Heterogenous Networks Can Be Unstable at Arbitrarily Low Injection Rates

A distinguishing feature of today's large-scale platforms for distributed computation and communication, such as the Internet, is their heterogeneity, predominantly manifested by the fact that a wide variety of communication protocols are simultaneously running over different distributed hosts. A fundamental question that naturally poses itself for such common settings of heterogeneous distributed systems concerns their ability to preserve or restore an acceptable level of performance during link failures. In this work, we address this question for the specific case of stability properties of greedy, contention-resolution protocols operating over a packet-switched communication network that suffers from link slowdowns. We focus on the Adversarial Queueing Theory framework, where an adversary controls the rates of packet injections and determines packet paths. In addition, the power of the adversary is enhanced to include the manipulation of link slowdowns. Within this framework, we show that the composition of LIS (Longest-in-System) with any of SIS (Shortest-in-System), NTS (Nearest-to-Source) and FTG (Furthest-to-Go) protocols is unstable at rates ρ > 0 when the network size and the link slowdown take large values. These results represent the current record for instability bounds on injection rate for compositions of greedy protocols over dynamic adversarial models, and also suggest that the potential for instability incurred by the composition of two greedy protocols may be worse than that of some single protocol.