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A polling model with threshold switching

Author

Listed:
  • 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)

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.

Suggested Citation

  • Onno Boxma & David Perry & Rachel Ravid & Uri Yechiali, 2025. "A polling model with threshold switching," Queueing Systems: Theory and Applications, Springer, vol. 109(4), pages 1-24, December.
    • Handle: RePEc:spr:queues:v:109:y:2025:i:4:d:10.1007_s11134-025-09954-1
    DOI: 10.1007/s11134-025-09954-1
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