| The reliability modeling for K-terminal network can serve as a basis for the reliability evaluation and risk analysis of some network systems,such as transportation,computers,communication and so on.Under the assumption that all edges are statistically independent,the calculation for traditional K-terminal network reliability requires knowledge of network structure and edge reliability.However,in practice,edge reliability is not easily obtained due to limited testing time or failure data.On the other hand,cascading failures or edge load sharing can result in the loss of independence of edge failure.For these reasons,traditional reliability modeling for K-terminal network faces with severe challenges.In practical application,a network is in a state of operation or failure,whose total number of failed edges is a random variable and has a certain probability distribution because the edge failures appear stochastically.Assume that the nodes are absolutely reliable and all the edges have identical distribution,this dissertation establishes a K-terminal network reliability model under the condition that the probability distribution for the total number of failed edges is given.Based on this model,the Bayesian Importance measures(IMs)for individual edge and joint IMs for a pair of edges are mainly developed.Two numerical examples are provided to demonstrate the application of these IMs for identifying weakness in transportation network.The main contents of this dissertation can be summarized as follows:1)Reliability modeling for K-terminal networkFor quantifying the characteristics of network structure concerning reliability,some concepts of spectra are introduced.Several relationships concerning the spectra are established:(i)between the D-spectrum and C-spectrum for network;(ii)between the D-spectrum and C-spectrum for individual edge;(iii)among the joint D-spectrum for two edges,the D-spectrum for individual edge,and the D-spectrum for network;(iv)among the joint C-spectrum for two edges,the C-spectrum for individual edge,and the C-spectrum for network.Then,the Monte-Carlo(MC)algorithms for evaluating these spectra are described.Once the values for these spectra are available,they open a way for calculating the network reliability and IMs.Finally,we establish the reliability model for the K-terminal network on the basis of the knowledge of the probability distribution for the total number of failed edges and network spectra.Some probabilistic equations for the reliability model are developed,including the network reliability,the network failure probability,the conditional reliability of the network,and the conditional failure probability of the network.2)The calculation of IMs for K-terminal networkBased on the above mentioned reliability model,we derive the equations for calculating the Bayesian IMs which evaluate the impact of individual edge within a network.Based on these equations,we determine the conditions under which Bayesian IMs reach the maximum,and discover the relationships between the Bayesian IMs and other common IMs.These IMs can provide numerical indexes for determining which edges are more critical to network failure or more important for network reliability improvement.Based on a given importance measure,network edges can be ranked concerning the impact they have on network failure probability or reliability.It is shown that(i)for arbitarary probability distribution,from the two standpoints of the D-spectum and C-spectrum,the two kinds of Bayesian IMs generate consistent rankings,and the H-IMs rankings imply the Bayesian IMs rankings;(ii)under some conditons on the probability distribution,the Birnbaum structural IM and Bayesian IMs generate an identical rankings;(iii)under some conditons on the probability distribution,the Barlow-Proschan structural IM and Bayesian IMs generate identical rankings.Finally,based on the reliability model established in 1),we derive the equations for calcuting the joint IMs which evaluate the extent of the interaction of two edges in a network based on their contribution to the network reliability or failure pobability.From joint IMs,the definitions for positive correlation and negative correlation are introduced.3)The IMs for K-terminal network under a saturated Lagrangian Poisson distributionWhen the failures of edges appear according to a branching process in which the number of the failed edges follows a saturated Lagrangian Poisson distribution(SLPD)with parameters θ and λ,we determine some sufficient and necessary conditions for ranking these edges according to the Bayesian IMs,where θ is mean value for the number of initial edge failures and λ is mean value for offspring distribution in branching process.It is shown that when θ and λ are sufficiently small,or θis sufficiently large,the Bayesian IMs rankings are structural rankings(i.e.the rankings depend only on network structure,regardless of the values of the parameters).Moreover,under the condition that the number of failed edges follows the SLPD,the joint IMs are investigated.Our research results show that when θ and λ are sufficiently small,or θ is sufficiently large and λ is sufficiently small,the correlation for any two edges is quite weak.Finally,numerical results of a transportation network show that Bayesian IMs and joint IMs can be used as effective tools to identify network weakness.4)The IMs for K-terminal network under a saturated non-homogeneous Poisson processesUnder the condition that the edge failures occur according to a saturated non-homogeneous Poisson processes(SNHPP)with intensity function λ(t),the dynamic rankings according to the Bayesian IMs and asymptotic property for joint IMs are explored.When the time t go to zero or positive infinity,our research results show that i)the rankings generated by Bayesian IMs are structural rankings;ii)joint IMs approach zero.Finally,Experimental results concerning a transportation network show that Bayesian IMs and joint IMs can be effectively used as tools for dynamically identifying network weaknesses when the edge failures occur according to a SNHPP. |
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