Journal of Advanced Transportation

Volume 2019, Article ID 5282879, 20 pages

https://doi.org/10.1155/2019/5282879

## A Traffic Assignment Model Based on Link Densities

^{1}School of Industrial Engineering, Diego Portales University, Santiago de Chile, Chile^{2}Institute of Basic Sciences, Faculty of Engineering, Diego Portales University, Santiago de Chile, Chile

Correspondence should be addressed to Louis de Grange; lc.pdu@egnarged.siuol

Received 31 December 2018; Revised 23 April 2019; Accepted 2 June 2019; Published 27 June 2019

Academic Editor: Francesco Viti

Copyright © 2019 Louis de Grange et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A new model is presented that determines the traffic equilibrium on congested networks using link densities as well as costs in a manner consistent with the fundamental traffic equation. The solution so derived satisfies Wardrop’s first principle. This density-based approach recognizes traffic flow reductions that may occur when network traffic congestion is high; also, it estimates queue lengths (i.e., the number of vehicles on saturated links), and it explicitly takes into account the maximum flow a link can handle, which is defined by the fundamental traffic equation. The model is validated using traffic microsimulations and implemented on a typical Nguyen-Dupuis network to compare it with a flow-based approach. System optimal assignment model based on link densities is also presented.

#### 1. Introduction

This paper develops and implements a deterministic model that solves the traffic equilibrium problem for a congested road network using network link densities. More specifically, the proposed density-based model solves a variational inequality whose cost vector is a function of the number of vehicles seeking to travel on the network at a given instant, consistent with the relationship between flow, cost (the inverse of speed), and density given by the fundamental traffic equation (*flow *=* speed*×*density*) for each network link. The solution arrived at gives a network traffic equilibrium that satisfies Wardrop’s first principle [1] and whose costs depend on link densities.

Using the link densities to obtain the traffic equilibrium on a congested network has several important advantages. In general terms, the modelling of the problem is more realistic than that achieved by the classical flow-based traffic assignment formulations employing monotonically increasing cost functions. More specifically, the density-based approach has the following desirable features.(i)It recognizes that link capacity (or maximum flow) is neither fixed nor exogenous but rather depends on the density level. In other words, the maximum flow that can cross a link varies as a function of link density. This prevents the flow from exceeding the link’s physical capacity. The capacities or maximum flows of the links in a network are determined by each link’s traffic speed and density as determined by the fundamental traffic equation.(ii)It determines whether a reduced flow level on a given link is due to low latent demand for its use (e.g., low density) or rather to the presence of traffic congestion (e.g., high density) limiting the amount of flow able to use the link and generating queues and longer delays.(iii)The average queue length on each link can be estimated.(iv)By generating estimates of the impact of densities on the flow levels that can circulate on network links, the approach provides important data for use in the design of road networks, highway entrance and exit ramps, and road pricing systems based on traffic saturation levels. This is a distinct advantage over flow-based models, which estimate only flows.

However, the density-based approach has a disadvantage: to find a solution which satisfies the flow conservation along the network is, in general, complicated. The implementation of numerical methods which allows for solutions which satisfy the flow conservation to be found will be the topic of future works.

The remainder of this paper is divided into four sections and three appendices. Section 2 contains a brief review of the literature on traffic assignment models, emphasizing the limits of classical deterministic flow-based models that assume a monotonically increasing relationship between flow and cost on each network link. Section 3 introduces our density-based traffic equilibrium model satisfying Wardrop’s first principle and discusses the existence of an equilibrium solution. This section also suggests an analytic expression for estimating the benefits of a road system project or policy using the proposed model in terms of consumer surplus before and after implementation. Section 4 uses a numerical example to compare our density-based model with a flow-based model that assumes a monotonically increasing relationship between link flows and costs. Section 5 summarizes our main conclusions.

Appendix A validates the proposed model using microsimulations of a small road network; Appendix B gives a more formal demonstration in support of the existence theorems given in Section 3; and finally, Appendix C presents the system optimal assignment model based on link densities, which will allow us to advance in future new lines of research.

