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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.

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 (

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.

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.

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.

The average queue length on each link can be estimated.

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

Appendix

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 [

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. [

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 [

Both UE and SUE models typically assume a monotonically increasing relationship between cost and flow (see [

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 [

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

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 [

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 [

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. [

For this reason, microsimulation is an approach that allows us to validate our new model, as we explain in Appendix

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 (

For roads or highways with multiple lanes, flow is expressed as vehicles per time unit per number of lanes [

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:

From the fundamental traffic equation, we can relate the flow

An advantage of using the expression

Relatively realistic flow-delay relationship on road link.

Relatively unrealistic flow-delay relationship on road link.

Another advantage of using density instead of flow is that it permanently incorporates a restriction on the maximum flow of the links or routes in the network. Such a restriction is illustrated in Figure

Finally, the cost-density relationship is as shown in Figure

Density-delay relationship on road link.

Let

Let

For all

For all

For all

It is important to observe that if

Actually the travel time

An expression for

The solution to this system of equations gives the traffic equilibrium based on densities that we are seeking. Since we assume

To prove the existence of a solution to the system of equations set forth in (

It should be noted that

Since the set

The following theorem gives an existence result for the inequality

If

Recall that vector field

Let

However, being a solution of

Before setting out the formal proof, however, we define

If

Given that

If

Assume that

Let

Thus, only Case 1 can occur, which implies the existence of at least one solution of

A proposition for a general example of a network that satisfies condition (

About the uniqueness, the following theorem gives us that the quantity of vehicles of arc is unique and, then, the cost of the route is also unique.

Let

We have

We deduce that, for each arc

A big issue for this model is the fact that if we consider

We consider the function

We construct the sequence

Step 2 tests whether the relative error in the flow conservation law is small enough. This algorithm has been implemented to obtain the results of the following section.

The proposed model can be used to conduct a social cost-benefit analysis of a road system project. If we let

A summary of the main characteristics of the classical flow-based and proposed density-based models and the differences between them is laid out in Table

Summary of Main Differences between Flow-Based and Density-Based Models.

Characteristic | Flow-Based Model | Density-Based Model |
---|---|---|

General concept | Assigns vehicles per hour among the different network links for a representative time period. | Assigns vehicles per kilometre among the different network links for a representative instant in time. |

| ||

Equilibrium condition | Satisfies Wardrop’s first principle for all network routes. | Satisfies Wardrop’s first principle for all network routes. |

| ||

State of system | Assumes that all trips in the trip matrix are assigned to the network and that all trips enter and exit within a given modelled period. | Assumes that the total number of vehicles on the network at a given instant is the same at the instant immediately preceding and immediately following. |

| ||

Link capacity | Each link is assumed to have a fixed, exogenous capacity or maximum flow. This capacity will often be exceeded. | Each link’s maximum flow, rather than being a fixed parameter, depends on vehicle density and speed across it and will vary with traffic conditions. Maximum flow is finite. |

| ||

Trip matrix | Assumes a trip matrix (flows between O-D pairs) for the period modelled that is fixed and exogenous to equilibrium cost. | Assumes the number of vehicles for the instant modelled is fixed and exogenous but the trip matrix (flows between O-D pairs) is a result of the equilibrium. |

| ||

Conservation of flow | By construction, assumes conservation of flow at all network nodes including cases where route time or cost is greater than the period modelled (i.e., there are travellers who do not arrive at their destination). | With |

| ||

Densities and queue lengths under congestion | Does not satisfactorily estimate either densities or queue lengths under high congestion. | Always correctly estimates densities and queue lengths, whether under low or high congestion. |

| ||

Fundamental traffic equation | Under high congestion, the | By construction, the |

| ||

Estimation of benefits of road system project or policy | Benefits are estimated in terms of network user time savings. | Benefits are estimated as the change in consumer surplus. |

In this section we discuss the application of the classical flow-based and proposed density-based models to a numerical example of a road network and compare their results. Both low and high congestion scenarios are considered. The two approaches are also used to estimate the benefits of a road system project. Notice that, in the absence of congestion, the density-based model will always provide identical results as the flow-based model.

