I will be investigating Traffic flow and optimization in an urban setting. This includes traffic jams, traffic modeling and finding ways to optimize traffic flow. Through the use of traffic modeling and optimization it is possible to alleviate traffic jams without having to build new roads and lanes. This is an effective way to provide rapid passage on roads and highways without the burden of high construction costs. I am interested in this topic because I enjoy applied mathematics and enjoy the study and development of man-made systems. The potential to make traffic flow more efficient intrigues me because it has the potential to cut carbon emissions. I am very interested in preserving and protecting the planet. Furthermore this topic has personal real world implications as I make use of my cities roads, driving across town many times a day. I believe that a more streamlined traffic system would save me hours of sitting in traffic as I often do. This investigation will feature a wide variety of math such as modeling, systems of linear equations and optimization to explore the fundamentals of traffic flow in a search for the perfect optimization of traffic flow. First developed by Frank Kite in the 1920s traffic flow flow theory and optimization aims to record, model and optimizer traffic flow in a given area. This can be taken to the next level when factors such as weather, time of day, road conditions, construction status and road obstructions are taken into account. Thus creating a multi-variable system that can be manipulated with the goal of yielding a seemingly uninterrupted traffic flow.Exploration This first model serves as a starting point from which the investigation will develop. In this fictional model two sets of parallel one way streets cross creating four intersections. The number of cars that pass through the four given intersection are given in terms of vph or vehicles per hour. Through the application of traffic flow theory and linear equations it is possible to find the optimal amount of traffic volume to keep the roads clear. Next we can model the traffic flow in and out of intersections with a simple in out function. In (X)Out (Y)Input/Outputx+w540+723=1263x+w=1263(X)315+540=765x+yx+y=765(Y)z+y230+110z+y=1020(X)840+708=1548w+zw+z=1548(X) Next we need to solve the systems of linear equations:x+w=1263 x+y=765z+y=1020w+z=15481) {x+w=1263,x+y=765} x+w= 1263 x+w-w=1263-w1263-w+y=765 1263-w+y-y=765-y 1263-w=765-y 1263-w-1263=765-y-1263 -w=-y-498w=y+498 x=1263-w x=1263-(w=y+498) x=765-ySolutions to {x+w=1263,x+y=765} w=y+498, x=765-y2) {z+y=1020,w+z=1548} z+y=1020 z+y-y=1020-yw+1020-y=1548 w+1020-y=1548 w+1020-y+y=1548+y w+1020=1548+y w+1020-1020=1548+y-1020 w=y+582Solutions to {z+y=1020,w+z=1548}w=y+528, z=1020-y The previous example modeled a simple four way intersection to find the optimal amount of vehicles that must pass through to keep the traffic flowing. However it is possible to model other factors such as speed density, flow density and speed flow. In this way it is modeling the interaction between traffic flow and traffic density. This is done using MFD’s (Macroscopic Flow diagrams). These are modeled: (Flow) (Density)(speed)Speed Density describes a roadway in terms of how fast the cars on it are going. In this way it is a linear negative slope. As the density increases the speed decreases this is because if there are more cars on the road the driver has to travel at a lower velocity. Flow-Density has two schools of thought one of which is purely theoretical and the other more practical in nature. The practical one describes traffic flow as a triangular curve made up of two vectors. One vector represents free flowing traffic the other represents congested traffic and therefore has a negative slope as it describes traffic that is being limited by the density of a given roadway. In this way the higher the density the lower the flow rate of traffic. The intersecting point is described as being the optimum point in which the most amount of cars can pass at a given time without causing traffic delay. Speed-Flow describe the optimum speed at which vehicles can go to sustain equilibrium in traffic flow, meaning the speed a vehicle has to go to keep traffic flowing. Similarly to Flow-Density, speed flow is made up of free flow and congested vectors. It is not a function means speed and density are not necessarily dependent on each other, meaning different speeds for different densities can produce equally optimal flow rates.