# Asymptotically Optimal Status Update Compression in Multi-Source System: Age–Distortion Tradeoff

## TL;DR

This paper studies how to optimally compress status updates from multiple sources to minimize delay while controlling distortion.

## Contribution

The paper introduces two compression schemes and proves their asymptotic optimality for minimizing average age under distortion constraints.

## Key findings

- Multi-quantizer compression with dedicated quantizers achieves asymptotic optimality for minimizing average age.
- Single-quantizer compression with a shared quantizer also achieves asymptotic optimality.
- The optimal average age increases linearly with the logarithm of distortion for both schemes.

## Abstract

We consider a compression problem in a multi-source status-updating system through a representative two-source scenario. The status updates are generated by two independent sources following heterogeneous Poisson processes. These updates are then compressed into binary strings and sent to the receiver via a shared, error-free channel with a unit rate. We propose two compression schemes—a multi-quantizer compression scheme, where a dedicated quantizer–encoder pair is assigned to each source for compression, and a single-quantizer compression scheme, employing a unified quantizer–encoder pair shared across both sources. For each scheme, we formulate an optimization problem to jointly design quantizer–encoder pairs, with the objective of minimizing the sum of the average ages subject to a distortion constraint of symbols, respectively. The following three theoretical results are established: (1) The combination of two uniform quantizers with different parameters, along with their corresponding AoI-optimal encoders, provides an asymptotically optimal solution for the multi-quantizer compression scheme. (2) The combination of a piecewise uniform w-quantizer with an AoI-optimal encoder provides an asymptotically optimal solution for the single-quantizer compression scheme. (3) For both schemes, the optimal sum of the average ages is asymptotically linear with respect to the log distortion, with the same slope determined by the sources’ arrival rates.