3.31 Assigning commuters to subway stations Figure P3.31 shows an urban area which is served only by a mass-transit rail line with four stops ~A, B, C, and D) as shown. There is a single destination at point T for all trips generated from this area (assume that all trips go exactly to point T) during each day's morning rush hour.

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    In order to get to a transit line station or to walk directly to T, the residents of the area must walk on an "infinitely dense" grid of urban streets whose directions run parallel to the boundaries of the area.
    The following information is now given:

1. During the morning rush hour the area generates 200 trips per km^2 with trip origins distributed uniformly.

2. Headways between trains are constant and equal to 6 minutes, and each train rider is equally likely to arrive at a station at any time between two successive departures of trains from that station. (All riders are assumed to be able to ride on the first train to leave a station after their arrival there.) Stops at each station are I minute long.

3. Trains travel between stations at a speed of 30 km/hr (this includes an adjustment for acceleration and deceleration periods). People walk at a constant speed of 5 km/hr.

4. The sole criterion that each individual uses to determine his/her route is to minimize the expected total trip time to T (including time spent waiting for and riding on trains). Each individual is assumed to know all the information given above concerning travel speeds, headways, and so on.

a. Determine the number of riders who will be using each of stations A, B, C, and D each day, as well as the number of those that will be walking directly to T.

b. Compute the expected travel time for a random resident of this area each day.

c. Draw the boundary of the region whose residents are 9 minutes or less away from T. Repeat for the 20-minute boundary. (Be careful in your work.)

d. Repeat part (a) by making the change in the initial data indicated below, while keeping everything else the same as before. (Each part below is separate.)

1 . The train speed increases to 40 km/hr.

2. Train headways are increased to 10 minutes.

3. Train speed >> walking speed.

e. Repeat part (a) by assuming that headways between trains are described by a negative exponential pdf with a mean of 6 minutes.