{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T16:21:32Z","timestamp":1770481292436,"version":"3.49.0"},"reference-count":31,"publisher":"Wiley","issue":"7","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":5428,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[1991,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Consider a flow network <jats:italic>G<\/jats:italic> = (\ud835\udcb1,\u2130) with node set \ud835\udcb1 and arc set \u2130 = {1, \u2026, <jats:italic>e<\/jats:italic>}. Assume that the nodes do not restrict flow transmission and the arcs have random, discrete and independent capacities <jats:italic>B<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>B<jats:sub>e<\/jats:sub><\/jats:italic>, and let <jats:italic><jats:bold>B<\/jats:bold> = {B<jats:sub>1<\/jats:sub>, \u2026, B<jats:sub>e<\/jats:sub>}<\/jats:italic>. Also, let <jats:italic>s<\/jats:italic> and <jats:italic>t<\/jats:italic> be a pair of nodes in \ud835\udcb1, let \u039b(<jats:bold><jats:italic>B<\/jats:italic><\/jats:bold>) denote the value of a maximum <jats:italic>s\u2013t<\/jats:italic> flow, and let \u0393 denote a set of <jats:italic>s\u2013t<\/jats:italic> cuts. This work describes a highly efficient Monte Carlo sampling plan for estimating the probability that <jats:italic>l<\/jats:italic> &lt; \u039b(<jats:bold><jats:italic>B<\/jats:italic><\/jats:bold>) \u2a7d <jats:italic>u<\/jats:italic>, the probability that a cut in \u0393 is critical and <jats:italic>l<\/jats:italic> &lt; \u039b(<jats:bold><jats:italic>B<\/jats:italic><\/jats:bold>) \u2a7d <jats:italic>u<\/jats:italic>, and the probability that a cut in \u0393 is critical, given that <jats:italic>l<\/jats:italic> &lt; \u039b(<jats:bold><jats:italic>B<\/jats:italic><\/jats:bold>) \u2a7d <jats:italic>u<\/jats:italic>. The proposed method takes advantage of an easily computed upper bound on the probability that <jats:italic>l<\/jats:italic> &lt; \u039b(<jats:bold><jats:italic>B<\/jats:italic><\/jats:bold>) \u2a7d <jats:italic>u<\/jats:italic>, which is a function of both <jats:italic>l<\/jats:italic> and <jats:italic>u<\/jats:italic> to gain its computational advantage. The article also describes techniques for computing confidence intervals that are valid for any sample size. Algorithms for implementing the proposed sampling experiment are included, and an example illustrates the efficiency of the proposed method.<\/jats:p>","DOI":"10.1002\/net.3230210706","type":"journal-article","created":{"date-parts":[[2007,5,12]],"date-time":"2007-05-12T11:49:45Z","timestamp":1178970585000},"page":"775-798","source":"Crossref","is-referenced-by-count":14,"title":["Characterizing stochastic flow networks using the monte carlo method"],"prefix":"10.1002","volume":"21","author":[{"given":"Christos","family":"Alexopoulos","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"George S.","family":"Fishman","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,10,11]]},"reference":[{"key":"e_1_2_1_2_2","unstructured":"C.Alexopoulos Maximum flows and critical cutsets in stochastic networks with discrete arc capacities. Ph.D. Thesis Department of Operations Research University of North Carolina at Chapel Hill (1988)."},{"key":"e_1_2_1_3_2","unstructured":"C.Alexopoulos Confidence intervals for conditional probabilities and ratios of expectations. Technical Report Series No. J\u201089\u20103 School of Industrial and Systems Engineering Georgia Institute of Technology. (1989 revised July1991)."},{"issue":"3","key":"e_1_2_1_4_2","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1109\/TR.1986.4335422","article-title":"Computational complexity of network reliability analysis: An overview","volume":"35","author":"Ball M. O.","year":"1987","journal-title":"IEEE Trans. Reliability"},{"key":"e_1_2_1_5_2","volume-title":"Statistical Theory of Reliability and Life Testing Probability Models To Begin With","author":"Barlow R. 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