The goal of this article is to analyze a repairable double hot standby system equipped with a single repair facility, employing Marked Markov Processes. It is assumed that elements’ life- and repair times follow arbitrary distributions. The proposed approach facilitates the calculation of the system’s reliability characteristics and allows investigating their sensitivity to the shape of input distributions. Building on prior research that investigated the reliability function and mean system lifetime, the current article focuses on analyzing the steady-state probabilities of the system. For this, two system repair scenarios are considered. To validate the new method, numerical examples that are compared to previously obtained analytical results, demonstrating a high degree of accuracy, are presented. Additionally, further numerical experiments illustrate the sensitivity of steady-state reliability measures to the initial parameters of the system.