The road network assumed for this application is an adaptation of the one developed by Nguyen and Dupuis [

Possible Routes for Origin-Destination Pairs.

Route No. | O-D Pair | Route (link sequence) |
---|---|---|

1 | A-K | 1-7-12 |

2 | A-K | 2-4-5-6-12 |

3 | A-K | 2-4-5-11-15 |

4 | A-K | 2-4-10-14-15 |

5 | A-K | 2-9-13-14-15 |

6 | A-K | 2-9-13-14-18-19 |

7 | A-K | 2-9-16-17-19 |

8 | A-M | 2-4-5-11-18 |

9 | A-M | 2-4-10-14-18 |

10 | A-M | 2-9-13-14-18 |

11 | A-M | 2-9-16-17 |

12 | C-K | 3-4-5-6-12 |

13 | C-K | 3-4-5-11-15 |

14 | C-K | 3-4-10-14-15 |

15 | C-K | 3-9-13-14-15 |

16 | C-K | 8-13-14-15 |

17 | C-K | 8-13-14-18-19 |

18 | C-K | 8-16-17-19 |

19 | C-M | 3-4-5-11-18 |

20 | C-M | 3-4-10-14-18 |

21 | C-M | 3-9-13-14-18 |

22 | C-M | 8-13-14-18 |

23 | C-M | 8-16-17 |

Nguyen-Dupuis Network (Adaptation).

We define the relationship between speed and density on each link defined by (

For the proposed density-based model, the flow of vehicles travelling between each of the four O-D pairs is defined by matrices

The gap function that we consider is

We solved (

A function that meets both conditions is the black curve shown in Figure

Flow-Delay Curves for Flow-Based and Density-Based Approaches.

Using the same

Equilibrium Route Assignments, Density-Based Approach (high congestion).

Route | O-D Pair | _{ p } | _{ p } |
---|---|---|---|

1 | A-K | 224.3485 | 0.3592 |

2 | A-K | 0.0000 | 0.4872 |

3 | A-K | 36.4514 | 0.3592 |

4 | A-K | 0.0000 | 0.4919 |

5 | A-K | 31.0230 | 0.3592 |

6 | A-K | 0.0000 | 0.3937 |

7 | A-K | 0.0000 | 0.3937 |

8 | A-M | 41.6268 | 0.3537 |

9 | A-M | 0.0000 | 0.4864 |

10 | A-M | 34.6943 | 0.3537 |

11 | A-M | 68.8369 | 0.3537 |

12 | C-K | 0.0000 | 0.8222 |

13 | C-K | 144.2187 | 0.6942 |

14 | C-K | 0.0000 | 0.8269 |

15 | C-K | 85.1820 | 0.6942 |

16 | C-K | 58.0120 | 0.6942 |

17 | C-K | 0.0000 | 0.7288 |

18 | C-K | 0.0000 | 0.7288 |

19 | C-M | 126.0871 | 0.6888 |

20 | C-M | 0.0000 | 0.8214 |

21 | C-M | 78.5134 | 0.6888 |

22 | C-M | 54.8398 | 0.6888 |

23 | C-M | 253.6105 | 0.6888 |

For high congestion scenario, the gap function for density-based model is equal at the equilibrium to

Equilibrium Link Assignments, Density-Based Approach (high congestion).

