We consider a single-server two-queue Markovian polling system with the following special feature. If the server is serving the infinite-buffer queue <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$Q_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> and the single-buffer queue <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$Q_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> is empty, then it stays at <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$Q_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> until it has become empty; but if a customer joins an empty <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$Q_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>, then the server only stays at <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$Q_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> as long as that queue has at least N customers (the threshold). If that customer joins <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$Q_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> while <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$Q_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> has less than N customers, then service at <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$Q_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> is preempted and the server instantaneously switches to <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$Q_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>. Arrivals to <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$Q_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> 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.

A polling model with threshold switching

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2025年10月16日

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Onno Boxma,
David Perry,
Rachel Ravid,
Uri Yechiali

摘要

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.