Using Mathematics to predict traffic flow - Essay Example

Using Mathematics Models to Predict Traffic Jams Introduction Traffic has been researched widely since the start of the twentieth century. With the introduction of the mass-produced cars, increasing figures of vehicles started to filter onto our street infrastructure. Through the increase of automobiles came a growth in traffic congestion challenges. From the 1920’s through 1950’s the number of researchers tried to model this new occurrence. Based on these early traffic representations, a new field of research developed and continues to this day.

We use macroscopic and microscopic models to understand modern techniques in the traffic flow. In this paper, we study microscopic model in detail, which utilizes ordinary differential equations. We also provide the reasons for studying the traffic flow and the current applications in science.Reasons for Studying TrafficThe basic aim of studying traffic is to comprehend traffic jamming and find ways to prevent it. The aim is to provide well-organized movement of traffic while reducing congestion problems.

Traffic modeling is studied in different disciplines and each of the fields explains why we study traffic.In Engineering, civil engineers are interested in traffic to be able to forecast and model traffic flow to generate operative and safe road systems and intersections. In addition, electrical engineers may create intelligent electronic appliances, which will monitor and get used to traffic conditions. Besides, environmental engineers may study how traffic jamming affects fuel consumption and air pollution.

For city planners, they study traffic to determine the most effectively use and apply traffic systems. A city planner may need to decide the effectof the addition of an on-ramp to a freeway or if to construct a bypass. Moreover, a city planner may need to determine certain types of cars that should be prevented from a certain roadway. The traffic models can assist state government and city planners to determine speed limits. Through creation of accurate simulations of traffic models, computer scientists enable city planners and engineers to rapidly test new designs.

Simulations can define how a planned change in infrastructure will influence traffic before any building is initiated.In mathematics, mathematicians construct new traffic models and study the effectiveness of modern models. They may attempt to clarify traffic occurrences like traffic congestion waves and jams.Traffic ModelsThere are two current types of models namely macroscopic and microscopic. Both models have modern state-of-the-art equations. Macroscopic models research on traffic from a general or average perspective whereas microscopic models research on the motion of individual vehicles.

The traffic jams can be studied with both types of models. The models can simulate common traffic behavior with some amount of accuracy. Macroscopic ModelsThey were the first models to be derived by scientists for studying traffic. These models were selected because traffic flow originally seemed to be similar to fluid flow through a pipe system or a river. These models study the system in terms of the movement of vehicles into and out of the flux (system), average speed of cars, or overall vehicles.

They are normally modeled using partial differential equations. Current macroscopic models use hyperbolic partial differential equations.Macroscopic Model Equation Examples: LWR modelThe LWR model is considered non-linear, a scalar, time varying, hyperbolic partial differential equation. Velocity depending on traffic density is one of its basic assumptions. LWR model agrees by pt(x,t) + ((x,t)V((x,t)))x = 0p= density of carsv=average speed of carsx,t= the intervals between 2 pointsp(x,t)= the vehicle density in the regionP(p)= equalibrium relationshipExampleThe traffic arriving at the upsteam of a hihgway was initially at condition A. At 8.00 AM, the condition swtiches to conditon B.

After an hour the condtion switches back to condtion B. The capacity at the bottleneck is 1400 veh/hr. Find how long the queue is and how long it persits.Conditionq(veh/hr)K(veh/mi)V(mi/hr)A6008.5770B20004050D140021.565D140013010.8Usng grapgical means the rate at the queue grows is UAD = = = = - 6.67 mi/hrThe rate at which the queue dissipates isUAD = = = = 6.69 mi/hrAR modelAR model is a more current model, which tries to move away from a river or fluid flow based model. it is described by the following equationst + (v)x = 0(v + P())t + v(v +P())x = 0P represents pseudo pressure, whose derivative with respect to p is (p-1)P´(p).

The last model referenced is known as the Zhang Model. This model moves fully away from fluid behavior. It applies a second equation derived from a microscopic model that creates a macro-micro link. It is given by the following equationst+ (v)x = 0vt + vvx + Vt()vx = 0Advantages and Disadvantages of Macroscopic ModelsThe main problem of macroscopic models is that they have comparatively simple calculations when related to microscopic models. They have lesser parameters than their microscopic equivalents.

The parameters required in the equation include model density, velocity and flow. The shortcoming of a macroscopic model is the loss of minor details or underlying forces, which can be modeled with microscopic models.Microscopic ModelsThese models try to model individual vehicles motion within a system. Microscopic models are generally functions of positions, acceleration, and velocity. They are usually formulated using ordinary differential equations, where each vehicle has its own equation.

Since a lead car typically dictates these models behavior, they are referred to as “car-following” models. The figure below represents microscopic models number cars in car-following conditions.These models were developed to attempt to match the way people behave in traffic conditions. In order to achieve this, the models comprise different driving states to define classic driving responses faced. There are three types of driving states.The first is the Free Traffic state. This condition is met with low vehicle density, and different vehicles can accelerate to their preferred velocity.

There is no lead car present to affect car position, acceleration, or velocity. The other driving state is the Following state. This state is met in everyday traffic flow, with medium to high car density. In this situation, a car’s velocity and acceleration are greatly decided by the lead vehicle. Thus, a driver tries to maintain maximum and minimum time gap between his vehicle and the lead vehicle.The last state is the Braking state or Emergency Response. It becomes dynamic if the existing vehicle is approaching a significantly slower vehicle.

