Macroscopic relations of urban traffic variables: Bifurcations, multivaluedness and instability

Recent experimental work has shown that the average flow and average density within certain urban networks are related by a unique, reproducible curve known as the Macroscopic Fundamental Diagram (MFD). For networks consisting of a single route this MFD can be predicted analytically; but when the networks consist of multiple overlapping routes experience shows that the flows observed in congestion for a given density are less than those one would predict if the routes were homogeneously congested and did not overlap. These types of networks also tend to jam at densities that are only a fraction of their routes’ average jam density.This paper provides an explanation for these phenomena. It shows that, even for perfectly homogeneous networks with spatially uniform travel patterns, symmetric equilibrium patterns with equal flows and densities across all links are unstable if the average network density is sufficiently high. Instead, the stable equilibrium patterns are asymmetric. For this reason the networks jam at lower densities and exhibit lower flows than one would predict if traffic was evenly distributed.Analysis of small idealized networks that can be treated as simple dynamical systems shows that these networks undergo a bifurcation at a network-specific critical density such that for lower densities the MFDs have predictably high flows and are univalued, and for higher densities the order breaks down. Microsimulations show that this bifurcation also manifests itself in large symmetric networks. In this case though, the bifurcation is more pernicious: once the network density exceeds the critical value, the stable state is one of complete gridlock with zero flow. It is therefore important to ensure in real-world applications that a network’s density never be allowed to approach this critical value.Fortunately, analysis shows that the bifurcation’s critical density increases considerably if some of the drivers choose their routes adaptively in response to traffic conditions. So far, for networks with adaptive drivers, bifurcations have only been observed in simulations, but not (yet) in real life. This could be because real drivers are more adaptive than simulated drivers and/or because the observed real networks were not sufficiently congested.     

The problem of cyclic flows in traffic assignment

A fundamental property of traffic assignment is that cyclic flows from a common origin or to a common destination cannot exist in an equilibrium solution. However, cyclic flows can easily be created by the Frank-Wolfe (F-W) assignment procedure, especially during its first several iterations. The PARTAN technique—a more rapidly converging derivative of the F-W method—can also create cyclic flows during its procedure. We show in this paper that once cyclic flows become part of a combined assignment, they are difficult to correct, thus presenting one impediment to convergence. We then present modifications to the F-W and PARTAN procedures that prevent cyclic flows from being created between adjacent pairs of nodes. The avoidance of cyclic flows in test problems is shown to accelerate the convergence of both the F-W and PARTAN techniques, particularly in the first several iterations. While the impossibility of cyclic flows in a true equilibrium solution is an important property of traffic assignment, this paper shows that (1) the F-W and PARTAN procedures eventually reduce cyclic flows to zero if they occur, (2) avoiding cyclic flows can be most helpful in the early iterations of these procedures, and (3) avoiding cyclic flows in large networks is very difficult because of large computational requirements.     

Speed or spacing? Cumulative variables,and convolution of model errors and time in traffic flow models validation and calibration

This paper proves that in traffic flow model calibration and validation the cumulative sum of a variable has to be preferred to the variable itself as a measure of performance. As shown through analytical relationships, model residuals dynamics are preserved if discrepancy measures of a model against reality are calculated on a cumulative variable, rather than on the variable itself. Keeping memory of model residuals occurrence times is essential in traffic flow modelling where the ability of reproducing the dynamics of a phenomenon – as a bottleneck evolution or a vehicle deceleration profile – may count as much as the ability of reproducing its order of magnitude. According to the aforesaid finding, in a car-following models context, calibration on travelled space is more robust than calibration on speed or acceleration. Similarly in case of macroscopic traffic flow models validation and calibration, cumulative flows are to be preferred to flows. Actually, the findings above hold for any dynamic model.     

