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If the server is serving the infinite-buffer queue\n \n \n $$Q_2$$<\/jats:tex-math>\n \n \n Q<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and the single-buffer queue\n \n \n $$Q_1$$<\/jats:tex-math>\n \n \n Q<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is empty, then it stays at\n \n \n $$Q_2$$<\/jats:tex-math>\n \n \n Q<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n until it has become empty; but if a customer joins an empty\n \n \n $$Q_1$$<\/jats:tex-math>\n \n \n Q<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , then the server only stays at\n \n \n $$Q_2$$<\/jats:tex-math>\n \n \n Q<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n as long as that queue has at least\n N<\/jats:italic>\n customers (the threshold). If that customer joins\n \n \n $$Q_1$$<\/jats:tex-math>\n \n \n Q<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n while\n \n \n $$Q_2$$<\/jats:tex-math>\n \n \n Q<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n has less than\n N<\/jats:italic>\n customers, then service at\n \n \n $$Q_2$$<\/jats:tex-math>\n \n \n Q<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is preempted and the server instantaneously switches to\n \n \n $$Q_1$$<\/jats:tex-math>\n \n \n Q<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Arrivals to\n \n \n $$Q_1$$<\/jats:tex-math>\n \n \n Q<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n when it is occupied are blocked and lost. This threshold discipline contrasts with the classical multi-queue polling model, where switching instants are typically determined by the length of the queue being served. We (i) derive explicit expressions for the joint queue length distribution; (ii) analyze the busy period distribution by employing an original approach that uses taboo states; and (iii) determine the sojourn time distribution for customers in both queues.\n <\/jats:p>","DOI":"10.1007\/s11134-025-09954-1","type":"journal-article","created":{"date-parts":[[2025,10,16]],"date-time":"2025-10-16T12:32:00Z","timestamp":1760617920000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A polling model with threshold switching"],"prefix":"10.1007","volume":"109","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4317-5380","authenticated-orcid":false,"given":"Onno","family":"Boxma","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David","family":"Perry","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rachel","family":"Ravid","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Uri","family":"Yechiali","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,10,16]]},"reference":[{"key":"9954_CR1","doi-asserted-by":"publisher","first-page":"541","DOI":"10.1007\/s11750-015-0408-6","volume":"24","author":"K Avrachenkov","year":"2016","unstructured":"Avrachenkov, K., Perel, E., Yechiali, U.: Finite-buffer polling systems with threshold-based switching policy. 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