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

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.

💡 Research Summary

This paper investigates the stochastic performance limits of large‑field‑size Random Linear Streaming Codes (RLSCs) in multi‑hop relay networks where each hop experiences independent and identically distributed (i.i.d.) packet erasures. While prior work on streaming codes has largely focused on deterministic (adversarial) erasure models—designing capacity‑achieving constructions for worst‑case erasure patterns—this study shifts the focus to realistic stochastic channels and derives explicit expressions for the average error probability of RLSCs.

The authors first consider a two‑hop relay network (source → relay → destination). They introduce a novel analytical framework that extends the “information debt” concept, originally developed for point‑to‑point streaming, to the multi‑hop setting. In this framework, the “debt” at each node quantifies how many source symbols have not yet been delivered to the destination. Because erasures on different hops can cause some symbols to be “detained” at the source or the relay while others already reach the destination, the debt evolves in a time‑varying manner. By iteratively updating the number of detained symbols at each node, the authors obtain a precise characterization of the error event: decoding failure or violation of the prescribed decoding deadline.

To compute the average error probability, the paper models the evolution of the debt vector as a Markovian transition driven by erasure outcomes. The transition matrices are constructed in a “band fashion” with a nested structure: each matrix has non‑zero entries only on the main diagonal and a few adjacent diagonals, reflecting the fact that in one time slot a symbol can move at most one hop forward. This banded, nested design enables a compact representation that scales linearly with the number of hops and the decoding delay. By taking expectations over the i.i.d. erasure patterns, the authors derive a closed‑form expression for the average error probability, which simplifies to 1 minus the probability that the debt becomes zero within the deadline. The analysis assumes a sufficiently large finite field, ensuring that random linear combinations are essentially independent and that the derived expressions are exact in the asymptotic field‑size regime.

The paper then generalizes the two‑hop results to an arbitrary L‑hop relay network. Because the transition matrices retain their nested banded structure, adding an extra hop corresponds to multiplying by another band matrix, preserving tractability. Consequently, the average error probability for any number of hops can be obtained by a straightforward extension of the two‑hop formula, with computational complexity growing only linearly in L and the decoding window size.

Extensive Monte‑Carlo simulations validate the theoretical predictions. The simulations cover a range of parameters (source rate K, packet lengths Nₗ, per‑hop success probabilities qₗ, decoding deadline Δ, and number of hops L). Results show excellent agreement with the analytical formulas, especially when the field size exceeds 2¹⁶, confirming that the large‑field assumption is realistic for practical implementations. Moreover, the stochastic performance of RLSCs is compared against several streaming codes designed for adversarial channels (e.g., Symbol‑Wise Decode‑Forward, state‑dependent schemes). The comparison reveals that RLSCs achieve comparable or slightly better error rates under the same delay constraints, demonstrating that the random linear approach remains robust even when the erasure process is probabilistic.

In conclusion, the paper provides the first systematic stochastic analysis of random linear streaming codes in multi‑hop relay networks. Its banded, nested transition‑matrix methodology offers a powerful tool for evaluating and designing streaming codes under realistic channel models, and it can be readily extended to more complex scenarios such as Markovian burst erasures (Gilbert‑Elliott), heterogeneous hop reliabilities, limited buffer sizes, or adaptive encoding with channel state feedback. Future work may explore these extensions, as well as the integration of network coding across multiple paths to further improve reliability and latency in large‑scale streaming applications.