On customer flows in Jackson queuing networks

TL;DR

This paper extends Melamed's theorem for Jackson queuing networks by characterizing customer flows as Poisson cluster processes and providing a comprehensive approximate version applicable for all probabilities of customers revisiting links.

Contribution

It introduces a new Poisson cluster process model for customer flows and generalizes Melamed's theorem to all revisit probabilities in Jackson networks.

Findings

Customer flow process is a Poisson cluster process.

Established a general approximate version of Melamed's theorem.

Applicable for all revisit probabilities 0 ≤ p < 1.

Abstract

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 $0\le p<1$.