On the Optimal Delay Growth Rate of Multi-Hop Line Networks: Asymptotically Delay-Optimal Designs and the Corresponding Error Exponents

Multi-hop line networks have emerged as an important abstract model for modern and increasingly dense communication networks. In addition, the growth of real-time and mission-critical services has created high demand for and increased research interest in low-latency communications. The combination of these facts motivates a new investigation of data transmission schemes for $L$ -hop line networks from a delay-vs-throughput perspective. To this end, this work defines a metric called the delay amplification factor for a target throughput $R$ , denoted by ${\mathsf {DAF}}(R)$ , which characterizes the growth rate of the (asymptotic) delay with respect to the number of hops. We show that all existing relay schemes, e.g., Decode-&-Forward (DF), have $\lim _{R\nearrow C} {\mathsf {DAF}}(R)=\Omega (L)$ , which is consistent with the decades-old perception that delay grows linearly with respect to $L$ . We then design a scheme satisfying $\lim _{R\nearrow C} {\mathsf {DAF}}(R)=1$ , if the bottleneck hop is the last hop, i.e., its asymptotic delay does not grow with respect to $L$ . The results imply that this linearly growing delay is an artifact of the existing DF designs, and it is possible to surpass it and attain the true fundamental limit with a new delay-centric solution. In the second half of this work, we further show that if variable-length coding and one-bit stop-feedback are allowed, we can relax the condition bottleneck being the last hop and attain $\lim _{R\nearrow C} {\mathsf {DAF}}(R)= 1$ for any arbitrary line networks.