Hysteresis phenomena of a Macroscopic Fundamental Diagram in freeway networks

Observations of traffic pairs of flow vs. density or occupancy for individual locations in freeways or arterials are usually scattered about an underlying curve. Recent observations from empirical data in arterial networks showed that in some cases by aggregating the highly scattered plots of flow vs. density from individual loop detectors, the scatter almost disappears and well-defined macroscopic relations exist between space-mean network flow and network density. Despite these findings for the existence of well-defined relations with low scatter, these curves should not be universal. In this paper we investigate if well-defined macroscopic relations exist for freeway network systems, by analyzing real data from Minnesota’s freeways. We show that freeway network systems not only have curves with high scatter, but they also exhibit hysteresis phenomena, where higher network flows are observed for the same average network density in the onset and lower in the offset of congestion. The mechanisms of traffic hysteresis phenomena at the network level are analyzed in this paper and they have dissimilarities to the causes of the hysteresis phenomena at the micro/meso level. The explanation of the phenomenon is dual. The first reason is that there are different spatial and temporal distributions of congestion for the same level of average density. Another reason is the synchronized occurrence of transitions from individual detectors during the offset of the peak period, with points remain beneath the equilibrium curve. Both the hysteresis phenomenon and its causes are consistently observed for different spatial aggregations of the network.     

Dynamic blocking for railyards: Part II. Heterogeneous traffic

Traffic through a railroad yard is said to be heterogeneous if all the destinations are not served with the same frequency. This paper presents a study of railroad yards when traffic is heterogeneous and stationary, that is, without rush hour (or rush day) phenomena. Destinations (blocks) are grouped into categories that are served with the same frequency. The number of blocks and the frequency corresponding to each class are the only traffic characteristics that need to be known to analyze a given yard. With this information it is easy to determine the minimum number of tracks needed, and the number of switches per car; a numerical example is provided. The minimum number of required tracks is always larger than that for homogeneous traffic, but under the right conditions, the number of switches can be smaller. An approximation for large numbers of blocks is used to isolate the traffic factors that make the number of switches either large or small.     

Simulating pedestrian movements at signalized crosswalks in Hong Kong

This paper presents a new pedestrian simulation (PS) model for signalized crosswalks in Hong Kong. This PS model is capable of estimating the variations of walking speed particularly on the effects of bi-directional pedestrian flows so as to determine the minimum required duration of pedestrian crossing time. Video records taken from the observational surveys at the selected crosswalk in urban area were used to extract the required data for model calibration. It was found that the design walking speed for signalized crosswalks should be varied by the effects of the bi-directional pedestrian flows. It was also interesting to note that the negative impact of the bi-directional flow effects (ranging from uni-directional to bi-directional pedestrian flows) on the chance of pedestrian crossing the crosswalk is increasing from free-flow to at-capacity flow conditions. The new PS model is also validated using an independent data set so as to examine the reliability of the simulation results. The validation results show that the new PS model can provide an accurate evaluation on the changes of walking speed and its standard deviation under different scenarios with particular emphasis on the effects of the bi-directional pedestrian flows. The advancement of this PS model can be applied to assess the effects of each improvement measure and to evaluate the benefits of each scenario in practice.     

Assessment of economic impacts from unexpected events with an interregional commodity flow and multimodal transportation network model

Loss and damage caused by unscheduled events, especially earthquakes, have sudden and significant impacts not only on the region’s economy where the event occurs but also on other regions. The New Madrid Seismic Zone, located in the center of the United States, could have great impacts on economic activities related to this area, if a major earthquake occurred. Based on the 1993 US Commodity Flow Survey [US Commodity Flow Survey, 1993. Available from: <http://www.bts.gov/ntda/cfs/prod.html>], more than 42% of total commodity flows in the US are related to the greater Midwest, which includes the New Madrid Seismic Zone. If a catastrophic earthquake occurred in this area, the indirect damages could spread far beyond the region, and could have sizable impacts on other regions. A model of interregional commodity flows, incorporating regional input–output relationships, and the corresponding transportation network flows, was applied to assess the economic impacts of such an unexpected event. The economic impacts from the event are described for three hypothetical scenarios, analyzing the magnitude and the extent of the direct and indirect impacts. These analytical results may be used to propose strategic management of the recovery and reconstruction efforts after the event.     

