Average long-lived binary consensus: Quantifying the stabilizing role played by memory

Abstract

Consider a system composed of n sensors operating in synchronous rounds. In each round an input vector of sensor readings x is produced, where the i -th entry of x is a binary value produced by the i -th sensor. The sequence of input vectors is assumed to be smooth: exactly one entry of the vector changes from one round to the next one. The system implements a fault-tolerant averaging consensus function f . This function returns, in each round, a representative output value v of the sensor readings x . Assuming that at most t entries of the vector can be erroneous, f is required to return a value that appears at least t + 1 times in x . We introduce the definition of instability of the system, which consists in the number of output changes over a random sequence of input vectors. We first design optimal (with respect to the instability measure) consensus systems: D 0 without memory, and D 1 with memory. Then we quantify the gain factor due to memory by computing c n ( t ) , the number of decision changes performed by D 0 per decision change performed by D 1 .