A Polymorphic Dynamic Network Loading Model

Transcription

1 Computer-Aided Civil and Infrastructure Engineering 23 (2008) A Polymorphic Dynamic Network Loading Model Yu (Marco) Nie Department of Civil and Environmental Engineering, Northwestern University & Jingtao Ma & H. Michael Zhang Department of Civil and Environmental Engineering, University of California, Davis, CA, USA Abstract: A polymorphic dynamic network loading (PDNL) model is developed and discretized to integrate a variety of macroscopic traffic flow and node models. The polymorphism, realized through a general node-link interface and proper discretization, offers several prominent advantages. First of all, PDNL allows road facilities in the same network to be represented by different traffic flow models based on the tradeoff of efficiency and realism and/or the characteristics of the targeted problem. Second, new macroscopic link/node models can be easily plugged into the framework and compared against existing ones. Third, PDNL decouples links and nodes in network loading, and thus opens the door to parallel computing. Finally, PDNL keeps track of individual vehicular quanta of arbitrary size, which makes it possible to replicate analytical loading results as closely as desired. PDNL, thus, offers an ideal platform for studying both analytical dynamic traffic assignment problems of different kinds and macroscopic traffic simulation. 1 INTRODUCTION Dynamic network loading (DNL) is an underlying component of many dynamic network problems in which path costs depend on temporal path flows in ways governed by traffic propagation and interaction. In the past CKS Professor, Tongji University, Shanghai, China. To whom correspondence should be addressed. y-nie@ northwestern.edu. two decades, DNL has been extensively studied owing to the needs of simulating urban traffic and solving dynamic traffic assignment (DTA) problems. In this article we are concerned with the following DNL problem: Definition 1 (Dynamic Network Loading): The DNL problem determines, on a congested network and over a fixed time period, link cumulative arrival/departure curves (hence time-dependent link/path travel times) corresponding to a given set of temporal path flow rates. In other words, DNL represents a mapping from path inflows to experienced travel times on the paths. According to how they model traffic propagation and interaction, existing DNL models may be classified into three groups: macroscopic, microscopic, and mesoscopic models. Providing a complete list of all existing DNL models/packages is difficult. Table 1 attempts to cover the best-known works in each of the three categories. 1 Although microscopic models provide the most detailed representation of traffic, they share two major limitations: the curse of high computational overhead, which often prohibits large-scale applications, and the analytical intractability often inherited from the Monte Carlo nature of the models. We are not suggesting, however, that macroscopic models can solve the DNL problem analytically. Indeed, even with the simplest assumptions, the mapping from path flow to path cost still cannot be cast in a closed form for general networks. Nevertheless, for macroscopic models, analytical formulas for such a mapping may be obtained for special networks (e.g., with a single or two tandem links). C 2008 Computer-Aided Civil and Infrastructure Engineering. Published by Blackwell Publishing, 350 Main Street, Malden, MA 02148, USA, and 9600 Garsington Road, Oxford OX4 2DQ, UK.

2 PDNL model 87 Table 1 Existing dynamic network loading models Macroscopic Mesoscopic Microscopic Traffic is represented collectively as continuum flows, whose evolution over networks are governed through macroscopic variables such as link in/out flows, capacities, and densities etc. FREFLO (Payne, 1979), KRONOS (Michalopoulos et al., 1984; Michalopoulos and Plum, 1986), METANET (Papageorgiou, 1990), CTM (Daganzo, 1994, 1995a), DYNALOAD (Xu et al., 1999), MCKW (Jin, 2003) Vehicles are represented individually, like in the microscopic case, but their movements are determined by macroscopic rules. CONTRAM (Taylor, 1990), DYNASMART (Mahmassani, 2000), DYNAMIT (Ben-Akiva et al., 1998), METROPOLIS (Palma and Marchal, 1998), CellNetLoad (Velan, 2000), DYNAMEQ (Mahut et al., 2003) Vehicles are represented individually, and move according to car-following logic and lane-changing rules, which may be stochastic and behavior-specific. CORSIM (FHWA, 1996), MITSIM (Yang and Koutsopoulos, 1996), INTEGRATION (Van-Aerde and Yagar, 1988), PARAMICS (Quadstone, 2002), VISSIM (Fellendorf, 1994), AIMSUM2 (Barcelo, 1998), TRANSIM (Bush, 2000) The ability of analyzing the properties of the mapping even for the very special cases may still offer useful insights to the general problem. Moreover, the results of macroscopic DNL models are more tractable and predictable, and such models are also easier to calibrate, because they employ fewer parameters. Particularly, enforcing the first-in-first-out (FIFO) principle into DNL is simpler in macroscopic/mesoscopic models. Explicitly or implicitly, a DNL model consists of two building blocks: the link model that describes traffic evolution within links, and the node model accounting for the interaction across links. We categorize existing link models into four classes: 1. Speed-density function models: the moving of traffic is determined by the speed-density relationship; 2. Bottleneck/queueing models: traffic congestion takes the form of queuing behind bottlenecks; 3. Hydrodynamic models: vehicles are modeled with functions of density and flow in space and time, analogous to fluid flow; 4. Whole-link function models: the link outflow is assumed to depend on link volume and/or inflow through a deterministic exit-flow or traversal time function. The majority of analytical DTA research falls into this category. The speed-density function model is not considered in the rest of this article because typically it has to be combined with bottleneck-type models to correctly capture the temporal and spatial behavior of vehicular queues (e.g., DYNAMSART in Mahmassani, 2000). Node models in DNL can be classified as follows according to how link interactions are modeled: 1. Competition-free nodes: Only the flow conservation law is obeyed at such nodes. The competitionfree node is often seen in the analytical DTA research (e.g., Merchant and Nemhauser, 1978a), accompanied with whole-link function models. 2. Uncontrolled competition nodes: traffic from different incoming links and/or heading to different outgoing links would have to compete against each other for the limited capacities (e.g., Daganzo, 1995b). A typical example is freeway junctions (on- and off-ramps) without metering facilities. 3. Controlled competition nodes: the competition among different traffic streams is managed by a control logic, such as signalized intersections. In this research we develop a polymorphic DNL model (PDNL). The most important feature of this new member in the DNL family is its polymorphism, namely, a general scheme that integrates various link and node models together. Specifically, polymorphism offers the following advantages: Flexibility: Road facilities in the same network may be represented by different models based on various criteria, such as the tradeoff of efficiency and realism, and the characteristics of the targeted problem. Extensibility: New macroscopic link/node models can be easily plugged into the framework and compared against existing ones. Parallelizability: The realization of polymorphism decouples links and nodes in network loading, and thus opens the door to parallel computing.