Visco‐elastic traffic flow model

To increase our understanding of the operations of traffic system, a visco‐elastic traffic model was proposed in analogy of non‐Newtonian fluid mechanics. The traffic model is based on mass and momentum conservations, and includes a constitutive relation similar to that of linear visco‐elastic fluids. The further inclusion of the elastic effect allows us to describe a high‐order traffic model more comprehensively because the use of relaxation time indicates that vehicle drivers adjust their time headway in a reasonable and safe range. The self‐organizing behaviour is described by introducing the effects of pressure and visco‐elasticity from the point of view in fluid mechanics. Both the viscosity and elasticity can be determined by using the relaxation time and the traffic sound speed. The sound speed can be approximately represented by the road operational parameters including the free‐flow speed, the jam density, and the density of saturation if the jam pressure in traffic flows is identical to the total pressure at the flow saturation point. A linear stability analysis showed that the traffic flow should be absolutely unstable for disturbances with short spatial wavelengths. There are two critical points of regime transition in traffic flows. The first point happens at the density of saturation, and the second point occurs at a density relating on the sound speed and the fundamental diagram of traffic flows. By using a triangular form flow–density relation, a numerical test based on the new model is carried out for congested traffic flows on a loop road without ramp effect. The numerical results are discussed and compared with the result of theoretical analysis and observation data of traffic flows. Copyright © 2011 John Wiley & Sons, Ltd.     

Forecasting containerized traffic for the port of Montreal (1981–1995)

This paper presents an approach to making long-term forecasts of overseas container traffic in the port of Montreal. The paper discusses, first, why performance models explaining traffic variations in ports when postulating a fixed hinterland may not be appropriate for the port of Montreal. Then, the forecasting approach itself is presented. Using 1981 as the base year, projections of container traffic through the port of Montreal until 1995 are developed by considering separately the traffic flows associated with the Canadian and the United States hinterlands. Our approach takes account of anticipated changes in Canadian and U.S. international trade volume, containerization rates and regional growth within Canada. It yields detailed forecasts of containerized traffic, by origin and destination, for 78 commodity groups, 7 world regions and 11 North American regions (i.e. the 10 Canadian provinces plus the United States). Two forecasting scenarios are considered and the aggregate results for 1995 are reported. Finally, a comparison of forecasts with recent data available for 1984 is made, suggesting that the internation trade projections used to generate our forecasts may be too conservative.     

Entropy weighted average method for the determination of a single representative path flow solution for the static user equilibrium traffic assignment problem

The formulation of the static user equilibrium traffic assignment problem (UETAP) under some simplifying assumptions has a unique solution in terms of link flows but not in terms of path flows. Large variations are possible in the path flows obtained using different UETAP solution algorithms. Many transportation planning and management applications entail the need for path flows. This raises the issue of generating a meaningful path flow solution in practice. Past studies have sought to determine a single path flow solution using the maximum entropy concept. This study proposes an alternate approach to determine a single path flow solution that represents the entropy weighted average of the UETAP path flow solution space. It has the minimum expected Euclidean distance from all other path flow solution vectors of the UETAP. The mathematical model of the proposed entropy weighted average method is derived and its solution stability is proved. The model is easy to interpret and generalizes the proportionality condition of Bar-Gera and Boyce (1999). Results of numerical experiments using networks of different sizes suggest that the path flow solutions for the UETAP using the proposed method are about identical to those obtained using the maximum entropy approach. The entropy weighted average method requires low computational effort and is easier to implement, and can therefore serve as a potential alternative to the maximum entropy approach in practice.     

