# Theater-Level Stochastic Air-to-Air Engagement Modeling via Event Occurrence Networks Using Piecewise Polynomial Approximation

#### Abstract

This dissertation investigates a stochastic network formulation termed an event occurrence network (EON). EONs are graphical representations of the superposition of several terminating counting processes. An EON arc represents the occurrence of an event from a group of (sequential) events before the occurrence of events from other event groupings. Events between groups occur independently, but events within a group occur sequentially. A set of arcs leaving a node is a set of competing events, which are probabilistically resolved by order relations. An important EON metric is the probability of being at a particular node or set of nodes at time t. Such a probability is formulated as an integral expression (generally a multiple integral expression) involving event probability density functions. This integral expression involves several stochastic operators: subtraction; multiplication; convolution, and integration. For the EON probability metric, simulation is generally computationally costly to obtain accurate estimates for large EONs, transient nodes, or "rare" states. Instead, using research with probabilistic activity networks, a numerical approximation technique using piecewise polynomial functions is developed. The dissertation's application area is air-to-air combat modeling.

*This paper has been withdrawn.*