On Transient Analysis of $$\Delta _N$$-Markov Chains

Abstract

AbstractConsider a cyber-physical system which has the purpose of maintaining the security of some physical asset. A particularly relevant scenario is one in which the system must perform its functions while at the same time enduring malicious attacks for an extended amount of time without energy renewal. We will study a theoretical model analogous to this cyber-physical system with an arbitrary number of nodes. The nodes in this system are able to communicate with one another and have the task of maintaining the security of the system. Initially all nodes comprising this system are in working order, however, due to attacks they can eventually become corrupted and then work against the security of the system. Of particular interest is the calculation of a number of metrics related to reliability of the system including the reliability function and the mean-time to failure. A particular case of this model has been examined by many researchers in the past, mostly using numerical methods. The contribution of this paper is the presentation of explicit expressions for the transient-state transition probabilities, reliability function, and the mean-time to failure along with rigorous proofs for the validity of these expressions. A result such as this is stronger than previous results via numerical methods and is an extension of previous analytical results where additional constraints on the model were imposed.