# A sequential update algorithm for computing the stationary distribution vector in upper block-Hessenberg Markov chains

## TL;DR

This paper introduces a new sequential update algorithm for accurately computing the stationary distribution in upper block-Hessenberg Markov chains, leveraging LBCL-augmented truncations and linear fractional programming.

## Contribution

It develops a novel algorithm using LBCL-augmented truncations and LFP problems to compute the exact stationary distribution without approximation.

## Key findings

- Algorithm effectively computes stationary distributions for complex queues.
- Demonstrated applicability to BMAP/M/∞ and M/M/s retrial queues.
- Provides bounds and convergence guarantees for the method.

## Abstract

This paper proposes a new algorithm for computing the stationary distribution vector in continuous-time upper block-Hessenberg Markov chains. To this end, we consider the last-block-column-linearly-augmented (LBCL-augmented) truncation of the (infinitesimal) generator of the upper block-Hessenberg Markov chain. The LBCL-augmented truncation is a linearly-augmented truncation such that the augmentation distribution has its probability mass only on the last block column. We first derive an upper bound for the total variation distance between the respective stationary distribution vectors of the original generator and its LBCL-augmented truncation. Based on the upper bound, we then establish a series of linear fractional programming (LFP) problems to obtain augmentation distribution vectors such that the bound converges to zero. Using the optimal solutions of the LFP problems, we construct a matrix-infinite-product (MIP) form of the original (i.e., not approximate) stationary distribution vector and develop a sequential update algorithm for computing the MIP form. Finally, we demonstrate the applicability of our algorithm to BMAP/M/$\infty$ queues and M/M/$s$ retrial queues.