Network Reliability and Resilience pp 1-50 | Cite as

# Theory

## Abstract

Sections 1 and 2 present brief summary of a standard material on reliability of monotone binary systems and their applications to networks. The definition of the binary system is extended to the case of more than two states. Section 3 contains the definition of the D-spectrum and the marginal D-spectra. D-spectrum is a multidimensional discrete probability distribution, whose *r*th coordinate is a discrete distribution of the number of the component whose failure causes the transition of the network from state \(j+1,\) into state *j* computed under assumption that components fail in random order. D-spectrum is a combinatorial characteristic of the system which in particular case of two-state system with i.i.d. components coincides with Samaniego’s signature. Network probabilistic resilience presented in Section 2 is the (\(1-\beta\))-quantile of the cumulative marginal D-spectrum. Sections 3, 4 show how to compute D-spectra for recurrent networks and series-parallel connection of network-type systems. Section 5 considers networks with multi-state edges which we call networks with colored links. Section 6 deals with Birnbaum Importance Measure (BIM) of network components for networks with identical and independent components. It is shown that the BIM of component *j* can be estimated via a network combinatorial parameter, so-called BIM-spectrum, which is closely related to the network D-spectrum. An extension of the BIM-spectrum allows to obtain a combinatorial formula for another important index, so-called *Joint Reliability Index*. Section 7 discusses reliability gradient function which allows to compute the increase of system reliability as a function of component reliability increase. In general case of arbitrary component reliabilities the calculation of gradient function becomes more involved but nevertheless can be carried out by using another combinatorial characteristic of the network, so-called *border states*. Border state is a network *DOWN* state which has a unit Manhattan distance from the network *UP* state. As a rule, most of computations aimed at network reliability estimation are NP-complete, and the Monte Carlo approximations remain our main computation tool. Section 1.8 describes in a non formal way three principal Monte Carlo procedures: estimation of network connectivity, estimation of the D-spectra and component BIM’s, and the estimation of the gradient function. It discusses also the question of accuracy provided by a limited number of Monte Carlo replications which guarantee a numerical result with an error small enough for engineering calculations. Finally, Sect. 9 discusses the construction of D-spectra for the case when a single component failure (e.g. node failure) leads the network to change its state *J* to state \(J - k,\)\(k \geq 2.\)

## Keywords

Multi-state network Multi-state D-spectra Component importance measure Gradient function Border states Monte Carlo accuracy## References

- 1.Barlow, R.E., and Proschan, F. 1975.
*Statistical theory of reliability and life testing*. NY: Holt, Rinehart and Winston, Inc.Google Scholar - 2.Birnbaum, Z.W. 1969. On the importance of different components in multicomponent system.
*Multivariate Analysis-II*, ed. P.R. Krishnaiah, 581–592. New York: Academic Press.Google Scholar - 3.Brandes, U., and T. Erlebach (eds.) 2005.
*Network analysis—Methodological foundations*. Berlin: Springer-Verlag.Google Scholar - 4.Burtin, Y., and B.G. Pittel. 1972. Asymptotic estimates of the reliability of complex systems.
*Engineering Cybernetics*10(3):445–451.Google Scholar - 5.David, H.A. 1981.
*Order Statistics*, second edition. NY: Wiley.Google Scholar - 6.Elperin, T., Gertsbakh, I., and M. Lomonosov. 1991. Estimation of network reliability using graph evolution models.
*IEEE Transactions on Reliability*R-40:572–581.Google Scholar - 7.Elperin, T., Getsbakh, I., and M. Lomonosov. 1992. An evolution model for Monte Carlo estimation of equilibrium network renewal parameters.
*Probability in Engineering and Informational Sciences*6:457–469.Google Scholar - 8.Gertsbakh, I.B. 1989.
*Statistical reliability theory*. NY:Marcel Dekker, Inc.Google Scholar - 9.Gertsbakh, I., and Y. Shpungin. 2004. Combinatorial approaches to Monte Carlo estimation of network lifetime distribution.
*Applied Stochastic Models in Business and Industry*20:49–57.Google Scholar - 10.Gertsbakh, Ilya and Yoseph Shpungin. 2009.
*Models of network reliability: Analysis, combinatorics, and Monte Carlo*. Boca Raton: CRC Press.Google Scholar - 11.Gertsbakh, I., and Y. Shpungin. Stochastic models of network survivability. 2012.
*Quality Technology and Quantitative Management*, Special Issue devoted to S. Zacks 9(1) to appear.Google Scholar - 12.Gertsbakh, Ilya and Yoseph Shpungin. 2011. Multidimensional spectra of multi state systems with binary components. In
*Recent Advances in Reliability: Signatures, Multi-State Systems and Statistical Inference*, eds. Frenkel, I. and A. Lisniansky, Chap. 4. Heidelberg: Springer, in press.Google Scholar - 13.Hong, Jong, Silk, and Chang Hou Lie. 1993. Joint reliability importance of two edges in undirected network.
*IEEE Transactions on Reliability*42(1):17–23.Google Scholar - 14.Levitin, G., Gertsbakh, I., and Y. Shpungin. 2010. Evaluating the damage associated with intentional network disintegration.
*Reliability Engineering and System Safety*96(4):433–439.Google Scholar - 15.Lisniansky, Anatoly and Gregory Levitin. 2003.
*Multi-state system reliability*. NJ: World Scientific.Google Scholar - 16.Lisniansky, A., Ilia, Frenkel, and Y. Ding. 2010.
*Multi-state reliability analysis and optimization for engineers and industrial managers*. London: Springer.Google Scholar - 17.Navarro, J., and F. Spizzichino. 2011. Different definitions of the concept of signature and relevant properties of coherent systems. In
*Recent Advances in Reliability: Signatures, Multi-State Systems and Statistical Inference*, eds. Frenkel, I. and A. Lisniansky, Chap. 3. Heidelberg: Springer, in press.Google Scholar - 18.Samaniego, F.J. 1985. On closure of the IFR under formation of coherent systems.
*IEEE Transactions on Reliability*34:69–72.Google Scholar - 19.Samaniego F.J. 2007.
*System signatures and their applications in engineering reliability*. NY: Springer.Google Scholar - 20.Shpungin, Y., and I. Gertsbakh. 2012. Combinatorial approach to computing importance indices in coherent systems.
*Probability in Engineering and Informational Sciences*26(1) (to appear).Google Scholar - 21.Spizzichino, F., Shpungin, Y., and I. Gertsbakh. 2011. Signatures of coherent systems built with separate modules.
*Journal of Applied Probability*48(3).Google Scholar - 22.Wolfram, Stephen. 1992.
*MATHEMATICA*:*A system for Doing Mathematics by Computer*. Addison-Wesley Publishing Company.Google Scholar - 23.Xueli, G., Liring, C., and J. Li. 2007. Analysis for joint importance of components in a coherent system.
*European Journal of Operational Research*182:282–299.Google Scholar