Link | _{ a } | _{ a } | _{ a } | | _{ a } | _{ a } | _{ a } |
---|---|---|---|---|---|---|---|

1 | 126.5223 | 5 | | 25 | 24.6955 | 0.2025 | 624.9073 |

2 | 17.9292 | 1 | 17.9292 | 25 | 32.0708 | 0.0312 | 575.0042 |

3 | 226.9215 | 10 | 22.6921 | 25 | 27.3079 | 0.3662 | 619.6738 |

4 | 84.1265 | 3 | | 25 | 21.9578 | 0.1366 | 615.7452 |

5 | 37.1041 | 2 | 18.5520 | 25 | 31.4480 | 0.0636 | 583.4237 |

6 | 0.0000 | 9 | 0.0000 | 25 | 50.0000 | 0.1800 | 0.0000 |

7 | 50.6089 | 2 | | 25 | 24.6955 | 0.0810 | 624.9073 |

8 | 306.1953 | 10 | | 25 | 19.3805 | 0.5160 | 593.4209 |

9 | 83.0978 | 5 | 16.6196 | 25 | 33.3804 | 0.1498 | 554.7684 |

10 | 0.0000 | 10 | 0.0000 | 25 | 50.0000 | 0.2000 | 0.0000 |

11 | 36.7618 | 2 | 18.3809 | 25 | 31.6191 | 0.0633 | 581.1875 |

12 | 47.2174 | 2 | 23.6087 | 25 | 26.3913 | 0.0758 | 623.0643 |

13 | 26.1439 | 2 | 13.0720 | 25 | 36.9280 | 0.0542 | 482.7217 |

14 | 32.8045 | 2 | 16.4023 | 25 | 33.5977 | 0.0595 | 551.0790 |

15 | 38.0592 | 2 | 19.0296 | 25 | 30.9704 | 0.0646 | 589.3545 |

16 | 45.8212 | 3 | 15.2737 | 25 | 34.7263 | 0.0864 | 530.3997 |

17 | 45.8212 | 3 | 15.2737 | 25 | 34.7263 | 0.0864 | 530.3997 |

18 | 32.3097 | 2 | 16.1549 | 25 | 33.8451 | 0.0591 | 546.7634 |

19 | 0.0000 | 2 | 0.0000 | 25 | 50.0000 | 0.0400 | 0.0000 |

Equilibrium Route Assignments, Density-Based Approach (low congestion).

Route | O-D Pair | _{ p } | _{ p } |
---|---|---|---|

1 | A-K | 126.9098 | 0.2507 |

2 | A-K | 0.0000 | 0.4214 |

3 | A-K | 0.0000 | 0.3031 |

4 | A-K | 0.0000 | 0.4280 |

5 | A-K | 0.0000 | 0.3031 |

6 | A-K | 0.0000 | 0.3451 |

7 | A-K | 0.0000 | 0.3451 |

8 | A-M | 77.0756 | 0.3051 |

9 | A-M | 0.0000 | 0.4300 |

10 | A-M | 24.9098 | 0.3051 |

11 | A-M | 51.7650 | 0.3051 |

12 | C-K | 0.0000 | 0.6404 |

13 | C-K | 122.2635 | 0.5221 |

14 | C-K | 0.0000 | 0.6470 |

15 | C-K | 28.2865 | 0.5221 |

16 | C-K | 109.4890 | 0.5221 |

17 | C-K | 0.0000 | 0.5640 |

18 | C-K | 0.0000 | 0.5640 |

19 | C-M | 38.6385 | 0.5240 |

20 | C-M | 0.0000 | 0.6489 |

21 | C-M | 18.8462 | 0.5240 |

22 | C-M | 37.2549 | 0.5240 |

23 | C-M | 166.2343 | 0.5240 |

Table

For the low congestion case, Table

Equilibrium Link Assignments, Density-Based Approach (low congestion).

