Busy period and stationary distribution for the queueing system $\mathbf {M^{\theta }/G/1/\infty }$ with a threshold switching between service modes
Author:
K. Yu. Zhernovyĭ
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 87 (2013), 51-63
MSC (2010):
Primary 60K25; Secondary 60K20
DOI:
https://doi.org/10.1090/S0094-9000-2014-00904-7
Published electronically:
March 21, 2014
MathSciNet review:
3241446
Full-text PDF Free Access
Abstract |
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Additional Information
Abstract: We consider the queueing system $\mathrm {M^{\theta }/G/1/\infty }$ with two service modes. The server switches between the main service mode and threshold mode at the beginning of the current customer service when the number of customers present in the system is larger than $h$. The mean duration of the busy time and formulas for the stationary distribution of the number of customers in the system are obtained.
References
- K. Yu. Zhernovyĭ, An investigation of a $\text {M}^{\theta }\text {/G/1/m}$ queueing system with service mode switching, Teoriya Imovirnostei ta Matematichna Statistika 86 (2012), 56–68. (Ukrainian)
- Vladimir V. Anisimov, Switching processes in queueing models, Applied Stochastic Methods Series, ISTE, London; John Wiley & Sons, Inc., Hoboken, NJ, 2008. MR 2437051
- A. M. Bratiĭchuk, An Investigation of Queueing Systems with a Bounded Queue, Thesis of Candidate Dissertation, Kyiv National Taras Shevchenko University, Kyiv, 2008. (Ukrainian)
- K. Yu. Zhernovyĭ, An Investigation of a $\text {M}^{\theta }\text {/G/1/m}$ system with switches between service modes and with threshold blocking of the input flow, Inform. Process. 10 (2010), no. 2, 159–180. (Russian)
- V. D. Boev, System Simulation. GPSS World Tools, “BHV-Peterburg”, Sankt-Peterburg, 2004. (Russian)
- Yu. V. Zhernovyĭ, Simulation of Queueing Systems, Publishing Center of L’viv National Ivan Franko University, L’viv, 2007. (Ukrainian)
References
- K. Yu. Zhernovyĭ, An investigation of a $\text {M}^{\theta }\text {/G/1/m}$ queueing system with service mode switching, Teoriya Imovirnostei ta Matematichna Statistika 86 (2012), 56–68. (Ukrainian)
- V. Anisimov, Switching Processes in Queueing Models, John Wiley and Sons, ISTE, London, 2008. MR 2437051 (2009i:60158)
- A. M. Bratiĭchuk, An Investigation of Queueing Systems with a Bounded Queue, Thesis of Candidate Dissertation, Kyiv National Taras Shevchenko University, Kyiv, 2008. (Ukrainian)
- K. Yu. Zhernovyĭ, An Investigation of a $\text {M}^{\theta }\text {/G/1/m}$ system with switches between service modes and with threshold blocking of the input flow, Inform. Process. 10 (2010), no. 2, 159–180. (Russian)
- V. D. Boev, System Simulation. GPSS World Tools, “BHV-Peterburg”, Sankt-Peterburg, 2004. (Russian)
- Yu. V. Zhernovyĭ, Simulation of Queueing Systems, Publishing Center of L’viv National Ivan Franko University, L’viv, 2007. (Ukrainian)
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Additional Information
K. Yu. Zhernovyĭ
Affiliation:
Department of Higher Mathematics, Faculty for Mathematics and Mechanics, L’viv National Ivan Franko University, Universytets’ka Street, 1, L’viv, 79000, Ukraine
Email:
k_zhernovyi@yahoo.com
Keywords:
The queueing system $\mathrm {M^{\theta }/G/1/\infty }$ with two service modes,
busy period,
stationary distribution of the number of customers
Received by editor(s):
October 12, 2011
Published electronically:
March 21, 2014
Article copyright:
© Copyright 2014
American Mathematical Society
