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• W2742810376 abstract "This paper considers a <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M2><mml:mi>G</mml:mi><mml:mi>I</mml:mi><mml:mo>/</mml:mo><mml:mi>M</mml:mi><mml:mo>/</mml:mo><mml:mn mathvariant=normal>1</mml:mn></mml:math> queue with vacations and multiple service phases. Whenever the system becomes empty, the server takes a vacation, causing the system to move to vacation phase <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M3><mml:mrow><mml:mn mathvariant=normal>0</mml:mn></mml:mrow></mml:math>. If the server returns from a vacation to find no customer waiting, another vacation begins. Otherwise, the system jumps from phase <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M4><mml:mrow><mml:mn mathvariant=normal>0</mml:mn></mml:mrow></mml:math> to some service phase <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M5><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:math> with probability <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M6><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant=normal>1,2</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>.</mml:mo></mml:math> Using the matrix geometric solution method and semi-Markov process, we obtain the distributions of the stationary system size at both arrival and arbitrary epochs. The distribution of the stationary waiting time of an arbitrary customer is also derived. In addition, we present some performance measures such as mean waiting time of an arbitrary customer, mean length of the type-<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M7><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:math> cycle, and mean number of customers in the system at the end of phase <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M8><mml:mrow><mml:mn mathvariant=normal>0</mml:mn></mml:mrow></mml:math>. Finally, some numerical examples are presented." @default.
• W2742810376 created "2017-08-17" @default.
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• W2742810376 date "2017-01-01" @default.
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• W2742810376 title "On the <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M1><mml:mrow><mml:mtext mathvariant=italic>GI/M/1</mml:mtext></mml:mrow></mml:math> Queue with Vacations and Multiple Service Phases" @default.
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• W2742810376 doi "https://doi.org/10.1155/2017/3246934" @default.
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