Abstract
This paper develops a continuous approximation model for analyzing the effect of road pricing on the spatial distribution of traffic flow. The traffic flow density, which describes traffic flow as a function of position, is derived for a rectangular city with a grid network. The analytical expression for the traffic flow density demonstrates how the toll level, the size and shape of the toll area, and the shape of the city affect the spatial distribution of traffic flow. As the size of the toll area increases, reducing the traffic flow density at the city center becomes difficult. As the aspect ratio of the toll area increases, the traffic flow density at the city center increases. The shape of the city has less impact on the traffic flow density than the shape of the toll area.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Road pricing, which is regarded as one of the most effective means to reduce traffic congestion, has been introduced in several cities such as London, Singapore, and Stockholm. A toll is imposed on vehicles per crossing of the cordon line surrounding a toll area (cordon-based pricing), per distance of travel (distance-based pricing), per time of travel (time-based pricing), or per day in a toll area (area-based pricing). Road pricing encourages travelers to adjust the number of trips and the travel route, thus affecting the spatial distribution of traffic flow. Examining how road pricing affects the spatial distribution of traffic flow provides a fundamental understanding of the effect of road pricing.
Two major approaches for addressing road pricing are based on discrete network models and continuous approximation models. The former rely on detailed traffic data and focus on developing efficient algorithms for large-scale problems, whereas the latter rely on approximated functions of travel demand and focus on finding fundamental relationships between variables. By providing analytical solutions, the continuous approximation models can help reveal managerial insights and supplement the discrete network models. The discrete network models on road pricing have studied cordon-based pricing [1,2,3,4,5,6] and area-based pricing [7,8,9,10,11,12,13] on actual networks. May and Milne [14] compared cordon-based, distance-based, time-based, and delay-based pricing schemes. The continuous approximation models have studied cordon-based and area-based pricing in idealized structures such as a linear city [15,16,17,18,19], a circular city [20], a grid network [21], and a radial-arc network [22, 23].
In this paper, we present a continuous approximation model for analyzing the effect of road pricing on the spatial distribution of traffic flow. The continuous approximation yields an analytical expression for the spatial distribution of traffic flow. The analytical expression leads to a better understanding of how the toll level, the size and shape of the toll area, and the shape of the city affect the spatial distribution of traffic flow. The model is thus useful for determining the size and shape of the toll area and the toll level. The model assumes a grid network, which can be found in Kyoto, Beijing, and many cities in North America. Miyagawa [21] dealt with the effect of road pricing in a grid network on the total amount of traffic flow inside the toll area. The effect of road pricing, however, varies according to the location. We then extend the scope to the spatial distribution of traffic flow, which describes traffic flow as a function of position. This extension allows us to examine the locational variation of traffic flow and identify potential congestion areas.
To describe the spatial distribution of traffic flow, the concept of traffic flow density was introduced by Smeed [24] and Holroyd [25]. The traffic flow density expresses the amount of traffic flow passing through a point as a density function. The traffic flow density has been derived for a circular city with the Euclidean distance [26], a circular city with a grid network [27] and a radial-arc network [28], and a square city with a grid network [28]. Subsequent works have considered a sector-shaped city with a radial-arc network [29], a rectangular city with a grid network and a barrier to travel [30], and a circular city with a radial-arc network and road pricing [23]. The time-dependent traffic flow density has also been derived [31, 32]. The traffic flow density in a rectangular city with a grid network and road pricing has not been derived previously.
The remainder of this paper is organized as follows. The next section introduces a grid network model. The following section derives the spatial distribution of traffic flow without road pricing. The penultimate section derives the spatial distribution of traffic flow in road pricing. The final section presents concluding remarks.
2 Grid network model
Consider a rectangular city with side lengths \(a_1\) and \(a_2\), as shown in Fig. 1. Any point in the city is expressed as \((x, y)\,(0\le x\le a_1, 0\le y\le a_2)\), where the origin of the coordinate system O is at the southwest corner of the city. The city has a dense grid road network. The shortest distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is then given by the rectilinear distance \(|x_1-x_2|+|y_1-y_2|\). A toll area is represented as a rectangle with side lengths \(b_1\) and \(b_2\) located at the center of the city \((a_1/2,a_2/2)\). Let \((x_b, y_b)\) be the southwest corner of the toll area, that is, \(x_b=(a_1-b_1)/2, y_b=(a_2-b_2)/2\). The road pricing system in the city is area pricing where all vehicles driving inside the toll area are charged a fixed toll t.
