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October  2012, 8(4): 939-968. doi: 10.3934/jimo.2012.8.939

M/M/c multiple synchronous vacation model with gated discipline

Zsolt Saffer 1, and  Wuyi Yue 2,

1. 

Department of Telecommunications, Budapest University of Technology and Economics, Budapest
2. 

Department of Intelligence and Informatics, Konan University, 8-9-1 Okamoto, Kobe 658-8501

Received  September 2011 Revised  July 2012 Published  September 2012

In this paper we present the analysis of an M/M/c multiple synchronous vacation model. In contrast to the previous works on synchronous vacation model we consider the model with gated service discipline and with independent and identically distributed vacation periods. The analysis of this model requires different methodology compared to those ones used for synchronous vacation model so far. We provide the probability-generating function and the mean of the stationary number of customers at an arbitrary epoch as well as the Laplace-Stieljes transform and the mean of the stationary waiting time. The stationary distribution of the number of busy servers and the stability of the system are also considered. In the final part of the paper numerical examples illustrate the computational procedure.
    This vacation queue is suitable to model a single operator controlled system consisting of more machines. Hence the provided analysis can be applied to study and optimize such systems.
Keywords: Queueing theory, multi-server queue, service discipline., synchronous vacation model.
Mathematics Subject Classification: Primary: 60K25, 68M20; Secondary: 90B2.
Citation: Zsolt Saffer, Wuyi Yue. M/M/c multiple synchronous vacation model with gated discipline. Journal of Industrial & Management Optimization, 2012, 8 (4) : 939-968. doi: 10.3934/jimo.2012.8.939
References:
[1]

A. Begum and M. Nadarajan, Multiserver markovian queueing system with vacation, Optimization, 41 (1997), 71-78. doi: 10.1080/02331939708844326.  Google Scholar

[2]

S. C. Borst and O. J. Boxma, Polling models with and without switch over times, Operations Research, 45 (1997), 536-543. doi: 10.1287/opre.45.4.536.  Google Scholar

[3]

X. Chao and Y. Zhao, Analysis of multi-server queues with station and server vacations, European Journal of Operational Research, 110 (1998), 392-406. doi: 10.1016/S0377-2217(97)00253-1.  Google Scholar

[4]

B. T. Doshi, Queueing systems with vacations-a survey, Queueing Systems, 1 (1986), 29-66. doi: 10.1007/BF01149327.  Google Scholar

[5]

M. Kuczma, "Functional Equations in a Single Variable," PWN-Polish Scientific Publishers, Warsaw, 1968.  Google Scholar

[6]

Y. Levy and U. Yechiali, An M/M/s queue with server's vacations, In INFOR 14, (1976), 153-163.  Google Scholar

[7]

Z. Saffer, An introduction to classical cyclic polling model, In Proc. of the 14th Int. Conf. on Analytical and Stochastic Modelling Techniques and Applications (ASMTA'07), (2007), 59-64. Google Scholar

[8]

H. Takagi, "Analysis of Polling Systems," MIT Press, 1986. Google Scholar

[9]

H. Takagi, "Queueing Analysis - A Foundation of Performance Evaluation, Vacation and Prority Systems," North-Holland, New York, 1991.  Google Scholar

[10]

N. Tian and G. Zhang, "Vacation Queueing Models: Theory and Applications. Series: International Series in Operations Research & Management Science," Springer-Verlag, New York, 2006.  Google Scholar

[11]

N. Tian and L. Li, The M/M/c queue with PH synchronous vacations, Journal of Systems Science and Complexity, 13 (2000), 007-016.  Google Scholar

[12]

N. Tian and G. Zhang, Stationary distributions of GI/M/c queue with PH type vacations, Queueing Systems, 44 (2003), 183–-202. doi: 10.1023/A:1024424606007.  Google Scholar

[13]

R. W. Wolff, Poisson arrivals see times averages, Operations Research, 30 (1982), 223-231. doi: 10.1287/opre.30.2.223.  Google Scholar

