Two-Dimensional Models of Markov Processes for System Availability

System availability is a mission-critical issue. Availability is mainly due to system or component failures and recovery/repair. Both of these processes are stochastic. They are associated with two random variables: mean time between failures and mean time to repair. Models have been developed to estimate the availability of certain systems, especially computer-based processing systems. Mostly, one-dimensional mathematical models are used for this purpose. However, certain applications, such as the availability of consensus protocols, may require two-dimensional models. This is because of security threats that bring another dimension to availability models. Hence, two dimensions are needed for such analysis. An example of this case is Blockchain, which is prone to failures and may have security vulnerabilities. In this paper, following a brief review of known approaches, two-dimensional approaches to availability estimation are introduced. These approaches find applications in the availability of distributed systems using consensus protocols such as PAXOS and IBFT/IBFT2.