Mathematical Model of the Reliability Analysis of a Heterogeneous Duplicate Data Transmission System
Abstract
In this work, we consider the mathematical model of the repaired data transmission system as a model of a closed heterogeneous cold standby system with one repair device with an arbitrary number of data sources with an exponential distribution function of uptime and an arbitrary distribution function of the repair time of its elements. We study the reliability of the system, defined as the steady-state probability of failure-free system operation. The proposed analytical methodology made it possible to evaluate the reliability of the entire system in case of failure of its elements. Explicit analytic and asymptotic expressions are obtained for the steady-state probabilities of system and the steady-state probability of failure-free system operation. The problem of analyzing the sensitivity of the reliability characteristics of the system under consideration to the types of repair time distributions was also studied. The obtained formulas showed the presence of a clear dependence of these characteristics on the types of distribution functions of the recovery time of system elements. However, numerical studies and analysis of the constructed graphs showed that this dependence becomes vanishingly small with a “quick” restoration of system elements.
References
[2] Houankpo H.G.K., Kozyrev D. Analytical Modeling and Simulation of Reliability of a Closed Homogeneous System with an Arbitrary Number of Data Sources and Limited Resources for their Processing. Sovremennye informacionnye tehnologii i IT-obrazovanie = Modern Information Technologies and IT-Education. 2018; 14(3):552-559. (In Russ., abstract in Eng.) DOI: https://doi.org/10.25559/SITITO.14.201803.552-559
[3] Houankpo H.G.K., Kozyrev D.V. Sensitivity Analysis of Steady State Reliability Characteristics of a Repairable Cold Standby Data Transmission System to the Shapes of Lifetime and Repair Time Distributions of its Elements. In: Samouilov K. E., Sevastianov L. A., Kulyabov D. S. (ed.) CEUR Workshop Proceedings. Selected Papers of the VII Conference "Information and Telecommunication Technologies and Mathematical Modeling of High-Tech Systems". 2017; 1995:107-113. Available at: http://ceur-ws.org/Vol-1995/paper-15-970.pdf (accessed 16.08.2020). (In Eng.)
[4] Liu Z., Hu L., Liu S., Wang Yu. Reliability analysis of general systems with bi-uncertain variables. Soft Computing. 2020; 24(9):6975-6986. (In Eng.) DOI: https://doi.org/10.1007/s00500-019-04331-6
[5] Li Y.L., Xu G.Q. Analysis of two components parallel repairable system with vacation. Communications in Statistics - Theory and Methods. 2019. (In Eng.) DOI: https://doi.org/10.1080/03610926.2019.1670847
[6] Ge X., Sun J., Wu Q. Reliability analysis for a cold standby system under stepwise Poisson shocks. Journal of Control and Decision. 2019. (In Eng.) DOI: https://doi.org/10.1080/23307706.2019.1633961
[7] Liu Y., Qu Z., Li X., An Y., Yin W. Reliability Modeling for Repairable Systems With Stochastic Lifetimes and Uncertain Repair Times. IEEE Transactions on Fuzzy Systems. 2019; 27(12):2396-2405. (In Eng.) DOI: https://doi.org/10.1109/TFUZZ.2019.2898617
[8] Liu Y., Ma Y., Qu Z., Li X. Reliability Mathematical Models of Repairable Systems With Uncertain Lifetimes and Repair Times. IEEE Access. 2018; 6:71285-71295. (In Eng.) DOI: https://doi.org/10.1109/ACCESS.2018.2881210.
[9] Wolstenholme L.C. Reliability Modelling: A Statistical Approach. New York, Routledge, Taylor & Francis, 2018. (In Eng.) DOI: https://doi.org/10.1201/9780203740958.
