the recipient of the previous @xmath177-wave interaction is the source of the next @xmath177-wave interaction . for each @xmath337 , we let @xmath338 be the recipient of the last interacting @xmath177-wave during @xmath327 . this gives rise to the sequence @xmath339 ; see figure [ figikfig ] for an illustration . s represent the times when @xmath176-waves interact with the junction , changing the turning ratios @xmath153 and @xmath154 . the crosses represent the times at which @xmath177-waves interact with the junction . the circles indicate the last @xmath177-wave interaction within the interval during which @xmath176 is constant . @xmath340 s denote the recipients of the last interacting @xmath177-waves . ] we make the following crucial observation . consider any three elements in the sequence of the form @xmath341 , @xmath342 and @xmath343 where @xmath344 . by definition , the product of the multiplication factors within @xmath345 is @xmath346    where we use the superscripts to indicate dependence on a specific time interval . the significance of is that the multiplication factor can be decomposed into a term @xmath347 with a very specific structure ( to be further elaborated below ) , and a term that would have been the multiplication factor as if the middle link @xmath348 were removed . clearly , the argument above applies equally to the case where @xmath349 , and we use @xmath350 to represent the term factored out if the middle link @xmath349 is removed . by repeating this procedure , one may eliminate all links @xmath147 and @xmath148 from the sequence @xmath351 except when they are the first or possibly the last in this sequence . as a result , the multiplicative terms @xmath352 and @xmath353 are factored out . therefore , the entire multiplication factor for the tangent norm is bounded from above by @xmath354 where @xmath355 and @xmath356 are subsequences of @xmath357 such that @xmath358 and @xmath359 . the first two terms in are results of eliminating @xmath147 and @xmath148 from the sequence @xmath351 ; the third term takes care of the first element and possibly the last element in the sequence , and is derived from the observation that @xmath360 for all @xmath344 and @xmath361 . it remains to show that the first and second terms of are bounded , and hence the entire is bounded . we write , without referring explicitly to the multiplication indices , that @xmath362 this leads to the following . @xmath363}{1\over |\alpha| } \cdot \sum_k \big|\alpha_{1,2}^{k+1}-\alpha_{1,2}^{k+2}\big| \right ) } \cdot \exp{\left ( \sup_{\alpha\in[\gamma,\,1-\gamma ] } { 1\over |1-\alpha| } \cdot \sum_k \big|\alpha_{1,2}^{k+2}-\alpha_{1,2}^{k+1}\big|   \right ) } \\ ~\leq~ & \exp{\left({1\over \gamma } \cdot tv(\mu ) \right)}\cdot \exp{\left ( { 1\over \gamma}\cdot tv(\mu )   \right ) } ~<~\infty\end{aligned}\ ] ] where @xmath364 is the bounded total variation of the path disaggregation variable @xmath176 . + * [ step 4 ] . * in this part of the proof we deal with the situation where @xmath365 ; the other case where @xmath366 is entirely similar . we begin with the following observation . @xmath367    since @xmath368 . in other words , the minimum is never attained at @xmath369 , and any @xmath177-wave interacting from @xmath147 will not generate new waves on @xmath146 or @xmath148 since @xmath370 does not change before and after the interaction . so the only interactions that may change the tangent vector norm are @xmath371 and @xmath372 , and the corresponding multiplication factors are respectively @xmath373 and @xmath374 . consider any sequence of interaction times @xmath375 of the @xmath176-waves . since the interactions @xmath376 or @xmath377 do not generate any increase in the tangent norm , we only need to consider the following sequence of interactions : @xmath378 . the resulting multiplication factor is bounded by constant of the form @xmath379 where the second multiplicative term deals with the first and the last element in the sequence since @xmath380 for all @xmath106 . we may then proceed in the same way as * step 3 * and conclude that this multiplication factor is bounded , provided that @xmath381 .      since @xmath382 is defined in terms of the supplies , it suffices for us to focus only on those densities such that @xmath383 for @xmath141 , @xmath76\times[a_i,\,b_i]$ ] . it is also useful to keep in mind that densities beyond the critical density always propagate backwards in space . the proof is divided into several steps . + * [ step 1 ] . * @xmath384 . we have @xmath385 . since the network is initially empty , all the supplies @xmath386 , @xmath387 $ ] , are maximal and equal to the respective flow capacities at @xmath388 . afterwards , a higher - than - critical density or a lower - than - maximum supply can only emerge from the downstream end of a link and propagate backwards along this link . moreover , these backward waves can never reach the upstream end of the link within @xmath389 since @xmath390 is the minimum link traversal time for backward waves .              * * case ( 1 ) . * we use the same notation shown in figure [ figtwojunc ] . according to the reason provided above , @xmath391 , @xmath392 for @xmath385 . let the time of the wave interaction be @xmath393 . recall from that @xmath394 if the minimum is attained at @xmath395 , then the entrance of @xmath146 will remain in the uncongested phase , i.e. @xmath396 after the interaction . hence , no lower - than - maximum supply is generated . on the other hand , if say @xmath397 is the smallest among the three , then @xmath398 in summary , the supply values corresponding to the backward waves generated at @xmath146 , if any , are uniformly above @xmath399 . * * case ( 2 ) . * we turn to the merge junction shown in figure [ figtwojunc ] , and note @xmath400 for @xmath385 . as usual , we let @xmath393 be the time of interaction . there are two further cases for the merge junction model as shown in figure [ figmerge ] . we first consider the situation illustrated in figure [ figmerge](a ) . clearly , we have that @xmath401 that is , the supply values of the backward waves generated at the downstream ends of @xmath159 and @xmath160 are uniformly above @xmath402 . + for the situation depicted in figure [ figmerge](b ) , we first note that the coordinates of @xmath403 is @xmath404 , and the coordinates of the solution @xmath405 either satisfy @xmath406 as shown exactly in figure [ figmerge](b ) , or @xmath407 taking as an example ( the case can be treated similarly ) , we have @xmath408 and no increase in density is produced at the downstream end of link @xmath159 since its exit flow is equal to the demand . + to sum up , the supply values corresponding to the backward waves generated at @xmath159 or @xmath160 , if any , are uniformly above @xmath402 . * * case ( 3 ) . * for each destination @xmath223 , let @xmath225 be its supply and denote by @xmath409 the virtual link connected to @xmath224 . if @xmath410 then no backward waves can be generated on this link and the supply value on the link is always equal to @xmath411 . if @xmath412 , then only one higher - than - critical density can exist on this virtual link , that is , @xmath177 such that @xmath413 and @xmath414 . clearly , its supply value is equal to @xmath225 . + to sum up , the supply value corresponding to the backward waves generated at any virtual link , if any , is bounded below by @xmath415 .    finally , we notice that all the backward waves exhaustively described above originate from the downstream end of a link , and they can not reach the upstream end of the same link within period @xmath389 . thus , these backward waves can not bring further reduction in the supply values by means of interacting with the junctions . we conclude that during period @xmath416 , the minimum supply in the network , @xmath417 , is bounded below ; that is , @xmath418    * [ step 2 ] . * we move on to @xmath337 . in addition to case ( 1)-(3 ) , which do not bring any supply values below @xmath419 , two more cases may arise in which higher - than - critical densities may be generated as a result of backward waves interacting with junctions :          * * case ( 4 ) . * without loss of generality , we assume that the backward wave that interacts with the diverge junction is coming from @xmath147 , and has the density value @xmath420 $ ] . let @xmath393 be the time of the interaction . in view of , if the minimum is attained at @xmath395 , then the interaction does not bring any increase in density at the downstream end of @xmath146 , hence no decrease in the supply there . + if the minimum is attained at @xmath421 , we deduce in a similar way as that @xmath422 the last inequality is due to the fact that a backward wave such as @xmath423 must be created at the downstream end of @xmath147 at a time earlier than @xmath424 , thus its supply value @xmath425 must be bounded below by @xmath426 . if the minimum is attained at @xmath427 , we have @xmath428 the last inequality is because any lower - than - maximum supply on @xmath148 must be created in the previous time interval . + to sum up , the supply values corresponding to the backward waves generated at @xmath146 , if any , are bounded below by @xmath426 . * * case ( 5 ) . * for the merge junction , we begin with the case illustrated in figure [ figmerge](a ) . assuming the backward wave that interacts with the merge junction from @xmath161 has the density value @xmath429 $ ] . similar to - , we have @xmath430 where the last inequalities are due to the same reason provided in * case ( 4)*. the case illustrated in figure [ figmerge](b ) can be treated in the same way as - . + to sum up , the supply values corresponding to the backward waves generated at @xmath159 or @xmath160 , if any , must be bounded below by @xmath431 . so far , we have shown that for @xmath432 where @xmath337 , the presence of higher - than - critical densities , as exhaustively illustrated through * case ( 1)- case ( 5 ) * , brings supply values throughout the entire network that are bounded below by @xmath433        the proof is divided into a few steps . + * [ step 1 ] . * notice that assumption 3 implies that the departure rates @xmath259 are non - zero on a finite set of intervals . indeed , let @xmath30 be the number of intervals where @xmath261 does not vanish then @xmath441 , which implies that @xmath30 is finite . + * [ step 2 ] . * using the assumption that each fundamental diagram @xmath59 has nonvanishing derivative , we have , at each origin , that the bounded total variation of the density must imply bounded total variation of the flow , and vise versa . therefore , the assumption on the bounded variation of @xmath261 