User-Equilibrium Traffic Flows

1. Two routes connect an origin and a destination. Routes 1 and 2 have performance functions tl = 2 + x1 and t2 = 1 + x2, where the t’s are in minutes and the x’s are in thousands of vehicles per hour. The travel times on the routes are known to be in user equilibrium. If an observation for route 1 finds that the gaps between 30% of the vehicles are less than 6 seconds, estimate the volume and average travel times for the two routes. (Hint: Assume a Poisson distribution of vehicle arrivals.)


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2. Three routes connect an origin and a destination with performance functions tl = 8 + 0.5×1, t2 = 1 + 2×2, and t3 = 3 + 0.75×3, with the x’s expressed in thousands of vehicles per hour and the t’s expressed in minutes. If the peak-hour traffic demand is 3400 vehicles, determine user-equilibrium traffic flows.