A multi-orthogonal polynomials' approach to bulk queueing theory

TL;DR

This paper introduces a novel approach using multi-orthogonal polynomials to analyze the transition probabilities and steady state of bulk queueing systems modeled by a stationary Markov process.

Contribution

It develops an integral expression for transition probabilities using multi-orthogonal polynomials and explores the steady state of the queueing model.

Findings

Derived integral expression for transition probabilities

Analyzed the steady state of the queueing system

Connected multi-orthogonal polynomials with queueing theory

Abstract

We consider a stationary Markov process that models certain queues with a bulk service of a fixed number $m$ of admitted customers. We find an integral expression of its transition probability function in terms of certain multi-orthogonal polynomials with respect to a system of distributions that contain measures supported on starlike subsets of the complex plane. We also study the corresponding steady state.