Continuity of the path delay operator for dynamic network loading with spillback

# Continuity of the path delay operator for dynamic network loading with spillback

Ke Han Benedetto Piccoli Terry L. Friesz Department of Civil and Environmental Engineering, Imperial College London, United Kingdom. Department of Mathematical Sciences and CCIB, Rutgers University - Camden, USA Department of Industrial and Manufacturing Engineering, Pennsylvania State University, USA.
###### Abstract

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.

###### keywords:
journal: Transportation Research Part B

Han, K., Piccoli, B., Friesz, T.L., 2015. Continuity of the path delay operator for dynamic network loading with spillback. Transportation Research Part B, DOI: 10.1016/j.trb.2015.09.009.

## 1 Introduction

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 Peeta and Ziliaskopoulos (2001) 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 (Friesz et al., 1993). 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 Han et al. (2013c); Smith and Wisten (1995); Wie et al. (2002) and Zhu and Marcotte (2000). 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 (Friesz et al., 2013), the route-swapping algorithm (Huang and Lam, 2002), the descent method (Han and Lo, 2003), the projection method (Han and Lo, 2002; Ukkusuri et al., 2012), and the proximal point method (Han et al., 2015a)

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. Zhu and Marcotte (2000) use the link delay model (Friesz et al., 1993) 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 (Han et al., 2012). In Bressan and Han (2013), continuity of the delay operator is shown for networks whose dynamics are described by the LWR-Lax model (Bressan and Han, 2011; Friesz et al., 2013), 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. Han et al. (2013c) consider Vickrey’s point queue model (Vickrey, 1969) 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, Szeto (2003) 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 111Well-posedness of a model refers to the property that the behavior of the solution hardly changes when there is a slight change in the initial/boundary conditions.: a merge and a diverge model, both originally presented by Daganzo (1995). The underpinning analytical framework employs the wave-front tracking methodology (Dafermos, 1972; Holden and Risebro, 2002) and the technique of generalized tangent vectors (Bressan, 1993; Bressan et al., 2000). 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 Garavello and Piccoli (2006) 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 Lebacque and Khoshyaran (1999). 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 4.2.1. This has also been confirmed by the earlier study of Szeto (2003), 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 5.4, 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 (Daganzo, 1995). 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 2 recaps some essential knowledge and notions regarding the LWR network model and the DNL procedure. Section 3 articulates the mathematical contents of the DNL model by formulating it as a PDAE system. Section 4 introduces the merge and diverge models and establishes their well-posedness. Section 5 provides a final proof of continuity and some discussions. Finally, we offer some concluding remarks in Section 6.

Throughout this paper, the time interval of analysis is a single commuting period expressed as for some . We let be the set of all paths employed by travelers. For each path we define the path departure rate which is a function of departure time :

 hp(⋅): [0,T] → R+

where denotes the set of non-negative real numbers. Each path departure rate is interpreted as a time-varying path flow measured at the entrance of the first arc of the relevant path, and the unit for is vehicles per unit time. We next define to be a vector of departure rates. Therefore, can be viewed as a vector-valued function of , the departure time 222For notation convenience and without causing any confusion, we will sometimes use instead of to denote path flow vectors..

The single most crucial ingredient t is the path delay operator, which maps a given vector of departure rates to a vector of path travel times. More specifically, we let

 Dp(t,h)∀t∈[t0,tf],∀p∈P

be the path travel time of a driver departing at time and following path , given the departure rates associated with all the paths in the network. We then define the path delay operator by letting , which is a vector consisting of time-dependent path travel times .

### 2.2 The Lighthill-Whitham-Richards model on networks

We recap the network extension of the LWR model (Lighthill and Whitham, 1955; Richards, 1956), 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 . The partial differential equation (PDE) representation of the LWR model is the following scalar conservation law

 ∂tρ(t,x)+∂xf(ρ(t,x)) = 0(t,x)∈[0,T]×[a,b] (2.1)

with appropriate initial and boundary conditions, which will be discussed in detail later. Here, denotes vehicle density at a given point in the space-time domain. The fundamental diagram expresses vehicle flow at as a function of , where denotes the jam density, and denotes the flow capacity. Throughout this paper, we impose the following mild assumption on :

(F). The fundamental diagram is continuous, concave and vanishes at and .

