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

Dingli Yuan, Zhiquan Tan, Zhongyi Huang

TL;DR

A novel merging approach to encode two message streams of different urgency levels into a single flow is proposed and present explicit constructions for encoding, contributing to the establishment of the achievability of a non-trivial achievable region in burst channel scenarios.

Abstract

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.

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

TL;DR

A novel merging approach to encode two message streams of different urgency levels into a single flow is proposed and present explicit constructions for encoding, contributing to the establishment of the achievability of a non-trivial achievable region in burst channel scenarios.

Abstract

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 , or random erasures in any fixed-sized sliding window. The message with a short decoding deadline is referred to as an urgent message, while the other one with a decoding deadline () is referred to as a less urgent message. We consider the scenario where and propose a non-trivial achievable region 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 . 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 , proving that it is the largest one obtained from all the rate pairs under the merging method.
Paper Structure (19 sections, 11 theorems, 58 equations, 14 figures, 2 tables)

This paper contains 19 sections, 11 theorems, 58 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notations
  3. Background knowledge
  4. Preliminaries
  5. Revisit of the Explicit Construction of $(W,B,N)$-Achievable Rate-Optimal Block Code in domanovitz2021anexplicit
  6. Main results
  7. Construction
  8. Merging Method
  9. Explicit Constructions Based on Merging Method
  10. The construction when $B\geq 2N-1$
  11. The construction when $B<2N-1$
  12. An Example
  13. Comparison
  14. Further Discussion on Sum Rate
  15. Proof of Theorem 2
  16. ...and 4 more sections

Key Result

Lemma 1

Given an $\left(n, k_{\mathrm{v}}, k_{\mathrm{u}}, T_{\mathrm{v}}, T_{\mathrm{u}}\right)_{\mathbb{F}}$-block code that is $(W, B, N)$-achievable, there exits a corresponding $\left(n, k_{\mathrm{v}}, k_{\mathrm{u}}, T_{\mathrm{v}}, T_{\mathrm{u}}\right)_{\mathbb{F}}$-streaming code that is $(W, B, N We note that $\mathbf{G}= \sum_{l=0}^{n-1} \mathbf{G}_l$. In particular, if we let $\mathbf{s}_i \t

Figures (14)

  • Figure 1: Generator matrix of the block code in domanovitz2021anexplicit
  • Figure 3: The flow chart of the encoding steps based on merging method. The shadow blocks indicate the messages that overlap with others.
  • Figure 4: The generator matrix $\mathbf{G}$ of the proposed code when $B\geq 2N-1$
  • Figure 5: The generator matrix $\mathbf{G}$ of the proposed code when $B<2N-1$
  • Figure 6: Example: The generator matrix of $(T_{\mathrm{v}},T_{\mathrm{u}},B,N)=(12,6,4,2)$ in channel with burst
  • ...and 9 more figures

Theorems & Definitions (34)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 1
  • Definition 5
  • Definition 6
  • Definition 7
  • Lemma 1
  • proof
  • ...and 24 more