{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,22]],"date-time":"2023-10-22T06:10:50Z","timestamp":1697955050290},"reference-count":18,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":7223,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[1987,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Consider an undirected network <jats:italic>G<\/jats:italic> with node set <jats:italic>V<\/jats:italic> and arc set <jats:italic>E<\/jats:italic> = {1, \u2026, <jats:italic>n<\/jats:italic>} where arcs fail randomly and independently. Let <jats:italic>T<\/jats:italic> be a subset of <jats:italic>V<\/jats:italic> and let <jats:italic>M<\/jats:italic><jats:sub>k<\/jats:sub> denote the number of ways that all the nodes of <jats:italic>T<\/jats:italic> are connected (<jats:italic>T<\/jats:italic>\u2010connectivity) with exactly <jats:italic>k<\/jats:italic> operating arcs and <jats:italic>n<\/jats:italic> \u2010 <jats:italic>k<\/jats:italic> failed arcs. This paper describes a sampling plan for estimating {<jats:italic>M<\/jats:italic><jats:sub>k<\/jats:sub>} and linear functions of these parameters, including the <jats:italic>T<\/jats:italic>\u2010connectedness reliability function <jats:italic>g<\/jats:italic>(<jats:italic>p<\/jats:italic>) for common failure probability 1 \u2010 <jats:italic>p<\/jats:italic>. Point and simultaneous interval estimates are derived for {<jats:italic>M<\/jats:italic><jats:sub>k<\/jats:sub>\/(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/tex2gif-stack-1.gif\" xlink:title=\"urn:x-wiley:00283045:media:NET3230170204:tex2gif-stack-1\" \/>}, where the interval estimates meet a fixed width criterion in <jats:italic>O<\/jats:italic>(<jats:italic>n<\/jats:italic> log <jats:italic>n<\/jats:italic>) time as the size of the network <jats:italic>n<\/jats:italic> grows. Whereas all previously proposed Monte Carlo sampling plans enable one to estimate <jats:italic>g<\/jats:italic>(<jats:italic>p<\/jats:italic>) for a fixed <jats:italic>p<\/jats:italic>, the proposed method allows one to estimate the entire function {<jats:italic>g<\/jats:italic>(<jats:italic>p<\/jats:italic>), <jats:italic>p<\/jats:italic> \u2208 <jats:italic>P<\/jats:italic>}, where either <jats:italic>P<\/jats:italic> = <jats:italic>P<\/jats:italic><jats:sup>*<\/jats:sup> = {<jats:italic>q<\/jats:italic><jats:sub>i<\/jats:sub>: O \u2266 <jats:italic>q<\/jats:italic><jats:sub>i<\/jats:sub> \u2266 1, 1 \u2266 <jats:italic>i<\/jats:italic> = \u2266 <jats:italic>v<\/jats:italic>} or <jats:italic>P<\/jats:italic> = <jats:italic>P<\/jats:italic><jats:sup>**<\/jats:sup> = {[<jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>]: 0 \u2266 <jats:italic>a<\/jats:italic> \u2266 <jats:italic>b<\/jats:italic> \u2266 1}. Here <jats:italic>P<\/jats:italic><jats:sup>*<\/jats:sup> denotes a finite set of points in [0, 1] and <jats:italic>P<\/jats:italic><jats:sup>**<\/jats:sup> an interval in [0, 1]. Simultaneous interval estimates are derived that meet a fixed width criterion in <jats:italic>O<\/jats:italic>(<jats:italic>n<\/jats:italic>) time for <jats:italic>P<\/jats:italic> = <jats:italic>P<\/jats:italic><jats:sup>*<\/jats:sup> and in <jats:italic>O<\/jats:italic>(<jats:italic>n<\/jats:italic><jats:sup>2<\/jats:sup>) time for <jats:italic>P<\/jats:italic> = <jats:italic>P<\/jats:italic><jats:sup>**<\/jats:sup> as <jats:italic>n<\/jats:italic> \u2192 \u221e. A <jats:italic>priori<\/jats:italic> bounds are also derived for <jats:italic>g<\/jats:italic> (<jats:italic>p<\/jats:italic>) and it is shown how these can be used to give guidance on the performance of the sampling plan. An example based on a network with 44 nodes and 85 arcs illustrates the proposed method.<\/jats:p>","DOI":"10.1002\/net.3230170204","type":"journal-article","created":{"date-parts":[[2007,5,11]],"date-time":"2007-05-11T22:17:57Z","timestamp":1178921877000},"page":"169-186","source":"Crossref","is-referenced-by-count":17,"title":["A monte carlo sampling plan for estimating reliability parameters and related functions"],"prefix":"10.1002","volume":"17","author":[{"given":"George S.","family":"Fishman","sequence":"first","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,11]]},"reference":[{"key":"e_1_2_1_2_2","series-title":"Applied Mathematics Series 55","volume-title":"Handbook of Mathematical Functions","author":"Abramowitz M.","year":"1964"},{"key":"e_1_2_1_3_2","volume-title":"The Design and Analysis of Computer Algorithms","author":"Aho A. V.","year":"1974"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1080\/00401706.1980.10486208"},{"key":"e_1_2_1_5_2","volume-title":"Statistical Theory of Reliability and Life Testing: Probability Models","author":"Barlow R. E.","year":"1975"},{"key":"e_1_2_1_6_2","unstructured":"F. T.Boesch Combinatorial optimization problems in the analysis and design of probabilistic networks. IEEE Trans. on Reliability in press."},{"key":"e_1_2_1_7_2","article-title":"A Monte Carlo sampling plan for estimating network reliability","author":"Fishman G. S.","year":"1987","journal-title":"Operations Research"},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1109\/TR.1986.4335388"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177704242"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1137\/0109047"},{"key":"e_1_2_1_11_2","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1080\/00401706.1965.10490252","article-title":"On simultaneous confidence intervals for multinomial populations","volume":"1","author":"Goodman L. A.","year":"1965","journal-title":"Technometrics"},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.1137\/0202024"},{"key":"e_1_2_1_13_2","unstructured":"G.Katona A theorem of finite sets. Theory of Graphs Proc. Tihany Colloquim September 1966 P. Erdos and G. Katona (eds.) pp.187\u2013207."},{"key":"e_1_2_1_14_2","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1525\/9780520319875-014","volume-title":"The number of simplices in a complex. 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