{"id":427,"date":"2011-09-02T12:53:50","date_gmt":"2011-09-02T16:53:50","guid":{"rendered":"http:\/\/www.reliabilityanalytics.com\/blog\/?p=427"},"modified":"2012-09-21T09:30:42","modified_gmt":"2012-09-21T13:30:42","slug":"reliability-modeling-combination-of-series-and-parallel","status":"publish","type":"post","link":"https:\/\/reliabilityanalytics.com\/blog\/2011\/09\/02\/reliability-modeling-combination-of-series-and-parallel\/","title":{"rendered":"Reliability Modeling: Combination of Series and Parallel"},"content":{"rendered":"

Most practical equipments and systems are combinations of series and parallel components as shown below<\/p>\n

\"\"<\/a><\/p>\n

To solve this network, one merely uses series and parallel relationships to decompose and recombine the network step by step. For example<\/p>\n

Rad<\/sub> = R1<\/sub> \u00b7 R2<\/sub> = (0.9)(0.8) = 0.72<\/p>\n

Rbd<\/sub> = R3<\/sub> \u00b7 R4<\/sub> \u00b7 R5<\/sub> = (0.8)(0.8)(0.9) = 0.576<\/p>\n

but Rad<\/sub> and Rbd<\/sub> are in parallel; thus, the unreliability of this parallel subsystem (S1<\/sub>) is<\/p>\n

QS1<\/sub> = Qad<\/sub> \u00b7 Qbd<\/sub> = (1 – Rad<\/sub>)(1 – Rbd<\/sub>) = (1 – 0.72)(1 – 0.576) = (0.28)(0.424) = 0.119<\/p>\n

and its reliability is<\/p>\n

RS1<\/sub> = 1 – QS1<\/sub> = 1 – 0.119\u00a0 = 0.881<\/p>\n

Now the network has been decomposed to<\/p>\n

\"\"<\/a><\/p>\n

Letting RS2<\/sub> equal the combined reliability of RS1<\/sub> and R6<\/sub> in series<\/p>\n

RS2<\/sub> = RS1<\/sub>\u00a0\u00b7 R6<\/sub> = (0.881)(0.9) = 0.793<\/p>\n

QS2<\/sub> = 1 – RS2<\/sub> = 1 – 0.793 = 0.207<\/p>\n

Q7<\/sub> = 1 – R7<\/sub> = 1 – 0.7 = 0.3<\/p>\n

Since QS2<\/sub> and Q7<\/sub> are in parallel, the total system unreliability is<\/p>\n

QAC<\/sub> = QS2<\/sub>\u00a0\u00b7 Q7<\/sub> = (0.207)(0.3) = 0.0621 and the total network reliability is<\/p>\n

RAC<\/sub> = 1 – QAC<\/sub> = 1 – 0.0621 = 0.938<\/p>\n

thus, the reliability of the combined network is 0.94, rounded to two decimal places.<\/p>\n

The System State Enumeration tool from the Reliability Analytics Toolkit<\/a> can be easily be applied to solve this and similar problems, using similar series-parallel decomposition methods.<\/p>\n

 <\/p>\n

References:<\/p>\n

1. MIL-HDBK-338, Electronic Reliability Design Handbook<\/a>, 15 Oct 84
\n2. Bazovsky, Igor, Reliability Theory and Practice<\/a>
\n3. O\u2019Connor, Patrick, D. T., Practical Reliability Engineering<\/a>
\n4. Birolini, Alessandro, Reliability Engineering: Theory and Practice<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Most practical equipments and systems are combinations of series and parallel components as shown below To solve this network, one merely uses series and parallel relationships to decompose and recombine the network step by step.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[59,55],"tags":[58],"class_list":["post-427","post","type-post","status-publish","format-standard","hentry","category-reliability-modeling","category-system-modeling","tag-system-modeling-2"],"_links":{"self":[{"href":"https:\/\/reliabilityanalytics.com\/blog\/wp-json\/wp\/v2\/posts\/427","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/reliabilityanalytics.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/reliabilityanalytics.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/reliabilityanalytics.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/reliabilityanalytics.com\/blog\/wp-json\/wp\/v2\/comments?post=427"}],"version-history":[{"count":6,"href":"https:\/\/reliabilityanalytics.com\/blog\/wp-json\/wp\/v2\/posts\/427\/revisions"}],"predecessor-version":[{"id":722,"href":"https:\/\/reliabilityanalytics.com\/blog\/wp-json\/wp\/v2\/posts\/427\/revisions\/722"}],"wp:attachment":[{"href":"https:\/\/reliabilityanalytics.com\/blog\/wp-json\/wp\/v2\/media?parent=427"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/reliabilityanalytics.com\/blog\/wp-json\/wp\/v2\/categories?post=427"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/reliabilityanalytics.com\/blog\/wp-json\/wp\/v2\/tags?post=427"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}