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Reliability Model of a Homogeneous Hot-Standby k-Out-of-n: G System

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Distributed Computer and Communication Networks: Control, Computation, Communications (DCCN 2021)

Part of the book series: Lecture Notes in Computer Science ((LNCCN,volume 13144))

Abstract

We consider the mathematical model of a closed homogenous redundant hot-standby system. This system consists of an arbitrary number of data links with an exponential distribution function (DF) of uptime and a general independent (GI) distribution function of the repair time of its elements with a single repair unit. We also consider the simulation model for the cases where it’s not always possible to carry out the mathematical model (to obtain explicit analytic expressions). The system with n components works until k components fail. With introduction of additional variable, explicit analytic expressions are obtained for the steady-state probabilities (SSP) of the system and the steady-state probability of failure-free system operation (PFFSO) using the constant variation method to solving the Kolmogorov differential equations systems. The SSP of the system are obtained with general independent distribution function of uptime using a simulation approach. We built dependence plots of the probability of system uptime against the fast relative speed of recovery; also plots of the reliability function relative to the reliability assessment. Five different distributions were selected for numerical and graphical analysis as Exponential (M), Weibull-Gnedenko (WG), Pareto (PAR), Gamma (G) and Lognormal (LN) distributions. The simulation algorithms were performed in the R programming language.

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Abbreviations

A –:

Random variable, time to failure of the main component,

B –:

Random variable, recovery time of a failed component,

A(x) –:

DF of the random variable A,

B(x) –:

DF of the random variable B,

b(x) –:

Probability density function (PDF) of the random variable B,

\(\tilde{b}(\lambda _i) = \int _{0}^{\infty } e^{-\lambda _ix}\cdot b(x)dx\) –:

Laplace transform of the PDF b(x) ; \(i=\overline{0,k-1}\) ,

EA –:

Mean uptime of a working component,

\(EB=b\) –:

Mean repair time of a failed component,

DB –:

Recovery time variance,

\(c=\frac{\sqrt{DB}}{EB}\) –:

Coefficients of variation,

\(\rho =\frac{EA}{EB}\) –:

Relative recovery rate,

\(\beta (x)=\frac{b(x)}{1-B(x)}\) –:

Conditional PDF of the residual repair duration of the element being repaired at time t (recovery rate) [15] ,

\(\lambda _i=(n-i)\cdot \alpha \) –:

Parameter of the exponential distribution of the uptime of components; \(i=\overline{0,k-1}\).

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Acknowledgments

The publication has been supported by the RUDN University Strategic Academic Leadership Program. The reported study was funded by RFBR, project No. 20-37-90137 (recipient Dmitry Kozyrev, formal analysis, validation, and recipient H.G.K. Houankpo, methodology and numerical analysis)

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Authors and Affiliations

  1. Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street, Moscow, 117198, Russia

    H. G. K. Houankpo & Dmitry Kozyrev

  2. V. A. Trapeznikov Institute of Control Sciences of RAS, 65 Profsoyuznaya Street, Moscow, 117997, Russia

    Dmitry Kozyrev

Authors
  1. H. G. K. Houankpo
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Correspondence to Dmitry Kozyrev .

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Editors and Affiliations

  1. V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia

    Vladimir M. Vishnevskiy

  2. Applied Probability and Informatics Department, RUDN University, Moscow, Russia

    Konstantin E. Samouylov

  3. V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia

    Dmitry V. Kozyrev

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Houankpo, H.G.K., Kozyrev, D. (2021). Reliability Model of a Homogeneous Hot-Standby k-Out-of-n: G System. In: Vishnevskiy, V.M., Samouylov, K.E., Kozyrev, D.V. (eds) Distributed Computer and Communication Networks: Control, Computation, Communications. DCCN 2021. Lecture Notes in Computer Science(), vol 13144. Springer, Cham. https://doi.org/10.1007/978-3-030-92507-9_29

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  • DOI: https://doi.org/10.1007/978-3-030-92507-9_29

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