3 88 Nie, Ma & Zhang The PDNL model takes a mesoscopic approach to trace individual vehicular quanta. A vehicular quantum is an indivisible flow element that is treated in PDNL like a vehicle in microscopic simulation. However the size of vehicular quanta can be set arbitrarily small, which makes it possible to replicate analytical loading results as closely as desired, whenever a need to compare numerical and analytical results arises. Section 4.2 explains in detail why the size of quanta is important to the stability of loading results. In a nutshell, PDNL offers a desirable tool for studying both DTA problems of different kinds and macroscopic traffic simulation. The next section briefly reviews analytical link models. Section 3 presents a node model for general intersections. Section 4 discusses the discretization scheme and the realization of polymorphism, and how the FIFO behavior is enforced. Numerical results are reported in Section 6, and Section 7 concludes the article. 2 LINK MODELS This section reviews bottleneck, whole-link function, and hydrodynamic models. Each link is assumed to be homogenous, i.e., the road characteristics remain the same everywhere in a link. 2.1 Bottleneck models In bottleneck-type models, vehicles always move along a link at the free-flow speed before they arrive at the exit node, where they form a FIFO queue if the outflow rate they induce exceeds the maximum discharge rate (bottleneck capacity) of the link. A continuous mathematical form reads { dλ 0 if λ(t) = 0 and u(t τ0 ) < C = dt u(t τ 0 ) C otherwise (1) τ(t) = τ 0 + λ(t + τ 0 )/C (2) where u(t) is the entry rate at time t; λ is the total number of queuing vehicles at the exit node; τ 0 is the free flow travel time; τ(t) is the link traversal time corresponding to the entry time t; and C is the bottleneck capacity. A bottleneck model that ignores the physical length of vehicles is called a point-queue (P-Q) model, which never predicts a queue spillback. To allow spillback, a simple remedy is to block inflow whenever the following condition is met: t t C l u(w) dw v(w) dw, t (3) 0 0 where C l is the holding capacity, i.e., the maximum number of vehicles that a link can accommodate; v(t) isthe exit rate at time t. We call a bottleneck model equipped with the condition (3) a spatial-queue (S-Q) model. Although the S-Q model captures the propagation of queues across links, it presumes that any queue is at jam density (Zhang and Nie, 2006), which does not accord with existing empirical evidence. We note that the S-Q model may be viewed as a simplified formulation of Newell s trilogy (Newell, 1993), which was later shown to be fully equivalent to the LWR model for triangle fundamental diagrams (Daganzo, 2005b). 2.2 Whole-link function models In general this class of models trade traffic realism with analytical tractability. The earliest known example, due to the pioneering work of Merchant and Nemhauser (1978a, 1978b), is the exit-flow model. This model assumes that the link exit flow rate at any time is a nondecreasing and concave function of current link volume, i.e., dx(t) dt = u(t) v(t) (4) v(t) = g e (x(t)) x(t) (5) where x(t) is the link volume at time t; g e ( ) is the link exit flow function. Equation (4) expresses the flow conservation condition, i.e., the net change of link volume equals the difference between inflow and outflow rates. The exit-flow model has been extensively used for studying DTA problems. Another widely used model in this category is a direct extension of the static link performance function (e.g., Friesz et al., 1993; Astarita, 1996; Carey and McCartney, 2002). Known as the delay-function model, it assumes that τ(t), the actual traverse time experienced by travelers entering a link at t, is a function of the link volume x(t). The model reads dx(t) dt v(t + τ(t)) = = u(t) v(t) (6) τ(t) = g d (x(t)) (7) u(t) 1 + τ(t) (8)