Link | _{ a } | _{ a } | _{ a } | | _{ a } | _{ a } | _{ a } |
---|---|---|---|---|---|---|---|

1 | 70.5055 | 5 | 14.1011 | 25 | 35.8989 | 0.1393 | 506.2138 |

2 | 13.7323 | 1 | 13.7323 | 30 | 36.2677 | 0.0276 | 498.0380 |

3 | 94.3981 | 10 | 9.4398 | 30 | 40.5602 | 0.2465 | 382.8805 |

4 | 55.1230 | 3 | 18.3743 | 25 | 31.6257 | 0.0949 | 581.1005 |

5 | 36.7487 | 2 | 18.3743 | 25 | 31.6257 | 0.0632 | 581.1005 |

6 | 0.0000 | 9 | 0.0000 | 30 | 50.0000 | 0.1800 | 0.0000 |

7 | 28.2022 | 2 | 14.1011 | 25 | 35.8989 | 0.0557 | 506.2138 |

8 | 226.1471 | 10 | 22.6147 | 25 | 27.3853 | 0.3652 | 619.3104 |

9 | 39.2293 | 5 | 7.8459 | 25 | 42.1541 | 0.1186 | 330.7354 |

10 | 0.0000 | 10 | 0.0000 | 25 | 50.0000 | 0.2000 | 0.0000 |

11 | 36.7487 | 2 | 18.3743 | 30 | 31.6257 | 0.0632 | 581.1005 |

12 | 28.2022 | 2 | 14.1011 | 30 | 35.8989 | 0.0557 | 506.2138 |

13 | 22.1236 | 2 | 11.0618 | 25 | 38.9382 | 0.0514 | 430.7267 |

14 | 22.1236 | 2 | 11.0618 | 30 | 38.9382 | 0.0514 | 430.7267 |

15 | 26.1893 | 2 | 13.0947 | 30 | 36.9053 | 0.0542 | 483.2632 |

16 | 36.7122 | 3 | 12.2374 | 30 | 37.7626 | 0.0794 | 462.1156 |

17 | 36.7122 | 3 | 12.2374 | 25 | 37.7626 | 0.0794 | 462.1156 |

18 | 28.7754 | 2 | 14.3877 | 30 | 35.6123 | 0.0562 | 512.3795 |

19 | 0.0000 | 2 | 0.0000 | 30 | 50.0000 | 0.0400 | 0.0000 |

We now compare the above results for the density-based model with those of a classical flow-based model.

We would expect the link flow and cost results obtained for_{1} under this approach to differ significantly from those obtained above for_{1} under the density-based approach, reflecting the difference between the two approaches when congestion is high. When_{2} and_{2} are compared, however, the link flow and cost results should be relatively similar, mirroring the greater similarity between the two approaches when congestion is low.

The link costs and flows under the two approaches for the high congestion case are brought together in Table

Equilibrium Link Assignments, Both Approaches (high congestion).

Link | Maximum flow for Density-Based model | Flow-Based Model | Density-Based Model | ||
---|---|---|---|---|---|

| _{ a } | _{ a } | _{ a } | _{ a } | |

1 | 625 | 0.2002 | | 0.2025 | 624.9073 |

2 | 625 | 0.0306 | 546.7128 | 0.0312 | 575.0042 |

3 | 625 | 0.3090 | 551.9899 | 0.3662 | 619.6738 |

4 | 625 | 0.1122 | | 0.1366 | 615.7452 |

5 | 625 | 0.0748 | | 0.0636 | 583.4237 |

6 | 625 | 0.1848 | 0.0028 | 0.1800 | 0.0000 |

7 | 625 | 0.0801 | | 0.0810 | 624.9073 |

8 | 625 | 0.4422 | | 0.5160 | 593.4209 |

9 | 625 | 0.1332 | 461.0855 | 0.1498 | 554.7684 |

10 | 625 | 0.2053 | 0.0056 | 0.2000 | 0.0000 |

11 | 625 | 0.0748 | | 0.0633 | 581.1875 |

12 | 625 | 0.0801 | | 0.0758 | 623.0643 |

13 | 625 | 0.0643 | 572.2073 | 0.0542 | 482.7217 |

14 | 625 | 0.0643 | 572.2129 | 0.0595 | 551.0790 |

15 | 625 | 0.0678 | 596.6946 | 0.0646 | 589.3545 |

16 | 625 | 0.0996 | 586.8883 | 0.0864 | 530.3997 |

17 | 625 | 0.0996 | 586.8883 | 0.0864 | 530.3997 |

18 | 625 | 0.0705 | 613.1272 | 0.0591 | 546.7634 |

19 | 625 | 0.0411 | 0.0154 | 0.0400 | 0.0000 |

Link Costs, Both Approaches (high congestion).