[14]

W. Yue, Y. Takahashi and H. Takagi, "Advances in Queueing Theory and Network Applications," Springer Science + Business Media, New York, 2010.  Google Scholar

[15]

G. Zhang and N. Tian, Analysis of queueing systems with synchronous single vacation for some servers, Queueing System, 45 (2003), 161-175. doi: 10.1023/A:1026097723093.  Google Scholar

[16]

G. Zhang and N. Tian, An analysis of queueing systems with multi-task servers, European Journal of Operational Research, 156 (2004), 375-389. doi: 10.1016/S0377-2217(03)00015-8.  Google Scholar

[17]

R. W. Wolff, "Stochastic Modeling and the Theory of Queues," Prentice-Hall, Englewood Cliffs, NJ, 1989.  Google Scholar

show all references

References:
[1]

A. Begum and M. Nadarajan, Multiserver markovian queueing system with vacation, Optimization, 41 (1997), 71-78. doi: 10.1080/02331939708844326.  Google Scholar

[2]

S. C. Borst and O. J. Boxma, Polling models with and without switch over times, Operations Research, 45 (1997), 536-543. doi: 10.1287/opre.45.4.536.  Google Scholar

[3]

X. Chao and Y. Zhao, Analysis of multi-server queues with station and server vacations, European Journal of Operational Research, 110 (1998), 392-406. doi: 10.1016/S0377-2217(97)00253-1.  Google Scholar

[4]

B. T. Doshi, Queueing systems with vacations-a survey, Queueing Systems, 1 (1986), 29-66. doi: 10.1007/BF01149327.  Google Scholar

[5]

M. Kuczma, "Functional Equations in a Single Variable," PWN-Polish Scientific Publishers, Warsaw, 1968.  Google Scholar

[6]

Y. Levy and U. Yechiali, An M/M/s queue with server's vacations, In INFOR 14, (1976), 153-163.  Google Scholar

[7]

Z. Saffer, An introduction to classical cyclic polling model, In Proc. of the 14th Int. Conf. on Analytical and Stochastic Modelling Techniques and Applications (ASMTA'07), (2007), 59-64. Google Scholar

[8]

H. Takagi, "Analysis of Polling Systems," MIT Press, 1986. Google Scholar

[9]

H. Takagi, "Queueing Analysis - A Foundation of Performance Evaluation, Vacation and Prority Systems," North-Holland, New York, 1991.  Google Scholar

[10]

N. Tian and G. Zhang, "Vacation Queueing Models: Theory and Applications. Series: International Series in Operations Research & Management Science," Springer-Verlag, New York, 2006.  Google Scholar

[11]

N. Tian and L. Li, The M/M/c queue with PH synchronous vacations, Journal of Systems Science and Complexity, 13 (2000), 007-016.  Google Scholar

[12]

N. Tian and G. Zhang, Stationary distributions of GI/M/c queue with PH type vacations, Queueing Systems, 44 (2003), 183–-202. doi: 10.1023/A:1024424606007.  Google Scholar

[13]

R. W. Wolff, Poisson arrivals see times averages, Operations Research, 30 (1982), 223-231. doi: 10.1287/opre.30.2.223.  Google Scholar

[14]

W. Yue, Y. Takahashi and H. Takagi, "Advances in Queueing Theory and Network Applications," Springer Science + Business Media, New York, 2010.  Google Scholar

[15]

G. Zhang and N. Tian, Analysis of queueing systems with synchronous single vacation for some servers, Queueing System, 45 (2003), 161-175. doi: 10.1023/A:1026097723093.  Google Scholar

[16]

G. Zhang and N. Tian, An analysis of queueing systems with multi-task servers, European Journal of Operational Research, 156 (2004), 375-389. doi: 10.1016/S0377-2217(03)00015-8.  Google Scholar

[17]

R. W. Wolff, "Stochastic Modeling and the Theory of Queues," Prentice-Hall, Englewood Cliffs, NJ, 1989.  Google Scholar

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