## 5.3.3 Nonhomogeneous Rectangular City ExampleWe consider as an example the rectangular city shown in Figure 5.2.
This city runs 2 miles east-west and 1 mile north-south with an
x) is given by the Manhattan
metric,
_{j}, y_{j}
_{ij} = | x_{i} - x_{j} | + | y_{i} -
y_{j} |
The home location for unit 1 is It is decided that the primary response areas for the two units will
be determined by a boundary line located a distance w west of facility 1
(and 1 - Using the general formulas given above, we arrive at expressions as
functions of T_{1}(A), and T_{2}(B) -
A) (see Table 5-3). As just indicated, the travel time formulas
are providing mean travel distances in this case. These functions
behave as we expect. For instance, _{1} = _{1},
grows linearly with w, whereas _{2} = _{2},
decreases linearly with w. The mean intraresponse area travel
times are nonlinear functions of w, with the two limiting
conditions (w = 0 and w = 1) checking with our intuition.
As we might expect, T_{2}(B) = 11/14 is greater
than T_{1}(B) = 1/2 since facility 2 is located
relatively far from the high-demand region, 0.75 x 1.0.
w = 1/2, equidistant from facilities 1 and 2,
yielding response areas A_{w = 1/2} and B -
A_{w = 1/2}. With such a partitioning, requests
are always assigned to the closest available unit, thereby minimizing
the "immediate cost of response," where cost is measured by mean travel
time. One might think that such a policy would minimize overall system
mean travel time. Substituting all the numerical values describing this
problem into (5.17), we obtain the system mean travel time,
A_{w =
1/2}) = 0.46246
_{1} = 0.49010_{2} = 0.43773
yielding a workload imbalance
_{1} - _{2}| = 0.05236
w =
1/2 yields minimum system-wide mean travel time, we should be able to
verify that by differentiation of with respect
to w. That is, solving the equation d/dw = 0 in terms of w should yield the optimal value
for w, implying minimum mean travel time. (Of course, we must be
careful to assure that 0 < w < 1 and that
d^{2}/dw^{2} > 0, implying that a minimum rather than
a maximum is obtained.)
The differentiation is straightforward since d/dw = 0 = w -
53/126
implying that the optimal value for w^{*}= 53/126 < 1/2
Response area 1, now denoted w 53/126, the result states that if unit 1 were to
respond instead of unit 2, the immediate savings gained by assigning the
closest unit (unit 1) are not sufficiently large to compensate for the
likelihood of another request causing unit 2 to be dispatched on a
distant request (perhaps into unit 1's high-demand zone) because of unit
1's unavailability. This means that minimal system-wide travel time is
obtained by occasionally incurring a travel time slightly greater than
the immediate minimum possible in order to leave the system in a state
that best anticipates future requests for service. This is a fundamental
property of many stochastic systems: In selecting a decision
alternative at any given time, it is often optimal to select an
alternative that does not yield the minimum possible immediate cost (or
the maximum possible immediate reward) in order to facilitate the most
javorable future system states and decision alternatives.
The interested student may well wonder what savings in mean travel time are obtained in this example by following an optimal boundary-line policy. The result is
A) =
0.46166
_{w*}or only 0.173 percent less than the equal-travel-time boundary line.
This result is consistent with other analysis of spatially distributed
probabilistic systems, which show that mean travel time is rather
insensitive to slight changes in system boundaries. What is surprising
about the boundary line that minimizes mean travel time is its effect on
workload imbalance. With
_{1} = 0.44189_{2} = 0.48594
yielding a workload imbalance of 0.04405, compared to the earlier
value of 0.05236, or a 15.87 percent improvement in this measure. Thus,
by designing the system to minimize one performance measure, we obtain
an improvement in another performance measure. This is unusual in
operations research, where one usually must confront trade-offs
requiring degradation in one measure of performance to achieve
improvement in another. In applications, Carter, Chaiken, and Ignall,
who originally analyzed the two-server model, and later Jarvis, who
extended the ideas to |