Link Flows, Both Approaches (high congestion).

A metric which can compare both models is the relative distance between the link costs and the link flows of the density-based model and the flow-based model. More precisely, we can consider the following quantities, where _{2}) and density-based (matrix_{2}) approaches are given in Table

Equilibrium Link Assignments, Both Approaches (low congestion).

Link | Maximum flow for Density-Based model | Flow-Based Model | Density-Based Model | ||
---|---|---|---|---|---|

| _{ a } | _{ a } | _{ a } | _{ a } | |

1 | 625 | 0.1410 | 499.9880 | 0.1393 | 506.2138 |

2 | 900 | 0.0282 | 500.0120 | 0.0276 | 498.0380 |

3 | 900 | 0.2426 | 380.2809 | 0.2465 | 382.8805 |

4 | 625 | 0.0947 | 562.9694 | 0.0949 | 581.1005 |

5 | 625 | 0.0631 | 562.9647 | 0.0632 | 581.1005 |

6 | 900 | 0.1848 | 0.0024 | 0.1800 | 0.0000 |

7 | 625 | 0.0564 | 499.9880 | 0.0557 | 506.2138 |

8 | 625 | 0.3579 | 619.7191 | 0.3652 | 619.3104 |

9 | 625 | 0.1151 | 317.3234 | 0.1186 | 330.7354 |

10 | 625 | 0.2053 | 0.0048 | 0.2000 | 0.0000 |

11 | 900 | 0.0631 | 562.9623 | 0.0632 | 581.1005 |

12 | 900 | 0.0564 | 499.9904 | 0.0557 | 506.2138 |

13 | 625 | 0.0530 | 456.7820 | 0.0514 | 430.7267 |

14 | 900 | 0.0530 | 456.7867 | 0.0514 | 430.7267 |

15 | 900 | 0.0564 | 499.9982 | 0.0542 | 483.2632 |

16 | 900 | 0.0821 | 480.2606 | 0.0794 | 462.1156 |

17 | 625 | 0.0821 | 480.2606 | 0.0794 | 462.1156 |

18 | 900 | 0.0583 | 519.7509 | 0.0562 | 512.3795 |

19 | 900 | 0.0411 | 0.0115 | 0.0400 | 0.0000 |

Link Costs, Both Approaches (low congestion).

Link Flows, Both Approaches (low congestion).

The relative difference obtained for costs and flows in both scenarios are reported in Table

Costs and Flows Relative Difference, Both Approaches (two congestion scenarios).

Scenario | N° of vehicles | Flows | Link Cost Relative distance | Link Flow Relative distance |
---|---|---|---|---|

high congestion | _{1} = 1237.4 | _{1} =2460 | 0.1382 | 0.0994 |

low congestion | _{2} = 801.67 | _{2} = 2000 | 0.0219 | 0.0303 |

Finally, the gap function for flow-based model with high congestion is

Both the flow matrix containing the

Trip Diagram from a Survey.

On the vertical axis are the 7 individuals surveyed after expanding the representative sample while on the horizontal axis is the time of day. As can be seen, Individual 1 departed at about 6:45 and arrived at 8:00. Individual 2 set out at 6:30 and arrived between 7:15 and 7:30. Individual 3 made two trips, the first one starting shortly after 7:00 and ending at 7:45 and the second one beginning soon after 8:15 and ending at about 8:45. By proceeding in this manner, the individuals or vehicles in the network at a given moment can be successively identified. For example, at 7:30 there were 6 individuals in the network. Since the survey also gathered the trip origins and destinations, an estimate of

This paper developed, validated, and implemented a deterministic traffic assignment model based on link densities. The proposed formulation solves a variational inequality in a manner consistent with the relationship between flow, cost (the inverse of speed), and density given by the fundamental traffic equation for each network link. The solution thus derived is a network traffic equilibrium of link densities that satisfies Wardrop’s first principle.

