{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,4,27]],"date-time":"2023-04-27T04:10:28Z","timestamp":1682568628354},"reference-count":13,"publisher":"Cambridge University Press (CUP)","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2012,6]]},"abstract":"<jats:p>Let<jats:italic>X<\/jats:italic>= {<jats:italic>X<\/jats:italic><jats:sub><jats:italic>t<\/jats:italic><\/jats:sub>:<jats:italic>t<\/jats:italic>\u2265 0} be a stationary piecewise continuous<jats:bold>R<\/jats:bold><jats:sup><jats:italic>d<\/jats:italic><\/jats:sup>-valued process that moves between jumps along the integral curves of a given continuous vector field, and let<jats:italic>S<\/jats:italic>\u2282<jats:bold>R<\/jats:bold><jats:sup><jats:italic>d<\/jats:italic><\/jats:sup>be a smooth surface. The aim of this paper is to derive a multivariate version of Rice's formula, relating the intensity of the point process of (localized) continuous crossings of<jats:italic>S<\/jats:italic>by<jats:italic>X<\/jats:italic>to the distribution of<jats:italic>X<\/jats:italic><jats:sub>0<\/jats:sub>. Our result is illustrated by examples relating to queueing networks and stress release network models.<\/jats:p>","DOI":"10.1017\/s002190020000913x","type":"journal-article","created":{"date-parts":[[2016,3,29]],"date-time":"2016-03-29T14:49:00Z","timestamp":1459262940000},"page":"351-363","source":"Crossref","is-referenced-by-count":0,"title":["On Rice's Formula for Stationary Multivariate Piecewise Smooth Processes"],"prefix":"10.1017","volume":"49","author":[{"given":"K.","family":"Borovkov","sequence":"first","affiliation":[]},{"given":"G.","family":"Last","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2016,2,4]]},"reference":[{"key":"S002190020000913X_ref9","volume-title":"Foundations of Modern Probability","year":"2002"},{"key":"S002190020000913X_ref8","volume-title":"Ordinary Differential Equations","year":"2002"},{"key":"S002190020000913X_ref7","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1017\/S0021900200015539","volume":"37","year":"2000","journal-title":"J. Appl. Prob."},{"key":"S002190020000913X_ref6","doi-asserted-by":"publisher","DOI":"10.1239\/aap\/1222868187"},{"key":"S002190020000913X_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/s00024-003-2354-8"},{"key":"S002190020000913X_ref4","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.1972.1054748"},{"key":"S002190020000913X_ref10","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-842X.2004.00322.x"},{"key":"S002190020000913X_ref2","volume-title":"Level Sets and Extrema of Random Processes and Fields","year":"2009"},{"key":"S002190020000913X_ref1","volume-title":"Random Fields and Geometry","year":"2007"},{"key":"S002190020000913X_ref13","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4149(84)90005-X"},{"key":"S002190020000913X_ref12","doi-asserted-by":"publisher","DOI":"10.1023\/A:1017942408501"},{"key":"S002190020000913X_ref11","first-page":"282","volume":"24","year":"1944","journal-title":"Bell System Tech. J."},{"key":"S002190020000913X_ref3","volume-title":"Elements of Queueing Theory","year":"1994"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002190020000913X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,26]],"date-time":"2023-04-26T06:43:29Z","timestamp":1682491409000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002190020000913X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,6]]},"references-count":13,"journal-issue":{"issue":"02","published-print":{"date-parts":[[2012,6]]}},"alternative-id":["S002190020000913X"],"URL":"https:\/\/doi.org\/10.1017\/s002190020000913x","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"value":"0021-9002","type":"print"},{"value":"1475-6072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,6]]}}}