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 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
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
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
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
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.