# Departure Process of Actively Managed Queue with Dependent Job Sizes

## TL;DR

This paper studies a queueing system where job sizes are dependent and service is actively managed, analyzing how jobs depart and are rejected over time.

## Contribution

The paper introduces a novel analysis of departure and rejection processes in queues with dependent job sizes and active management.

## Key findings

- The expected number of job departures and rejections within a time interval is derived.
- The impact of job size dependence and rejection probabilities on system performance is quantified.
- Numerical examples show how system load and dependence levels affect departure and rejection rates.

## Abstract

We focus on a queueing model in which the sizes of arriving jobs are stochastically dependent and each job may be denied service with a probability determined by the queue size (active management). Both of these effects are known to occur in computer networking and many other real-world realizations of queueing systems. For such a model, we perform a thorough transient and stationary analysis of the job departure process and the job rejection process. The results include theorems on the expected number of jobs that depart within a specified time interval, the departure intensity at a given time, the stationary departure rate, the expected number of jobs rejected within a specified interval, the transient rejection intensity and the stationary rejection rate. Sample numerical calculations are provided for illustration. They include various settings of the level of dependence between jobs, job rejection probabilities, and system load, as well as their impact on the departure and rejection processes.