Three-phase traffic theory and two-phase models with a fundamental diagram in the light of empirical stylized facts

Despite the availability of large empirical data sets and the long history of traffic modeling, the theory of traffic congestion on freeways is still highly controversial. In this contribution, we compare Kerner’s three-phase traffic theory with the phase diagram approach for traffic models with a fundamental diagram. We discuss the inconsistent use of the term “traffic phase” and show that patterns demanded by three-phase traffic theory can be reproduced with simple two-phase models, if the model parameters are suitably specified and factors characteristic for real traffic flows are considered, such as effects of noise or heterogeneity or the actual freeway design (e.g. combinations of off- and on-ramps). Conversely, we demonstrate that models created to reproduce three-phase traffic theory create similar spatiotemporal traffic states and associated phase diagrams, no matter whether the parameters imply a fundamental diagram in equilibrium or non-unique flow-density relationships. In conclusion, there are different ways of reproducing the empirical stylized facts of spatiotemporal congestion patterns summarized in this contribution, and it appears possible to overcome the controversy by a more precise definition of the scientific terms and a more careful comparison of models and data, considering effects of the measurement process and the right level of detail in the traffic model used.     

Do Public Transport Improvements Increase Agglomeration Economies? A Review of Literature and an Agenda for Research

Public transport improvements may increase economic productivity if they enable the growth and densification of cities, downtowns, or industrial clusters and thereby increase external agglomeration economies. It has been argued that the potential agglomeration benefits are large; if so, understanding them better would be useful in making funding decisions about public transport improvements. We reviewed theoretical and empirical literature on agglomeration as well as a small number of articles on transportation's role in agglomeration. The theoretical literature is useful in understanding possible avenues by which transportation improvements might affect agglomeration, although there is little discussion of public transport specifically. Relevant empirical studies tend to focus on metropolitan regions and use a generalized measure of transportation cost. But public transport impacts on agglomeration are likely to be different from road investment impacts. We identified several ways of conducting research building on this literature that would help evaluate the agglomeration impacts of public transport proposals: tracing the links between transport, agglomeration, and productivity; better motivating research using theories of agglomeration mechanisms; taking scale and redistribution into account; exploring the functional form of agglomeration economies; accounting for endogeneity in model structure; and considering development context.     

Estimation of origin-destination matrices from link traffic counts on congested networks

Conventional methods for estimating origin-destination (O-D) trip matrices from link traffic counts assume that route choice proportions are given constants. In a network with realistic congestion levels, this assumption does not hold. This paper shows how existing methods such as the generalized least squares technique can be integrated with an equilibrium traffic assignment in the form of a convex bilevel optimization problem. The presence of measurement errors and time variations in the observed link flows are explicitly considered. The feasibility of the model is always guaranteed without a requirement for estimating consistent link flows from counts. A solution algorithm is provided and numerical simulation experiments are implemented in investigating the model's properties. Some related problems concerning O-D matrix estimation are also discussed.     

Selection of a trip table which reproduces observed link flows

In urban traffic management and planning, an important problem is estimating the number of drivers traveling between each origin-destination zone. We review a model due to Nguyen for estimating these numbers of drivers, based on counts of the traffic flows on each street, and develop an effective algorithm for solving it. The multiplicity of solutions of this model poses the additional question of which solution to use; we introduce a secondary optimization problem to overcome this difficulty. Efficient solution techniques are described for these problems and computational results are reported. It is noted that the most efficient solution methods involve user interaction to specify values of parameters which improve the convergence rates.     

Modelling rain effects on risk‐taking behaviours of multi‐user classes in road networks with uncertainty

This paper proposes a new travel time reliability‐based traffic assignment model to investigate the rain effects on risk‐taking behaviours of different road users in networks with day‐to‐day demand fluctuations and variations in travel time. A generalized link travel time function is used to capture the rain effects on vehicle travel times and road conditions. This function is further incorporated into daily demand variations to investigate those travel time variations arising from demand uncertainty and rain condition. In view of these rain effects, road users' perception errors on travel times and risk‐taking behaviours on path choices are incorporated in the proposed model with the use of a logit‐based stochastic user equilibrium framework. This new model is formulated as a variational inequality problem in terms of path flows. A numerical example is used to illustrate the application of the proposed model for assessment of the rain effects on road networks with uncertainty.     

