Increasing traffic demands and technological developments provide novel design opportunities for future air traffic management (ATM). In order to evaluate current air traffic operations and future designs, over the past decades, several mathematical models have been proposed for air traffic conflict and collision probability estimation. However, few comparative evaluations of these models with respect to their mathematical core exist. Such comparative evaluations are particularly difficult since different authors employ different model definitions, notations, and assumptions, even when using the same modeling techniques. The aim of this paper is: 1) to present the mathematical core of the existing approaches for air traffic conflict and collision probability estimation using the same body of notations and definitions; 2) to outline the advances in estimating the probability of air traffic conflict and collision using a unified mathematical framework; 3) to various air traffic applications and their use of directed mathematical models for air traffic conflict and collision probability estimation; and 4) to provide insight into the capabilities and restrictions of the mathematical models in the evaluation of future ATM designs.

There is broad consensus that situation awareness (SA) plays a key role in agent-based modelling of complex sociotechnical systems. However in the social sciences and human factors literature there are different views on what SA is and how it could be modelled. More specifically, one school of research considers SA as the process of gaining awareness, another school refers to it as to the product of gaining awareness, whereas the third school sees SA as a combination of the process and product. Typically, agent-based modelling of SA is done from the second view for each individual agent, possibly with additional social components to enable interaction. Current developments in multiagent systems indicate that social abilities and relations between agents should be not an addition, but at the core of any model of a sociotechnical system. To address this issue, we develop a mathematical modelling framework of SA relations between agents which supports all three views. The use of the framework is demonstrated by an example of retrospective accident modelling from the aviation domain.

This paper presents our experiences of exploring how i* modelling can be used as a bridge between informal air traffic management (ATM) operational concepts and formal agent-based simulation. We report our work on an EU airspace design project that describes a revised concept of operations for lower level airspace around airports, and includes simulation based safety analysis of the critical scenarios. We describe our research towards using i* to address two challenges-how to model the revised concept from the informal concept of operations document, and how to present safety critical scenarios to operational experts. Modelling strategic aspects of a concept of operations is new to ATM, and we draw upon our experiences to provide lessons learned and directions for future work.

The hybrid SIR joint particle filter has been developed as an effective approximation of the exact Bayesian filter for maintaining tracks of multiple maneuvering targets from unassociated measurements. This paper further develops this approach for the situation of limited sensor resolution and two maneuvering targets. For this problem the exact Bayesian filter recursion is characterized, and is subsequently used to develop a novel version of the hybrid SIR joint particle filter. For a simple example, the effectiveness of the novel approach in handling possibly merged measurements is demonstrated through Monte Carlo simulations.

Joint PDA has proven to be effective in tracking multiple targets from measurements amidst clutter and missed detections. Joint IPDA has built upon this by including the probability of target existence as a track quality measure to enable automatic tracking and track maintenance. Both JPDA and JIPDA suffer from the problem of track coalescence of near target tracks. JPDA* is an extension of JPDA which avoids coalescence by pruning specific permutation hypotheses prior to hypothesis merging. Following JPDA*'s descriptor system derivation, this paper developes JIPDA*, an extension of JIPDA which avoids track coalescence. JIPDA* updates the probability of target existence as the track quality measure. An initial simulation study corroborates the effectiveness of this approach for tracking crossing targets in heavy clutter

The problem of evaluating the performability density and distribution of degradable computer systems is considered. A generalized model of performability is considered, wherein the dynamics of configuration modes are modeled as a nonhomogeneous Markov process, and the performance rate in each configuration mode can be time dependent. The key to the development of a unifying mathematical framework is the introduction of two related performability processes: the forward performability process over the interval (0,t), and the performability-to-go process over the interval (t,T), where T is the mission time. Stochastic differential equations techniques show that the joint density of the forward performability and configuration states satisfies a linear, hyperbolic partial differential equation (PDE) with time-dependent coefficients that runs forward in time, while the performability-to-go process satisfies an adjoint PDE running reverse in time. A numerical method for solving the PDEs is presented and is illustrated with examples.>

The reversion in time of a stochastic difference equation in a hybrid space with a Markovian solution is presented. The reversion is obtained by a martingale approach, which previously led to reverse time forms for stochastic equations with Gauss-Markov or diffusion solutions. The reverse time equations follow from a particular noncanonical martingale decomposition, while the reverse time equations for Gauss-Markov and diffusion solutions followed from the canonical martingale decomposition. The need for this noncanonical decomposition stems from the hybrid state-space situation. Moreover, the nonGaussian discrete-time situation leads to reverse time equations that incorporate a Bayesian estimation step. The latter step is carried out for linear systems with Markovian switching coefficients, and the result is shown to provide the solution to the problem of fixed-interval smoothing. For an application of this smoothing approach to a trajectory with sudden maneuvers, simulation results are given to illustrate the practical use of the reverse time equations obtained.>

The problem of evaluating the performability and instantaneous availability of gracefully degradable computer systems is considered. The key to the development of a unifying mathematical framework is the introduction of two related performability processes: forward performability process over the interval (0,t) and the performability-to-go process over the interval (t,T), where T is the mission time. Using the techniques of partial differential equations, it is shown that the joint distribution of the forward performability and configuration states satisfies a linear, hyperbolic partial differential equation (PDE) running forward in time, while the performability-to-go process satisfies a similar PDE running in reverse time. The method includes nonhomogeneous Markov models of the configuration mode. A numerical method for solving the PDEs is presented and illustrated with examples. Practical implications of the reverse-time equations for the sensitivity analysis of instantaneous availability is discussed.>