Computing the Exponential of Large Block-Triangular Block-Toeplitz Matrices Encountered in Fluid Queues

TL;DR

This paper introduces algorithms that efficiently compute the exponential of large block-triangular block-Toeplitz matrices, crucial for fluid queue analysis, by exploiting their structure and decay properties.

Contribution

The paper presents novel algorithms leveraging the Toeplitz structure and decay properties to compute matrix exponentials of large block-triangular matrices efficiently.

Findings

Algorithms enable computation of large matrix exponentials otherwise infeasible.

Decay properties of the exponential are established, aiding in approximation.

Efficient methods improve analysis of fluid queues with large state spaces.

Abstract

The Erlangian approximation of Markovian fluid queues leads to the problem of computing the matrix exponential of a subgenerator having a block-triangular, block-Toeplitz structure. To this end, we propose some algorithms which exploit the Toeplitz structure and the properties of generators. Such algorithms allow to compute the exponential of very large matrices, which would otherwise be untreatable with standard methods. We also prove interesting decay properties of the exponential of a generator having a block-triangular, block-Toeplitz structure.