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 (Garavello and Piccoli, 2006; Holden and Risebro, 1995), but also reflect certain behavioral and operational considerations, such as vehicle turning preferences (Daganzo, 1995), driving priorities (Coclite et al., 2005), and signal controls (Han et al., 2014). 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 (Holden and Risebro, 1995). In another line of research, the boundary conditions are defined using link demand and supply (Lebacque and Khoshyaran, 1999), which represent the link’s sending and receiving capacities. Models following this approach include Daganzo (1995); Jin and Zhang (2003) and Jin (2010). The solution of a Riemann Problem is given by the Riemann Solver (RS), to be defined below.

#### 2.2.1 The Riemann Solver

We consider a general road junction with incoming roads and outgoing roads, as shown in Figure 1.

We denote by the incoming links and by the outgoing links. In addition, for every , the dynamic on is governed by the LWR model

 ∂tρi(t,x)+∂xfi(ρi(t,x)) = 0(t,x)∈[0,T]×[ai,bi] (2.2)

where link is expressed as the spatial interval ], and we always use the subscript ‘’ to indicate dependence on the link . The initial condition for this conservation law is

 ρi(0,x) = ^ρi(x)x∈[ai,bi] (2.3)

Notice that the above 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.

###### Definition 2.1.

(Riemann Problem) The Riemann Problem at the junction is defined to be an initial value problem on a network consisting of the single junction with incoming links and outgoing links, all extending to infinity, such that the initial densities are constants on each link:

 {ρi(0,x) ≡ ^ρix∈(−∞,bi],i∈{1,…,m}ρj(0,x) ≡ ^ρjx∈[aj,+∞),j∈{m+1,…,m+n}

where are constants, .

A Riemann Solver for the junction is a mapping that, given any -tuple of Riemann initial conditions , provides a unique -tuple of boundary conditions 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 333We refer the reader to Holden and Risebro (1995) for a definition of weak entropy solution at a junction. A precise definition of the Riemann Solver is given as follows.

###### Definition 2.2.

(Riemann Solver) A Riemann Solver for the junction with incoming links and outgoing links is a mapping

 RS:  m+n∏k=1[0,ρjamk] ⟶ m+n∏k=1[0,ρjamk] (^ρ1,…,^ρm+n) ↦ (¯¯¯ρ1,…,¯¯¯ρm+n)

which relates Riemann initial data to boundary conditions , such that the following hold.

1. The solution of the Riemann Problem restricted on each link is given by the solution of the initial-boundary value problem with initial condition and boundary condition , .

2. The Rankine-Hugoniot condition (flow conservation) holds:

 m∑i=1fi(¯¯¯ρi) = m+n∑j=m+1fj(¯¯¯ρj) (2.4)
3. The consistency condition holds:

 RS[RS[^ρ]] = RS[^ρ] (2.5)

Three conditions must be satisfied by the Riemann Solver (RS). Item 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. (2.4) simply stipulates the conservation of flow across the junction. (2.5) is a desirable property and is sometimes referred to as the invariance property (Jin, 2010).

###### Remark 2.3.

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 (Coclite et al., 2005; Han et al., 2014; Holden and Risebro, 1995; Jin and Zhang, 2003) and Jin (2010).

#### 2.2.2 The link demand and supply

For each link , we let be the critical density at which the flow is maximized. The demand for incoming links and the supply for outgoing links are defined in terms of the density near the exit and entrance of the link, respectively (Lebacque and Khoshyaran, 1999):

 Di(t) = Di(ρi(t,bi−)) = {Ciif  ρi(t,bi−) ≥ ρcifi(ρi(t,bi−))if  ρi(t,bi−) < ρci (2.6) Sj(t) = Sj(ρj(t,aj+)) = {Cjif  ρj(t,aj+) ≤ ρcjfj(ρj(t,aj+))if  ρj(t,aj+) > ρcj (2.7)

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 or ; see Figure 2 for an illustration. In our subsequent presentation, without causing confusion we will use notations and 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 expressed as a directed graph with being the set of links and being the set of nodes. Let be the set of paths employed by travelers, and be the set of origin-destination pairs. Each path is expressed as an ordered set of links it traverses:

 p = {I1,I2,…,Im(p)}

where 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 , we denote by the set of incoming links and the set of outgoing links.

