# Rare Events of Transitory Queues

## TL;DR

This paper establishes a large deviations principle for the workload process in a transitory queue with finite jobs, analyzing rare events using ordered statistics and self-normalized sums, marking the first such study in this context.

## Contribution

It introduces the first large deviations analysis for rare events in transitory queueing models, connecting ordered statistics with exponential sums.

## Key findings

- Derived the large deviations principle for the workload process.
- Connected ordered statistics with self-normalized sums of exponential variables.
- Provided foundational analysis for rare events in transitory queues.

## Abstract

We study the rare event behavior of the workload process in a transitory queue, where the arrival epochs (or points) of a finite number of jobs are assumed to be the ordered statistics of independent and identically distributed (i.i.d.) random variables. The service times (or marks) of the jobs are assumed to be i.i.d. random variables with a general distribution, that are jointly independent of the arrival epochs. Under the assumption that the service times are strictly positive, we derive the large deviations principle (LDP) satisfied by the workload process. The analysis leverages the connection between ordered statistics and self-normalized sums of exponential random variables to establish the LDP. This paper presents the first analysis of rare events in transitory queueing models, supplementing prior work that has focused on fluid and diffusion approximations.