A driver attempts to apply various levels of braking force to avoid colliding with the lead vehicle.Microscopic Model Equation Examples: Gipp’s ModelThe basic microscopic model is Gipp’s Model. It uses driving states to model traffic movement. The model is defined by the following equationn(t) Where nth vehicle position is denoted as xn(t). this implies that the acceleration of the recent vehicle ẍn(t) relies on the velocity and location of the vehicle in front. C is a sensitivity parameter.

Model notation denotes maximum acceleration with which the driver of vehicle  wishes to undertake, denotes the most severe braking that the driver of vehicle  wishes to undertake , denotes the effective size of vehicle , that is, the physical length plus a margin into which the following vehicle is not willing to intrude, even when at rest, is the speed at which the driver of vehicle  wishes to travel, I denotes the location of the front of vehicle  at time *,  denotes the speed of vehicle  at time , and denotes the apparent reaction time, a constant for all vehicles.

[2]Intelligent Driver ModelIntelligent Driver Model (IDM) is a recent state-of-the-art model. The model improves other models and it comprises an acceleration strategy with a braking approach to cover the above three driving states. It is defined by the following equationsThe s* term under the primary function is an extension of s* in the numerator of the primary function. s represents vehicle gap; v is the velocity, and ∆v is the velocity difference. IDM uses many more parameters unlike other models.

The free traffic state of IDM model equation dictates if velocity(s) is very large, resulting to the interaction term to become insignificant. Therefore, the free traffic term isIt is definitely seen that as v→v0, the acceleration vfree(v)→0. This simulates the tendency for a driver to steadily lower their acceleration as they tend to their preferred velocity v0.The interaction or braking term of the IDM model equation dominates the braking and following driving state. The interaction term is defined by the following equationsv= maximum speed allowed on the roada= maximum accelerationssi = xi−1 − xi = actual gap from vehicle in fronts*= desired gap∆vi = vi − vi−1 = speed difference with car in frontT = safe time headwayb = comfortable deceleration= minimum bumper to bumper distance to the front vehicle a= acceleration During normal driving situations, the vT term governs.

The term vT tries to sustain a particular time gap T from the car being trailed. The term v∆v/2 governs if approaching a vehicle at high-speed rate. The model tries to brake within the limit b, though will exceed b’s value if needed to prevent a collision. ExampleAssume that has ring road with 40 cars. Vehicle 1 will be behind vehicle 40. For this example, the following values the equations parameters. Original speeds are given and since all cars are considered equal, vector ODEs are further simplified to:The two ordinary differential question can be solved using Renge-Kutta methods  of orders 1, 3, and 5 , to show the effects of calculation accuracy in the results.

Microscopic Model AssumptionsMany assumptions are accounted for by increasing the quantity, complexity, or variation of parameters. Macroscopic models postulate a homogenous driver reaction. This implies that a driver simulated by these equations responds just the same way in all conditions. Another assumption is that, in most models there are no accidents or collisions occurring. In addition, homogeneous cars are assumed, for instance, they have the same acceleration and braking reaction and vehicle length.

Lastly, the assumption is that ideal driving conditions are postulated.Advantages and Disadvantages of Microscopic ModelsThe main advantage of microscopic models is the capability to study different vehicle motion. The flow and density macroscopic ideas can be as well studied with the microscopic models.The major disadvantage with these models is that one ODE equation is needed for each car. Microscopic models develop to very computationally expensive using large systems of equations. Thus, they require modern computers to make them convenient.

In addition to disadvantages, these models can suffer from extreme values, for example, emergency interaction/braking responses that challenge physical possibility. However, modern models have greatly reduced this type of behavior.Applications of Traffic ModelsPresent applications of microscopic models are only restricted by the imaginings. Traffic jamming is a persistent problem with no vivid answer. These traffic models are being used in several ways to attempt to get answers to the jamming problem.

A number of examples are listed below. Vehicle following systems are being used in technology called adaptive-cruise-control. In usingseveral sensors, a car fitted out with an adaptive-cruise-control system can sense the lead vehicles’velocity. This enables the vehicle to travel at an agreed velocity until it meets a slower car. The vehicle will then actively adjust its velocity automatically to match that of the slower car. Cars with this technology are being worked on to decide if adaptive-cruise-control can raise traffic density without causingtraffic jamming problems.

Some car manufacturers providecars with innovative crash-avoidance characteristics. These vehicles use several sensors to check for inevitable accidents by using modified car-following models. When the algorithm senses an inevitable accident before the driver responds, it can immediate begin applying braking force to minimize the crash impact. Various commercial adjustments with traffic models are accessible for engineers and city planners to use. These systems usually have both macroscopic and microscopic workings.

An intelligent traffic system can change traffic indication programing; close or open lanes of traffic, or even dynamically change different speed limit signs. Checking systems can also alert traffic management departments when traffic patterns specify a deviance from normal levels. Joined with commercial traffic modeling program, these kinds of systems can raise vehicle density without reducing velocity.Works CitedOrosz, Gbor, R. Eddie Wilson, RbertSzalai, &GborStpn (2009).. "Exciting Traffic Jams: Nonlinear Phenomena behind Traffic Jam Formation on Highways.

" Physical Review E 80.4.Print.Treiber, Martin."Microsimulation of Road Traffic Flow."TechnischeUniversitat Dresden. Web. 15 Apr. 2011.http://vwisb7.vkw.tu-dresden.de/~treiber/MicroApplet/index.htmlTreiber, Martin, AnsgarHennecke, &Dirk Helbing (2000): Congested Traffic States in Empirical Observations and Microscopic Simulations. Physical Review E 62.2 1805- 824. Print.