For each , the density dynamic is governed by the scalar conservation law

 (3.8)

subject to initial condition and boundary conditions to be determined in Section 3.2. The fundamental diagram satisfies condition (F) stated at the beginning of Section 2.2. In order to explicitly incorporate drivers’ route choices, for every such that we introduce the function , , which represents, in every unit of flow , the fraction associated with path . We call these variables path disaggregation variables (PDV). For each car moving along the link , its surrounding traffic can be distinguished by path (e.g. 20% following path , 30% following path , 50% following path ). 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, , are constants along the trajectories of cars in the space-time diagram, where is the trajectory of a moving car on link . That is,

 ddtμpi(t,x(t)) = 0∀p  %suchthat  Ii∈p,

which, according to the chain rule, becomes

 ∂tμpi(t,x(t))+∂xμpi(t,x)⋅ddtx(t) = 0,

which further leads to another set of partial differential equations on link :

 ∂tμpi(t,x)+vi(ρi(t,x))⋅∂xμpi(t,x) = 0∀p  such that  Ii∈p (3.9)

Here, is the solution of (3.8). The following obvious identity holds

 ∑p∋Iiμpi(t,x) = 1whenever   ρi(t,x) > 0 (3.10)

where means “path contains (or traverses) link ”, and the summation appearing in (3.10) is with respect to all such . By convention, if , then for all .

### 3.2 Boundary conditions at an ordinary node

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 to represent the set of ordinary nodes in the network.

As mentioned earlier, the partial differential equations on links incident to 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 (Coclite et al., 2005), which relies on knowledge of the PDVs for all . We define the time-dependent flow distribution matrix associated with :

 AJ(t) = {αJij(t)}∈[0,1]|IJ|+|OJ| (3.11)

where by convention, we use subscript to indicate incoming links, and to indicate outgoing links. Each element represents the turning ratios of cars discharged from that enter downstream link . Then, for all that traverses , the following holds.

 αJij(t)=∑p∋Ii,Ijμpi(t,bi)∀ Ii∈IJ, Ij∈OJ (3.12)

It can be easily verified that and according to (3.10).

We are now ready to express the boundary conditions for the ordinary junction . Let

 RSAJ:|IJ|+|OJ|∏k=1[0,ρjamk]→|IJ|+|OJ|∏k=1[0,ρjamk]

be a given Riemann Solver. Notice that the dependence of the Riemann Solver on has been indicated with a superscript. The boundary conditions for PDEs (3.8) read

 ρk(t,bk) = RSAJk[(ρi(t,bi−))Ii∈IJ , (ρj(t,aj+))Ij∈OJ]∀Ik∈IJ (3.13) ρl(t,al) = RSAJl[(ρi(t,bi−))Ii∈IJ , (ρj(t,aj+))Ij∈OJ]∀Il∈OJ (3.14)

where denotes the -th component of the mapping, .

###### Remark 3.1.

Intuitively, (3.13)-(3.14) mean that, given the current traffic states and adjacent to the junction , the Riemann Solver specifies, for each incident link or , the corresponding boundary conditions or . 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 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 (3.9) are:

 μpj(t,aj) = fi(ρi(t,bi))⋅μpi(t,bi)fj(ρj(t,aj))∀p  such that  {Ii,Ij}⊂p,∀Ij∈OJ (3.15)

where the numerator expresses the exit flow on link associated with path , which, by flow conservation, is equal to the entering flow of link associated with the same path ; the denominator represents the total entering flow of link .

###### Remark 3.2.

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.

