On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels

The abstract model of the multi-hop relay channel is fundamental to a vast variety of modern communication systems. This fact, coupled with the demand for ultra-reliable-low-latency communication (URLLC), motivates a new investigation of relay channels from a delay-vs-throughput perspective. This work seeks to analyze this tradeoff in the regime of asymptotically large, yet still finite delay. A new metric called the Delay Amplification Factor (DAF) is introduced, which allows analytic comparison of the asymptotic delay across different relay solutions, e.g. decode-&-forward (DF), compress-&-forward, etc. The optimal DAF (over all possible existing/future designs) is then characterized for two special settings, one with fixed-length coding and one with variable-length coding and 1-bit stop feedback. The results show that under some general conditions, the optimal end-to-end delay over an L-hop line network is asymptotically comparable to the delay over the single bottleneck hop, and it does not grow linearly with respect to L. The linearly growing delay penalty commonly encountered in DF and other schemes is thus an artifact rather than a fundamental limit of multi-hop relay communication.