Metric  DP Logic  TCAS Logic 
NMACs  3  169 
Alerts Strengthenings Reversals  690,406 92,946 9569  994,317 40,470 197,315 

model assumed by the MDP. The simulation of TCAS (Version 7.1) used a constant 500 ft/s hor-izontal closure rate. To generate the initial state of the encounter, h˙0 and h˙1 are chosen from a uniform distribution from .1000ft/min to 1000ft/min. The variable τ is set to 40s and sRA is set to COC. So that a signi.cant fraction of encounters (～13%) have a vertical miss distance of less than 100ft, h is set to τ(h˙0 . h˙1)+ η, where η is chosen from a zero-mean Gaussian with 25ft standard deviation.
Table
4
summarizes
the
results
of
evaluating
the
DP
and
TCAS
logics
on
one
million
encoun-
ters. As the table shows, TCAS results in 50 times more NMACs than the DP logic, even though it alerts 45% more frequently. Although TCAS strenghtens half as frequently as the DP logic, it reverses 20 times more frequently. It should be emphasized, however, that the simple model was designed to stress test the logic, not provide accurate estimates of performance in the current airspace. High-.delity encounter models based on recorded surveillance data have been developed for
this
purpose
and
will
be
discussed
in
the
next
section
[39].

3.8 SAFETY CURVE
A safety curve is one way to visualize the e.ect of varying a system parameter on safety and alert rate.
Figure
6
shows
the
safety
curves
for
the
DP
logic
and
TCAS.
The
DP
logic
safety
curve
was
produced by varying the cost of alerting from zero to one. The other event costs were held .xed. The upper-right part of the curve corresponds to costs of alerting near zero and the lower-left part corresponds to costs near one.
The sensitivity of TCAS, in terms of alerting thresholds and other parameters, varies discretely with altitude. Above 42,000ft, TCAS is at its highest sensitivity level and will maneuver earlier and more aggressively to prevent collision. The safety curve for TCAS was generated by varying the initial altitude. The upper-right part of the curve corresponds to greater altitudes.
The safety curve shows that the DP logic can exceed or meet the level of safety provided by TCAS while alerting far less frequently. Plotting
safety
curves
like
the
one
in
Fig.
6
can
aid
in
choosing the relative cost for alerting. The alert cost can be chosen to provide the lowest rate of alerting for a given required safety level. Alternatively, one can choose the maximum acceptable alert rate and use the appropriate alert cost to maximize safety.

0 0.20.40.60.81 Pr(Alert)
Figure 6. Safety curve. Each point on the curve was estimated from 100,000 simulations.

3.9 EFFICIENTLY EVALUATING PERFORMANCE METRICS
A DP algorithm known as iterative policy evaluation can e.ciently evaluate the logic on metrics that
are
di.erent
from
those
used
to
optimize
the
logic
[35].
During
the
process
of
validating
the
logic, it may be useful to determine where in the state space the system has di.culty preventing NMAC. It may also be useful to determine the probability that the system will eventually alert from
di.erent
starting
states.
Figure
7
shows
these
metrics
evaluated
on
a
slice
　
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