Adaptive timeliness of consensus in presence of crash and timing faults

The ∆-timed uniform consensus is a stronger variant of the traditional consensus and it satisfies the following additional property: Every correct process terminates its execution within a constant time ∆ (∆-timeliness), and no two processes decide differently (Uniformity). In this paper, we consider the ∆-timed uniform consensus problem in presence of fc crash processes and ft timing-faulty processes, and propose a ∆-timed uniform consensus algorithm. The proposed algorithm is adaptive in the following sense: It solves the ∆-timed uniform consensus when at least ft + 1 correct processes exist in the system. If the system has less than ft + 1 correct processes, the algorithm cannot solve the ∆-timed uniform consensus. However, as long as ft + 1 processes are non-crashed, the algorithm solves (non-timed) uniform consensus. We also investigate the maximum number of faulty processes that can be tolerated. We show that any ∆-timed uniform consensus algorithm tolerating up to ft timing-faulty processes requires that the system has at least ft + 1 correct processes. This impossibility result implies that the proposed algorithm attains the maximum resilience about the number of faulty processes. We also show that any ∆-timed uniform consensus algorithm tolerating up to ft timing-faulty processes cannot solve the (non-timed) uniform consensus when the system has less than ft + 1 non-crashed processes. This impossibility result implies that our algorithm attains the maximum adaptiveness.