4 PDNL model 89 where g d ( ) denotes the delay function. Equation (8) (see, e.g., Astarita, 1996) ensures the FIFO behavior by forcing vehicles that enter the link at t to be pushed out at t + τ(t). Carey and Ge (2005) showed that the delay function model converges to the LWR model on a single link when the length of discrete segments approaches zero. However, in general the whole link model lacks a mechanism to capture the backward shockwaves, which makes it difficult to reflect vehicular queuing in a realistic way. It is established (Friesz et al., 1993; Daganzo, 1995b; Carey and McCartney, 2002) that any affine function can guarantee FIFO in the delay-function model. Whether or not there exists any nonlinear functional form that ensures FIFO, however, remains an unresolved issue. Xu et al. (1999) gave an example of FIFO violation when the delay function is quadratic. 2.3 Hydrodynamic models Hydrodynamic models view traffic as a continuous fluid represented by density (k), speed (s), and flow-rate (q). They are also known as kinematic wave models (KW) because their solutions can be characterized by combinations of kinematic waves in either of the three quantities. The movement of traffic on a homogeneous link is governed by the following flow conservation law: q x + k t = 0 (9) where x and t denote space and time, respectively. A fundamental assumption in hydrodynamic models is that the flow rate q is a function of traffic density k, i.e., q = F(k) (10) Combining (10) into (9) yields the KW model of traffic flow, which is widely known as the LWR model (Lighthill and Whitham, 1955; Richards, 1956): q k x + k t = 0, q = F (11) k The LWR model can be solved numerically by the finite difference method such as the first-order Gudunov scheme (Godunov, 1959). For the triangle fundamental diagram, Daganzo (1994) proposed a streamlined scheme known as the cell transmission model (CTM). Luke (1972) and Newell (1993) showed that, in a system governed by the one-dimensional conversation law such as the LWR model, cumulative counts at any spacetime point is the lower-envelope of those derived from boundary conditions. This result was recently formalized by Daganzo (2005a, 2005b) as a variational formulation of the KW model. The Luke Newell Daganzo (LND) theory of traffic flow can generate exact wave solutions for the triangle fundamental diagram (Daganzo, 2005b). This latest development is not included in this study, however, because the corresponding node models are yet to be developed. 3 NODE MODELS This section only considers nodes with competition. The free-competition node is not further discussed because its model is flow conservation. Following Daganzo (1995a), this section first examines the diverge and merge on freeways, and then extends to general intersections. Let us first define the demand and supply of a link. Definition 2 (Link Demand and Supply): The demand of a link, D, is the maximum possible exit flow rate, i.e., D = min{c, Q}, the supply of a link, S, is the maximum possible receiving flow rate, i.e., S = min{c, R} where C is the flow capacity depending on road characteristics and/or control strategies; Q is the rate of the flow that is ready to exit; R is the maximum entry flow rate to the link permitted by the current traffic condition. How R may be computed determines largely the behavior of queue spillback and varies in different link models. We will discuss this in detail in the next section. A merge problem can be formulated as a maximization problem as follows (see Figure 1) max v = v i3, subject to 0 v i3 D i, i = 1, 2, i v i3 S 3 (12) i where D i is the demand of link i, i = 1, 2. S 3 is the supply of link 3. The solution to this problem has the following simple structure: v i3 = a i3 v, i,v = min{d 1 + D 2, S 3 } (13) where a i3, i is called distribution ratio. Daganzo (1995a) suggested that a i3 is proportional to some priority parameters p i3 (which is determined by geometry design and other properties), a i3 = constant i, p i3 = 1 (14) p i3 i

5 90 Nie, Ma & Zhang v 12 v 13 v 13 v 23 a 12 a 13 a 13 a 23 Fig. 1. Merge and diverge nodes. Daganzo s merge model requires priority parameters as exogenous inputs. To simplify the issue, Jin and Zhang (2001) proposed the following demand-based distribution scheme a i3 = D i i D (15) i Lebacque and Khoshyaran (2005) showed that this scheme does not respect the invariance principle, which means that the distribution ratio is subject to discontinuous changes in some cases. The violation, nevertheless, does not affect discretized models (Lebacque and Khoshyaran, 2005). In this research, we adopt a distribution scheme similar to (15), as follows: a i3 = D i, D i = min{d i, S 3 }, i (16) i D i Note that the proposed scheme departs from (15) when D i > S 3, i. This behavior is intended based on the conjecture that the traffic demand on any upstream branch no longer influences its share in the total outflow as long as it exceeds the downstream supply. Also, (16) allows one to develop formulas for general intersection as described later in this section. Similarly, a diverge problem can be formulated as max v = v 12 + v 13, subject to a 12 v S 2, a 13 v S 3,v D 1 (17) where a 1i is called turning proportion. In general, a 1i depends on traffic composition, and varies with time and demand pattern. Thus, turning proportions have to be derived from the destinations of the vehicles ready to advance into each diverge at any given time. The solution to the above mathematical program is simply ( v 12 = a 12 v, v 13 = a 13 v, v = min D 1, S 2, S ) 3 (18) a 12 a 13 A challenging issue raised at a diverge node is how to enforce the FIFO discipline. In Daganzo (1995a), FIFO is enforced by first transmitting through a node those vehicles that have been waiting for a longer time. We will elaborate how this is implemented in Section 4.3. Now we present a general model that handles nodes with multiple upstream and downstream links. A similar formula was given in Jin and Zhang (2004) but the flow distribution (15) was used in that work. Consider an intersection with multiple incoming and outgoing links as shown in Figure 2. The demand of an incoming link I i is D i, and the supply of an outgoing link O i is S i. For each I i the proportion of vehicles heading for each direction j, a ij, is also given. This general layout is a bit more complicated because the merges and diverges entangle with each other. Let us first define the virtual demands and supplies as follows: ( ( )) Sj D i = min D i, max a j = 1,...,l, j i (19) ij ( ) l S i = min S i, a ji D j (20) j=1 The virtual demand D i represents the maximum possible outflow rate dictated by the standard diverge formula (one upstream link and multiple downstream links), whereas the virtual supply S i is the maximum possible receiving flow rate determined from a merge analysis

