Optimal queue placement in dynamic system optimum solutions for single origin-destination traffic networks

The Dynamic System Optimum (DSO) traffic assignment problem aims to determine a time-dependent routing pattern of travellers in a network such that the given time-dependent origin-destination demands are satisfied and the total travel time is at a minimum, assuming some model for dynamic network loading. The network kinematic wave model is now widely accepted as such a model, given its realism in reproducing phenomena such as transient queues and spillback to upstream links. An attractive solution strategy for DSO based on such a model is to reformulate as a set of side constraints apply a standard solver, and to this end two methods have been previously proposed, one based on the discretisation scheme known as the Cell Transmission Model (CTM), and the other based on the Link Transmission Model (LTM) derived from variational theory. In the present paper we aim to combine the advantages of CTM (in tracking time-dependent congestion formation within a link) with those of LTM (avoiding cell discretisation, providing a more computationally attractive with much fewer constraints). The motivation for our work is the previously-reported possibility for DSO to have multiple solutions, which differ in where queues are formed and dissipated in the network. Our aim is to find DSO solutions that optimally distribute the congestion over links inside the network which essentially eliminate avoidable queue spillbacks. In order to do so, we require more information than the LTM can offer, but wish to avoid the computational burden of CTM for DSO. We thus adopt an extension of the LTM called the Two-regime Transmission Model (TTM), which is consistent with LTM at link entries and exits but which is additionally able to accurately track the spatial and temporal formation of the congestion boundary within a link (which we later show to be a critical element, relative to LTM). We set out the theoretical background necessary for the formulation of the network-level TTM as a set of linear side constraints. Numerical experiments are used to illustrate the application of the method to determine DSO solutions avoiding spillbacks, reduce/eliminate the congestion and to show the distinctive elements of adopting TTM over LTM. Furthermore, in comparison to a fine-level CTM-based DSO method, our formulation is seen to significantly reduce the number of linear constraints while maintaining a reasonable accuracy.     

Spillback congestion in dynamic traffic assignment: A macroscopic flow model with time-varying bottlenecks

In this paper, we propose a new model for the within-day Dynamic Traffic Assignment (DTA) on road networks where the simulation of queue spillovers is explicitly addressed, and a user equilibrium is expressed as a fixed-point problem in terms of arc flow temporal profiles, i.e., in the infinite dimension space of time’s functions. The model integrates spillback congestion into an existing formulation of the DTA based on continuous-time variables and implicit path enumeration, which is capable of explicitly representing the formation and dispersion of vehicle queues on road links, but allows them to exceed the arc length. The propagation of congestion among adjacent arcs will be achieved through the introduction of time-varying exit and entry capacities that limit the inflow on downstream arcs in such a way that their storage capacities are never exceeded. Determining the temporal profile of these capacity constraints requires solving a system of spatially non-separable macroscopic flow models on the supply side of the DTA based on the theory of kinematic waves, which describe the dynamic of the spillback phenomenon and yield consistent network performances for given arc flows. We also devise a numerical solution algorithm of the proposed continuous-time formulation allowing for “long time intervals” of several minutes, and give an empirical evidence of its convergence. Finally, we carry out a thorough experimentation in order to estimate the relevance of spillback modeling in the context of the DTA, compare the proposed model in terms of effectiveness with the Cell Transmission Model, and assess the efficiency of the proposed algorithm and its applicability to real instances with large networks.     

Dynamic network loading: A stochastic differentiable model that derives link state distributions

We present a dynamic network loading model that yields queue length distributions, accounts for spillbacks, and maintains a differentiable mapping from the dynamic demand on the dynamic queue lengths. The model also captures the spatial correlation of all queues adjacent to a node, and derives their joint distribution. The approach builds upon an existing stationary queueing network model that is based on finite capacity queueing theory. The original model is specified in terms of a set of differentiable equations, which in the new model are carried over to a set of equally smooth difference equations. The physical correctness of the new model is experimentally confirmed in several congestion regimes. A comparison with results predicted by the kinematic wave model (KWM) shows that the new model correctly represents the dynamic build-up, spillback and dissipation of queues. It goes beyond the KWM in that it captures queue lengths and spillbacks probabilistically, which allows for a richer analysis than the deterministic predictions of the KWM. The new model also generates a plausible fundamental diagram, which demonstrates that it captures well the stationary flow/density relationships in both congested and uncongested conditions.     

