Removing logarithms from Poisson process error bounds

We present a new approximation theorem for estimating the error in approximating the whole distribution of a finite-point process by a suitable Poisson process. The metric used for this purpose regards the distributions as close if there are couplings of the processes with the expected average distance between points small in the best-possible matching. In many cases, the new bounds remain constant as the mean of the process increases, in contrast to previous results which, at best, increase logarithmically with the mean. Applications are given to Bernoulli-type point processes and to networks of queues. In these applications the bounds are independent of time and space, only depending on parameters of the system under consideration. Such bounds may be important in analysing properties, such as queueing parameters which depend on the whole distribution and not just the distribution of the number of points in a particular set.