# A discrete-time GI^X/Geo^Y/1 queue with early arrival system

## TL;DR

This paper analyzes a discrete-time GI^X/Geo^Y/1 queue with early arrivals, deriving steady-state distributions and connecting to continuous-time models, relevant for telecommunication systems.

## Contribution

It extends existing models by considering early arrival systems, providing new steady-state distribution results and their relation to continuous-time queues.

## Key findings

- Derived steady-state queue length distributions for EAS.
- Showed limiting case converges to continuous-time queue results.
- Numerical results illustrate the model's behavior.

## Abstract

A discrete-time batch service queue with batch renewal input and random serving capacity rule under the late arrival delayed access system (LAS-DA), has recently appeared in the literature [2]. In this paper, we consider the same model under the early arrival system (EAS), since it is more applicable in telecommunication systems where an arriving batch of packets needs to be transmitted in the same slot in which it has arrived. In doing so, we derive the steady-state queue length distributions at various epochs and show that in limiting case the result gets converted to the continuous-time queue [1]. We discuss a few numerical results as well.