Multiplexed Streaming Codes for Messages with Different Decoding Delays in Channel with Burst and Random Erasures

In a real-time transmission scenario, messages are transmitted through a channel that is subject to packet loss. The destination must recover the messages within the required deadline. In this paper, we consider a setup where two different types of messages with distinct decoding deadlines are transmitted through a channel model that introduces either one burst erasure of length at most $B$, or $N$ random erasures in any fixed-sized sliding window. The message with a short decoding deadline $T_{\mathrm{u}}$ is referred to as an urgent message, while the other one with a decoding deadline $T_{\mathrm{v}}$ ($T_{\mathrm{v}} > T_{\mathrm{u}}$) is referred to as a less urgent message. We consider the scenario where $T_{\mathrm{v}} > T_{\mathrm{u}} + B$ and propose a non-trivial achievable region $\mathcal{R}$ for the aforementioned channel model. We propose a novel merging approach to encode two message streams of different urgency levels into a single flow and present explicit constructions for encoding, contributing to the establishment of the achievability of region $\mathcal{R}$. Our comprehensive analysis demonstrates that this region encompasses the rate pairs of existing encoding schemes and coincides with the capacity region in burst channel scenarios. Lastly, we investigate the property of the achievable region $\mathcal{R}$, proving that it is the largest one obtained from all the rate pairs under the merging method.