Vol. 10 No 5 - Pi Mu Epsilon

BENNINK. RED LIGHT, GREEN LIGHT 357block distance. The horizontal separation between the legs of the " X is the timeinterval A?, which is shown as a triangular-looking function of s in the right halfof Figure 4. As one can see from the right half of the figure, the time intervalbetween northbound and southbound trajectories is never more than half a periodsince the space "wraps around". The challenge is to determine if and how greenlight regions (solid horizontal bars marked with a "G") may be placed so as tointersect both trajectories at each intersection (thin horizontal lines).If green lights were to last at least half a period, the problem would betrivial; no matter how the intersections were translated, the green light durationwould always exceed A?. The fact that tà is short of half a period by the amount/+means there are two forbidden zones (shaded horizontal bars) where the greenregion is not long enough to catch both trajectories; i.e., where t_ < A?. Theproblem has a solution if we can translate the intersections so that none of themfalls in a forbidden zone. Geometrical analysis of Figure 4 reveals that each358 PI MU EPSILON JOURNALforbidden zone extends over a distance vmJy., on the s axis. Clearly theintersection spacing b must be at least as large as this. But since two forbiddenzones must be avoided, the condition which is sufficient to guarantee a solutionbecomes b > 2v,,,ofyet- Solving this inequality for,&,equation (3), we find that a solution exists as long asdaop s b - 2(w + L).and combining it withFigure 4.Figure 3.In practice, this is never a concern. (If you cannot stop your car in a little undera block, either you are driving too fast or you need new brakes!) For a concreteexample, letN= 4, b = 3309, UJ = 3OJ, L = 151, vmw = 30mihr = 44ft/s, andubr~ = 15p. Then T = 30s and from equation (3) we have&, s 2.5s, whichmeans tgm s 12.5s. Over every 4-block stretch of road there are two sectionsvm&, = 1 lo/? long which periodic northbound and southbound cars traverse attimes more than 12.5s apart. If the lights are scheduled properly, such cars willbe able to proceed &om intersection to intersection without having to stop for ared light.