# Stable Throughput and Delay Analysis of a Random Access Network With Queue-Aware Transmission

## TL;DR

This paper analyzes the stability and delay performance of multi-user slotted ALOHA networks with multi-packet reception, providing new mathematical tools and bounds for delay and stability regions, including explicit formulas for symmetric cases.

## Contribution

It introduces a comprehensive mathematical framework for stability and delay analysis in queue-aware multi-user ALOHA networks with MPR capabilities, including explicit delay bounds and optimal transmission probabilities.

## Key findings

- Derived stability regions for two- and three-user cases.
- Provided tight bounds for average delay in symmetric MPR scenarios.
- Obtained explicit delay expressions and optimal transmission probabilities for symmetric capture models.

## Abstract

In this work we consider a two-user and a three-user slotted ALOHA network with multi-packet reception (MPR) capabilities. The nodes can adapt their transmission probabilities and their transmission parameters based on the status of the other nodes. Each user has external bursty arrivals that are stored in their infinite capacity queues. For the two- and the three-user cases we obtain the stability region of the system. For the two-user case we provide the conditions where the stability region is a convex set. We perform a detailed mathematical analysis in order to study the queueing delay by formulating two boundary value problems (a Dirichlet and a Riemann-Hilbert boundary value problem), the solution of which provides the generating function of the joint stationary probability distribution of the queue size at user nodes. Furthermore, for the two-user symmetric case with MPR we obtain a lower and an upper bound for the average delay without explicitly computing the generating function for the stationary joint queue length distribution. The bounds as it is seen in the numerical results appear to be tight. Explicit expressions for the average delay are obtained for the symmetrical model with capture effect which is a subclass of MPR models. We also provide the optimal transmission probability in closed form expression that minimizes the average delay in the symmetric capture case. Finally, we evaluate numerically the presented theoretical results.