A traffic signal is said to "fail" if it does not provide a sufficiently long green signal for the accumulated cars to enter the intersection. It has been measured that a maximum of six vehicles can enter an intersection in the 20 sec after a light changes. If a light is set with a cycle of 20-sec red and 20-sec green (ignore the yellow phase) and the average traffic flow is 400 vehicles per hour in the direction of interest, what is the probability that the signal will fail during any particular cycle? (Assume that failure occurs if seven or more vehicles arrive during the cycle. Assume that the arrivals are Poisson.) Notice that the first of the preceding two assumptions is not strictly correct since the failure on the cycle following a failure is more likely owing to the leftover vehicles. If the preceding cycle has not failed, find the probability that the next two cycles in a row will fail? What is the conditional probability of failure of the second given that the first failed? The failure probability can be reduced by increasing cycle times, but this also increases the likelihood that cars will be unnecessarily delayed. A properly designed signal balances these two effects.