On customer flows in Jackson queuing networks

Melamed’s theorem states that for a Jackson queuing network, the equilibrium flow along a link follows Poisson distribution if and only if no customers can travel along the link more than once. Barbour & Brown~(1996) considered the Poisson approximate version of Melamed’s theorem by allowing the customers a small probability $p$ of travelling along the link more than once. In this paper, we prove that the customer flow process is a Poisson cluster process and then establish a general approximate version of Melamed’s theorem accommodating all possible cases of $0\le p<1$.