# Tail Asymptotics for the $M_1,M_2/G_1,G_2/1$ Retrial Queue with Priority

## TL;DR

This paper analyzes the tail asymptotics of a priority retrial queue with two customer types, providing detailed asymptotic behavior of customer numbers under heavy-tailed service times, aiding performance evaluation.

## Contribution

It introduces a stochastic decomposition method to derive tail asymptotics for a complex priority retrial queue with heavy-tailed service times, advancing understanding of its steady-state behavior.

## Key findings

- Tail asymptotics for queue and orbit customer numbers are derived.
- Heavy-tailed service times significantly influence the tail behavior.
- The approach facilitates performance approximation and numerical analysis.

## Abstract

Stochastic networks with complex structures are key modelling tools for many important applications. In this paper, we consider a specific type of network: the retrial queueing systems with priority. This type of queueing system is important in various applications, including telecommunication and computer management networks with big data. For this type of system, we propose a detailed stochastic decomposition approach to study its asymptotic behaviour of the tail probability of the number of customers in the steady-state for retrial queues with two types (Type-1 and Type-2) of customers, in which Type-1 customers (in a queue) have non-preemptive priority to receive service over Type-2 customers (in an orbit). Under the assumption that the service times of Type-1 customers have a regularly varying tail and the service times of Type-2 customers have a tail lighter than Type-1 customers, we obtain tail asymptotic properties for the number of customers in the queue and in the orbit, respectively, conditional on the server's status, in terms of a detailed stochastic decomposition approach. Tail asymptotic properties are often used as key tools for approximating various performance metrics and constructing numerical algorithms for computations.