On the morning commute problem with bottleneck congestion and parking space constraints

Morning commuters choose their departure times based on a combination of factors—the chances of running into bottleneck congestion, the likely schedule delays, and parking space availability. This study investigates the morning commute problem with both bottleneck congestion and parking space constraints. In particular, it considers the situation when some commuters have reserved parking spots while others have to compete for public ones on a first-come-first-served basis. Unlike the traditional pure bottleneck model, the rush-hour dynamic traffic pattern with a binding parking capacity constraint varies with the relative proportions of the two classes of commuters. It is found that an appropriate combination of reserved and unreserved parking spots can temporally relieve traffic congestion at the bottleneck and hence reduce the total system cost, because commuters without a reserved parking spot are compelled to leave home earlier in order to secure a public parking spot. System performance is quantified in terms of the relative proportions of the two classes of commuters and is compared with those in the extreme cases when all auto commuters have to compete for parking and when none of them have to compete for one.     

Quasi-dynamic traffic assignment with residual point queues incorporating a first order node model

Static traffic assignment models are still widely applied for strategic transport planning purposes in spite of the fact that such models produce implausible traffic flows that exceed link capacities and predict incorrect congestion locations. There have been numerous attempts to constrain link flows to capacity. Capacity constrained models with residual queues are often referred to as quasi-dynamic traffic assignment models. After reviewing the literature, we come to the conclusion that an important piece of the puzzle has been missing so far, namely the inclusion of a first order node model. In this paper we propose a novel path-based static traffic assignment model for finding a stochastic user equilibrium in general transportation networks. This model includes a first order (steady-state) node model that yields more realistic turn capacities, which are then used to determine consistent capacity constrained traffic flows, residual point (vertical) queues (upstream bottleneck links), and path travel times consistent with queuing theory. The route choice part of the model is specified as a variational inequality problem, while the network loading part is formulated as a fixed point problem. Both problems are solved using existing techniques to find a solution. We illustrate the model using hypothetical examples, and also demonstrate feasibility on large-scale networks.     

Ant Colony Routing algorithm for freeway networks

Dynamic traffic routing refers to the process of (re)directing vehicles at junctions in a traffic network according to the evolving traffic conditions. The traffic management center can determine desired routes for drivers in order to optimize the performance of the traffic network by dynamic traffic routing. However, a traffic network may have thousands of links and nodes, resulting in a large-scale and computationally complex non-linear, non-convex optimization problem. To solve this problem, Ant Colony Optimization (ACO) is chosen as the optimization method in this paper because of its powerful optimization heuristic for combinatorial optimization problems. ACO is implemented online to determine the control signal – i.e., the splitting rates at each node. However, using standard ACO for traffic routing is characterized by four main disadvantages: 1. traffic flows for different origins and destinations cannot be distinguished; 2. all ants may converge to one route, causing congestion; 3. constraints cannot be taken into account; and 4. neither can dynamic link costs. These problems are addressed by adopting a novel ACO algorithm with stench pheromone and with colored ants, called Ant Colony Routing (ACR). Using the stench pheromone, the ACR algorithm can distribute the vehicles over the traffic network with less or no traffic congestion, as well as reduce the number of vehicles near some sensitive zones, such as hospitals and schools. With colored ants, the traffic flows for multiple origins and destinations can be represented. The proposed approach is also implemented in a simulation-based case study in the Walcheren area, the Netherlands, illustrating the effectiveness of the approach.     

Real-time congestion pricing strategies for toll facilities

This paper analyzes the dynamic traffic assignment problem on a two-alternative network with one alternative subject to a dynamic pricing that responds to real-time arrivals in a system optimal way. Analytical expressions for the assignment, revenue and total delay in each alternative are derived as a function of the pricing strategy. It is found that minimum total system delay can be achieved with many different pricing strategies. This gives flexibility to operators to allocate congestion to either alternative according to their specific objective while maintaining the same minimum total system delay. Given a specific objective, the optimal pricing strategy can be determined by finding a single parameter value in the case of HOT lanes. Maximum revenue is achieved by keeping the toll facility at capacity with no queues for as long as possible. Guidelines for implementation are discussed.     

