Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Stochastic Analysis of Random Linear Streaming Codes in Multi-Hop Relay Networks

Kai Huang, Xinyu Xie, Chunpeng Chen, Wenjie Guan, Xiaoran Wang, Jinbei Zhang

TL;DR

This work develops a stochastic performance analysis for large-field random linear streaming codes (RLSCs) in multi-hop relay networks under i.i.d. packet erasures. It introduces an information-debt framework to track detained source symbols across hops and derives a closed-form-like expression for the end-to-end error probability p_e by constructing nested joint transition matrices that capture the evolution of detained symbols and decoding progress. The authors extend the two-hop TRN results to general L-hop MRN, providing a scalable approach to compute p_e via matrix embeddings and iHMM-like reasoning, and validate the theory with extensive simulations showing favorable performance of RLSCs relative to adversarial-channel codes. The work offers practical insight into the stochastic limits of streaming in relay networks and suggests techniques to manage complexity (small state caps and sparse matrix methods) for real-world deployments.

Abstract

In this paper, we aim to explore the stochastic performance limit of large-field-size Random Linear Streaming Codes (RLSCs) in multi-hop relay networks. In our model, a source transmits a sequence of streaming messages to a destination through multiple relays subject to a delay constraint. Most previous research focused on deterministic adversarial channel which introduces only restricted types of erasure patterns, and aimed to design the optimal capacity-achieving codes. In this paper, we focus on stochastic channel where each hop is subject to i.i.d. packet erasures, and carry out stochastic analysis on the error probability of multi-hop RLSCs. Our contributions are three-folds. Firstly, the error event of large-field-size RLSCs is characterized in two-hop relay network with a novel framework, which features quantification of information flowing through each node in the network. Due to the erasures in different hops, some source symbols can be "detained" at the source or relay while others have arrived at the destination. By iteratively computing the number of detained symbols at each node, this framework extends the concept "information debt" from point-to-point network [Pinwen Su et al. 2022] into two-hop relay networks. Secondly, based on the error event, the expression of average error probability in two-hop network is derived by carefully analyzing the expectation terms. To handle the expectation over all possible erasure patterns along two hops of the network, the transition matrices of the detained symbols are novelly constructed in a "band fashion" with nested structure. Thirdly, the derived results in two-hop network are further generalized into relay networks with arbitrary number of hops. Furthermore, simulations are conducted to verify the accuracy of our stochastic analysis, and compare with some existing streaming codes for the adversarial channels.

On the Stochastic Analysis of Random Linear Streaming Codes in Multi-Hop Relay Networks

TL;DR

This work develops a stochastic performance analysis for large-field random linear streaming codes (RLSCs) in multi-hop relay networks under i.i.d. packet erasures. It introduces an information-debt framework to track detained source symbols across hops and derives a closed-form-like expression for the end-to-end error probability p_e by constructing nested joint transition matrices that capture the evolution of detained symbols and decoding progress. The authors extend the two-hop TRN results to general L-hop MRN, providing a scalable approach to compute p_e via matrix embeddings and iHMM-like reasoning, and validate the theory with extensive simulations showing favorable performance of RLSCs relative to adversarial-channel codes. The work offers practical insight into the stochastic limits of streaming in relay networks and suggests techniques to manage complexity (small state caps and sparse matrix methods) for real-world deployments.

Abstract

In this paper, we aim to explore the stochastic performance limit of large-field-size Random Linear Streaming Codes (RLSCs) in multi-hop relay networks. In our model, a source transmits a sequence of streaming messages to a destination through multiple relays subject to a delay constraint. Most previous research focused on deterministic adversarial channel which introduces only restricted types of erasure patterns, and aimed to design the optimal capacity-achieving codes. In this paper, we focus on stochastic channel where each hop is subject to i.i.d. packet erasures, and carry out stochastic analysis on the error probability of multi-hop RLSCs. Our contributions are three-folds. Firstly, the error event of large-field-size RLSCs is characterized in two-hop relay network with a novel framework, which features quantification of information flowing through each node in the network. Due to the erasures in different hops, some source symbols can be "detained" at the source or relay while others have arrived at the destination. By iteratively computing the number of detained symbols at each node, this framework extends the concept "information debt" from point-to-point network [Pinwen Su et al. 2022] into two-hop relay networks. Secondly, based on the error event, the expression of average error probability in two-hop network is derived by carefully analyzing the expectation terms. To handle the expectation over all possible erasure patterns along two hops of the network, the transition matrices of the detained symbols are novelly constructed in a "band fashion" with nested structure. Thirdly, the derived results in two-hop network are further generalized into relay networks with arbitrary number of hops. Furthermore, simulations are conducted to verify the accuracy of our stochastic analysis, and compare with some existing streaming codes for the adversarial channels.

Paper Structure

This paper contains 18 sections, 20 theorems, 69 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Point-to-point Streaming Codes for Adversarial Channel
  3. Multi-hop Streaming Codes for Adversarial Channel
  4. Point-to-point Streaming Codes for Stochastic Channel
  5. Challenges and Our Contributions
  6. System Model and Definitions
  7. Characterization of Error Event of RLSCs in TRN
  8. Characterization of Error Probability of RLSCs in TRN
  9. Generalization to Multi-hop Relay Networks
  10. Numerical Results
  11. Relative error of $p_e$ in Theorem \ref{['theorem:TRN']}
  12. Performance of RLSCs with different decoding latency, erasure probability and coding rate
  13. Performance comparison to the streaming code proposed in SLF 3hop first paper
  14. Conclusion
  15. Proof of Proposition \ref{['Proposition:characterization of the error event']}
  16. ...and 3 more sections

Key Result

Proposition 1

Assume GMDS holds. For RLSCs in TRN, for any fixed index $i_0 \ge 0$, (a)$\mathbf{s}(t)$ is $\Delta$-decodable for all timeslots $t$ that (b)$\mathbf{s}(t)$ is not $\Delta$-decodable for the rest of $t$.

Figures (17)

  • Figure 1: Illustration of a $L$-hop relay network (with $L+1$ nodes).
  • Figure 2: The encoding process at the source $r_0$.
  • Figure 3: $\mathbf{x}_0(3),\mathbf{x}_0(4),\mathbf{x}_0(6),\mathbf{x}_0(7)$ are erased in $(r_0,r_1)$.
  • Figure 4: The encoding process at the relay $r_1$. The white blanks are due to the channel erasures in the first hop $(r_0,r_1)$.
  • Figure 5: The encoding process at the relay $r_1$ after eliminating the erased entries.
  • ...and 12 more figures

Theorems & Definitions (31)

  • Definition 1
  • Definition 2
  • Example 1
  • Definition 3
  • Definition 4
  • Proposition 1
  • Lemma 1
  • Definition 5
  • Definition 6
  • Proposition 2
  • ...and 21 more