#### 2. Literature Review

A widely accepted result in the study of vehicles on congested road networks is the so-called Wardrop equilibrium, also known as Wardrop’s first principle of route choice [1]. It is equivalent to the concept put forward by Knight [2] as a simple behavioural rule for describing the way trips are distributed across alternative routes under congested conditions.

Wardrop’s first principle states that the vehicle travel time or cost for every network route used will be equal to or less than the time or cost that would be experienced on any unused route. Each user attempts to minimize noncooperatively their trip cost or time. Traffic flows that satisfy this principle are generally referred to as “user equilibrium” (UE) flows since each user chooses the route they find best. In short, a user-optimal equilibrium is reached when no user can reduce their travel time or cost on the network through unilateral action.

The first mathematical model of user equilibrium on a congested road network assuming a monotonically increasing relationship between the flows and costs along the network links was formulated by Beckmann et al. [3], using a nonlinear optimization problem. However, traffic assignment problems based on link flows have generally been addressed using variational inequality, to include asymmetric cost functions, or multivariable cost functions. Notable among works that treat the formulation of the equilibrium problem, the existence of a solution and the solution algorithm are the following: Dafermos and Sparrow [4], Smith [5], Dafermos [6, 7], Florian and Spiess [8], Fisk and Nguyen [9], Fisk and Boyce [10], Nagurney [11], Hammond [12], Marcotte and Guelatt [13], Auchmuty [14], Gabriel and Bernstein [15], and Patriksson [16]. All these works represent mainly theoretical contributions and algorithmic implementation.

A variation on Wardrop is the stochastic user equilibrium (SUE), in which no user can unilaterally change routes to improve their perceived travel cost or time. Some stochastic or probabilistic approaches are used, under a similar theoretical framework, to represent different phenomena, such as uncertainty, randomness, and/or heterogeneity of users and route alternatives. The precise formulation depends on how these factors are incorporated. Surveys of this class of models are found in Daganzo and Sheffi [17], Hazelton [18], Ramming [19], Prashker and Bekhor [20], Karoonsoontawong and Lin [21], Li and Huang [22], and Batista et al. [23]. These formulations constitute an extension of the deterministic equilibrium models. Lim and Kim [24] combine trip distribution and route choice model into a single feedback process within a single framework.

Both UE and SUE models typically assume a monotonically increasing relationship between cost and flow (see [25, 26]). This ensures that they are practical to implement and their results are easy to analyze and interpret, but for heavily or hypercongested situations they are greatly lacking in realism.

Another major limitation of flow-based approaches is that they allow the assignment of flow levels that exceed link capacity. Notice that Beckmann model does not specify cost or capacity functions, but it does assume that these functions must be monotonous and growing. Depending on the cost function considered for network links, the flow may exceed the defined capacity, for example, with increasing monotonic functions [27]. This unrealistic result has been interpreted as the additional wait time (or queuing time to enter the network), but the main problem is that it overestimates the number of vehicles which can circulate on the network’s links in a given time period.

A third important drawback with flow-based approaches is that they assume link maximum flows are fixed, exogenous parameters, yet, as the fundamental traffic equation indicates, maximum link flow depends on density which in turn is related to the demand for link use. In other words, link maximum flow is more like a variable than a fixed parameter.

It is precisely these various shortcomings that are remedied by our density-based model, set out below in Section 3.

Note that extensions to flow-based assignment models incorporating an additional restriction barring each link’s flow from exceeding a fixed and exogenous capacity have been developed by Larsson and Patriksson [28], Ferrari [29], and Nie et al. [30]. These modifications do avoid the problem of physically overloaded links, but they still assume a monotonically increasing flow-delay relationship that, as already mentioned, is unrealistic when congestion is high.

Dynamic assignment models have been comprehensively studied in the specialized literature but are not directly related to the approach we propose here. Ran and Boyce [31], Peeta and Ziliakopoulus [32], and Tampere and Viti [33] provide extensive references and discuss developments in dynamic transportation network modelling and analysis and associated computational methods. Surveys have also been published by Boyce et al. [34] and Szeto and Lo [35] and, more recently, by Liu et al [36].

