Stochastic behavior of an n-node blockchain under cyber attacks from multiple hackers with random re-setting times

This paper investigates the stochastic behavior of an n-node blockchain which is continuously monitored and faces non-stop cyber attacks from multiple hackers. The blockchain will start being re-set once hacking is detected, forfeiting previous efforts of all hackers. It is assumed the re-setting process takes a random amount of time. Multiple independent hackers will keep attempting to hack into the blockchain until one of them succeeds. For arbitrary distributions of the hacking times, detecting times, and re-setting times, we derive the instantaneous functional probability, the limiting functional probability, and the mean functional time of the blockchain. Moreover, we establish that these quantities are increasing functions of the number of nodes, formalizing the intuition that the more nodes a blockchain has the more secure it is.