### 3.3 Flow distribution at origin or destination nodes

We consider a node that is either the origin or the destination of some path . One immediate observation is that the flow conservation constraint (2.4) no longer holds at such a node since vehicles either are ‘generated’ (if is an origin) or ‘vanish’ (if 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 3 for an illustration. By introducing virtual links to the original network, we obtain an augmented network in which all road junctions are ordinary, and hence fall within the scope of the previous section. Figure 3: Illustration of the virtual links. Left: a virtual link connecting an origin (s) to an ordinary junction J. Right: a virtual link connecting a destination (t) to an ordinary junction J.

Let us denote by the set of origins in the augmented network . For any , we denote by the set of paths that originate from , and by the virtual link incident to this origin. For each we denote by the departure rate (path flow) along . It is expected that a buffer (point) queue may form at in case the receiving capacity of the downstream is insufficient to accommodate all the departure rates . For this buffer queue, denoted , we employ a Vickrey-type dynamic (Vickrey, 1969); that is,

 (3.16)

where denotes the supply of the virtual link . The only difference between (3.16) and Vickrey’s model is the time-varying downstream receiving capacity provided by the virtual link. 444The right hand side of the ordinary differential equation (3.16) is discontinuous. An analytical treatment of this irregular equation is provided by Han et al. (2013a, b) using the variational formulation.

It remains to determine the dynamics for the path disaggregation variables (PDV). More precisely, we need to determine for the virtual link where , and is the upstream boundary of . This will be achieved using the Vickrey-type dynamic (3.16) and the FIFO principle. Specifically, we define the queue exit time function where denotes the time at which drivers depart and join the point queue, if any; expresses the time at which the same group of drivers exit the point queue. Clearly, FIFO dictates that

 ∫t0∑p∈Pshp(τ)dτ = ∫λs(t)0fs(ρs(τ,as))dτ (3.17)

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:

 μps(λs(t),as) = hp(t)∑q∈Pshq(t)∀p∈Ps (3.18)

Notice that, if , then the flow leaving the point queue at time is also zero; thus there is no need to determine the path disaggregation variables. Therefore, the identity (3.18) is well defined and meaningful.

### 3.4 Calculation of path travel times

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 for any is defined, in a way similar to (3.17), as

 ∫t0fi(ρi(τ,ai))dτ = ∫λi(t)0fi(ρi(τ,bi))dτ (3.19)

For a path expressed as , the time to traverse it is calculated as

 λs∘λ1∘λ2…∘λm(p)(t) (3.20)

where 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.

### 3.5 The PDAE system

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 and node set ; the set of virtual links; the augmented network including virtual links; the set of origins in ; the set of paths originating from ; the set of ordinary junctions in ; the set of incoming links of a junction ; the set of outgoing links of a junction ; the flow distribution matrix associated with junction ; the Riemann Solver for junction , which depends on .

We also list some key variables of the PDAE system below.

• the path departure rate along ;

• the vehicle density on link ;

• the proportion of flow on link associated with path (path disaggregation variable);

• the point queue at the origin ;

• the point queue exit time function at origin .

Given any vector of path departure rates , the proposed PDAE system for calculating path travel times is summarized as follows.

 ∀s∈S (3.21) ∫t0∑p∈Pshp(τ)dτ = ∫λs(t)0fs(ρs(τ,as))dτ ∀s∈S (3.22) ∫t0fi(ρi(τ,ai))dτ = ∫λi(t)0fi(ρi(τ,bi))dτ ∀Ii∈~A (3.23) ∂tρi(t,x)+∂x[ρi(t,x)⋅vi(ρi(t,x))]=0 (t,x)∈[0,T]×[ai,bi] (3.24) ∂tμpi(t,x)+vi(ρi(t,x))⋅∂xμpi(t,x)=0 (t,x)∈[0,T]×[ai,bi] (3.25) μps(λs(t),as) = hp(t)∑q∈Pshq(t) ∀s∈S,p∈Ps (3.26) ∀p⊃{Ii,Ij} (3.27) AJ(t)={αJij(t)},αJij(t)=∑p∋Ii,Ijμpi(t,bi) ∀Ii∈IJ,Ij∈OJ (3.28) ρk(t,bk) = RSAJk[(ρi(t,bi−))Ii∈IJ , (ρj(t,aj+))Ij∈OJ] ∀Ik∈IJ (3.29) ρl(t,al) = RSAJl[(ρi(t,bi−))Ii∈IJ , (ρj(t,aj+))Ij∈OJ] ∀Il∈OJ (3.30) Dp(t,h) = λs∘λ1∘λ2…∘λm(p)(t) ∀p∈P,∀t∈[0,T] (3.31)