Origins and destinations are assumed to be uniformly distributed in the city. The uniform distribution is important as the first approximation and serves as a basis for further analysis with more realistic distributions. The travel cost C for trips of length R is defined as
where \(\alpha \ (>0)\) is the travel cost per unit distance. Every traveler is assumed to minimize the number of turns in choosing the least cost route. If there exist two least cost routes with an equal number of turns, the amount of traffic flow is equally divided between the two routes. The travel demand D is assumed to decrease with the travel cost C and expressed as
where \(D_0\) is the travel demand when \(C=0\) and \(\beta \ (>0)\) is a parameter for elasticity. The exponential function has been widely used in spatial interaction models [33].
The traffic flow density is defined as the amount of traffic flow passing through a point in the city. Let \(f_x\) and \(f_y\) be the densities of traffic flow passing through a point (x, y) in the east–west and north–south directions, respectively. The amount of traffic flow passing through the small segment with length \(\textrm{d}y\) (\(\textrm{d}x\)) in the east–west (north–south) direction is then \(f_x\textrm{d}y\) (\(f_y\textrm{d}x\)). The amount of traffic flow passing through the segment between two points \((x,y_1)\) and \((x,y_2)\) in the east–west direction, denoted by \(V_x\), is given by
and the amount of traffic flow passing through the segment between two points \((x_1,y)\) and \((x_2,y)\) in the north–south direction, denoted by \(V_y\), is given by
as shown in Fig. 1.
3 Traffic flow density without road pricing
In this section, we derive the traffic flow density when no toll is charged, that is, \(t=0\). Although Vaughan [28] and Miyagawa [30] derived the traffic flow density without road pricing, they assumed inelastic travel demand (\(\beta =0\) in Eq. (2)). We relax the assumption to elastic travel demand.
First, we derive the traffic flow density in the east–west direction. Let \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\) be origin and destination of trips, respectively. The traffic passes through the infinitesimal segment between two points (x, y) and \((x, y+\textrm{d}y)\) in the easterly direction if
as shown in Fig. 2a, or
as shown in Fig. 2b. Since the amount of traffic flow passing through the segment in the westerly direction is the same as that in the easterly direction, the amount of traffic flow passing through the segment in the east–west direction is
Letting \(\textrm{d}y\rightarrow +0\) gives the traffic flow density in the east–west direction
Next, we derive the traffic flow density in the north–south direction. The traffic passes through the infinitesimal segment between two points (x, y) and \((x+\textrm{d}x, y)\) in the northerly direction if
as shown in Fig. 3a, or
as shown in Fig. 3b. Since the amount of traffic flow passing through the segment in the southerly direction is the same as that in the northerly direction, the amount of traffic flow passing through the segment in the north–south direction is
Letting \(\textrm{d}x\rightarrow +0\) gives the traffic flow density in the north–south direction
4 Traffic flow density in road pricing
In this section, we derive the traffic flow density in road pricing. We then examine how the toll level, the size and shape of the toll area, and the shape of the city affect the traffic flow density. To see the effect of road pricing on the traffic flow inside the toll area, we focus on the traffic flow density inside the toll area.
Road pricing can affect not only travel demand but also travel routes, reducing the amount of traffic flow passing through the toll area. The traveler passes through the toll area only if the travel cost of passing through the toll area is smaller than that of making a detour around the toll area. Miyagawa [21] showed that the traveler passes through the toll area if both origin and destination are inside the gray regions in Fig. 4. For example, the travel cost between origin
and destination
is given by
The traveler then passes through the toll area if \(y_1>y_b+t/(2\alpha ), y_2\le y_b+b_2-t/(2\alpha )\).
