# The Cost of Delay in Status Updates and their Value: Non-linear Ageing

## TL;DR

This paper introduces new metrics, CoUD and VoIU, to quantify the freshness and value of information in status update systems, analyzing their impact on system performance under linear and non-linear ageing costs.

## Contribution

It proposes the CoUD and VoIU metrics, expanding the analysis of information freshness by considering non-linear ageing costs and their effect on optimal update policies.

## Key findings

- CoUD performance varies with cost per time unit.
- Optimal policies are consistent for linear ageing, vary for non-linear.
- VoIU performance is affected only under non-linear cost increases.

## Abstract

We consider a status update communication system consisting of a source-destination link. A stochastic process is observed at the source, where samples are extracted at random time instances, and delivered to the destination, thus, providing status updates for the source. In this paper, we expand the concept of information ageing by introducing the cost of update delay (CoUD) metric to characterize the cost of having stale information at the destination. The CoUD captures the freshness of the information at the destination and can be used to reflect the information structure of the source. Moreover, we introduce the value of information of update (VoIU) metric that captures the reduction of CoUD upon reception of an update. Using the CoUD, its by-product metric called peak cost of update delay (PCoUD), and the VoIU, we evaluate the performance of an M/M/1 system in various settings that consider exact expressions and bounds. Our results indicate that the performance of CoUD differs depending on the cost assigned per time unit, however the optimal policy remains the same for linear ageing and varies for non-linear ageing. When it comes to the VoIU the performance difference appears only when the cost increases non-linearly with time. The study illustrates the importance of the newly introduced variants of age, furthermore supported in the case of VoIU by its tractability.