A Poisson limit theorem for sums of dissociated 0–1 random variables is refined by deriving the first terms in an asymptotic expansion. The most natural refinement does not remove all the first-order error in a number of applications to tests of clustering, and a further approximation is derived which gives excellent results in practice. The proofs are based on the technique of Stein and Chen.

We consider single-class Markovian queueing networks with state-dependent service rates (the immigration processes of Whittle (1968)). The distance of customer flows from Poisson processes is estimated in both the open and closed cases. The bounds on distances lead to simple criteria for good Poisson approximations. Using the bounds, we give an asymptotic, closed network version of the ‘loop criterion' of Melamed (1979) for an open network. Approximation of two or more flows by independent Poisson processes is also studied.

To obtain a limit with independent components in the superposition of m-dimensional point processes, a condition corresponding to asymptotic independence must be included. When this condition is relaxed, convergence to limits with dependent components is possible. In either case, convergence of finite distributions alone implies tightness and thus weak convergence in the function space D[0, ∞) × … × D[0, ∞).