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Restricted and Unrestricted Lanes

1. A freeway has six lanes, four of which are unrestricted (open to all vehicle traffic), and two of which are restricted lanes that can be used only by vehicles with two or more occupants. The performance function for the highway is t = 12 + (2/NL)x (with t in minutes, NL being the number of lanes, and x in thousands of vehicles). During the peak hour, 3000 vehicles with one occupant and 4000 vehicles with two occupants depart for the destination. Determine the distribution of traffic between restricted and unrestricted lanes such that total person-hours are minimized.

2. Two routes connect an origin–destination pair with performance functions t1 = 5 + (x1/2)2 and t2 = 7 + (x2/4)2 (with t’s in minutes and x’s in thousands of vehicles per hour). It is known that at user equilibrium, 75% of the origin–destination demand takes route 1. What percentage would take route 1 if a system-optimal solution were achieved, and how much travel time would be saved?

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3. Two routes connect an origin–destination pair with performance functions t1 = 5 + 3.5×1 and t2 = 1 + 0.5 22x (with t’s in minutes and x’s in thousands of

vehicles per hour). It is known that at x2 = 3, the difference between the first derivatives of the system- optimal and user-equilibrium math programs, evaluated with respect to x2, is 7 [dS(x)SO/dx2 − dS(x)UE/dx2 = 7]. Determine the difference in total vehicle travel times (in vehicle minutes) between user- equilibrium and system-optimal solutions.

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