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r
g(ω) pd (xj ). j
In this case, the acceptance probability for the standard Metropolis-Hastings algorithm is
r
h(˜ω, x˜1,x˜2,..., x˜J ) g(ω) j pd (xj )
r .
h(ω, x1,x2,...,xJ ) g(˜ω) j pd (˜xj )
By inserting (8) in the above expression one ω, ˜Under minimal
obtains ρ(ω, uJ , ˜uJ ). assumptions, the Markov Chain generated by the D(k) is uniformly ergodic with stationary distribution h(ω) given by (9). Therefore, after a burn in period, the extractions D(k) accepted by the algorithm will concentrate around the modes of h(ω), which, by (9) coincide with the optimal points of U (ω). Results that characterize the convergence rate to the stationary distribution can be found, for example, in [20].
A general guideline to obtain faster convergence is to concentrate the search distribution g(ω) where U (ω) assumes nearly optimal values. The algorithm represents a trade-off between computational effort and the “peakedness” of the target distribution. This trade-off is tuned by the parameter J which is the power of the target distribution and also the number of extractions of X at each step of the chain. Increasing J concentrates the distribution more around the optimizers of U (ω), but also increases the number of simulations one needs to perform at each step. Obviously if the peaks of U (ω) are already
7
ELEVATION VIEW x2
10000 ft
1500 ft
GLIDE PATH
3o RUNWAY
5 nmi
x1 PLAN VIEW x2
LEFT DOWNWIND
BASE LEG
30o FINALS GLIDE PATH RUNWAY
5 nmi x1
Figure 1: Schematic representation of approach maneuver
quite sharp, this implies some advantages in terms of computation, since there is no need to increase further the peakedness of the criterion by running more simulations. For the specific U (ω) proposed in the previous section, a trade-off exists between its peakedness and the parameter A, which is related to probability of constraint violation. In particular, the greater A is the less peaked the criterion U (ω) becomes, because the relative variation of u(ω, x) is reduced, and therefore more computational effort is required for the optimization of U (ω).
4 ATC in terminal and approach sectors
4.1 Current practice
Terminal Maneuvering Area (TMA) and Approach Sectors are perhaps the most difficult areas for ATC. The management of traffic, in this case, includes tasks such as determining landing sequences, issuing of “vector” maneuvers to avoid collisions, holding the aircraft in “stacks” in case of congested traffic, etc. Here, we give a schematic representation of
8
the problem as described in [2, 6].
During most of the flight, aircraft stay at cruising altitudes, above 30000 ft. In the current organization, the traffic at these altitudes has an en-route structure, which facilitates the action of ATC. Aircraft follow pre-specified corridors at different flight levels. Flight levels are given in 3 digit numbers, representing hundreds of feet; for example, the altitude of 30000 ft is denoted by FL300.
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Monte Carlo Optimization for Conflict Resolution in Air Traffic Control(7)
