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A mathematical model of a road block

01 Jan 2010-
TL;DR: The eect of a road block on the trac depends on three parameters, the density of the oncoming trac, the time T and the ratio of the speed limit in the road block to the speedlimit on the open road.
Abstract: The eect of a road block on trac ow is investigated using the model in which the trac velocity is a linear function of the trac density. The road consists of two lanes in one direction and one lane is closed for a short distance by a road block. The length of the tailback at the entrance to the road block, the time spent in the road block and the trac ux at the exit to the road block are calculated. The maximum length of the tailback is a linear function of the time T that the road block was in place. The eect of the road block on the trac depends on three parameters, the density of the oncoming trac, the time T and the ratio of the speed limit in the road block to the speed limit on the open road. The congestion caused by the road block can be managed by adjusting T and .

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Citations
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Journal ArticleDOI
TL;DR: To the best of our knowledge, there is only one application of mathematical modelling to face recognition as mentioned in this paper, and it is a face recognition problem that scarcely clamoured for attention before the computer age but, having surfaced, has attracted the attention of some fine minds.
Abstract: to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

3,015 citations

Book ChapterDOI
01 Jan 2005

8 citations

Proceedings ArticleDOI
03 Jul 2014
TL;DR: A prototype of a system that considers the real time traffic scenario on the road to implement dynamic vehicle routing and provides assistance to the driver in choosing the path that would take minimum time for travel.
Abstract: This paper proposes a prototype of a system that considers the real time traffic scenario on the road to implement dynamic vehicle routing. Conventionally, a path which is shorter in distance is considered to take minimum travel time. However, this need not always be true. Depending upon the traffic density prevailing on the road at the given instant, the travel time may vary. The proposed system is developed with the help of a data fusion algorithm. The data fusion algorithm is prepared based on the information obtained from a mathematical model developed using Dijkstra's Algorithm and from Image Processing based on real time scenario. The entire system is made stand alone by using eBox. At the end of computation, the system provides assistance to the driver in choosing the path that would take minimum time for travel.

4 citations


Additional excerpts

  • ...[4] [5] [6]....

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01 Jan 2012

1 citations


Cites background from "A mathematical model of a road bloc..."

  • ...1 Background Traffic flows in urban areas became in the last years a very active research field; see, for instance, [4, 5, 8] and references cited therein....

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01 Jan 2014
TL;DR: The results obtained show that for initial traffic densities exceeding a critical value the flux through the system will be increased if switching is allowed, and inevitably disruptions will be caused by lane changes which will propagatethrough the system and also into the oncoming stream.
Abstract: In order to facilitate the flow of traffic in Gauteng province, South Africa, during peak hours the transport authority is investigating the effect of allowing minibus taxis to use the lanes presently reserved for buses either throughout the day or just during peak hours. One would expect the reduction of flow in the normal lane to result in increased car speeds in this lane and also increased speeds for the minibus taxis in the bus lane, however bus speeds may be reduced and therefore timetables not adhered to. The results obtained show that for initial traffic densities exceeding a critical value the flux through the system will be increased if switching is allowed. Inevitably disruptions will be caused by lane changes which will propagate through the system and also into the oncoming stream. Such dynamics issues are also briefly discussed. ∗Mathematics Department, University of Western Australia, Crawley, WA 6009, Australia email: neville.fowkes@uwa.edu.au †School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa. email: Dario.Fanucchi@wits.ac.za ‡School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa

1 citations

References
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Book
01 Jan 1974
TL;DR: In this paper, a general overview of the nonlinear theory of water wave dynamics is presented, including the Wave Equation, the Wave Hierarchies, and the Variational Method of Wave Dispersion.
Abstract: Introduction and General Outline. HYPERBOLIC WAVES. Waves and First Order Equations. Specific Problems. Burger's Equation. Hyperbolic Systems. Gas Dynamics. The Wave Equation. Shock Dynamics. The Propagation of Weak Shocks. Wave Hierarchies. DISPERSIVE WAVES. Linear Dispersive Waves. Wave Patterns. Water Waves. Nonlinear Dispersion and the Variational Method. Group Velocities, Instability, and Higher Order Dispersion. Applications of the Nonlinear Theory. Exact Solutions: Interacting Solitary Waves. References. Index.

8,808 citations

Journal ArticleDOI
TL;DR: This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic, including microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models.
Abstract: Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ``phantom traffic jams'' even though drivers all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ``freeze by heating''? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to self-driven many-particle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for self-driven many-particle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.

