{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,11,13]],"date-time":"2023-11-13T00:14:00Z","timestamp":1699834440192},"reference-count":4,"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":10724,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[1977,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We begin with a three\u2010stage linear graph in which the first stage has a single node u and the third stage a single node v. The second stage has k independent nodes, each of which is connected by one link to u and to v. In general, we can form a (2n+1)\u2010stage linear graph recursively by letting each node in the second stage of a three\u2010stage linear graph be replaced by a copy of a (2n\u20101)\u2010stage linear graph.<\/jats:p><jats:p>A link can either be in the busy state or the idle state. We assume that the states of each link are mutually independent and that any link between stage i and stage i + 1 has the probability I.z of being idle. The nodes u and v are said to be connectable if there exists at least one path from u to v with no busy link. Let P(u, v) denote the probability of such a path existing. Further, let N(2n+1, k) denote the set of (2n+1)\u2010stage linear graphs whose center stages have k nodes.<\/jats:p><jats:p>In this paper, we determine the size of N(2n+1, k). We also give the linear graph in N(2n+1, k) which has the largest P(u, v) and the one which has the smallest. We then show how our results apply to a recent problem in connecting networks.<\/jats:p>","DOI":"10.1002\/net.3230070205","type":"journal-article","created":{"date-parts":[[2007,5,11]],"date-time":"2007-05-11T03:48:48Z","timestamp":1178855328000},"page":"185-192","source":"Crossref","is-referenced-by-count":4,"title":["A problem on blocking probabilities in connecting networks"],"prefix":"10.1002","volume":"7","author":[{"given":"F. R. K.","family":"Chung","sequence":"first","affiliation":[]},{"given":"F. K.","family":"Hwang","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,11]]},"reference":[{"key":"e_1_2_1_2_2","unstructured":"Chung F. R. K. \u201cOn Switching Networks and Block Designs \u201dConference Records of the Tenth Annual Asilomar Conference on Circuits Systems and Computers Pacific Grove California 1976 pp.212\u2013218."},{"key":"e_1_2_1_3_2","unstructured":"Hall M. Jr. Combinatorial Theory Blaisdell 1967."},{"key":"e_1_2_1_4_2","unstructured":"Hwang F. K. \u201cLink Designs and Probability Analysis for a Class of Connecting Networks \u201d to appear."},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1002\/j.1538-7305.1955.tb03799.x"}],"container-title":["Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnet.3230070205","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/net.3230070205","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,12]],"date-time":"2023-11-12T10:30:20Z","timestamp":1699785020000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/net.3230070205"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1977,6]]},"references-count":4,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1977,6]]}},"alternative-id":["10.1002\/net.3230070205"],"URL":"https:\/\/doi.org\/10.1002\/net.3230070205","archive":["Portico"],"relation":{},"ISSN":["0028-3045","1097-0037"],"issn-type":[{"value":"0028-3045","type":"print"},{"value":"1097-0037","type":"electronic"}],"subject":[],"published":{"date-parts":[[1977,6]]}}}