American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Stochastic method for power-aware checkpoint intervals in wireless environments: Theory and application
  • JIMO Home
  • This Issue
  • Next Article
    Stochastic decomposition in discrete-time queues with generalized vacations and applications
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.  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.  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.  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.  doi: 10.1007/BF01149327.  Google Scholar

[5]

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

[6]

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

[12]

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

[13]

R. W. Wolff, Poisson arrivals see times averages,, Operations Research, 30 (1982), 223.  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, (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.  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.  doi: 10.1016/S0377-2217(03)00015-8.  Google Scholar

[17]

R. W. Wolff, "Stochastic Modeling and the Theory of Queues,", Prentice-Hall, (1989).   Google Scholar

show all references

References:
[1]

A. Begum and M. Nadarajan, Multiserver markovian queueing system with vacation,, Optimization, 41 (1997), 71.  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.  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.  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.  doi: 10.1007/BF01149327.  Google Scholar

[5]

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

[6]

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

[12]

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

[13]

R. W. Wolff, Poisson arrivals see times averages,, Operations Research, 30 (1982), 223.  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, (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.  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.  doi: 10.1016/S0377-2217(03)00015-8.  Google Scholar

[17]

R. W. Wolff, "Stochastic Modeling and the Theory of Queues,", Prentice-Hall, (1989).   Google Scholar

[1]

Tzu-Hsin Liu, Jau-Chuan Ke. On the multi-server machine interference with modified Bernoulli vacation. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1191-1208. doi: 10.3934/jimo.2014.10.1191
[2]

Tao Jiang, Liwei Liu. Analysis of a batch service multi-server polling system with dynamic service control. Journal of Industrial & Management Optimization, 2018, 14 (2) : 743-757. doi: 10.3934/jimo.2017073
[3]

Willem Mélange, Herwig Bruneel, Bart Steyaert, Dieter Claeys, Joris Walraevens. A continuous-time queueing model with class clustering and global FCFS service discipline. Journal of Industrial & Management Optimization, 2014, 10 (1) : 193-206. doi: 10.3934/jimo.2014.10.193
[4]

Wai-Ki Ching, Sin-Man Choi, Min Huang. Optimal service capacity in a multiple-server queueing system: A game theory approach. Journal of Industrial & Management Optimization, 2010, 6 (1) : 73-102. doi: 10.3934/jimo.2010.6.73
[5]

Jeongsim Kim, Bara Kim. Stability of a queue with discriminatory random order service discipline and heterogeneous servers. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1237-1254. doi: 10.3934/jimo.2016070
[6]

Gábor Horváth, Zsolt Saffer, Miklós Telek. Queue length analysis of a Markov-modulated vacation queue with dependent arrival and service processes and exhaustive service policy. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1365-1381. doi: 10.3934/jimo.2016077
[7]

Ke Sun, Jinting Wang, Zhe George Zhang. Strategic joining in a single-server retrial queue with batch service. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020120
[8]

Zsolt Saffer, Wuyi Yue. A dual tandem queueing system with GI service time at the first queue. Journal of Industrial & Management Optimization, 2014, 10 (1) : 167-192. doi: 10.3934/jimo.2014.10.167
[9]

Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019106
[10]

Pikkala Vijaya Laxmi, Singuluri Indira, Kanithi Jyothsna. Ant colony optimization for optimum service times in a Bernoulli schedule vacation interruption queue with balking and reneging. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1199-1214. doi: 10.3934/jimo.2016.12.1199
[11]

Shaojun Lan, Yinghui Tang, Miaomiao Yu. System capacity optimization design and optimal threshold $N^{*}$ for a $GEO/G/1$ discrete-time queue with single server vacation and under the control of Min($N, V$)-policy. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1435-1464. doi: 10.3934/jimo.2016.12.1435
[12]

Wenjuan Zhao, Shunfu Jin, Wuyi Yue. A stochastic model and social optimization of a blockchain system based on a general limited batch service queue. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020049
[13]

Dequan Yue, Wuyi Yue. A heterogeneous two-server network system with balking and a Bernoulli vacation schedule. Journal of Industrial & Management Optimization, 2010, 6 (3) : 501-516. doi: 10.3934/jimo.2010.6.501
[14]

Gopinath Panda, Veena Goswami, Abhijit Datta Banik, Dibyajyoti Guha. Equilibrium balking strategies in renewal input queue with Bernoulli-schedule controlled vacation and vacation interruption. Journal of Industrial & Management Optimization, 2016, 12 (3) : 851-878. doi: 10.3934/jimo.2016.12.851
[15]

Yoshiaki Inoue, Tetsuya Takine. The FIFO single-server queue with disasters and multiple Markovian arrival streams. Journal of Industrial & Management Optimization, 2014, 10 (1) : 57-87. doi: 10.3934/jimo.2014.10.57
[16]

Dhanya Shajin, A. N. Dudin, Olga Dudina, A. Krishnamoorthy. A two-priority single server retrial queue with additional items. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019085
[17]

Yi Peng, Jinbiao Wu. Analysis of a batch arrival retrial queue with impatient customers subject to the server disasters. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020067
[18]

Dequan Yue, Wuyi Yue, Zsolt Saffer, Xiaohong Chen. Analysis of an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy. Journal of Industrial & Management Optimization, 2014, 10 (1) : 89-112. doi: 10.3934/jimo.2014.10.89
[19]

Pikkala Vijaya Laxmi, Obsie Mussa Yesuf. Analysis of a finite buffer general input queue with Markovian service process and accessible and non-accessible batch service. Journal of Industrial & Management Optimization, 2010, 6 (4) : 929-944. doi: 10.3934/jimo.2010.6.929
[20]

Michiel De Muynck, Herwig Bruneel, Sabine Wittevrongel. Analysis of a discrete-time queue with general service demands and phase-type service capacities. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1901-1926. doi: 10.3934/jimo.2017024

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (23)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences