A polling model with threshold switching

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 $$Q_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> and the single-buffer queue $$Q_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> is empty, then it stays at $$Q_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> until it has become empty; but if a customer joins an empty $$Q_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> , then the server only stays at $$Q_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> as long as that queue has at least N customers (the threshold). If that customer joins $$Q_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> while $$Q_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> has less than N customers, then service at $$Q_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> is preempted and the server instantaneously switches to $$Q_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> . Arrivals to $$Q_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> 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.