6 PDNL model 91 4 POLYMORPHISM AND DISCRETIZATION S i We are now ready to discuss how the continuous macroscopic models can be discretized and how these discrete link and node models can be bound together to achieve the polymorphism. ~ S 4 v 34 v 14 v 13 ~ D1 Fig. 2. A general uncontrolled-competition node. (multiple upstream links and one downstream link). The flux between each incoming outgoing link pair ij can then be computed by the following formula ( ) D i a ij v ij = min D i a ij, S j l D k=1 k a kj v 12 v 24 D i (21) Proposition 1: The merge (Equations (13) and (16)) and diverge (Equation (18)) models are special cases of the general uncontrolled-competition node mode described by Equations (19) (21). Proof: The result is obvious and thus omitted. Readers may refer to chapter 5 (Nie, 2006) for details. Remark: Controlled-competition nodes can be modeled using the supply demand approach. The only difference is that the demands from upstream links should be adjusted based on control strategies. For a general intersection with signals, for example, the demand of each incoming link may be given by 1 if the link is assigned D = min{bc, Q}, b = the right of way 0 otherwise (22) 4.1 A link-node interface Figure 3 illustrates an interface that bridges links and nodes while maintaining their relative independence. As shown, a link is divided into three parts: a core traffic propagation section (TPS), which implements corresponding flow dynamics, an entry boundary (ENB), and an exit boundary (EXB), elements that handle the data exchange between a link and a node. ENB is a fictitious element that temporarily holds the traffic flow ready to enter the link at the current time. Vehicles do not, however, spend time to traverse an ENB. Conversely EXB holds vehicles that are about to leave the link in the next time step provided the associated node model would allow them to do so. In other words, any vehicle presenting in an EXB unit is considered to have demanded to leave the link, and the request will be collectively handled by a node in charge in the form of link demands. Unlike ENB, EXB is considered part of the physical link and thus vehicles do spend at least one time step in it. The actual waiting time in EXB, however, is determined by the node model. Furthermore, notice that EXB consists of a list of sub elements called movements, each corresponding to a downstream link. Vehicles will be classified upon their arrival at an EXB and sent to the movement according to the next link in their journey. This procedure is termed destination classification. By default each vehicle should have a predetermined path in the context of DNL described in Definition 1. However, the framework also allows us to implement any adaptive (reactive) rerouting strategy. ENB and EXB together offer a data-exchange interface between links and nodes. At each time step, node operations are first performed to update link boundary conditions. Specifically, node operations can be divided into three steps: 1. Collect demands (from upstream EXB) and supplies (from downstream links) as depicted in Definition Compute in-out fluxes using Equations (19) (21). 3. Move flows from EXB to ENB, based on the calculated fluxes. where b can be obtained from either a fixed timing plan or an adaptive strategy. As we noted before, different node and link models may require different calculations of demands, but the

7 92 Nie, Ma & Zhang Entry Boundary Traffic Propagation Section Exit Boundary Link Left turn movement Through movement vehicular quantum Right turn movement Node Supply Demand Supply Flux for each movement Demand Node Models with competition Demand Supply Supply Demand Note: the color of the vehicular quantum is used to distinguish turning movment. Fig. 3. An illustration of the link-node interface. overall procedure stated above remains the same. After all nodes are processed, boundary conditions are updated and so links can propagate traffic forward based on the built-in rules of traffic dynamics. In this process, a link will absorb all traffic in its ENB and send traffic ready to leave into its EXB. The proposed interface has two important implications. First of all, because links and nodes are decomposed, and only communicate with adjacent ENBs/EXBs, the operations of each link/node are encapsulated and independent of others. That is to say, various node and link models can work together in a network as long as they work for the interface. This is how the polymorphism is achieved. Second, all nodes can be processed simultaneously in each time step because they only affect the adjacent link boundaries. This argument applies to link models, but note that a link should not be processed until its boundaries are updated by adjacent nodes. Consequently, parallel computing can be applied to enhance the computational performance as needed. We do not, however, provide a parallel implementation in this work. 4.2 Discrete link and node models The length of each time step is denoted as φ l (called the loading interval). Traffic flow is measured in the unit of the vehicular quantum. A vehicular quantum is similar to an individual vehicle except that a quantum may carries an arbitrary amount of traffic flow. We assume that each vehicular quantum carries an identical amount of traffic, denoted by δ f. Generally the fidelity of the discretization scheme depends on the time resolution φ l. However, as φ l decreases, the amount of flows that can be transmitted in each time step becomes smaller, and when this quantity gets close to or smaller than δ f, numerical instability is the result. To see this, consider a link with a capacity of 18,00 vph and set φ l to 1 second. Clearly,