The traffic flow density in the east–west direction is obtained by considering the traffic passing through the infinitesimal segment between two points (x, y) and \((x, y+\textrm{d}y)\), as shown in Fig. 5. If \(y_b\le y\le y_b+t/(2\alpha )\) or \(y_b+b_2-t/(2\alpha )\le y\le y_b+b_2\), travelers whose origin and destination are outside the toll area do not pass through the segment. The traffic flow density in the east–west direction is given by
where
Note that \(f_x^\textrm{a}, f_x^\textrm{b}, f_x^\textrm{c}, f_x^\textrm{d}\) correspond to the four cases shown in Fig. 5. If \(y_b+t/(2\alpha )<y<y_b+b_2-t/(2\alpha )\), travelers whose origin and destination are outside the toll area can pass through the segment, and \(f_x^\textrm{a}\) and \(f_x^\textrm{d}\) are replaced by
The traffic flow density in the north–south direction is obtained by considering the traffic passing through the infinitesimal segment between two points (x, y) and \((x+\textrm{d}x, y)\), as shown in Fig. 6. If \(x_b\le x\le x_b+t/(2\alpha )\) or \(x_b+b_1-t/(2\alpha )\le x\le x_b+b_1\), travelers whose origin and destination are outside the toll area do not pass through the segment. The traffic flow density in the north–south direction is given by
where
Note that \(f_y^\textrm{a}, f_y^\textrm{b}, f_y^\textrm{c}, f_y^\textrm{d}\) correspond to the four cases shown in Fig. 6. If \(x_b+t/(2\alpha )<x<x_b+b_1-t/(2\alpha )\), travelers whose origin and destination are outside the toll area can pass through the segment, and \(f_y^\textrm{a}\) and \(f_y^\textrm{d}\) are replaced by
The traffic flow densities \(f=f_x+f_y\) inside the toll area for the three toll levels (\(t=0, 0.1, 0.5\)) are shown in Fig. 7, where \(a_1=a_2=1, b_1=b_2=0.6, D_0=1, \alpha =1, \beta =1\). As the toll level increases, the traffic flow density inside the toll area decreases. Note that the traffic flow density jumps at \(x=x_b+t/(2\alpha ), x_b+b_1-t/(2\alpha )\) and \(y=y_b+t/(2\alpha ), y_b+b_2-t/(2\alpha )\). This is because for \(x_b+t/(2\alpha )<x<x_b+b_1-t/(2\alpha )\) and \(y_b+t/(2\alpha )<y<y_b+b_2-t/(2\alpha )\), some travelers whose origin and destination are outside the toll area pass through the toll area, as shown in Fig. 4.
To examine the effect of the size of the toll area on the traffic flow density, we consider two sizes of the toll area: \(b_1=b_2=0.4\) and \(b_1=b_2=0.6\). The traffic flow densities inside the toll area at \(y=a_2/2\) for the two sizes are shown in Fig. 8, where \(a_1=a_2=1, D_0=1, \alpha =1, \beta =1\). Note that if the toll area is small, the traffic flow density at the city center decreases rapidly with the toll level t. This is because as the size of the toll area becomes smaller, the detour distance to avoid the toll area decreases and much traffic makes a detour around the toll area. It follows that to reduce the amount of traffic flow at the city center, the toll area should be small and the toll level should be high.
To examine the effect of the shape of the toll area on the traffic flow density, we consider two shapes of the toll area: \(b_1/b_2=2\,(b_1=0.4\sqrt{2}, b_2=0.4/\sqrt{2})\) and \(b_1/b_2=3\,(b_1=0.4\sqrt{3}, b_2=0.4/\sqrt{3})\). The traffic flow densities inside the toll area at \(y=a_2/2\) for the two shapes are shown in Fig. 9, where \(a_1=a_2=1, D_0=1, \alpha =1, \beta =1\). The traffic flow density for \(b_1/b_2=1\,(b_1=b_2=0.4)\) is shown in Fig. 8a. Note that as the aspect ratio of the toll area \(b_1/b_2\) becomes larger, the traffic flow density at the city center increases. Thus, reducing the amount of traffic flow at the city center is easier when the toll area is a square.
