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Keywords = decomposable semi-regenerative processes

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21 pages, 484 KiB  
Article
Renewable k-Out-of-n System with the Component-Wise Strategy of Preventive System Maintenance
by Vladimir Rykov, Olga Kochueva and Elvira Zaripova
Mathematics 2023, 11(9), 2158; https://doi.org/10.3390/math11092158 - 4 May 2023
Cited by 3 | Viewed by 1576
Abstract
At the SMARTY-22 conference, a review of the regenerative methods development was presented, including its application to the study of a non-renewable k-out-of-n system. This paper develops the previous study for the renewable k-out-of-n system, including an investigation different [...] Read more.
At the SMARTY-22 conference, a review of the regenerative methods development was presented, including its application to the study of a non-renewable k-out-of-n system. This paper develops the previous study for the renewable k-out-of-n system, including an investigation different preventive maintenance strategies based on the system state observation. We also include the review of Smith’s regeneration idea development. Some new results are presented that form the basis for an algorithm for comparing preventing maintenance strategies with respect to the maximization of the availability factor. A numerical study was conducted for the 4-out-of-6 and 4-out-of-8 models. The study demonstrates the sensitivity of decision making to the shape of the repair time distribution. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Analytics, Asymptotics, Estimation and Applications)
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23 pages, 388 KiB  
Article
Application of Decomposable Semi-Regenerative Processes to the Study of k-out-of-n Systems
by Vladimir Rykov, Nika Ivanova and Dmitry Kozyrev
Mathematics 2021, 9(16), 1933; https://doi.org/10.3390/math9161933 - 13 Aug 2021
Cited by 13 | Viewed by 2107
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. [...] Read more.
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. Full article
(This article belongs to the Special Issue Stochastic Modeling and Applied Probability)
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18 pages, 1211 KiB  
Article
Approximations in Performance Analysis of a Controllable Queueing System with Heterogeneous Servers
by Dmitry Efrosinin, Natalia Stepanova, Janos Sztrik and Andreas Plank
Mathematics 2020, 8(10), 1803; https://doi.org/10.3390/math8101803 - 16 Oct 2020
Cited by 9 | Viewed by 3164
Abstract
The paper studies a controllable multi-server heterogeneous queueing system where servers operate at different service rates without preemption, i.e., the service times are uninterrupted. The optimal control policy allocates the customers between the servers in such a way that the mean number of [...] Read more.
The paper studies a controllable multi-server heterogeneous queueing system where servers operate at different service rates without preemption, i.e., the service times are uninterrupted. The optimal control policy allocates the customers between the servers in such a way that the mean number of customers in the system reaches its minimal value. The Markov decision model and the policy-iteration algorithm are used to calculate the optimal allocation policy and corresponding mean performance characteristics. The optimal policy, when neglecting the weak influence of slow servers, is of threshold type defined as a sequence of threshold levels which specifies the queue lengths for the usage of any slower server. To avoid time-consuming calculations for systems with a large number of servers, we focus here on a heuristic evaluation of the optimal thresholds and compare this solution with the real values. We develop also the simple lower and upper bound methods based on approximation by an equivalent heterogeneous queueing system with a preemption to measure the mean number of customers in the system operating under the optimal policy. Finally, the simulation technique is used to provide sensitivity analysis of the heuristic solution to changes in the form of inter-arrival and service time distributions. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)
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18 pages, 729 KiB  
Article
On Reliability of a Double Redundant Renewable System with a Generally Distributed Life and Repair Times
by Vladimir Rykov, Dmitry Efrosinin, Natalia Stepanova and Janos Sztrik
Mathematics 2020, 8(2), 278; https://doi.org/10.3390/math8020278 - 19 Feb 2020
Cited by 7 | Viewed by 2581
Abstract
The paper provides reliability analysis of a cold double redundant renewable system assuming that both life-time and repair time distributions are arbitrary. The proposed approach is based on the theory of decomposable semi-regenerative processes. We derive the Laplace–Stieltjes transform of two main reliability [...] Read more.
The paper provides reliability analysis of a cold double redundant renewable system assuming that both life-time and repair time distributions are arbitrary. The proposed approach is based on the theory of decomposable semi-regenerative processes. We derive the Laplace–Stieltjes transform of two main reliability measures like the distribution of the time between failures and the time to the first failure. The transforms are used to calculate corresponding mean times. It is further derived in closed form the time-dependent and time stationary state probabilities in terms of the Laplace transforms. Numerical results illustrate the effect of the type of distributions as well as their parameters on the derived reliability and probabilistic measures. Full article
(This article belongs to the Special Issue Queueing Methods in Reliability Theory and Information Technology Systems)
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