On the Analysis of Random Linear Streaming Codes in Stochastic Channels

Abstract

Random Linear Streaming Codes (RLSCs) can dramatically reduce the queuing delay of block codes in real-time services. In this paper, we aim to explore the fundamental limit of large-field-size RLSCs in stochastic symbol erasure channels (SEC). The Non-systematic RLSCs (NRLSCs) in i.i.d. SEC has been analyzed in [Pinwen Su et al. 2022]. In this work, we first derive the closed-form expression on the exact error probability of NRLSCs in Gilbert-Elliott symbol erasure channels (G-ESEC). Compared to i.i.d SEC, the erasure probability of G-ESEC depends on channel state, thus transitions between the states should be considered. To deal with the stochastic state transitions, we introduce two novel techniques. (i) To account for the impact of switching states on probability terms, we find and leverage the recursive structure of the state transition traces. (ii) To obtain the expected number of error timeslots, we derive the stationary initial distribution of the states, and formulate iterative equation to characterize the expectation terms. Then we analyze the Systematic RLSCs (SRLSCs) in a special SEC, i.e., the packet erasure channel (PEC). In this scenario, SRLSCs could save some source symbols which should have exceeded the decoding delay in NRLSCs, and thus could significantly reduce the error probability. To this point, our contributions are two-folds. (i) Through a case study, we find a counter-intuitive phenomenon that SRLSCs can cause unexpected error events comparing to NRLSCs in some erasure patterns. Then we fully characterize the error event of SRLSCs for any erasure pattern. (ii) For i.i.d. PEC, we derive an analytical expression on exact error probability of SRLSCs when length of memory approaches infinity and coding rate equals to 1/2. Simulations are conducted to verify the accuracy of our analysis and compare the performance of NRLSCs, SRLSCs, and existing streaming codes.