8 PDNL model 93 ENB v 01 v v TPS i 1 i..., i i-1 i i+1 L-1 Fig. 4. Discretization of the LWR model. no more than 0.5 vehicles should be moved through any given location on that link during each φ l to respect the capacity. However, if the δ f = 1 vehicle, the realized capacity could only alternate between 0 (no quantum is moved) or 3,600 vph (one quantum is moved). Such sharp jumps can introduce not only instability, but also unexpected shockwaves as shown in Nie (2006). These problems, however, can be overcome by using a sufficiently small δ f Link models. To discretize the LWR model, we first divide the TPS of a link into L 1 cells, with an identical length (see Figure 4) δx = s f φ l where s f is the free-flow speed and δx is the spatial step. The number of cells L is calculated by L = [dist/δx] where dist is the link length and [a] is the largest integer less than a. Note that the EXB element is also a cell with a length δx l = dist (L 1)δx δx. It is easy to verify that our discretization scheme satisfies the CFL (Courant, Friedrichs, and Lewy) condition (Courant et al., 1928) s φ l δx s φ l f δx 1 (23) The CFL condition is important to ensure the stability of finite difference methods for hyperbolic PDEs. Intuitively, this condition requires that the distance s f φ l traveled by a vehicle in one time step does not exceed δx. The number of vehicular quanta in each cell i is denoted by l i and the corresponding density is given by v, δx k i l i v i EXB k i = l iδ f (24) δx In an EXB, vehicular quanta are classified by movements. Let l m L denote the number of quanta in movement m, we have M ll m = l L m=1 where M is the total number of movements in the EXB. Given any initial state, we first need to update the flux L v i,i+1 across the boundary between cells i and i + 1 (see Figure 4). This is called a Riemann problem, i.e., finding the flux at a density discontinuity that separates two flow states with constant density. The solution to the Riemann problem in the LWR case can be nicely interpreted, again, using the following supply demand approach (Lebacque, 1996). 2 v i,i+1 = min{d i, S i+1 } (25) where D i and S i+1 are demand of cell i and supply of cell i + 1, respectively, and { { F(ki ) if k i k c C if ki k c D i = S i = C if k i k c F(k i ) if k i k c (26) Now we are ready to summarize the flow propagation procedure (FPP) in the discrete LWR model. Assume that l i (t), i = 0,..., Lat time t (i.e., the number of vehicular quanta in each cell, including ENB and EXB) are given, the FPP reads ALGORITHM FPP-LWR Step 0: Compute densities k i using (24) for i = 0, 1,..., L. Step 1: Calculate the boundary flux v i,i+1 for i = 1,..., L 1 using Equations (25) and (26). Set v 0,1 = l 0. Step 2: Transform v i,i+1 into an integer number of quantum n i,i+1, i = 1,..., L 1. Step 3: Move n i,i+1 vehicular quanta from cell i into cell i + 1, i = 1,..., L 1. Move all quanta in ENB l 0 (t) into cell 1. Step 4: Update l i (t + 1) = l i (t) + n i 1,i n i,i+1, i = 1,..., L 1. Set l 0 (t + 1) = 0, update ll m(t + 1) = l L(t) + n m L 1,L, m = 1,..., M. Here n m L 1,L is the number of vehicular quanta that enter EXB and head for the downstream link associated with movement m. We note that each vehicular quantum will maintain its own path information that will only be used for classification when it enters EXB. The Step 3 of the above algorithm does not explain how vehicular quanta are stored and moved. This will be covered when we discuss the FIFO issue in Section 4.3. The bottleneck-type models can be discretized in the same manner as the LWR model (see Figure 4) but the implementation is much simpler. The FPP for the bottleneck models is summarized as follows: ALGORITHM FPP-BOTTLENECK Step 0: Update l i (t + 1) = l i 1 (t), i = 1,..., L 1. Update two boundaries with l 0 (t + 1) = 0, l L (t + 1) = l L (t) + l L 1 (t).

9 94 Nie, Ma & Zhang Floating cells EXB ENB TPS... i t e l i... L+1 L i 1 Earlier leaving time 0 Fig. 5. Discretization of the delay-function model. Step 1: Move l L 1 (t) vehicular quanta from ENB to EXB. That is, vehicular quanta are not actually moved from one cell to another as in the LWR model. Rather, they stay in ENB until they directly jump into EXB. Discretizing the delay-function model is a little bit different. As shown in Figure 5, TPS is not discretized into a fixed number of cells. Rather, we group vehicular quanta that are about to leave at the same time into floating cells.thefloating cells are ordered by their leaving times and the total number of floating cells for each link is changed according to traffic conditions. Note that vehicular quanta entering the link at different times may leave at the same time, and thus merge into the same floating cell. Let l i (t) and te i, i = 1,..., L denote the number of quanta in the floating cell i and the time they should enter the EXB at time t, respectively. We also use l 0 (t) and l L+1 (t) to denote the number of quanta in EXB and ENB, respectively. The FPP for the delay-function model is as follows: ALGORITHM FPP-DELAY FUNCTION Step 0: Compute the leaving time for the group of vehicles in ENB. Step 0.1: Update total link volume x(t) = δ f L i=0 l i(t). Step 0.2: Calculate the travel time τ(t) = g d (x(t)). The time when these quanta will enter EXB is t e = t + [ τ(t) φ l ] 1. 3 Step 1: Check the designated leaving time of the last floating cell L. Ifte L = t e, update l L(t + 1) = l L (t) + l L+1 (t); otherwise (i.e., t e > tl 4 e ), create a new floating cell with quanta l L+1 and leaving time t e. Step 2: Check if the first floating cell 1 is ready to enter EXB. If the current time t = t 1 e, do the following: Step 2.1: update l 0 (t + 1) = l 0 (t) + l 1 (t) Step 2.2: Move l 1 (t) vehicular quanta from ENB to EXB. Similar to the discrete bottleneck models, vehicles are held in ENB until they are transmitted into EXB. Step 2.3: Destroy the floating cell 1. Step 3: Update the index for current floating cells. We emphasize that vehicles should not stay in EXB longer than one time step in this model, because otherwise the actual exit time will be later than that predicted from the delay function. This implies that the node in charge should be free of restraints (capacity, limited downstream supplies, and the competition from other links) other than the conservation law Node models. We now discretize the node model, which carries traffic from upstream EXBs to downstream ENBs. Consider an intersection with n approaches. Each approach i has an incoming and outgoing link denoted by I i and O i, respectively. Let l ii (t) be the number of quanta in EXB of approach i at t, and l oi (t) be the number of quanta in ENB of approach i at t. Further, l j ii (t) isthe number of quanta in EXB of approach i heading for the approach j. The discrete FPP reads ALGORITHM FPP-NODE Step 0: Calculate the turning proportion in each EXB using the following: a ij = l j ii (t) l ii (t) Calculate the demand D i for each EXB of approach i and supply S i for each downstream link of approach i. This largely depends on the characteristic of link models, as explained below.

