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.