[10] Varde P.V., Pecht M.G. System Reliability Modeling. In: Risk-Based Engineering. Springer Series in Reliability Engineering. Springer, Singapore; 2018. p. 71-113. (In Eng.) DOI: https://doi.org/10.1007/978-981-13-0090-5_4
[11] Gullo L.J. Reliability Models. In: Raheja D., Gullo L. J. (ed.) Design for Reliability; John Wiley & Sons, Inc.; 2012. p. 53-65. (In Eng.) DOI: https://doi.org/10.1002/9781118310052.ch4
[12] Leemis L. Reliability Modelling and Applications. Technimetrics. 1989; 31(2):268. (In Eng.) DOI: https://doi.org/10.1080/00401706.1989.10488534
[13] Fathizadeh M., Khorshidian K. An Alternative Approach to Reliability Analysis of Cold Standby Systems. Communication in Statistics - Theory and Methods. 2016; 45(21):6471-6480. (In Eng.) DOI: https://doi.org/10.1080/03610926.2014.944660
[14] Xu J.J., Xiao Z-j. Reliability Analysis of Cold Standby Compound System. Advanced Materials Research. 2014; 945-949:1116-1119. (In Eng.) DOI: https://doi.org/10.4028/www.scientific.net/AMR.945-949.1116
[15] Wu Q. Reliability analysis of a cold standby system attacked by shocks. Applied Mathematics and Computation. 2012; 218(23):11654-11673. (In Eng.) DOI: https://doi.org/10.1016/j.amc.2012.05.051
[16] Vanderperre E.J. Reliability analysis of a renewable multiple cold standby system. Operations Research Letters. 2004; 32(3):288-292. (In Eng.) DOI: https://doi.org/10.1016/j.orl.2003.10.002
[17] Kendall D.G. Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain. Annals of Mathematical Statistics. 1953; 24(3):338-354. (In Eng.) DOI: https://doi.org/10.1214/aoms/1177728975
[18] Parshutina S.A., Bogatyrev V.A. Models to support design of highly reliable distributed computer systems with redundant processes of data transmission and handling. In: 2017 International Conference "Quality Management,Transport and Information Security, Information Technologies" (IT&QM&IS). St. Petersburg; 2017. p. 96-99. (In Eng.) DOI: https://doi.org/10.1109/ITMQIS.2017.8085772
[19] Teh J., Lai C., Cheng Y. Impact of the Real-Time Thermal Loading on the Bulk Electric System Reliability. IEEE Transactions on Reliability. 2017; 66(4):1110-1119. (In Eng.) DOI: https://doi.org/10.1109/TR.2017.2740158
[20] Sevast’yanov B.A. An Ergodic Theorem for Markov Processes and Its Application to Telephone Systems with Refusals. Theory of Probability & Its Applications. 1957; 2(1):104-112. (In Eng.) DOI: https://doi.org/10.1137/1102005
[21] Lisnianski A., Laredo D., BenHaim H. Multi-state Markov Model for Reliability Analysis of a Combined Cycle Gas Turbine Power Plant. In: 2016 Second International Symposium on Stochastic Models in Reliability Engineering, Life Science and Operations Management (SMRLO). Beer-Sheva; 2016. p. 131-135. (In Eng.) DOI: https://doi.org/10.1109/SMRLO.2016.31
[22] Billinton R., Ge J. A comparison of four-state generating unit reliability models for peaking units. IEEE Transactions on Power Systems. 2004; 19(2):763-768. (In Eng.) DOI: https://doi.org/10.1109/TPWRS.2003.821613
[23] Doob J.L. Asymptotic properties of Markoff transition prababilities. Transactions of the American Mathematical Society. 1948; 63(3):393-421. (In Eng.) DOI: https://doi.org/10.1090/S0002-9947-1948-0025097-6
[24] Trivedi K. Probability and Statistics with Reliability Queuing and Computer Science Applications. New York: Wiley; 2002. (In Eng.)
[25] Petrovsky I.G. Lekcii po teorii obyknovennyh differencial'nyh uravnenij [Lectures on the Theory of Ordinary Differential Equations]. Moscow, GITTL; 1952. (In Russ.)
This work is licensed under a Creative Commons Attribution 4.0 International License.
Publication policy of the journal is based on traditional ethical principles of the Russian scientific periodicals and is built in terms of ethical norms of editors and publishers work stated in Code of Conduct and Best Practice Guidelines for Journal Editors and Code of Conduct for Journal Publishers, developed by the Committee on Publication Ethics (COPE). In the course of publishing editorial board of the journal is led by international rules for copyright protection, statutory regulations of the Russian Federation as well as international standards of publishing.
Authors publishing articles in this journal agree to the following: They retain copyright and grant the journal right of first publication of the work, which is automatically licensed under the Creative Commons Attribution License (CC BY license). Users can use, reuse and build upon the material published in this journal provided that such uses are fully attributed.
