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A Study On The Dynamic Network Reliability ZOHREH ABDEYAZDAN z abdeyazdan@math iut ac ir January 11 2019 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Sareh Goli Advisor Dr Zahra Saberi 2000 MSC 05C25 90B25 05C60Keywords Reliability function Network Conditional hazard rate function Link TerminalAbstract Analyzing networks as one of the most basic systems in the communications telecommunication and internet net works has always been a concern for many researchers and craftsmen so today we are witnessing a new evolution inthe field of network security and reliability A network is a collection of nodes and links that some particular nodes are called terminals Rail station road in tersections are some example for nodes and railways connection between the components of a network are someexamples for links A network is defined by the triplet N V E T that V and E represents the node and thelink set respectively and T V is a subset of V that called terminal set we suppose V m and E n Infact each network can be modeled using a graph of nodes and edges For example a tree network is a collection ofsubnetworks which every two nodes are connected with a single edge Failure of the components of a network linksor nodes occurs on the basic of a stochastic mechanism and the failure of a link means the erase of the link and thefailure of a node means that all links incident to that node are erased In many cases networks have two states up and down but in reality there are networks such as gas production net works transport networks and which may have more than two states If for a network with L 1 states whereL 2 determine the network state with index K K 0 1 L then K L represents the complete performance andK 0 represents the complete failure and K 1 2 L 1 indicates that the network is in state of gradual disintegration There are different perspectives on the reliability function of the network One of these views is the study of thenetwork under two states up and down and another view of the network is based on three states of complete perfor mance partial performance and complete failure Also in evaluation the aging of networks and comparing them witheach other various criteria and dimensions have been investigated the most important of which are the reliabilityfunction hazard rate function and stochastic ordering In this thesis we are interested in the studying different types of networks and providing different methods for findingreliability function and stochastic ordering This study can be used on a network with n component or two networkswith different components Another objective of this thesis is to study the conditional hazard rate function of thenetwork when network is functioning at time t although some of the link of the network is failed and all of nodes areabsolutely reliable The structure of the thesis is as follows In chapter 1 the definition of the network and the history of studies carried out on the reliability function and networkcharacteristics are discussed In chapter 2 after a brief overview of the introductory concepts we introduce the basic concepts of network reliabil ity in this chapter a history of studies on stochastic ordering and order statistics that are required in the next chaptersare discussed In chapter 3 in order to examine the networks aging indicators under the assumption that the failure of the compo nents occurs according to the nonhomogeneous poisson process or the renewal process the reliability function themean residual lifetime function and the hazard rate function are studied

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