Application of Decomposable Semi-Regenerative Processes to the Study of k-out-of-n Systems

Abstract

This paper aimed to demonstrate the capabilities of decomposable semi-regenerative processes for the investigation of the k-out-of-n system. Proposed in 1955 by W. Smith, the regeneration idea has come a long way in terms of development and has found widespread applications. First, we briefly recall the history of the development of the regeneration idea and the main results of the theory of regenerative, semi-regenerative, and decomposable semi-regenerative processes. Then, the methods of the theory of decomposable semi-regenerative processes are used for the study of a k-out-of-n renewable system with exponentially distributed life and generally distributed repair times of its components. This system is very important for practice and its special cases have previously been considered (including by the authors); however, only special cases and using other methods are considered herein. In the current paper, two scenarios of system repair after its failure are considered for the first time: the partial and the full system repair scenarios. For both scenarios, the time-dependent system state probabilities are calculated in terms of their Laplace transforms. The closed form representation of the stationary probabilities for both scenarios are also presented. These latest results represent a new contribution to the study of this system.

On the Positive Recurrence of Finite Regenerative Stochastic Models

“…This property explains the name of this model. There are numerous papers devoted to the analysis of this model see, for instance, the papers mentioned above [6,7], which contain a brief review of the research on this topic. In addition, various aspects of the analysis of this system can be found in Refs.…”

“…In this research, we demonstrate how this approach allows to simplify the proof of the positive recurrence of the finite k-out-of-n repairable system studied in Ref. [6] on the base of the so-called semi-regenerative processes. The analysis in Ref.…”

“…The analysis in Ref. [6] allowed us to calculate both stationary and non-stationary performance indexes in such a model, while the problem of the existence of the stationary distribution and hence the stationary performance characteristics received less attention. We note that the basic results of the theory of semi-regenerative processes can be found in the survey paper [7].…”

“…[6] allowed us to calculate both stationary and non-stationary performance indexes in such a model, while the problem of the existence of the stationary distribution and hence the stationary performance characteristics received less attention. We note that the basic results of the theory of semi-regenerative processes can be found in the survey paper [7]. In particular, we first show how the regenerative method allows to to prove, relatively simply, both the positive recurrence of the basic regenerative process and the existence of its stationary distribution when lifetimes are exponential and then we extend the stability analysis to the repairable system with non-exponential lifetimes, provided the repair times are assumed to be unbounded.…”

On Queues with Working Vacation and Interdependence in Arrival and Service Processes

“…The k-out-of-n models have a wide range of practical applications in many spheres of human activity, including engineering, telecommunication, industry, biology, etc., and are effectively used to study the reliability of complex real-world redundant systems, such as aircraft multi-engine systems, multi-pump hydraulic control systems, unmanned underwater vehicles, unmanned multi-rotor flight modules of high-altitude telecommunications platforms, etc. In a series of works by Rykov, Kozyrev, et al [24,25] a range of k-out-of-ntype models have been proposed for the reliability study of systems taking into account the dependence of system failure not only on the number of failed components but also on their location in the system, on their proximity to each other, on the state of the random environment and other dependence aspects. For example, in [26] Rykov et al performed the reliability study of a k-out-of-n system, whose components' residual lifetimes depend on the increase in load after the failure of any component.…”