Air traffic collision risk modelling

In view of the high safety standards required from air traffic management (ATM), the validation of an advanced ATM design can only be done with the help of mathematical models for the collision risk of controlled air traffic in route networks, that allow for numerical evaluation. Hence, we need a model for collision risk between controlled aircraft in an arbitrary route network, the numerical evaluation of which compares with the well known Reich model for uncontrolled aircraft. We first consider the well-known theory on the (first) hitting of an absorbing boundary by a Markov process. Since the characterization of (first) hitting reads in terms of a partial differential equation with boundary conditions, its numerical evaluation is too complex for a situation of many aircraft in a network. In view of this finding, we next consider the in-crossing of a transient boundary, to develop an extended version of the Reich model which is general enough to model air traffic in arbitrary route networks