The model’s use of link densities to determine traffic equilibrium has a number of advantages over the traditional flow-based approach. Firstly, it recognizes that link capacity (or maximum flow) is not fixed but rather is a function of density levels. In other words, the maximum flow that can cross a link is variable and depends on demand, that is, density. Secondly, and, as a consequence of the foregoing, the proposed approach prevents flows from exceeding the links’ theoretical maximum capacities. These maximum flows or capacities are determined as a function of the speed and density on each link as given by the fundamental traffic equation. Thirdly, the density-based model identifies whether a reduced flow level on a given link is due to low latent demand for its use (e.g., low density) or, on the contrary, to high congestion (e.g., high density) reducing the flow that can use the link, thereby generating traffic queues and longer delays.

An added benefit is that the estimates the model generates of link densities and their impacts on flow levels that can effectively use network links provide important data for planning road networks, calculating toll road revenues, and designing road pricing mechanisms based on traffic saturation levels.

The proposed model was validated using a traffic microsimulator (see Appendix A) and applied to a numerical example based on the well-known Nguyen-Dupuis network. It was then compared to the classical flow-based approach for a road system project involving the addition of a link. The results showed that flow-based models tend to underestimate network equilibrium cost and allow flows that often exceed the links’ theoretical capacities even in low congestion scenarios. Furthermore, by underestimating costs the traditional approach also underestimates the net benefits of road system expansion projects or policies such as road pricing that lower vehicles use. These benefits arise from the reduction in the use of hypercongested networks attendant upon the implementation of a new link or a vehicle use disincentive policy, which as well as cutting costs increases road network capacity. The density-based approach, by contrast, takes these improvements into account.

In light of the above, we believe our proposed model based on densities constitutes a new and innovative methodology for analyzing and evaluating road and other transport projects and policies involving networks subject to congestion. The model’s density-based approach overcomes a number of limitations inherent in the classical methodologies based on route flows.

A disadvantage is the difficulty in applying an algorithm whose solution satisfies the flow conservation property. A future work on this topic could consist of minimizing the relative error in flow conservation over the set of solutions of

The validation of the density-based model was performed for a small road network, the same as the one developed in De Grange et al. [

Road Network Microsimulated in Aimsun.

The microsimulations were used to calibrate the relationship between speed and density, which we defined as

Microsimulation of Cost-Density Relationship.

To generate the input data, we assumed two O-D pairs under different demand levels between pair

O-D Trips and Routes for Microsimulation.

O-D pair | Route link sequences |
---|---|

| p1: |

| p2: |

| p3: |

| p4: |

Using the

Link Flow Dispersion: Microsimulation vs Density-Based Model.

Dispersion of No. of Vehicles on Links: Microsimulation vs Density-Based Model.

Link Cost Dispersion: Microsimulation vs Density-Based Model.

O-D Pair Dispersion: Microsimulation vs Density-Based Model.

Finally, the maximum gap function for flow-based model (considering all scenarios) was

In what follows we develop a general example of a road network under congestion that satisfies the hypothesis (

Suppose that, for all

For each origin-destination pair

Given an

First, we choose an

Second, we show that no link’s cost is infinite with vector

Now consider a link

For greater clarity, in what follows we denote the vector

Given a sequence

The quantity

Thus, we have demonstrated

The optimization problem that gives the system’s optimal equilibrium assignment is as follows:

We now prove that the constraint

There exists a number

Let

Having thus proved that, for all sufficiently small

The exercises and numerical analyses that support the results presented in this article are based on large amounts of data. The reader can personally contact the authors if these data are required.

The research presented corresponds to the academic work carried out by the authors within their academic commitment at the School of Industrial Engineering of the Universidad Diego Portales. No additional resources were used.

The authors declare that there are no conflicts of interest regarding the publication of this paper.