Relationship between logistics infrastructure and trade: Evidence from Spanish regional exports

Geographical factors and transport infrastructure are two of the key determinants that influence international competitiveness. In this sense, the quality of such infrastructure and how widespread it is, the distribution and capacity of logistics facilities in a country, as well as the number of private operators and their degree of specialisation, all play an increasingly important role in the design of business strategies aimed at increasing a country’s share of the international market. Until recently, however, availability and access to logistics services have been considered secondary factors when defining business competitiveness. This paper estimates an augmented gravity model of trade that specifically includes logistics and transport infrastructure indicators as explanatory variables. The model is estimated by using bilateral exports from 19 Spanish regions to 64 destinations (45 countries and 19 Spanish regions) with data for the period 2003–2007. The findings show that logistics is indeed important for the analysis of trade flows in goods and they highlight the importance of logistics measures at the regional level. In particular, the number, size and quality of logistics facilities positively influence export flows.     

Dynamics of heterogeneity in urban networks: aggregated traffic modeling and hierarchical control

Real traffic data and simulation analysis reveal that for some urban networks a well-defined Macroscopic Fundamental Diagram (MFD) exists, which provides a unimodal and low-scatter relationship between the network vehicle density and outflow. Recent studies demonstrate that link density heterogeneity plays a significant role in the shape and scatter level of MFD and can cause hysteresis loops that influence the network performance. Evidently, a more homogeneous network in terms of link density can result in higher network outflow, which implies a network performance improvement. In this article, we introduce two aggregated models, region- and subregion-based MFDs, to study the dynamics of heterogeneity and how they can affect the accuracy scatter and hysteresis of a multi-subregion MFD model. We also introduce a hierarchical perimeter flow control problem by integrating the MFD heterogeneous modeling. The perimeter flow controllers operate on the border between urban regions, and manipulate the percentages of flows that transfer between the regions such that the network delay is minimized and the distribution of congestion is more homogeneous. The first level of the hierarchical control problem can be solved by a model predictive control approach, where the prediction model is the aggregated parsimonious region-based MFD and the plant (reality) is formulated by the subregion-based MFDs, which is a more detailed model. At the lower level, a feedback controller of the hierarchical structure, tries to maximize the outflow of critical regions, by increasing their homogeneity. With inputs that can be observed with existing monitoring techniques and without the need for detailed traffic state information, the proposed framework succeeds to increase network flows and decrease the hysteresis loop of the MFD. Comparison with existing perimeter controllers without considering the more advanced heterogeneity modeling of MFD highlights the importance of such approach for traffic modeling and control.     

Breakthrough innovations in intermodal freight transport

Breakthrough innovations, whether technological, organizational or both, are a necessity if the market share of intermodal freight transport is to expand. The main growth potential lies in the markets for flows over short distances, for perishable and high-value commodities, for small consignments, and for flows that demand speed, reliability and flexibility. It will take radical innovations to produce a breakthrough in the modal split and allow these new markets to be conquered. This special issue is based on papers presented at an international conference on freight transport automation and multimodality, held in Delft in May 2002, that are illustrative of the direction of breakthrough research and development (R&D) aimed at increasing the market share for intermodal transport.     

Marginal cost congestion pricing based on the network fundamental diagram

Congestion pricing schemes have been traditionally derived based on analytical representations of travel demand and traffic flows, such as in bottleneck models. A major limitation of these models, especially when applied to urban networks, is the inconsistency with traffic dynamics and related phenomena such as hysteresis and the capacity drop. In this study we propose a new method to derive time-varying tolling schemes using the concept of the Network Fundamental Diagram (NFD). The adopted method is based on marginal cost pricing, while it also enables to account realistically for the dynamics of large and heterogeneous traffic networks. We derive two alternative cordon tolls using network-aggregated traffic flow conditions: a step toll that neglects the spatial distribution of traffic by simply associating the marginal costs of any decrease in production within the NFD to the surplus of traffic; and a step toll that explicitly accounts for how network performance is also influenced by the spatial variance in a 3D-NFD. This pricing framework is implemented in the agent-based simulation model MATSim and applied to a case study of the city of Zurich. The tolling schemes are compared with a uniform toll, and they highlight how the inhomogeneous distribution of traffic may compromise the effectiveness of cordon tolls.