To examine the shape of the city on the traffic flow density, we consider two shapes of the city: \(a_1/a_2=2\,(a_1=\sqrt{2}, a_2=1/\sqrt{2})\) and \(a_1/a_2=3\,(a_1=\sqrt{3}, a_2=1/\sqrt{3})\). The traffic flow densities inside the toll area at \(y=a_2/2\) for the two shapes are shown in Fig. 10, where \(b_1=b_2=0.4, D_0=1, \alpha =1, \beta =1\). The traffic flow density for \(a_1/a_2=1\,(a_1=a_2=1)\) is shown in Fig. 8a. Note that the shape of the city has less impact on the traffic flow density than the shape of the toll area.
5 Conclusions
This paper has developed a continuous approximation model for analyzing the spatial distribution of traffic flow in road pricing. The traffic flow density has been derived for a rectangular city with a grid network. The model provides a fundamental understanding of the effect of road pricing on the spatial distribution of traffic flow.
The traffic flow density is useful for designing road pricing systems. The analytical expression for the traffic flow density demonstrates how the toll level, the size and shape of the toll area, and the shape of the city affect the spatial distribution of traffic flow. As the size of the toll area increases, reducing the traffic flow density at the city center becomes difficult. As the aspect ratio of the toll area increases, the traffic flow density at the city center increases. The shape of the city has less impact on the traffic flow density than the shape of the toll area. Note that finding these relationships by using discrete network models requires the computation of traffic flow for various combinations of the parameters. These relationships help planners determine the size and shape of the toll area and the toll level.
Not only area-based pricing but also other pricing schemes such as cordon-based and distance-based pricing have been studied. In particular, marginal cost pricing is the first-best pricing in that the social welfare is maximized by internalizing externalities of congestion [34]. Future research should compare these pricing schemes in terms of the spatial distribution of traffic flow.
References
Sumalee, A.: Optimal road user charging cordon design: A heuristic optimization approach. Computer-Aided Civil and Infrastructure Engineering 19, 377–392 (2004)
Zhang, X., Yang, H.: The optimal cordon-based network congestion pricing problem. Transportation Research Part B 38, 517–537 (2004)
Ekström, J., Sumalee, A., Lo, H.K.: Optimizing toll locations and levels using a mixed integer linear approximation approach. Transportation Research Part B 46, 834–854 (2012)
Zheng, N., Waraich, R.A., Axhausen, K.W., Geroliminis, N.: A dynamic cordon pricing scheme combining the macroscopic fundamental diagram and an agent-based traffic model. Transportation Research Part A 46, 1291–1303 (2012)
Kristoffersson, I.: Impacts of time-varying cordon pricing: Validation and application of mesoscopic model for Stockholm. Transport Policy 28, 51–60 (2013)
Simoni, M.D., Pel, A.J., Waraich, R.A., Hoogendoorn, S.P.: Marginal cost congestion pricing based on the network fundamental diagram. Transportation Research Part C 56, 221–238 (2015)
Maruyama, T., Harata, N.: Difference between area-based and cordon-based congestion pricing: Investigation by trip-chain-based network equilibrium model with nonadditive path costs. Transportation Research Record 1964, 1–8 (2006)
Maruyama, T., Sumalee, A.: Efficiency and equity comparison of cordon- and area-based road pricing schemes using a trip-chain equilibrium model. Transportation Research Part A 41, 655–671 (2007)
Zhang, L., Liu, H., Sun, D.: Comparison and optimization of cordon and area pricings for managing travel demand. Transport 29, 248–259 (2014)
Zheng, N., Rérat, G., Geroliminis, N.: Time-dependent area-based pricing for multimodal systems with heterogeneous users in an agent-based environment. Transportation Research Part C 62, 133–148 (2016)
Gu, Z., Shafiei, S., Liu, Z., Saberi, M.: Optimal distance-and time-dependent area-based pricing with the network fundamental diagram. Transportation Research Part C 95, 1–28 (2018)