A link-node complementarity model and solution algorithm for dynamic user equilibria with exact flow propagations

In this paper, we propose a link-node complementarity model for the basic deterministic dynamic user equilibrium (DUE) problem with single-user-class and fixed demands. The model complements link-path formulations that have been widely studied for dynamic user equilibria. Under various dynamic network constraints, especially the exact flow propagation constraints, we show that the continuous-time dynamic user equilibrium problem can be formulated as an infinite dimensional mixed complementarity model. The continuous-time model can be further discretized as a finite dimensional non-linear complementarity problem (NCP). The proposed discrete-time model captures the exact flow propagation constraints that were usually approximated in previous studies. By associating link inflow at the beginning of a time interval to travel times at the end of the interval, the resulting discrete-time model is predictive rather than reactive. The solution existence and compactness condition for the proposed model is established under mild assumptions. The model is solved by an iterative algorithm with a relaxed NCP solved at each iteration. Numerical examples are provided to illustrate the proposed model and solution approach. We particularly show why predictive DUE is preferable to reactive DUE from an algorithmic perspective.     

Congestion spill effects of Heathrow and Frankfurt airports on connection traffic in European and Gulf hub airports

We develop two stage fixed-effects single-spill and double-spill models for congestion connection spills of London Heathrow and Frankfurt airports on 9 hub airports in Europe and the Gulf. Our panel data covers connection traffic from 1997 to 2013 for Heathrow and 1997 to 2011 for Frankfurt. The single-spill results support strongly that the connection spills from Heathrow’s capacity limitations do strengthen competing hub airports of major alliance groups and to a lesser degree one Gulf hub. The double-spill model for Heathrow and Frankfurt shows nearly asymmetric overall spill characteristics between the two airports. Our results underline the influence of airline network strategies on congestion spills as European airline networks are shaped by alliances and umbrella mergers. Thus, the airline network perspective in airport capacity expansion decisions needs to play a greater role, as indicated by our asymmetric results for overall spill effects between Heathrow and Frankfurt airports.     

A generic class of first order node models for dynamic macroscopic simulation of traffic flows

Node models for macroscopic simulation have attracted relatively little attention in the literature. Nevertheless, in dynamic network loading (DNL) models for congested road networks, node models are as important as the extensively studied link models. This paper provides an overview of macroscopic node models found in the literature, explaining both their contributions and shortcomings. A formulation defining a generic class of first order macroscopic node models is presented, satisfying a list of requirements necessary to produce node models with realistic, consistent results. Defining a specific node model instance of this class requires the specification of a supply constraint interaction rule and (optionally) node supply constraints. Following this theoretical discussion, specific macroscopic node model instances for unsignalized and signalized intersections are proposed. These models apply an oriented capacity proportional distribution of the available supply over the incoming links of a node. A computationally efficient algorithm to solve the node models exactly is included.     

Modeling and optimization of multimodal urban networks with limited parking and dynamic pricing

Cruising-for-parking constraints mobility in urban networks. Car-users may have to cruise for on-street parking before reaching their destinations. The accessibility and the cost of parking significantly influence people's travel behavior (such as mode choice, or parking facility choice between on-street and garage). The cruising flow causes delays eventually to everyone, even users with destinations outside limited parking areas. It is therefore important to understand the impact of parking limitation on mobility, and to identify efficient parking policies for travel cost reduction. Most existing studies on parking fall short in reproducing the dynamic spatiotemporal features of traffic congestion in general, lack the treatment of dynamics of the cruising-for-parking phenomenon, or require detailed input data that are typically costly and difficult to collect. In this paper, we propose an aggregated and dynamic approach for modeling multimodal traffic with the treatment on parking, and utilize the approach to design dynamic parking pricing strategies. The proposed approach is based on the Macroscopic Fundamental Diagram (MFD), which can capture congestion dynamics at network-level for single-mode and bi-modal (car and bus) systems. A parsimonious parking model is integrated into the MFD-based multimodal modeling framework, where the dynamics of vehicular and passenger flows are considered with a change in the aggregated behavior (e.g. mode choice and parking facility choice) caused by cruising and congestion. Pricing strategies are developed with the objective of reducing congestion, as well as lowering the total travel cost of all users. A case study is carried out for a bi-modal city network with a congested downtown region. An elegant feedback dynamic parking pricing strategy can effectively reduce travel delay of cruising and the generic congestion. Remarkably, such strategy, which is applicable in real-time management with limited available data, is fairly as efficient as a dynamic pricing scheme obtained from system optimum conditions and a global optimization with full information about the future states of the system. Stackelberg equilibrium is also investigated in a competitive behavior between different parking facility operators. Policy indications on on-street storage capacity management and pricing are provided.     

