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We start our exposition by a discussion of the model\u2019s stability condition. Writing $$Y_i=B_i-A_i$$<\/jats:tex-math>\n \n \n Y<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n =<\/mml:mo>\n \n B<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n -<\/mml:mo>\n \n A<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, for independent sequences of nonnegative i.i.d. random variables $$\\{A_i\\}_{i\\in {\\mathbb N}_0}$$<\/jats:tex-math>\n \n \n {<\/mml:mo>\n \n A<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n }<\/mml:mo>\n <\/mml:mrow>\n \n i<\/mml:mi>\n \u2208<\/mml:mo>\n \n N<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$\\{B_i\\}_{i\\in {\\mathbb N}_0}$$<\/jats:tex-math>\n \n \n {<\/mml:mo>\n \n B<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n }<\/mml:mo>\n <\/mml:mrow>\n \n i<\/mml:mi>\n \u2208<\/mml:mo>\n \n N<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and assuming $$\\{V_i\\}_{i\\in {\\mathbb N}_0}$$<\/jats:tex-math>\n \n \n {<\/mml:mo>\n \n V<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n }<\/mml:mo>\n <\/mml:mrow>\n \n i<\/mml:mi>\n \u2208<\/mml:mo>\n \n N<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is an i.i.d. sequence as well (independent of $$\\{A_i\\}_{i\\in {\\mathbb N}_0}$$<\/jats:tex-math>\n \n \n {<\/mml:mo>\n \n A<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n }<\/mml:mo>\n <\/mml:mrow>\n \n i<\/mml:mi>\n \u2208<\/mml:mo>\n \n N<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$\\{B_i\\}_{i\\in {\\mathbb N}_0}$$<\/jats:tex-math>\n \n \n {<\/mml:mo>\n \n B<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n }<\/mml:mo>\n <\/mml:mrow>\n \n i<\/mml:mi>\n \u2208<\/mml:mo>\n \n N<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>), we then consider three special cases (i) $$V_i$$<\/jats:tex-math>\n \n V<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> equals a positive value a<\/jats:italic> with certain probability $$p\\in (0,1)$$<\/jats:tex-math>\n \n p<\/mml:mi>\n \u2208<\/mml:mo>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and is negative otherwise, and both $$A_i$$<\/jats:tex-math>\n \n A<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$B_i$$<\/jats:tex-math>\n \n B<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> have a rational LST, (ii) $$V_i$$<\/jats:tex-math>\n \n V<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> attains negative values only and $$B_i$$<\/jats:tex-math>\n \n B<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> has a rational LST, (iii) $$V_i$$<\/jats:tex-math>\n \n V<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is uniformly distributed on [0,\u00a01], and $$A_i$$<\/jats:tex-math>\n \n A<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is exponentially distributed. In all three cases, we derive transient and stationary results, where the transient results are in terms of the transform at a geometrically distributed epoch.\n<\/jats:p>","DOI":"10.1007\/s11134-021-09698-8","type":"journal-article","created":{"date-parts":[[2021,3,13]],"date-time":"2021-03-13T05:02:46Z","timestamp":1615611766000},"page":"225-245","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["A multiplicative version of the Lindley recursion"],"prefix":"10.1007","volume":"98","author":[{"given":"Onno","family":"Boxma","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andreas","family":"L\u00f6pker","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michel","family":"Mandjes","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9257-1115","authenticated-orcid":false,"given":"Zbigniew","family":"Palmowski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,3,13]]},"reference":[{"key":"9698_CR1","volume-title":"Applied Probability and Queues","author":"S Asmussen","year":"2008","unstructured":"Asmussen, S.: Applied Probability and Queues. 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