{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,25]],"date-time":"2025-02-25T13:47:47Z","timestamp":1740491267050,"version":"3.38.0"},"reference-count":19,"publisher":"Cambridge University Press (CUP)","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2012,9]]},"abstract":"<jats:p>In this paper we study the distribution of the location, at time <jats:italic>t<\/jats:italic>, of a particle moving <jats:italic>U<\/jats:italic> time units upwards, <jats:italic>V<\/jats:italic> time units downwards, and <jats:italic>W<\/jats:italic> time units of no movement (idle). These are repeated cyclically, according to independent alternating renewals. The distributions of <jats:italic>U<\/jats:italic>, <jats:italic>V<\/jats:italic>, and <jats:italic>W<\/jats:italic> are absolutely continuous. The velocities are <jats:italic>v<\/jats:italic> = +1 upwards, <jats:italic>v<\/jats:italic> = -1 downwards, and <jats:italic>v<\/jats:italic> = 0 during idle periods. Let <jats:italic>Y<\/jats:italic>\n               <jats:sup>+<\/jats:sup>(<jats:italic>t<\/jats:italic>), <jats:italic>Y<\/jats:italic>\n               <jats:sup>\u2212<\/jats:sup>(<jats:italic>t<\/jats:italic>), and <jats:italic>Y<\/jats:italic>\n               <jats:sup>0<\/jats:sup>(<jats:italic>t<\/jats:italic>) denote the total time in (0, <jats:italic>t<\/jats:italic>) of movements upwards, downwards, and no movements, respectively. The exact distribution of <jats:italic>Y<\/jats:italic>\n               <jats:sup>+<\/jats:sup>(<jats:italic>t<\/jats:italic>) is derived. We also obtain the probability law of <jats:italic>X<\/jats:italic>(<jats:italic>t<\/jats:italic>) = <jats:italic>Y<\/jats:italic>\n               <jats:sup>+<\/jats:sup>(<jats:italic>t<\/jats:italic>) - <jats:italic>Y<\/jats:italic>\n               <jats:sup>\u2212<\/jats:sup>(<jats:italic>t<\/jats:italic>), which describes the particle's location at time <jats:italic>t<\/jats:italic>. Explicit formulae are derived for the cases of exponential distributions with equal rates, with different rates, and with linear rates (leading to damped processes).<\/jats:p>","DOI":"10.1017\/s002190020000958x","type":"journal-article","created":{"date-parts":[[2016,3,29]],"date-time":"2016-03-29T10:49:46Z","timestamp":1459248586000},"page":"850-865","source":"Crossref","is-referenced-by-count":3,"title":["Generalized Telegraph Process with Random Delays"],"prefix":"10.1017","volume":"49","author":[{"given":"Daoud","family":"Bshouty","sequence":"first","affiliation":[]},{"given":"Antonio","family":"Di Crescenzo","sequence":"additional","affiliation":[]},{"given":"Barbara","family":"Martinucci","sequence":"additional","affiliation":[]},{"given":"Shelemyahu","family":"Zacks","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2016,2,4]]},"reference":[{"key":"S002190020000958X_ref19","doi-asserted-by":"publisher","DOI":"10.1239\/jap\/1082999081"},{"volume-title":"Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables","year":"1992","key":"S002190020000958X_ref1"},{"first-page":"155","volume-title":"Mathematical Ecology","year":"1986","key":"S002190020000958X_ref17"},{"key":"S002190020000958X_ref16","doi-asserted-by":"publisher","DOI":"10.2307\/1427568"},{"key":"S002190020000958X_ref15","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4149(90)90056-X"},{"key":"S002190020000958X_ref9","doi-asserted-by":"publisher","DOI":"10.2307\/3215093"},{"key":"S002190020000958X_ref14","doi-asserted-by":"publisher","DOI":"10.1239\/aap\/1067436339"},{"first-page":"175","volume-title":"Mathematical and Statistical Methods for Actuarial Sciences and Finance","year":"2011","key":"S002190020000958X_ref8"},{"key":"S002190020000958X_ref13","doi-asserted-by":"publisher","DOI":"10.1239\/aap\/1093962242"},{"key":"S002190020000958X_ref7","doi-asserted-by":"publisher","DOI":"10.1002\/asmb.456"},{"key":"S002190020000958X_ref12","doi-asserted-by":"publisher","DOI":"10.1051\/ps:2006012"},{"key":"S002190020000958X_ref6","doi-asserted-by":"publisher","DOI":"10.1239\/jap\/1269610818"},{"volume-title":"A Table of Series and Products","year":"1975","key":"S002190020000958X_ref11"},{"key":"S002190020000958X_ref5","doi-asserted-by":"publisher","DOI":"10.1080\/10451120290019186"},{"volume-title":"Tables of Integrals, Series, and Products","year":"2007","key":"S002190020000958X_ref10"},{"key":"S002190020000958X_ref4","doi-asserted-by":"publisher","DOI":"10.1239\/aap\/1005091360"},{"first-page":"211","volume-title":"Advanced Special Functions and Integration Methods","year":"2001","key":"S002190020000958X_ref3"},{"key":"S002190020000958X_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/s00365-006-0643-6"},{"key":"S002190020000958X_ref18","doi-asserted-by":"publisher","DOI":"10.1239\/jap\/1091543417"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002190020000958X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,4,22]],"date-time":"2017-04-22T00:17:50Z","timestamp":1492820270000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002190020000958X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,9]]},"references-count":19,"journal-issue":{"issue":"03","published-print":{"date-parts":[[2012,9]]}},"alternative-id":["S002190020000958X"],"URL":"https:\/\/doi.org\/10.1017\/s002190020000958x","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"type":"print","value":"0021-9002"},{"type":"electronic","value":"1475-6072"}],"subject":[],"published":{"date-parts":[[2012,9]]}}}