10 PDNL model 95 Table 2 Calculation of supplies and demands for different links Demand Supply P-Q min{c ii φ l, l ii (t)δ f } S-Q min{c ii φ l, l ii (t)δ f } max {C h,ii x oi,0} LWR min{c ii φ l, l ii (t)δ f } the supply of the first cell (see Figure 4) S 1, computed as in Equation (26). Delay-function l ii (t)δ f Note: C ii is the flow capacity of link I i ; and C h,oi is the holding capacity of link O i ; x oi is the traffic volume on link O i. Step 1: Calculate fluxes v ij using the Equations (19) (21). Transform v ij into number of quanta n ij. Step 2: Update l j ii (t + 1) = lj ii (t) n ij, i, j, and l oj (t + 1) = n i=1 n ij, j. Step 3: Move n ij vehicular quanta from EXB of approach i into ENB of approach j. It should be noted that FIFO is not imposed in this discrete node model (see Section 4.3 for details). In Step 0, the calculation of demands and supplies is summarized in Table 2. An infinite supply means vehicular queue will never spill over. In the LWR model, however, queueing effects will impact the upstream links as soon as the last cell of the downstream link is congested. In traditional travel demand modeling, traffic is assumed to be released from the centroid of traffic analysis zones (TAZ). Each centroid connects with the physical network through centroid connectors. In our DNL, we define a dummy origin node (DON) and a dummy origin link (DOL) for each TAZ. DOL is an imaginary road facility that has zero length, infinite flow capacity, and infinite holding capacity. DOL channels traffic to- ward either a centroid (if created) or the physical network, and provides a place to hold them in the event of demand surge. By using DOL as a buffer, the waiting time at origins can be taken into account. As shown in Figure 6, DOL does not include a TPS component. All vehicular quanta in ENB should be directly moved into movements of EXB at each time step. Each DON dispatches vehicular quanta according to the predetermined assignment table (an assignment table specifies how much flows should leave the DON at the current time and what paths they use) as follows: ALGORITHM FPP-DON Step 1: Given the current loading interval t, for each destination s associated with current DON r, and for each path used. Step 1.1: Read from the assignment table f kh rs, where h is the current assignment interval. Step 1.2: Calculate the flows to be released at t f kt rs = frs kh i a f kh rs i a + γ k rs + 0.5δ f + γ k rs 0.5δ f if γ k rs δ f if γ k rs δ f frs kh + γrs k otherwise i a where i a is the ratio of the assignment and loading interval, and γ k rs is the accumulative residual flow for path k of O-D pair rs. Step 1.3: Calculate the number of vehicular quanta to be released at t on path kn kt rs = [ f rs kt/δ f ]. Update accumulative residual flow γ rs k = γ rs k + ( f kt δ f n kt rs ). rs Dummy Origin Link Dummy Origin Node ENB EXB Dummy Destination Link Departing flows ENB Dummy Destination Node Arriving Flows Fig. 6. Structure of dummy origins and destinations.

11 96 Nie, Ma & Zhang Step 1.4: If n kt rs > 0, create n kt rs vehicular quanta and insert them into ENB of the DOL corresponding to DON r. Step 2: Mix the quanta in ENB after all destinations are processed. This is to equalize the departure times for different destinations. Similarly each TAZ has a dummy destination node (DDN) and dummy destination link (DDL) to receive traffic destination for the zone (Figure 6). Each DDL only has an ENB component to handle the incoming quanta. DDL will simply delete and remove all quanta present in the ENB. 4.3 Enforce FIFO Imposing FIFO in DNL is crucial because otherwise one cannot ensure time-dependent link traversal times retrieved from cumulative curves are correct. Maintaining FIFO on links is relatively easy. It is sufficient to keep the order in which vehicular quanta enter the link. The simplest way of achieving this is to store quanta in a queue structure. In our implementation, vehicular quanta always form a queue according to the order they enter a link, whether they are held in ENB, EXB, or cells. We emphasize that only in the discrete LWR model, quanta will actually go through cells. Whenever quanta needs to be moved (based on the calculated fluxes), they are always taken out of the current queue from the front and inserted into the back of the next queue. In this way, no quantum can overtake others. Moreover, note that there is a separate queue in EXB for each movement. FIFO violation can occur at a diverge even if all vehicular quanta are stored in a queue for all movements of EXB. To avoid this, one needs to keep track of the entry time of each vehicular quantum to ensure only moving quanta that have waited for the longest time no matter which movement they belong to (see Daganzo, 1995a). The following procedure can be inserted between Step 2 and Step 3 in Algorithm FPP-Node to implement the idea. After Step 2, n ij (t), i.e., the number of vehicular quanta to be moved from link I i to O j, has been determined from the general node model. The procedure will update n ij (t) such that the resulting movement strictly obeys FIFO. ALGORITHM FIFO-NODE Step 0: Check the first n ij (t) quantum in the queue of movement j in EXB of link I i. Get the earliest arrival time t a min for all quanta to be moved. Step 1: Check the queues again, update n f ij (t) = n ij (t) ˆn ij (t). ˆn ij (t) is the number of quanta in the queue of movement j that arrive at the link later than t a min. Accordingly, Algorithm FPP-Node will move quanta based on n f ij (t) in Step A flowchart of PDNL Figure 7 presents an overall PDNL flowchart, which assumes the network is empty at the onset. The procedure is terminated either when the network clears out (all released vehicles have arrived at their destinations) or a gridlock is detected (no vehicle can be moved because they are trapped in an interlock situation). However, other termination criteria may also be considered. In each time step t, the first operation is to release new quanta from DON based on the time-dependent travel demand pattern (see Algorithm FPP-DON). Then a preprocess is followed. Incidents, for example, can be taken into account at this step. In case some vehicles do not have an assigned path and thereby need route guidance, this preprocess may also include the calculation of instantaneous shortest paths. The main course of PDNL is sequentially processing nodes and links to propagate vehicular quanta. Whenever a link FPP is called, the cumulative traffic counts up to t are recorded. Further, all quanta entering the ENB of a DDL at t will be labeled as arrived and thereby removed from the network. 5 NUMERICAL EXPERIMENTS 5.1 Experiment settings Consider a network shown in Figure 8, which has eight nodes, seven links, and two O-D pairs. The road properties of the four nondummy links are given in Table 3. The flow-density relationship on all links is assumed to follow the triangle fundamental diagram illustrated in Figure 9. The assignment horizon is from 6:00 to 6:30 am, during which the uniform departure rates for O-D pair 5 6 and 5 7 are 1,800 vph, and 2,400 vph, respectively. An incident occurs on link 3 (see Figure 8) at 6:10 am and the last 15 minutes during which the capacity is reduced to 25% of its regular value. Obviously, node 2 is a permanent bottleneck (with a capacity 1,200 vph) because of the downstream lane drop. The incident will cause a temporal bottleneck (with a capacity 1,200 vph) on link 3. Queues are expected to develop at both locations because the uniform demands (1,800 and 2,400 vph, respectively) exceed the capacities of the two bottlenecks. Unless specified