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

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. 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.     

Dynamic pricing for managed lanes with multiple entrances and exits

Priced managed lanes are increasingly being used to better utilize the existing capacity of the roadway to relieve congestion and offer reliable travel time to road users. In this paper, we investigate the optimization problem for pricing managed lanes with multiple entrances and exits which seeks to maximize the revenue and minimize the total system travel time (TSTT) over a finite horizon. We propose a lane choice model where travelers make online decisions at each diverge point considering all routes on a managed lane network. We formulate the problem as a deterministic Markov decision process and solve it using the value function approximation (VFA) method for different initializations. We compare the performance of the toll policies predicted by the VFA method against the myopic revenue policy which maximizes the revenue only at the current timestep and two heuristic policies based on the measured densities on the managed and general purpose lanes (GPLs). We test the results on four different test networks. The primary findings from our research suggest the usefulness of the VFA method for determining dynamic tolls. The best-found objective value from the method at its termination is better than other heuristics for all test networks with average improvements in the objective ranging between 10% and 90% for revenue maximization and 0–27% for TSTT minimization. Certain VFA initializations obtain best-found toll profiles within first 5–50 iterations which warrants computational time savings. Our findings also indicate that the revenue-maximizing optimal policies follow the “jam-and-harvest” behavior where the GPLs are pushed towards congestion in the earlier time steps to generate higher revenue in the later time steps, a characteristic not observed for the policies minimizing TSTT.     

Airport congestion pricing and terminal investment: Effects of terminal congestion,passenger types,and concessions

None of the airport-pricing studies have differentiated the congestion incurred in the terminals from the congestion incurred on the runways. This paper models and connects the two kinds of congestion in one joint model. This is done by adopting a deterministic bottleneck model for the terminal to describe passengers’ behavior, and a simpler static congestion model for the runway. We find that different from the results obtained in the literature, uniform airfare does not yield the first-best outcome when terminal congestion is explicitly taken into account. In particular, business passengers are at first-best charged a higher fare than leisure passengers if and only if their relative schedule-delay cost is higher. We further identify circumstances under which passengers are, given a uniform airport charge scheme, under- or over-charged with respect to the terminal charge. Furthermore, when concession surplus is added to the analysis, the airport may raise (rather than reduce) the airport charge in order to induce more business passengers who in turn will lengthen leisure passengers’ dwell time and hence increase their chance of purchasing concession goods. Finally, the impacts of terminal capacity expansion and time-varying terminal fine toll are discussed.     

A note on using Davidson's function in equilibrium assignment

This paper describes the use of the Davidson congestion function in modelling network flows using equilibrium assignment. A modification to the function is given, which defines the function over all flows values, and consequently removes computational difficulties noted in earlier studies. The modification requires the inclusion of an additional model parameter, and the selection of a suitable value for this parameter is studied for two sets of data. The modified Davidson function is also compared to two alternative functions; a step-wise linear function and a quartic polynomial function, which have both been proposed as congestion functions. Comparisons are made between observed link volumes and the assigned volumes from these models. It is concluded that the modified Davidson function is useful for inclusion in an equilibrium assignment model, given its ability to reflect differences in network link type (e.g. capacity and speed) and environment through its parameters, the conceptual advantage of the function through its derivation from queueing theory, and the previous discovery of reliable methods for estimating its parameters. A value of about 0.8–0.9 is suggested for the parameter (μ, 0 < μ < 1) introduced in the modification.     

