# Traffic Light Sequences Essay Research Paper MEI — страница 2

green light = 35.5m / 5m/s = 7.1 sec s This is the time for the last vehicle to clear the first set of lights, however, we have to consider the time delay between cars starting. 7.1+7=14.1 sec s Therefore the minimum time the green light must be illuminated is 14.1 seconds. Dead Time Time = period of one cycle – period of green light. = 20.9s – 14.1s = 6.8s Therefore, the system cannot be clear until 6.8 seconds after the green light has been extinguished. Period of red light To find the duration of the red light, I shall map out the two sets of lights and fill in the blanks! Light one Period/seconds Light two Green 14.1 Red Amber 3 Red Red 6.8 Red Red 3 Red/Amber Red 14.1 Green Red 3 Amber Red 6.8 Red Red/Amber 3 Red Therefore the period of the red light is (6.8+3+14.1+3+6.8)=33.7 seconds MODEL 2 In order to produce an equation for these traffic lights, I shall introduce some unknowns. Variable Symbol Distance between traffic signals. Dsig Mean speed of traffic. v Mean length of one vehicle. L Mean inter-vehicle gap (static). Gs Number of vehicles per flow. n Time delay between cars starting at first set of signals. t Duration of amber light. A Maximum distance. Dmax Maximum time. Tmax Dead Time. Tdead Period of green light. Tgreen Period of red light. Tred Dmax = Dsig+[(n-1)Gs]+nL Dmax = Dsig+nGs+nL-Gs Tmax = (Dmax/v)+(n-1)t Tmax = Dmaxv-1+nt-t Tgreen = (Dmax – Dsig)/v + t(n-1) Tgreen = Dmaxv-1 – Dsigv-1 + tn – t Tdead = Tmax – Tgreen Tred = 2Tdead +Tgreen + 2A Data collection The data I have used was collected in Broughton, Edinburgh. A construction site near the road is blocking one lane of this two-way, single carriageway road. The data was collected by the students in the Mechanics A-level maths set. Stop watches were used to time the traffic signals, and a wheel ruler to measure the dimensions of the road. The weather was clear and windy. Real Situation In a real situation it is even more complicated as not only would we have to bear in mind the acceleration and deceleration of the vehicles, but even these would not be constant. Many other factors also effect the results, such as if the traffic is already moving at the first set of signals, also factors such as weather conditions effect the style of driving. Factor Error bounds Car Length x0.75m Inter-car gap 2.6mx0.2m Time when system is empty 5secx2.5sec Speed of flow 4.26ms-1×0.75ms-1 List of other factors + Weather + Time of day + Presence of special vehicles, e.g. Articulated lorries, bicycles, horses etc. + Reckless drivers jumping lights. CONCLUSION This project has given me an interesting insight into the outwardly simple maths of everyday systems. I would consider these models to be anything but perfect as there are far to many random factors involved and this makes it too difficult to model accurately without chaos theory. 336

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