12 PDNL model 97 Start Traffic demand pattern The assignment table Initialize : set network empty, current time t = 0, activeveh = 0, movedveh = 0 Network Topology, Signal Timing, Incidents t=t+1 t<t or activeveh>0? No Terminate. Loading Succeed. Yes Release vehicles from each DON for current time t, Update activeveh Prepare flow propagation: Check current incidents, update enroute routing table for vehicles without a predetermined path etc Note: activeveh: vehicles on network movedveh: vehicles are moved at the current time FPP: flow propagation procedure NumLock: Number of consecutive time itnervals in which no vehicle can be moved. GridThresh: the maximum NumLock to be tolerated before a gridlock is proclaimed. No Call FPP algorithm for each nondummy node based on node type, Update movedveh Call FPP algorithm for each link based on link type, Update movedveh and activeveh Record the flows that enter and leave the link at current time Terminate. Loading failed with a gridlock Yes NumLock = 0 No MovedVeh =0? Yes NumLock+=1 numlock > GridThresh? Fig. 7. The overall PDNL flow chart. otherwise, (nondummy) links are implemented using the LWR model, and the loading interval φ i and the quantum size δ f are set to 5 seconds and 0.2 vehicle, respectively. 5.2 Base scenario The evolution of queues on three key links (1, 3, and 7) is reported in the density contour plots in Figure 10. In a density contour plot, the dark color represents high density, and thus identifies the accumulation and dissipation of vehicular queues. As shown in the figure, a queue built up on link 7 right after traffic arrives at node 2, and the shock wave traveled upstream and reached node 8 shortly after 6:16 am. A simple calculation indicates that the time when the queue end arrives at node 8 is about 6:17 am in theory, very close to the simulation results. In the meantime, a queue was induced by the scheduled incident at 6:10 am on link 3, and propagated backward at a speed s w 4.85 mph. When the end of this queue arrives at about 0.43 miles downstream at node 8 (at 6:17

13 98 Nie, Ma & Zhang Bottleneck Link Travel demand for 5-6: 1800 veh/hr uniformly in T 5-7: 2400 veh/hr uniformly in T Assignment horizon T: 6:00-6:30 am Link Link 1 Link 3 8 Incident 1.0 mile from entrance 75% capacity drop duration: 6:10-6:25 am 4 7 Fig. 8. Test network for PDNL. Table 3 Road properties of nondummy links in the test network Link ID Dist (miles) s f (mph) C (vphpl) C h (vpmpl) Lane , , , , am), the inflow rate of link 3 began to drop due to the spillback on the other branch. Using the diverge formula (18), it is easy to find that the inflow rate to link 3 was reduced from 2,400 vph to = 1,600 vph. As a consequence, the shock wave speed on link 3 decreases from mph to about 1.52 mph. At t = 6:25 am, the incident was removed and a rarefaction wave was generated and traveled upstream at s w = 18.0 mph. The queue on link 3 was cleared up after 3 minutes, when its end was about 0.1 miles from the link entrance. 5.3 Comparison of different link models In this section, link 7 was implemented using other link models for comparison. The results also highlight the polymorphism of the loading model. Spatial-queue, point-queue, and the delay-function (D-F) models are tested. For the D-F model, the following piecewise linear function is adopted to approximate the P-Q model: τ 0 if x(t) κc h τ(t) = τ 0 + x(t) κc h C otherwise (27) where C h is the holding capacity and 0 <κ<1isa parameter. For the reason to become clear, three values of κ (0.12, 0.24, and 0.36) are tested. For narrative convenience we name the scenarios to be tested as follows: SQ (link 7 is implemented with the S-Q model), PQ (the P-Q model), DF-1 (the D-F model with κ = 0.12), DF-2 (κ = 0.24), and DF-3 (κ = 0.36). The base scenario tested in the last section, in which link 7 is implemented with the LWR model is named BASE. The loading results of the SQ scenario are detailed in Figure 11. Unlike the BASE scenario, the queue spillback on link 7 was delayed because the queue on link 3 reached node 8 first and halved feeding flows into link 7. It can be seen that the queue shrank on link 7, corresponding to the spillback on link 3, and then grew again and eventually spillback after the rarefaction wave 4800 Link 2 q (vph) q (vph) mph mph Link k (vpm) k (vpm) Link 1 and 3 Link 2 and Fig. 9. Fundamental diagrams for nondummy links.

