On the power series approximations of a structured batch arrival two-class retrial system with weighted fair orbit queues

TL;DR

This paper develops power series approximations for a complex two-class retrial queue system with structured batch arrivals, providing recursive formulas and boundary value problem solutions, validated through numerical examples.

Contribution

It introduces a novel power series approximation method for a structured batch arrival retrial system with weighted fair orbit queues, including both exponential and arbitrary service times.

Findings

Power series expansions accurately approximate stationary distributions.

Recursive formulas facilitate computation of queue-length distributions.

Numerical results confirm the effectiveness of the proposed approach.

Abstract

We provide power series approximations for a structured batch arrival single server retrial system with two infinite capacity weighted fair orbit queues, i.e., the re-transmission rate of an orbit depends on the state of the other orbit queue. We consider both exponential and arbitrary distributed service times. In both cases we obtain power series expansions of the generating functions of the stationary joint orbit queue-length distributions, and provide a recursive approach to calculate their coefficients. We also show how to obtain the generating function of the stationary joint orbit queue-length distribution with the aid of a Riemann boundary value problem. Power series approximations are also provided for the model with two independent Poisson streams of jobs with single arrivals. Numerical illustrations are performed and show the accuracy of our approach.