Sending Perishable Information: Coding Improves Delay-Constrained Throughput Even for Single Unicast

Abstract

This paper considers network communications under a hard timeliness constraint , where a source node streams perishable information to a destination node over a directed acyclic graph subject to a hard delay constraint. Transmission along any edge incurs unit delay, and it is required that every information bit generated at the source at the beginning of time $t$ to be received and recovered by the destination at the end of time $t+ {\mathrm {D}}-1$ , where $ {\mathrm {D}}>0$ is the maximum allowed end-to-end delay. We study the corresponding delay-constrained unicast capacity problem. This paper presents the first example showing that network coding (NC) can achieve strictly higher delay-constrained throughput than routing even for the single unicast setting and the NC gain can be arbitrarily close to 2 in some instances. This is in sharp contrast to the delay-unconstrained ( $ {\mathrm {D}}=\infty $ ) single-unicast case where the classic min-cut/max-flow theorem implies that coding cannot improve throughput over routing. Motivated by the above findings, a series of investigation on the delay-constrained capacity problem is also made, including: 1) an equivalent multiple-unicast representation based on a time-expanded graph approach; 2) a new delay-constrained capacity upper bound and its connections to the existing routing-based results [Ying et al. 2011]; 3) an example showing that the penalty of using random linear NC can be unbounded; and 4) a counter example of the tree-packing Edmonds’ theorem in the new delay-constrained setting. Built upon the time-expanded graph approach, we also discuss how our results can be readily extended to cyclic networks. Overall, our results suggest that delay-constrained communication is fundamentally different from the well-understood delay-unconstrained one and call for investigation participation.