A bi-level model of dynamic traffic signal control with continuum approximation

This paper proposes a bi-level model for traffic network signal control, which is formulated as a dynamic Stackelberg game and solved as a mathematical program with equilibrium constraints (MPEC). The lower-level problem is a dynamic user equilibrium (DUE) with embedded dynamic network loading (DNL) sub-problem based on the LWR model (Lighthill and Whitham, 1955; Richards, 1956). The upper-level decision variables are (time-varying) signal green splits with the objective of minimizing network-wide travel cost. Unlike most existing literature which mainly use an on-and-off (binary) representation of the signal controls, we employ a continuum signal model recently proposed and analyzed in Han et al. (2014), which aims at describing and predicting the aggregate behavior that exists at signalized intersections without relying on distinct signal phases. Advantages of this continuum signal model include fewer integer variables, less restrictive constraints on the time steps, and higher decision resolution. It simplifies the modeling representation of large-scale urban traffic networks with the benefit of improved computational efficiency in simulation or optimization. We present, for the LWR-based DNL model that explicitly captures vehicle spillback, an in-depth study on the implementation of the continuum signal model, as its approximation accuracy depends on a number of factors and may deteriorate greatly under certain conditions. The proposed MPEC is solved on two test networks with three metaheuristic methods. Parallel computing is employed to significantly accelerate the solution procedure.     

Stockholm congestion charging: an assessment with METROPOLIS and SILVESTER

This paper reviews and compares the performance of two dynamic transportation models – METROPOLIS and SILVESTER – which are used to predict the impacts of congestion charging for Stockholm. Both are mesoscopic dynamic models treating accumulation and dissipation of traffic queues, route choice, modal split and departure time choice. The models are calibrated independently for the baseline situation without charges and applied to forecast the effects of congestion charging. The results obtained from the two models are mutually compared and validated against the actual outcome of the Stockholm congestion charging scheme. Both models successfully predict the outcomes of the congestion charging trial at both aggregate and disaggregate levels. Results of welfare analysis, however, differ substantially due to differences in model specification.     

Congestion-aware system optimal route choice for shared autonomous vehicles

We study the shared autonomous vehicle (SAV) routing problem while considering congestion. SAVs essentially provide a dial-a-ride service to travelers, but the large number of vehicles involved (tens of thousands of SAVs to replace personal vehicles) results in SAV routing causing significant congestion. We combine the dial-a-ride service constraints with the linear program for system optimal dynamic traffic assignment, resulting in a congestion-aware formulation of the SAV routing problem. Traffic flow is modeled through the link transmission model, an approximate solution to the kinematic wave theory of traffic flow. SAVs interact with travelers at origins and destinations. Due to the large number of vehicles involved, we use a continuous approximation of flow to formulate a linear program. Optimal solutions demonstrate that peak hour demand is likely to have greater waiting and in-vehicle travel times than off-peak demand due to congestion. SAV travel times were only slightly greater than system optimal personal vehicle route choice. In addition, solutions can determine the optimal fleet size to minimize congestion or maximize service.     

Traffic flooding the low countries: How the Dutch cope with motorway congestion

Motorway congestion is a common characteristic of the larger conurbations all over the world. Using the example of the Randstad region in The Netherlands, the causes and conditions for the growing congestion on main roads are presented and explained. From an international comparison it appears that the Randstad region is characterized by a relatively high density and high-access quality of its motorway network, giving rise to an extremely high level of usage. The Dutch policy of coping with the consequent congestion problems at the network design level is outlined, followed by a presentation of the dynamic traffic management approaches to these problems. Special attention is given to the so-called target group policy that aims at prioritizing specific user groups such as freight traffic and trucks. Finally, attention is given to the Dutch attempts to introduce congestion pricing as a means of tackling congestion.     

Access control policies without inside queues: Their properties and public policy implications

An access control policy that eliminates all queues beyond the entry points to a network has obvious benefits, which include smooth travel and predictable travel times inside the network. Yet it has never been proven, to the best of our knowledge, whether excluding inside queues yields sub-optimal network performance or, in other words, allowing inside queues can actually further reduce the system travel cost. Moreover, it is not clear whether an optimal control policy derived from efficiency considerations can also be a fair policy to all road users. This paper provide answers to these questions in the context of a monocentric network. By analyzing the structure of the access control problem considering all feasible policies (with/without inside queues), we show that the minimal system cost realizable by access control can be obtained without directly solving a non-convex optimization program, and can indeed always be achieved by a control policy excluding all of the inside queues. These optimal policies are defined by a polyhedral set and a Finite Generation Algorithm can be applied to derive the analytical form of this set. The optimal policies are not unique in general, thus making it possible to achieve both minimal system cost and fairness simultaneously.