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 depen-dent manner. The accepted arrivals remain in the systemfor a random amount of time, where the individual sojourntimes are i.i.d. For such systems, we develop semi-closedform expressions for the steady state probabilities that canbe seen to be insensitive to the sojourn time distribution,and that rely essentially on the static probabilities of markedPoisson objects meeting the state acceptance criteria. Thelatter observation is then exploited to yield straightforwardexact simulation algorithms to sample from the steady statedistribution. In addition, for the special case where the ar-rivals are spheres in a Euclidean space that are lost wheneverthey overlap with an existing sphere, we develop large devi-ations asymptotics for the probability of observing a largenumber of spheres in the system in steady state, under di-verse asymptotic regimes. Applications include modelinginterference in wireless networks and connectivity in ad-hocnetworks.