# On the Distribution of AoI for the GI/GI/1/1 and GI/GI/1/2* Systems: Exact Expressions and Bounds

## TL;DR

This paper derives exact expressions and bounds for the distribution of Age of Information in GI/GI/1/1 and GI/GI/1/2* queueing systems, enhancing understanding of information freshness in systems with limited queue capacity.

## Contribution

It provides the first closed-form expressions and a simple methodology for bounds on AoI violation probability in these queueing models, including non-preemptive scheduling.

## Key findings

- Exact AoI violation probability expressions for GI/GI/1/1 and M/GI/1/1 systems.
- Upper bounds for AoI violation probability that tighten with higher utilization.
- Bounds overestimate violation probability by decreasing amounts as departure rate increases.

## Abstract

Since Age of Information (AoI) has been proposed as a metric that quantifies the freshness of information updates in a communication system, there has been a constant effort in understanding and optimizing different statistics of the AoI process for classical queueing systems. In addition to classical queuing systems, more recently, systems with no queue or a unit capacity queue storing the latest packet have been gaining importance as storing and transmitting older packets do not reduce AoI at the receiver. Following this line of research, we study the distribution of AoI for the GI/GI/1/1 and GI/GI/1/2* systems, under non-preemptive scheduling. For any single-source-single-server queueing system, we derive, using sample path analysis, a fundamental result that characterizes the AoI violation probability, and use it to obtain closed-form expressions for D/GI/1/1, M/GI/1/1 as well as systems that use zero-wait policy. Further, when exact results are not tractable, we present a simple methodology for obtaining upper bounds for the violation probability for both GI/GI/1/1 and GI/GI/1/2* systems. An interesting feature of the proposed upper bounds is that, if the departure rate is given, they overestimate the violation probability by at most a value that decreases with the arrival rate. Thus, given the departure rate and for a fixed average service, the bounds are tighter at higher utilization.