implies that the density and path disaggregation variable are of bounded variation on the virtual link incident to that origin . + * [ step 3 ] . * assumption 1 guarantees that , on each link , all waves of the type @xmath442 or @xmath443 are contact discontinuities for @xmath177 sufficiently small and , in particular , travel with a constant speed . moreover , all @xmath176-waves travel with the same speed for low densities . therefore , taking into account * step 1 * , whenever @xmath176 is non - zero , it is uniformly bounded away from zero on the entire network . in particular there exists @xmath444 so that @xmath445 $ ] for every @xmath446 and @xmath14 . therefore , at diverging junctions , the coefficients @xmath447 and @xmath373 satisfy the same properties . + * [ step 4 ] . * let us now turn to the total variation of the path disaggregation variables . we know , from * step 2 * , that @xmath176 has bounded variation on virtual links incident to the origins . we also know from * step 3 * that @xmath176 is bounded away from zero whenever it is non - zero ; and the same holds for the turning ratios at diverge junctions . consider a @xmath176-wave @xmath448 , then its variation @xmath449 can change only upon interaction with diverge junctions . more precisely , denote by @xmath450 and @xmath451 the wave respectively before and after the interaction , we have @xmath452 where @xmath453 or @xmath454 and is bounded away from zero . since the @xmath176-waves travel only forward on the links with uniformly bounded speed , we must have that the interactions with diverge junctions can occur only finite number of times . thus we conclude that @xmath176 has bounded total variation . let @xmath455 be the shift of the @xmath96-th jump of @xmath261 , which occurs at time @xmath456 , then the expression of @xmath255 is possibly affected on the interval @xmath457 $ ] ( assuming @xmath458 ) . more precisely , if @xmath459 then no wave is generated on the virtual link while a shift in the queue is generated with @xmath460 , where @xmath461 is the jump in @xmath261 occurring at time @xmath456 . on the other hand , if @xmath462 then a wave @xmath463 is produced at time @xmath456 on the virtual link with shift @xmath464 where @xmath465 is the speed of the wave @xmath463 . we can then compute : @xmath466 moreover , @xmath467 , thus the norm of the tangent vector generated is the same as before .    to prove that the norm of the tangent vector @xmath252 is bounded for all times , let us first consider a backward - propagating wave @xmath463 interacting with the queue @xmath468 , and with a shift @xmath469 . then we can write : @xmath470 where @xmath471 and @xmath472 are the time - derivatives of @xmath255 before and after the interaction , respectively . therefore , denoting @xmath465 the speed of the wave @xmath463 , we get : @xmath473 thus the norm of the tangent vector is bounded . aboudolas , k. , papageorgiou , m. , komsmatopoulous , e. , 2009 . store - and - forward based methods for the signal control problem in large - scale congested urban road networks . transportation research part c , 17 ( 2 ) , 163 - 174 . han , k. , friesz , t.l . , szeto , w.y . , liu , h. , 2015a . elastic demand dynamic network user equilibrium : formulation , existence and computation . transportation research part b , doi : 10.1016/j.trb.2015.07.008 . han , k. , friesz , t.l . , yao , t. , 2013a . a partial differential equation formulation of vickrey s bottleneck model , part i : methodology and theoretical analysis . transportation research part b 49 , 55 - 74 . han , k. , piccoli , b. , szeto , w.y . continuous - time link - based kinematic wave model : formulation , solution existence , and well - posedness . transportmetrica b : transport dynamics , doi : 10.1080/21680566.2015.1064793 . yperman , i. , logghe , s. , immers , l. , 2005 . the link transmission model : an efficient implementation of the kinematic wave theory in traffic networks , advanced or and ai methods in transportation , proc . 10th ewgt meeting and 16th mini - euro conference , poznan , poland , 122 - 127 , publishing house of poznan university of technology .
this paper establishes the continuity of the path delay operators for _ dynamic network loading _ ( dnl ) problems based on the lighthill - whitham - richards model , which explicitly capture vehicle spillback . 
 the dnl describes and predicts the spatial - temporal evolution of traffic flow and congestion on a network that is consistent with established route and departure time choices of travelers . 
 the lwr - based dnl model is first formulated as a system of _ partial differential algebraic equations _ ( pdaes ) . 
 we then investigate the continuous dependence of merge and diverge junction models with respect to their initial / boundary conditions , which leads to the continuity of the path delay operator through the _ wave - front tracking _ methodology and the _ generalized tangent vector _ technique . as part of our analysis leading up to the main continuity result 
 , we also provide an estimation of the minimum network supply without resort to any numerical computation . 
 in particular , it is shown that gridlock can never occur in a finite time horizon in the dnl model . 
 ( 1,0)469    path delay operator , continuity , dynamic network loading , lwr model , spillback , gridlock