Spatial Loss Systems: Exact Simulation and Rare Event Behavior

We consider spatial marked Poisson arrivals in a Polish space. These arrivals are accepted or lost in a general state dependent manner. The accepted arrivals remain in the system for a random amount of time, where the individual sojourn times are i.i.d. For such systems, we develop semi-closed form expressions for the steady state probabilities that can be seen to be insensitive to the sojourn time distribution, and that rely essentially on the static probabilities of marked Poisson objects meeting the state acceptance criteria. The latter observation is then exploited to yield straightforward exact simulation algorithms to sample from the steady state distribution. In addition, for the special case where the arrivals are spheres in a Euclidean space that are lost whenever they overlap with an existing sphere, we develop large deviations asymptotics for the probability of observing a large number of spheres in the system in steady state, under diverse asymptotic regimes. Applications include modeling interference in wireless networks and connectivity in ad-hoc networks.