1999_HiMCM_Outstanding_Papers(6)

数模资料

6

EQUATIONS GOVERNING P, AND THEIR DERIVATIONS:It is possible to derive equations for the rate of cars makingeach of the three choices, from all four compass directions. Forinstance, if Pis the probability to turn away from the thorough-fare, the rate of cars per second that want to turn away from theeast or west is P(r(rE-W), and the rate of cars that want to be in,say, lane 0, is Pthe intersection going south is the rate turning right from westE-W)/2. The total rate of cars that are leavingplus the rate turning left from east, or, P(rP(rE-W)/2 + P(rE-W)/2 =E-W). Thus, P(rE-W)is the amount of cars that have to comefrom the south onto the thoroughfare to make up for those carslost, which can be split into half to accommodate east and left-bound cars. This is all mirrored by the southbound street, andso P(rE-W)/2is the rate of cars in every turning lane in everydirection.

For cars not turning, the rate approaching from the east or westminus the rate turning away is the total rate: (rE-W) – P(rE-W) =(1 – P)*(rE-W).

The remaining possibility is cars crossing thoroughfare. Pcarscoming at the intersection do not leave via the thoroughfare, sothe rate Aof cars going straight is 2P(A+ P(r(rE-W)) = A, which canbe solved for A= PE-W)/(1 – P).GREEN LIGHT COMBINATIONS

To decide the most efficient way to cycle through the differentcombinations of green lights, we came up with all of the list inAppendix B. We chose the necessary ones and found the bestcircular order: 1, 5, 2, 4, 3, 6, and back to 1 again. In an averagecycle, all of these states would be present for an amount of timeproportional to the average number of cars that need to useeach combination. We found the number of cars for each

combo, which is based on Pand rlane with the most cars to pass through is the lane that limits itsE-W. For each combination, thebrevity. By finding the lane with the greatest relative amount ofcars for each combination, we hoped to minimize the time thatno traffic is moving and maximize the time that traffic is flowing.We used two approaches to this problem. First, we assumedthat all traffic stops when changing from one green light combi-nation to another and improved our model to let traffic contin-ue to flow when it does not need to stop. For instance, betweengreen light combinations 1 and 2, lane 1 does not need to cometo a halt. In both cases, we divided this greatest flow rate by thetotal flow rate, which is the sum of the six greatest flow rates.By repeating this for the six different green light combinations,we arrived at expressions that, when evaluated, returned thefraction of a full light cycle that the corresponding green lightcombination would be lit. For example, in the first and less effi-cient model, the total flow was –1/(P– 1), and our limiting fac-tor expression for the first green light combination is (1 – P).Therefore, on average, the fraction of a light changing cycle thatgreen light combination 1 is active is estimated as (P– 1)2. All ofthese results are in Appendix B.

The graph in Figure 2gives a visual representation of how theoverlapping data is better. The more efficient model is darker.The lines represent the six green light configurations, of whichthere are three unique expressions. They are a plot of the proba-bility that each configuration is used in a cycle against thegreater likelihood that more people would want to turn onto

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side roads. It is a graph of time fraction to P. The bold linesintersect sooner and are closer to each other than the fine linesare, an indication of better total efficiency.

Figure 2. [0.38, 0.50] x[0.10, 0.33]

INTERESTING RELATIONSHIPS BETWEEN VARIABLES

We created an equation containing all three variables P, rE-W,and rN-S, First we added together the expressions that representthe output of lanes 4 and 5; this sum is equal to rtion is: (rN-S. The equa-E-WP) + (rE-WP2)/(1 – P) = rmore sense (and looks prettier) when solved for N-S. This equation makesP:

P= rN-S/(rN-S+ rE-W). It is then possible to substitute the rightside of this equation into the ratio equations in Appendix B toget applicable data. The graph in Figure 3, shows how rindirect proportion to rS-Nis inE-W, when Pis constant.

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Figure 3.

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DETERMINING THE OPTIMUM TIME PER LIGHT CYCLESeveral factors impact the problem. First, synchronization oflights must be avoided. This is because if cars get backed up atone light and the next light simultaneously turns green, therewill not be many cars to pass through the second green light.Synchronization is difficult to control because the length of alight cycle can vary depending on whether cars are on the sen-sor when the cycle is allowed to move on to a non-straight-thor-oughfare. In a city with differing block lengths, the probabilities