0 5 101520253035404550
Time (s)
Two seconds after the descend advisory is issued, the system strengthens the advisory. At the following time step, the probability distribution over sRA reveals that the own aircraft is un-responsive to the strengthening with approximately 3/4 probability, which again represents the probability of not executing a subsequent advisory. With approximately 1/4 probability the own aircraft is responding to the strengthening and with a very small probability (on the order of 10.6), not shown, the own aircraft is following the initial descend advisory. As the own aircraft begins to respond to the strengthening three seconds after its issuance, the distribution is updated to re.ect this change in pilot behavior. Ten seconds after the pilot begins responding, it is believed that the own aircraft is responding to the strengthening with probability 0.9. The entry time distribution falls o. quickly as the aircraft come within close proximity laterally.
The DP logic discontinues the advisory at t = 34s, and the belief state over sRA quickly changes in response to the own aircraft leveling o.. The aircraft are vertically separated by 441ft at the point of minimal horizontal separation (368ft), and an NMAC does not occur.
TCAS also initially issues a descend advisory, six seconds after the DP logic, and then strengthens four seconds later. The alerts come too late, and an NMAC results at t = 39 s.

6.5 SIMULATION RESULTS
Table
8
summarizes
the
results
of
evaluating
the
DP
and
TCAS
logics
on
500,000 encounters generated by the correlated encounter model. The performance of the DP logic optimized using three di.erent pilot response models—deterministic, linear, and quadratic—was assessed. These three systems, together with TCAS, were tested in three operating environments: one with a deterministic pilot response, one with a linear probabilistic pilot response, and one with a quadratic

0 5 101520253035404550
Time (s)
probabilistic pilot response. The table compares the probabilities of certain events, such as an NMAC and an alert. The mean number of times the advisory is changed, E[RA changes], and the mean amount of time the systems spend alerting, E[RA duration], are also shown. Ensuring these two metrics stay low is critical to operational acceptability.
The following observations are worth mentioning:
1.
The logic optimized through dynamic programming resolves more NMACs than TCAS while alerting less often. Although the DP logic does strengthen more frequently, the strengthening cost can be increased until a su.ciently low level of strengthening is achieved.

2.
The logic computed using the deterministic model has an increased probability of NMAC when operating in a di.erent environment than the one for which the logic is optimized. The logic computed using probabilistic pilot response appears less sensitive to unmodeled behavior.

3.
The linear and quadratic DP logics perform comparably in almost every category despite the structural di.erences in the pilot response models used to compute them. Quadratic DP strengthens
more
often,
as
might
be
expected
based
on
Fig.
22.

The
standard
errors
associated
with
the
estimates
of
Table
8
were
also
calculated
and
were
found to be small in relation to the actual estimates. The standard error, on average, was 10.08% the size of the estimate, indicating a reasonable level of accuracy in the estimation of the metrics.
Figure
26
shows
the
probability
distributions
for
the
number
of
advisory
changes
and
　
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