Exact tail asymptotics for a three dimensional Brownian-driven tandem queue with intermediate inputs

TL;DR

This paper derives exact tail asymptotics for the stationary distributions of a three-dimensional Brownian-driven tandem queue with intermediate inputs, extending known results from lower dimensions.

Contribution

It generalizes the kernel method and employs copula techniques to obtain precise tail asymptotics for a 3D SRBM, a less-explored higher-dimensional case.

Findings

Exact tail asymptotics for the third buffer's stationary distribution

Joint stationary distribution tail asymptotics

Extension of kernel method to 3D SRBM

Abstract

The semimartingale reflecting Brownian motion (SRBM) can be a heavy traffic limit for many server queueing networks. Asymptotic properties for stationary probabilities of the SRBM have attracted a lot of attention recently. However, many results are obtained only for the two-dimensional SRBM. There is only little work related to higher dimensional ($\geq 3$) SRBMs. In this paper, we consider a three dimensional SRBM: A three dimensional Brownian-driven tandem queue with intermediate inputs. We are interested in tail asymptotics for stationary distributions. By generalizing the kernel method and using copula, we obtain exact tail asymptotics for the marginal stationary distribution of the buffer content in the third buffer and the joint stationary distribution.