5.7 SAFETY CURVE
Figure
19
shows
the
safety
curves
for
the
DP
logic
and
TCAS
when
di.erent
parameters
are
varied.
Each point on the curves was estimated using 10,000 simulations from the white-noise encounter model. The DP logic safety curves were produced by varying the cost of alerting from zero to one while keeping the other event costs .xed. Separate curves were produced for the three methods of estimating the entry time distribution. The upper-right region of the plot corresponds to costs of alerting near zero and the lower-left region corresponds to costs near one.
The safety curve for TCAS was generated by varying the sensitivity level of TCAS. The sensitivity level of TCAS is a system parameter of the logic that increases with altitude. At higher sensitivity levels, TCAS will generally alert earlier and more aggressively to prevent NMAC.
The safety curves show that the DP logic can exceed or meet the level of safety provided by TCAS while alerting far less frequently. The safety curves can aid in choosing an appropriate value for the cost of alerting that satis.es a required safety threshold.
Figure
19
also
reveals
that
the
DP
and
Monte
Carlo
methods
for
estimating
τ o.er similar performance and that they both outperform the simple method, especially when the cost of alerting is high and the logic can only alert sparingly to prevent NMAC. In the upper-right region of the plot, the three methods are nearly indistinguishable. In this region, the systems always alert and are always successful in preventing NMAC.

5.8 TERMINAL CYCLING
Section
5.1
justi.ed
terminating
encounters
when
the
horizontal
separation
drops
below
500ft
because usually at that point the encounter has resulted in NMAC or has been resolved. In roughly head-on encounters, the horizontal separation will typically drop below 500ft for no more than one decision point. However, in slow-closure encounters where aircraft are .ying nearly parallel to each other, the aircraft may come within 500 ft for multiple decision points. In such situations, it may be desirable to issue an advisory to prevent collision.

0 0.10.20.30.40.50.60.70.80.91 Pr(Alert)
One remedy for this situation is to incorporate cycles at states where τ = 0. By cycling at τ = 0 up to some horizon, it allows the system to model the e.ect of advisories after the intruder comes within 500ft or some other chosen threshold. Consequently, the optimized logic will issue alerts even when τ = 0 because it can result in preventing collision.
One way to e.ciently compute J0(sc) with N cycles is to .rst initialize J0(sc)= C(sc) and
then apply the update rule 
.  . 
J0(sc) ← min a .C(sc) +  T (sc, a, sc)J0(sc).  (28) 
sc 

N times. Once J0 is obtained, J1,...,JK ,J ˉ may be computed as before using the procedure in
K
Section
5.2.

Figure
20
shows
example
policy
plots
using
.ve
terminal
cycles.
The
alert
region
shifts
entirely
to the left. The remainder of this report uses .ve cycles, corresponding to the typical .ve-second initial pilot response delay.

5.9 DISCUSSION
The experiments demonstrate that the collision avoidance logic that results from solving the MDP using the method presented in this report reduces the risk of collision by a factor of 50 while issuing fewer alerts than TCAS in the simulated encounters. The system reverses less than 1% of the time that TCAS reverses, and the system strengthens less frequently as well. This section assumed a
　
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