Bounds on the time to detect failures using bounded-capacity message links

We consider a system of distributed processors that communicate by passing messages and that have inexact information about time. Speciically, a processor knows that a single message is delayed by at most time d and the time between any two of its consecutive steps is at least c 1 and at most c 2 ; it has no other way of estimating elapsed time. This simple model is very close to traditional models used in distributed computing theory, and has been studied by Attiya and Lynch 2, 1] among others. We extend the model by making a realistic assumption about how the delay of messages is aaected by the rate at which they are sent. We deene a model of message links with bounded capacity, which are guaranteed to deliver messages at only a given rate. If a processor sends messages at a greater rate, they may incur greater delay. We quantify the eeect of this bounded capacity on the time necessary to detect processor failures. We consider a system of two processors connected by a bi-directional message link of (integral) capacity. First we give two very simple protocols that guarantee any stopping failure will be detected within time 2Cd + d and C 2 d== + Cd + d respectively, where C = c 2 =c 1. The main result is an almost-matching lower bound of 2Cd+d== or C 2 d==+Cd+d, whichever is less. If the link is uni-directional, our result specializes to give a matching upper and lower bound of C 2 d== + Cd + d.