3,117 citations


"A mathematical model of a road bloc..." refers background in this paper

  • ...More accurate quantitative predictions may require more sophisticated models in which the traffic velocity may depend on a power of the traffic density or on the density gradient [3, 4]....

    [...]

Journal ArticleDOI
TL;DR: To the best of our knowledge, there is only one application of mathematical modelling to face recognition as mentioned in this paper, and it is a face recognition problem that scarcely clamoured for attention before the computer age but, having surfaced, has attracted the attention of some fine minds.
Abstract: to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

3,015 citations


"A mathematical model of a road bloc..." refers background or methods in this paper

  • ...A continuum model for traffic flow ([1] to [5]) is used which assumes that the traffic is travelling steadily along a long road with no side turnings before the road block is set up....

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  • ...Using the theory of characteristic for first order quasi-linear partial differential equations ([1] to [5]) it can be verified that the characteristic projections on the (x, t) plane are the one-parameter family of straight lines t− 1 = x 1− 2f(σ) − σ 1− 2f(σ) , f(σ) 6= 1 2 (25) x = σ, f(σ) = 1 2 (26) and that on these lines ρ(s, σ) = f(σ) , (27) where σ is the parameter along the initial curve and s is the parameter along the characteristic curves....

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Book
Sam Howison1
27 Jun 2008
TL;DR: In this article, a wide variety of mathematical subjects are used to illustrate how applied mathematics is realized in practice in the everyday world, including distributions, ordinary and partial differential equations, and asymptotic methods as well as basics of modelling.
Abstract: Drawing from a wide variety of mathematical subjects, this book aims to show how mathematics is realised in practice in the everyday world. Dozens of applications are used to show that applied mathematics is much more than a series of academic calculations. Mathematical topics covered include distributions, ordinary and partial differential equations, and asymptotic methods as well as basics of modelling. The range of applications is similarly varied, from the modelling of hair to piano tuning, egg incubation and traffic flow. The style is informal but not superficial. In addition, the text is supplemented by a large number of exercises and sideline discussions, assisting the reader's grasp of the material. Used either in the classroom by upper-undergraduate students, or as extra reading for any applied mathematician, this book illustrates how the reader's knowledge can be used to describe the world around them.

95 citations

Book
01 Jan 1988
TL;DR: Transverse waves on strings, including d'Alembert's solution transverse vibrations of strings long waves on canals, including Bessel functions surface waves on relatively deep water characteristics and boundary conditions transmission lines, including use of Laplace transform for partial differential equations with constant coefficients, Kelvin's cable, Heaviside cables linear isentropic isotropic elastodynamics, including Rayleigh waves, Rayleigh-Lamb theory linear dilational waves in thermoelastic and Voigt solids unidirectional traffic flow, including Burgers' equation non-
Abstract: Transverse waves on strings, including d'Alembert's solution transverse vibrations of strings long waves on canals, including an introduction to Bessel functions surface waves on relatively deep water characteristics and boundary conditions transmission lines, including use of Laplace transform for partial differential equations with constant coefficients, Kelvin's cable, Heaviside cables linear isentropic isotropic elastodynamics, including Rayleigh waves, Rayleigh-Lamb theory linear dilational waves in thermoelastic and Voigt solids unidirectional traffic flow, including Burgers' equation non-linear plane dilatational waves in solids. Index.

73 citations


"A mathematical model of a road bloc..." refers background or methods in this paper

  • ...The velocity of the shock is determined from the Rankine-Hugoniot condition [1, 4]....

    [...]

  • ...A continuum model for traffic flow ([1] to [5]) is used which assumes that the traffic is travelling steadily along a long road with no side turnings before the road block is set up....

    [...]

  • ...It can be shown that if the density decreases across the shock then the shock is unstable while if it increases then the shock is stable [1]....

    [...]

  • ...Using the theory of characteristic for first order quasi-linear partial differential equations ([1] to [5]) it can be verified that the characteristic projections on the (x, t) plane are the one-parameter family of straight lines t− 1 = x 1− 2f(σ) − σ 1− 2f(σ) , f(σ) 6= 1 2 (25) x = σ, f(σ) = 1 2 (26) and that on these lines ρ(s, σ) = f(σ) , (27) where σ is the parameter along the initial curve and s is the parameter along the characteristic curves....

    [...]