Tight bounds for the performance of longest-in-system on DAGs

Abstract

A growing amount of work has been invested in recent years in analyzing packet-switching networks under worst-case scenarios rather than under probabilistic assumption. Most of this work makes use of the modelof “adversarialqueuing theory” proposed by Borodin et al. [6], under which an adversary is allowed to inject into the network any sequence of packets as long as – roughly speaking – it does not overload the network.We show that the protocolLongest-In-System, when applied to directed acyclic graphs, uses buffers of only linear size (in the length of the longest path in the network). Furthermore, we show that any packet incurs only linear delay as well. These results separate LIS from other common universally stable protocols for which there exist exponential lower bounds that are obtained on DAGs. Our upper bounds are complemented by linear lower bounds on buffer sizes and packet delays.