Delay Scaling Laws of Random Wireless Networks: Impact of Blocklength

We investigate the end-to-end delay of multipleunicast wireless networks. In contrast to previous works where the end-to-end delay is measured by the queuing delay, in this work we measure the delay by the total blocklength of the communication scheme. As the capacity characterization of multiple-unicast networks is open, we consider random wireless networks (the Gupta-Kumar model) and investigate the end-to-end delay scaling law with respect to the number of nodes. The end-to-end delay of delivering a file depends on the file size as well as the throughput. Our main contribution is the characterization of the end-to-end delay scaling law of the multihopping scheme, which depends on the file size. Our main finding is that if the file size is sufficiently large, the end-to-end delay scaling law is proportional to it. While this is expected, if the file size is not large enough, the end-to-end delay scaling law becomes independent of it. In particular, in a network with 2k randomly one-to-one paired users and area k and a source with F(k) bits to send, we show that the delay is omega(root kF(k)) if F(k) - Omega(root klog k), while it is omega(k log k) if F(k) = o(root klog k). Our result is derived by studying the multihopping scheme for large random wireless networks. Using ideas from moderate deviations theory and finite length bounds in the literature, we derive a lower bound on the required blocklength for the network.