Eqn (3.21) describes the (potential) queuing process at each origin. Eqns (3.22) and (3.23) express the queue exit time function for a point queue, and the link exit time function for a link, respectively. Eqns (3.24)-(3.25) express the link dynamics in terms of car density and PDV; Eqns (3.26)-(3.27) specifies the upstream boundary conditions for the PDV as these variables can only propagate forward in space. Eqns (3.28)-(3.30) determine the boundary conditions at junctions. Finally, Eqn (3.31) determines the path travel times.

The above PDAE system involves partial differential operators and . Solving such a system requires solution techniques from the theory of numerical partial differential equations (PDE) such as finite difference methods (Godunov, 1959; LeVeque, 1992) and finite element methods (Larsson and Thomée, 2005).

## 4 Well-posedness of two junction models

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 (Bressan, 2000), and the initial value problem for the Hamilton-Jacobi equations (Daganzo, 2006). 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 4, and the corresponding merge and diverge rules are proposed initially by Daganzo (1995) in a discrete-time setting with fixed vehicle turning ratios and driving priority parameters.

### 4.1 The two junction models

#### 4.1.1 The diverge model

We first consider the diverge junction shown on the left part of Figure 4, with one incoming link and two outgoing links and . The demand and the supplies, and , are defined by (2.6) and (2.7) respectively. The Riemann Solver for this junction relies on the following two conditions.

• Cars leaving advance to and according to some turning ratio which is determined by the PDV in the DNL model.

• Subject to (A1), the flow through the junction is maximized.

In the original diverge model (Daganzo, 1995), the vehicle turning ratios, denoted and 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 (3.28). The diverge junction model, described by (A1) and (A2), can be more explicitly written as:

 fout1(t) = min{D1(t), S2(t)α1,2(t), S3(t)α1,3(t)}fin2(t) = α1,2(t)⋅fout1(t)fin3(t) = α1,3(t)⋅fout1(t) (4.32)

where denotes the exit flow of link , and denotes the entering flow on link , .

#### 4.1.2 The merge model

We now turn to the merge junction in Figure 4, with two incoming links and and one outgoing link . In view of this merge junction, assumption (A1) becomes irrelevant as there is only one outgoing link; and assumption (A2) cannot determine a unique solution 555More generally, as pointed out by Coclite et al. (2005), when the number of incoming links exceeds the number of outgoing links, (A1) and (A2) combined are not sufficient to ensure a unique solution.. To address this issue, we consider a right-of-way parameter and the following priority rule:

(R1) The actual link exit flows satisfy . Figure 5: Illustration of the merge model. The shaded areas represent the feasible domain of the maximization problem (4.33), and the thick line segments (Ω) are the optimal solution sets of (4.33). Left: Rule (R1) is compatible with (A2); and there exists a unique point Q satisfying both (A2) and (R1). Right: (R1) is incompatible with (A2); in this case, the model selects point Q within the set Ω that is closest to the ray (R) from the origin with slope 1−pp.

Notice that (R1) may be incompatible with assumption (A2); and we refer the reader to Figure 5 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 among all the points yielding the maximum flow. Clearly, such a point is unique. Mathematically, we let be the set of points that solves the following maximization problem:

 {max fout4+fout5such that   0≤fout4≤D4(t),  0≤fout5≤D5(t),  fout4+fout5 ≤ S6(t) (4.33)

Moreover, we define the ray . Then the solution of the merge model is defined to be the projection of onto ; that is,

 (fout,∗4,qout,∗5) = argmin(fout4,fout5)∈Ωd[(fout4,fout5),R] (4.34)

where denotes the Euclidean distance between the point and the ray :

 d[(fout4,fout5),R] = min(x,y)∈R∥∥(fout4,fout5)−(x,y)∥∥2

### 4.2 Well-posedness of the diverge junction model

In this section, we investigate the well-posedness of the diverge junction model. Unlike previous studies (Garavello and Piccoli, 2006; Han et al., 2015b), a major challenge in this case is to incorporate drivers’ route choices, expressed by the path disaggregation variable , 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 and . As we shall see in Section 4.2.1 below, such a continuity does not hold in general. Following this, Section 4.2.2 provides sufficient conditions for the continuity to hold. These sufficient conditions are crucial for the desired continuity of the delay operator.