14 PDNL model 99 Fig. 10. Density contour plots in the base scenario.

15 100 Nie, Ma & Zhang Fig. 11. Density contour plots in the SQ scenario.

16 PDNL model 101 Fig. 12. Density contour and in-out flow rate in PQ scenario. initiated by the incident removal hit node 8. The PQ scenario (see Figure 12) is similar to the SQ scenario except that the queue on link 7 did not spillback at all. Figure 13 compares the time-dependent link traversal times on link 7 produced by three DF models (cor- responding to different κ) against those obtained from PQ, SQ, and BASE scenarios. As shown, DF-1 (κ = 0.1) overestimated the congestion on link 7 while DF-2 led to a slight underestimation. DF-3 (κ = 0.3) failed to capture congestion. Generally, the time-dependent travel times SQ PQ Travel Time (seconds) BASE DF-1 DF DF :00 08:08 08:16 08:25 08:33 08:41 08:50 08:58 Time (Hour:Minute) Fig. 13. Time-dependent traversal times on link 7 in different scenarios.

17 102 Nie, Ma & Zhang Table 4 Average travel delays and speeds in six scenarios Average delay (min) Average speed (mph) OD 5 6 OD5 7 OD5 6 OD 5 7 BASE SQ PQ DF DF DF produced by this D-F model (Equation (27)) are similar to those of the PQ model, as expected. However, the impact of κ is crucial and, unfortunately, is hard to predetermine because of its dependence on the inflow pattern. Essentially, this is due to the lack of a mechanism to estimate queue length. Table 4 compares the average delays and speeds for each O-D pair in all six scenarios. The results show that travel delays between one O D pair were underestimated by more than 30% compared to the Base scenario, only because the queue spillback effects on a single link is not correctly captured. Such a difference is hard to overlook in solving DTA problems, even for the coarse DTA solution procedures that are currently available. A recent study (Nie and Zhang, 2007) has demonstrated the impact of DNL results on the dynamic user-optimal solutions. 6 CONCLUSIONS This research proposed a new addition to the long list of the existing DNL models. The most important feature of the proposed DNL model is the polymorphism, namely the integration of a variety of macroscopic traffic flow models and node models. The polymorphism, realized through a general node-link interface and proper discretization, offers several prominent advantages. First of all, PDNL allows road facilities in the same network to be represented by different traffic flow models based on the tradeoff of efficiency and realism and/or the characteristics of the targeted problem. Second, PDNL is extendible because new macroscopic link/node models can be easily plugged into the framework and compared against existing ones. Third, PDNL decouples links and nodes in network loading, and thus opens the door to parallel computing, which holds promise in high-performance application. What is more, its capability of keeping track of individual vehicular quanta of arbitrary size makes it possible to replicate analytical loading results as closely as desired, whenever a need to compare numerical and analytical results arises. PDNL offers an ideal platform for studying both analytical DTA problems of different kinds and macroscopic traffic simulation. Our numerical experiments have clearly demonstrated the polymorphic feature of the proposed DNL platform. PDNL has been tested on large real networks (with thousands of links and hundreds of thousands of O D pairs) and has demonstrated competitive computational performance. We will present those results in a subsequent paper. The comparison of different link models highlighted the importance of modeling queue spillback effects. Ignoring (in point-queue and delay-function model) or misrepresenting (spatial-queue) queue spillback on a single link led to over 30% underestimation of travel delays for one O D pair in our experiments. Such a difference should not be overlooked when one selects a DNL platform for solving a DTA problem. ACKNOWLEDGMENT The authors would like to thank two anonymous reviewers for their constructive comments on an earlier version of the article. This research is supported in part by a grant from the National Science Foundation under the number CMS # NOTES 1. Recently, several attempts (e.g., Yang and Morgan, 2006; Burghout et al., 2005) were made to develop hybrid simulation platforms, which allow one to select any of the three approaches to simulate different parts of the network. 2. Daganzo (1994) termed it as a sending receiving method. 3. Operator [a] rounds a into the closest integer. 4. Note that t e < tl e is not possible because of FIFO. REFERENCES Astarita, V. (1996), A continuous time link model for dynamic network loading based on travel time function, in The Proceedings of the 13th International Symposium on Transportation and Traffic Theory, Lyon, France, Barcelo, J. (1998), Parallelization of microscopic traffic simulation for ATT systems analysis, in P. Marcotte and S. Nguyen (eds.), Equilibrium and Advanced Transportation Modelling, Kluwer Academic Publishers. Ben-Akiva, M., Bierlaire, M., Koutsopoulos, H. & Mishalani, R. (1998), DYNAMIT: A simulation-based system for traffic

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