Zheng, N., Geroliminis, N.: Area-based equitable pricing strategies for multimodal urban networks with heterogeneous users. Transportation Research Part A 136, 357–374 (2020)
Chen, Y., Zheng, N., Vu, H.L.: A novel urban congestion pricing scheme considering travel cost perception and level of service. Transportation Research Part C 125, 103042 (2021)
May, A.D., Milne, D.S.: Effects of alternative road pricing systems on network performance. Transportation Research Part A 34, 407–436 (2000)
Mun, S., Konishi, K., Yoshikawa, K.: Optimal cordon pricing. Journal of Urban Economics 54, 21–38 (2003)
Mun, S., Konishi, K., Yoshikawa, K.: Optimal cordon pricing in a non-monocentric city. Transportation Research Part A 39, 723–736 (2005)
Verhoef, E.T.: Second-best congestion pricing schemes in the monocentric city. Journal of Urban Economics 58, 367–388 (2005)
Li, Z.-C., Lam, W.H.K., Wong, S.C.: Modeling intermodal equilibrium for bimodal transportation system design problems in a linear monocentric city. Transportation Research Part B 46, 30–49 (2012)
Tsai, J.-F., Lu, S.-Y.: Reducing traffic externalities by multiple-cordon pricing. Transportation 45, 597–622 (2018)
Li, Z.-C., Wang, Y.-D., Lam, W.H.K., Sumalee, A., Choi, K.: Design of sustainable cordon toll pricing schemes in a monocentric city. Networks and Spatial Economics 14, 133–158 (2014)
Miyagawa, M.: Effect of road pricing on traffic volume and toll revenue in grid network. Journal of the Operations Research Society of Japan 66, 142–152 (2023)
Miyagawa, M.: Cordon and area road pricing in radial-arc network. Journal of the Operations Research Society of Japan 62, 121–131 (2019)
Miyagawa, M.: Road pricing and spatial distribution of traffic flow in radial-arc network Japan. Journal of Industrial and Applied Mathematics 40, 589–602 (2023)
Smeed, R.: Road development in urban areas. Journal of the Institution of Highway Engineering 10, 5–30 (1963)
Holroyd, E.: Routing traffic in a square city to minimize route-crossing. Beitrage zur Theorie des Verkehrsflusses, Strassenbau und Strassenverkehrstechnik 86, 166–175 (1968)
Ohtsu, S., Koshizuka, T.: A basic study on a flow distribution in an urban area. Journal of the City Planning Institute of Japan 33, 319–324 (1998). (in Japanese)
Vaughan, R.J., Doyle, E.: Rectangular routeing in Smeed’s city. Transportation Research Part B 13, 181–182 (1979)
Vaughan, R.J.: Urban Spatial Traffic Patterns. Pion, London (1987)
Tanaka, K., Kurita, O.: Spatial distribution of traffic in a sector-shaped city under radial-arc routing system: An analysis on the optimal assignment of road area for designing a congestion-free city. Journal of the City Planning Institute of Japan 36, 865–870 (2001). (in Japanese)
Miyagawa, M.: Spatial distribution of traffic flow in a rectangular city with a grid network and a rectangular barrier. FORMA 37, 21–33 (2022)
Tanaka, K., Kurita, O.: Mathematical analysis of the radial-ring road network in the light of the traffic flow density: Spatial and temporal distribution of traffic during the peak commuting period. Transactions of the Japan Society for Industrial and Applied Mathematics 13, 321–352 (2003). (in Japanese)
Tanaka, K., Kurita, O.: Time-dependent traffic flow in a rectangular city with rectilinear distance. FORMA 18, 165–186 (2003)
Roy, J.R.: Spatial Interaction Modelling. Springer, Berlin (2010)
Walters, A.A.: The theory and measurement of private and social cost of highway congestion. Econometrica 29, 676–699 (1961)
Acknowledgements
This research was supported by JSPS KAKENHI Grant Number JP21K04546. I am grateful to anonymous reviewers and the editor for their helpful comments and suggestions.
Funding
Open Access funding provided by University of Yamanashi.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflcit of interest
The author has no competing interests to declare that are relevant to the content of this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Miyagawa, M. Effect of road pricing on the spatial distribution of traffic flow in a grid network. Japan J. Indust. Appl. Math. 41, 1149–1163 (2024). https://doi.org/10.1007/s13160-023-00643-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13160-023-00643-1