Point Processes in Large Spaces: Poisson Approximations

#### R.F. Serfozo

1997, v.3, №4, 597-610

ABSTRACT

This study concerns queueing networks, spatial service systems and particle systems whose evolutions can be formulated by certain random transformations of point processes that represent the system inputs in time or space. The focus is on large systems in which the number of customers or particles in bounded regions tend to be small and the evolution of the system can be approximated by Poisson processes. The main result is a limit theorem for random transformations of point processes into a large space that converge to Poisson processes. This result is applied to several networks and particle systems. One example is for particles placed in a Euclidean space at time $0$ according to a point process that obeys a (spatial) weak law of large numbers and the particles evolve such that the probability of a particle being in a compact set tends to $0$ at a certain rate. Then as time tends to infinity, the point process of particle locations on a rescaled space converges in distribution to a Poisson process. This is a large-space analogue of Derman's classical stationary Markovian particle system where the particle locations at any time form a Poisson process.

Keywords: Poisson process,large networks,time-space transformations,particle systems,M/G/$\infty$ queueing system