Another approach that better captures the flow-delay relationship (which is increasing under low congestion and decreasing under high congestion) in the fundamental traffic equation uses traffic microsimulation models. A recent survey on the state of the art in traffic assignment models using microsimulation may be found in Calvert et al. [37]. These tools allow physical road network restrictions and the effects of density on traffic flow and speed to be incorporated into the modelling, but they are difficult to apply to large networks. Furthermore, microsimulation models do not use the traffic equilibrium concept since by definition they are dynamic formulations for individual behaviour that base route choice rather on heuristics.

For this reason, microsimulation is an approach that allows us to validate our new model, as we explain in Appendix A. Microsimulation considers each of the vehicles moving within the network as the unit of analysis. Each vehicle makes decisions based on its destination, the behavior of other vehicles, and the travel times perceived between alternative routes that are the result of vehicle interactions. Flow-delay functions in the links are not required; however, these flow-delay functions can be estimated as a result of the individual interactions of the vehicles traveling in the different arcs of the microsimulated network. These interactions include the effect of density on traffic circulation, crossing blocks, etc. Therefore, microsimulation is an approach that we can consider as a benchmark for our new model in small networks. In larger networks, with many vehicles circulating simultaneously, microsimulation has limitations, especially due to the increase in alternative routes, or convergence of results. It is in these cases where we consider that our new model presents the greatest advantages.

#### 3. Formulation of the Proposed Traffic Assignment Model

##### 3.1. General Definitions

Traffic flows typically are not uniform but rather vary across space and time, making them difficult to describe. Nevertheless, their behaviour has traditionally been explained in terms of the relationships between just three traffic variables: flow, speed, and density (the lattermost also known as concentration).

In a deterministic approach, mean speed () is defined as the average speed of multiple vehicles crossing a specific point or link. The flow or volume () is defined as the number of vehicles crossing a given road or highway segment during a given period of time. Finally, density () is defined as the number of vehicles occupying that segment.

For roads or highways with multiple lanes, flow is expressed as vehicles per time unit per number of lanes [38]. The same is true of density, which is also expressed in terms of the total width of the route. A good survey of these models may be consulted in Wang et al. [39] and Kucharski and Drabicki [40].

The oldest and probably simplest macroscopic traffic flow model was proposed by Greenshield (1935). It assumes that, under uninterrupted flow conditions, speed and density are linearly related. Although Greenshield’s formulation is considered to be the tool with the widest scope for traffic flow modelling due to its simplicity and reasonable goodness-of-fit, it has not been universally accepted given that it does not provide a good fit when congestion is low. The formal expression of the Greenshield model is as follows:where is the speed corresponding to the density level , is the free-flow speed, and is the so-called jam density. This last term refers to the extreme traffic density level associated with completely stopped traffic flow, usually in the range of 185–250 vehicles per mile per lane. In other words, it is the density when flow is zero. A very recent study relating speed to vehicles’ use of space in an equilibrium context is reported in Martínez-Díaz and Pérez [41]. There are other, more complex specifications of the relationship between speed and density, but for our purposes Equation (1) is sufficient.

From the fundamental traffic equation, we can relate the flow for a given network link with the flow’s speed and density () as . The speed can in turn be related with the link’s cost or time and its length as . Finally, if we define as the number of vehicles along , then .

An advantage of using the expression is that it allows the cost of a link to be defined as a function of its density rather than its flow and can thereby capture the typical situation depicted in Figure 1 in which congestion results in a reversal of the flow-delay curve so that for a given flow level there may be two travel cost or time alternatives, such as and in the figure. This relationship may be compared with the less realistic portrayal in Figure 2 that is typical of traffic equilibrium models incorporating congestion [3]. Clearly, the assumption in such models of a monotonically increasing flow-cost relationship limits their ability to accurately represent heavily congested conditions.