#### 4.2.1 An example of ill-posedness

It has been shown by Han et al. (2015b) that the diverge model (Section 4.1.1) with constant turning ratios, and , is well-posed. However, the assumption of fixed and exogenous turning ratios does not hold in DNL models; see (3.28). As a result the well-posedness is no longer true, which is demonstrated by the following counterexample.

We consider the diverge junction (Figure 4) and assume the same fundamental diagram for all the links for simplicity. We consider a series of constant initial data parameterized by on the three links , and :

 ρ1(0,x) ≡ ^ρ1∈(ρc,ρjam),ρ2(0,x) ≡ ^ρε2 ∈(ρc,ρjam],ρ3(0,x) ≡ ^ρε3∈(0,ρc) (4.35)

where and are the critical density and the jam density, respectively. , , and satisfy:

 f(^ρε2) = εf(^ρ1),f(^ρε3) = (1−ε)f(^ρ1) (4.36)

where is a parameter. This configuration of initial data implies that link and link are both in the congested phase, while link 3 is in the uncongested (free-flow) phase, see Figure 6 for an illustration. Figure 6: An example of ill-posedness of the initial value problem with endogenous route choices. Left: junction topology. Right: illustration of the constant initial densities on the links.

Two paths exist in this example: and . The initial conditions for and , which arise from travelers’ route choices, are as follows.

 μp11(0,x) ≡ ε,μp21(0,x) ≡ 1−ε

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 ; in particular, we have the following two cases.

• When , we claim that the initial conditions and satisfying (4.35)-(4.36) constitute a constant solution at the junction. To see this, we follow the junction model (4.32) and the definitions of demand and supply (2.6)-(2.7) to get

 fout1(t) = min{D1(t),S2(t)α1,2,S3(t)α1,3}=min{C,f(^ρε2)ε,C1−ε} = min{C,f(^ρ1),C1−ε} = f(^ρ1) (4.37) fin2(t) = α1,2⋅fout1(t) = εf(^ρ1) = f(^ρε2) (4.38) fin3(t) = α1,3⋅fout1(t) = (1−ε)f(^ρ1) = f(^ρε3) (4.39)

where denotes the flow capacity. Thus , and form a constant solution at the junction.

• When , the turning ratios satisfy , . Effectively, link is only connected to link . We easily deduce that

 fout1(t) ≡ C (4.40) fin2(t) ≡ 0 (4.41) fin3(t) ≡ C (4.42)

As a result, the solution on link is given by a backward-propagating rarefaction wave (or expansion wave, fan wave) with and on the two sides. On link , a forward-propagating rarefaction wave with and (that is, the limit of as ) on the two sides is created. Link remains in a completely jam state with full density . See Figure 7 for an illustration of the solutions. Figure 7: Comparison of solutions on I1, I2 and I3 for the two cases: ε>0 and ε=0. 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 I1 and I3.

The two sets of solutions corresponding to and are shown in Figure 7. We first notice that in these two cases, the boundary flows are very different, despite the infinitesimal difference in the PDV, namely from to . In particular, both and jump from and (when ) to (when ). 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 (4.37), the expression for when is

 fout1(t) = min{C,f(^ρε2)ε,C1−ε} = min{C,εf(^ρ1)ε}

As long as is positive, the fraction . However, when we have an expression of ‘’, which should be equal to since link has effectively no influence on the junction, and we have . The above argument amounts to the following statement:

 if  ε>0,C>εf(^ρ1)ε; if  ε=0,C<εf(^ρ1)ε,

which explains the jump in the solutions when tends to zero.

###### Remark 4.1.

The fact that tends to as plays a key role in this example. As we shall see later in Theorem 4.2, bounding