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

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