On the exact distribution of the number of perceived conflicts with prescribed duration in air traffic control

This paper derives the exact distribution-in the sense of not asymptotic-of the number of perceived conflicts with prescribed duration during a known time period. A perceived conflict is defined as an occurrence during which the separation between two airplanes on different air routes drops below a given safety level; and the subsumed geometric model consists of two intersecting routes of equal altitude. The duration of a perceived conflict varies over a given interval and depends, beside the problem’s parameters, on the approach pattern of the two aircraft to the intersection point of the two air paths. The underlying statistical assumption regarding aircraft arrivals is that they are independent, with respect to the air routes, and subject to Poisson forms with possibly different parameters. The resulting distribution, it is shown, is not a compound Poisson, as was previously surmised in the literature, but a Poisson sum of combinatorial number sof independent Poisson variables. It is studied in detail under a certain condition and compared with a compound Poisson and a simulated distribution.