# Martingale approach for tail asymptotic problems in the generalized Jackson network

## TL;DR

This paper develops a martingale-based method to analyze tail asymptotics of the stationary joint queue length distribution in generalized Jackson networks, extending previous results to more general settings and larger networks.

## Contribution

It introduces a martingale approach for tail asymptotics in GJNs with general and phase-type distributions, including networks with more than two stations.

## Key findings

- Tail asymptotics for joint queue length distributions are characterized.
- The method applies to networks with more than two stations, though results are less complete.
- The approach extends previous phase-type distribution results to more general distributions.

## Abstract

We study the tail asymptotic of the stationary joint queue length distribution for a generalized Jackson network (GJN for short), assuming its stability. For the two station case, this problem has been recently solved in the logarithmic sense for the marginal stationary distributions under the setting that arrival processes and service times are of phase-type. In this paper, we study similar tail asymptotic problems on the stationary distribution, but problems and assumptions are different. First, the asymptotics are studied not only for the marginal distribution but also the stationary probabilities of state sets of small volumes. Second, the interarrival and service times are generally distributed and light tailed, but of phase type in some cases. Third, we also study the case that there are more than two stations, although the asymptotic results are less complete. For them, we develop a martingale method, which has been recently applied to a single queue with many servers by the author.