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

¹ School of Statistics, Shandong University of Finance and Economics, Jinan, Shandong 250014, P.R. China.
² School of Mathematics and Statistics, Carleton University, Ottawa, ON, Canada K1S 5B6.

^* Corresponding author: mathdsh@gmail.com

Received: 22 September 2020
Accepted: 17 December 2021

Abstract

In this paper, we consider a three-dimensional Brownian-driven tandem queue with intermediate inputs, which corresponds to a three-dimensional semimartingale reflecting Brownian motion whose reflection matrix is triangular. For this three-node tandem queue, no closed form formula is known, not only for its stationary distribution but also for the corresponding transform. We are interested in exact tail asymptotics for stationary distributions. By generalizing the kernel method, and using extreme value theory and copula, we obtain exact tail asymptotics for the marginal stationary distribution of the buffer content in the third buffer and for the joint stationary distribution.