Optimal Message-Driven Implementations of Omega with Mute Processes

We investigate the complexity of algorithms in message-driven models. In such models, events in the computation can only be caused by message receptions, but not by the passage of time. Hutle and Widder [2005a] have shown that there is no deterministic message-driven self-stabilizing implementation of the eventually strong failure detector and thus Omega in systems with uncertainty in message delays and channels of unknown capacity using only bounded space. Under stronger assumptions it was shown that even the eventually perfect failure detector can be implemented in message-driven systems consisting of at least f + 2 processes (f being the upper bound on the number of processes that crash during an execution). In this article we show that f + 2 is in fact a lower bound in message-driven systems, even if nonstabilizing algorithms are considered. This contrasts time-driven models where f + 1 is sufficient for failure detector implementations. Moreover, we investigate algorithms where not all processes send message, that is, are active, but some (in a predetermined set) remain passive. Here, we show that the f + 2 processes required for message-driven systems must be active, while in time-driven systems it suffices that f processes are active. We also provide message-driven implementations of Omega. Our algorithms are efficient in the sense that not all processes have to send messages forever, which is an improvement to previous message-driven failure detector implementations.