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.

翻译：本文旨在探究大域尺寸随机线性流码（RLSCs）在多跳中继网络中的随机性能极限。在我们的模型中，源节点通过多个中继节点向目的节点传输一系列流式消息，并受到延迟约束。以往研究大多集中于确定性对抗信道，该类信道仅引入有限类型的擦除模式，并致力于设计最优的容量可达码。本文聚焦于随机信道，其中每一跳均遭受独立同分布的包擦除，并对多跳RLSCs的误码概率进行随机分析。我们的贡献主要体现在三个方面。首先，通过一种新颖的框架刻画了两跳中继网络中大域尺寸RLSCs的误码事件，该框架的核心在于量化网络中每个节点的信息流。由于不同跳的擦除，部分源符号可能被“滞留”在源节点或中继节点，而其他符号已到达目的节点。通过迭代计算每个节点的滞留符号数量，该框架将点对点网络中的“信息债务”概念[Pinwen Su等人，2022]扩展至两跳中继网络。其次，基于误码事件，通过细致分析期望项，推导出两跳网络中平均误码概率的表达式。为处理网络两跳所有可能擦除模式的期望，我们创新性地以“带状”嵌套结构构建了滞留符号的转移矩阵。第三，将两跳网络的推导结果进一步推广至任意跳数的中继网络。此外，通过仿真验证了随机分析结果的准确性，并与现有对抗信道流码进行了性能比较。