A polling model with threshold switching

Onno Boxma (), David Perry (), Rachel Ravid () and Uri Yechiali ()
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Onno Boxma: Eindhoven University of Technology
David Perry: Holon Institute of Technology
Rachel Ravid: Braude College of Engineering Karmiel
Uri Yechiali: School of Mathematical Sciences, Tel-Aviv University

Queueing Systems: Theory and Applications, 2025, vol. 109, issue 4, No 2, 24 pages

Abstract: Abstract We consider a single-server two-queue Markovian polling system with the following special feature. If the server is serving the infinite-buffer queue $$Q_2$$ Q 2 and the single-buffer queue $$Q_1$$ Q 1 is empty, then it stays at $$Q_2$$ Q 2 until it has become empty; but if a customer joins an empty $$Q_1$$ Q 1 , then the server only stays at $$Q_2$$ Q 2 as long as that queue has at least N customers (the threshold). If that customer joins $$Q_1$$ Q 1 while $$Q_2$$ Q 2 has less than N customers, then service at $$Q_2$$ Q 2 is preempted and the server instantaneously switches to $$Q_1$$ Q 1 . Arrivals to $$Q_1$$ Q 1 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.

Keywords: Polling model; Threshold switching policy; Busy period; Sojourn time (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s11134-025-09954-1

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