3.7.2 More Realistic Travel-Time Model
In most practical situations, the
speed of urban response units depends on travel distance: longer
trips, in general, are
taken at a higher average speed than are shorter trips. It is
therefore desirable to
develop expressions for E[T] that take into consideration some types
relationships between travel time and travel distance [unlike
expressions (3.81) and
(3.86), which assumed that effective travel speed remains constant
with distance]. One
plausible model is the following. Let us assume that urban service
vehicles responding to
a call, first go through an acceleration stage (perhaps while
maneuvering their way
through side streets, turns, etc.) until they reach a cruising speed
that they maintain
through the middle stage of the trip (while, perhaps, traveling on
thoroughfares, etc.) up to the final stage of it, during which they
decelerate to a stop.
Let us further assume that during the initial and final stages,
vehicles accelerate (or
decelerate) at a constant rate of a miles/min2 and that during the
middle stage, travel is
at a constant cruising speed of v, miles/min.
One can obviously think of many other physical scenarios that
would lead to different
expressions for E[T | D = d]. A considerable amount of field data,
however, suggests that
(3.88) and (3.89) often provide truly excellent approximations for
services-see, for instance, [KOLE 75, JARV 75, HAUS 75].
In order to evaluate the two integrals in (3.90) it is necessary to know the pdf for the travel distance, fd(x).
From (3.88) and (3.89) we then have (in minutes)
for the average travel time in responding to a fire alarm in this district.
Unfortunately, the pdf for the travel disance fD(x) is often difficult to obtain, either theoreticallly or from field data. The following approximate expression for E[T|D=d] is then often used in order to overcome this problem:
This expression is compared with
(3.88) and (3.89) in
Figure 3.30. Note that (3.92) is a "conservative" model
The example above is not
atypical. Estimates of E[T]
obtained through (3.92) and (3.93) are usually very close to
estimates obtained through
the time-consuming approach summarized by (3.88)-(3.90) for the
values of v, and a one
encounters in urban service applications.
The travel time/travel distance relationship (3.92) greatly simplifies the calculation of E[T] and provides excellent approximations to results produced through more complicated analyses.
It should be noted that our earlier results regarding variation of E[T] with district geometries apply unaltered in the context of (3.93). This is because of the continued linear relationship between D and T, which is augmented in (3.93) only by an additive constant, a term that does not affect variational analyses of district geometries. Even when cruising travel speed depends on direction of travel, i.e., when vx. vy., the addition of a constant to the travel time will not affect the variations of travel times with district designs. Most important, optimal designs remain unchanged.
9 It is also possible to have this effective travel speed depend on the direction of travel (e.g., as in the case of right-angle travel).