# ENGM680 lecture4 Delta star transformation

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Published on January 8, 2008

Author: Marietta1

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Network Reliability Analysis & Delta-Star Transformation:  Network Reliability Analysis & Delta-Star Transformation Xiaomin Zhao Oct. 9, 2007 Outline:  Outline 1 Instruction to Network Reliability Analysis 2 Parallel and Series Reduction 3 Delta-Star Transformation 3.1 one input node and two output nodes 3.2 each node is either an input or an output Reference: Chapter 5, W Kuo and M J Zuo, Optimal Reliability Modeling: Principles and Applications, John Wiley & Sons, New York, 2003 System description:  Reliability Network Diagram System description system description Reliability Block Diagram State Transition Diagram series structures, parallel structures, parallel-series structures… bridge structure, double-bridge structure… series system… … 1 Network Reliability Analysis:  1 Network Reliability Analysis Computer Text wide existence Telecommunication electricity Transportation … 1 Network Reliability Analysis:  1 Network Reliability Analysis Description node link 2-terminal network k-terminal network A network may be defined as working if a single source node is able to communicate with a single sink node. A~F A network is defined as working properly if k nodes are able to communicate with one another. A~B,A~C,A~D,… A~F We focus on network diagram 1 Network Reliability Analysis:  1 Network Reliability Analysis Assumption The system is coherent. all component are relevant; improvement of component performance does not degrade the performance of the system The system and its component may be in only two possible states: working or failed The states of the component are independent random variables The mission time of the system and its components are implicitly specified. we deal primarily with system and component reliabilities rather than their reliability functions of time t. e) Either links or nodes but not both are failure prone. Thus, a network diagram can be treated in a way similar to a reliability block diagram. 1 Network Reliability Analysis:  1 Network Reliability Analysis Pivotal Decomposition Inclusion-Exclusion Method Sum-of-Disjoint-Products Method …. “In network reliability evaluation, parallel and series reduction should always be applied first. When no more series and parallel reductions are possible, other techniques should be used.” “Delta-star and star-delta transformations can make further parallel and/or series reductions possible.” 2 Parallel and series reductions:  2 Parallel and series reductions Series structure Parallel structure reliability of component i unreliability of component i 2 Parallel and series reductions:  2 Parallel and series reductions Example 2 Parallel and series reductions:  2 Parallel and series reductions Slide11:  Using only parallel and series reductions, we are usually unable to simplify a general network into a single supercomponent. Other techniques should be applied to find the exact system reliability. 3 Delta-Star Transformation:  3 Delta-Star Transformation Assumption: The nodes of the network are perfect while the links are failure prone. 3.1 Star/Delta structure with one input node and two output nodes:  3.1 Star/Delta structure with one input node and two output nodes 3.1 Star/Delta structure with one input node and two output nodes:  3.1 Star/Delta structure with one input node and two output nodes Delta-star transformation: Star-Delta transformation: 3.1 Star/Delta structure with one input node and two output nodes:  3.1 Star/Delta structure with one input node and two output nodes Example1 delta->star 3.1 Star/Delta structure with one input node and two output nodes:  3.1 Star/Delta structure with one input node and two output nodes 3.1 Star/Delta structure with one input node and two output nodes:  3.1 Star/Delta structure with one input node and two output nodes Star-delta Star-delta transformation is not very efficient. delta->star star->delta 2 times 1 times delta->star 1 times Parallel reduction Slide18:  It is often impossible for one node to be only an input or only an output node for the identified delta structure. E.g. 3.2 Delta Structure in which each node may be either an input or an output:  3.2 Delta Structure in which each node may be either an input or an output When each node of the delta structure may be either an input node or an output node, we have to preserve all the probabilities in the delta-star transformation. To make the delta-star transformation possible, Rosenthal and Frisque allow the center node in the star structure to be failure prone 3.2 Delta Structure in which each node may be either an input or an output:  3.2 Delta Structure in which each node may be either an input or an output Delta-star transformation when the center node in failure prone 3 Delta Structure in which each node may be either an input or an output:  3 Delta Structure in which each node may be either an input or an output Example3 delta->star series Reliability of the standard bridge network Slide22:  Thanks ^_^

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