Seventeen seconds into the encounter, the DP logic issues a descend to pass below the intruder. The expected cost for issuing a descend advisory is approximately 0.00928, lower than the expected cost for issuing a climb advisory (0.0113) or for not issuing an advisory (0.00972). The DP entry time distribution at this time has a conditional mean E[τ | τ< 40s] of approximately 12.01s. This represents the mean time until horizontal NMAC inside the horizon. A considerable portion of the probability mass (～40%) is assigned to outside the horizon, when τ ≥ 40s. The Monte Carlo entry time distribution, in comparison, has less support but a comparable conditional mean of 17.12 s. Only 15 % of the probability mass is concentrated on τ ≥ 40s. The point estimate of τ using the simple method is 21.65 s.
After the descend advisory is issued, the intruder begins to increase its descent, causing the DP logic to reverse the descend to a climb 20s into the encounter. The pilot begins the climb maneuver three seconds later. Once the aircraft are safely separated, the DP logic discontinues the advisory at t = 31s. The minimum horizontal separation is 342ft, at which time the vertical separation is 595ft. No NMAC occurs.
TCAS initially issues a climb advisory four seconds into the encounter because it anticipates, using straight-line projection, that by climbing it can safely pass above the intruder. Nine seconds later, when the own aircraft is executing its climb advisory, TCAS reverses the climb to a descend because it projects that maintaining the climb will not provide the required separation. TCAS strengthens the advisory three seconds later, but fails to prevent the NMAC. The aircraft miss each other by 342ft horizontally and 44ft vertically. Although the TCAS logic alerts earlier and more often, the DP logic still outperforms it in this example encounter.

5.6 PERFORMANCE ASSESSMENT
Table
7
summarizes
the
results
of
simulating
the
DP
logic
and
the
TCAS
logic
on
one
million
encounters
generated
by
the
white-noise
encounter
model
of
Section
4.1.
The
performance
of
the
DP logic with the three entry distributions was assessed. The table summarizes the number of NMACs, alerts, strengthenings, and reversals.
As the table shows, the DP logic can provide a much lower NMAC rate while signi.cantly reducing the alert rate. The Monte Carlo entry time distribution results in more NMACs, but it alerts less frequently than the other methods. Increasing the number of samples used generally improves performance but increases online computation time. The DP logic using the simple point estimate of τ resolves all but one NMAC while rarely reversing or strengthening the advisory, but alerts more frequently than Monte Carlo and DP.
×104
Altitude (ft)

Time (s)
(a) Vertical pro.le.
×104

.1.5 .1 .0.50 0.511.5 East (ft) ×104
(b) Horizontal pro.le.
Figure 17. Example encounter comparing the system with the DP entry time distribution against TCAS.

0 10 20 30 ≥ 40
τ (s)
(a) t = 17s
0 10 20 30 ≥ 40
τ (s)
(b) t = 20s
Figure 18. Entry time distribution computed using dynamic programming (DP), Monte Carlo (MC), and the simple (Simple) methods
at
the
two
alerting
points
of
the
DP
logic
in
the
example
encounter
of
Fig.
17.

TABLE 7
Three-dimensional performance evaluation

DP Logic 
Metric  DP Entry  MC Entry  Simple Entry  TCAS Logic 
　
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