# A simple analysis of a D/GI/1 vacation queue with impatient customers

## TL;DR

This paper analyzes a D/GI/1 queue with vacations and impatient customers, providing conditions for the existence of a limit distribution and explicit solutions under specific distributions.

## Contribution

It introduces new conditions for the limit distribution existence and derives explicit solutions for certain vacation and service time distributions.

## Key findings

- Sufficient condition for limit distribution existence.
- Integral equations for reneging and balking scenarios.
- Explicit solutions for exponential and deterministic distributions.

## Abstract

In this paper, we deal with an $D/GI/1$ vacation system with impatient customers. We give a sufficient condition for the existence of a limit distribution of the waiting time and integral equations are derived in both reneging and balking scenarios. Explicit solutions are given when vacation times are exponentially distributed and service times are either exponentially distributed or deterministic.