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

Previous Article
Stochastic decomposition in discrete-time queues with generalized vacations and applications
JIMO Home
This Issue
Next Article
Stochastic method for power-aware checkpoint intervals in wireless environments: Theory and application

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

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

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

Abstract
Related Papers

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.
[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.
[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.
[4]	B. T. Doshi, Queueing systems with vacations-a survey,, Queueing Systems, 1 (1986), 29. doi: 10.1007/BF01149327.
[5]	M. Kuczma, "Functional Equations in a Single Variable,", PWN-Polish Scientific Publishers, (1968).
[6]	Y. Levy and U. Yechiali, An M/M/s queue with server's vacations,, In INFOR 14, (1976), 153.
[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.
[8]	H. Takagi, "Analysis of Polling Systems,", MIT Press, (1986).
[9]	H. Takagi, "Queueing Analysis - A Foundation of Performance Evaluation, Vacation and Prority Systems,", North-Holland, (1991).
[10]	N. Tian and G. Zhang, "Vacation Queueing Models: Theory and Applications. Series: International Series in Operations Research & Management Science,", Springer-Verlag, (2006).
[11]	N. Tian and L. Li, The M/M/c queue with PH synchronous vacations,, Journal of Systems Science and Complexity, 13 (2000), 007.
[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.
[13]	R. W. Wolff, Poisson arrivals see times averages,, Operations Research, 30 (1982), 223. doi: 10.1287/opre.30.2.223.
[14]	W. Yue, Y. Takahashi and H. Takagi, "Advances in Queueing Theory and Network Applications,", Springer Science + Business Media, (2010).
[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.
[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.
[17]	R. W. Wolff, "Stochastic Modeling and the Theory of Queues,", Prentice-Hall, (1989).

show all references

References:

[1]	A. Begum and M. Nadarajan, Multiserver markovian queueing system with vacation,, Optimization, 41 (1997), 71. doi: 10.1080/02331939708844326.
[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.
[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.
[4]	B. T. Doshi, Queueing systems with vacations-a survey,, Queueing Systems, 1 (1986), 29. doi: 10.1007/BF01149327.
[5]	M. Kuczma, "Functional Equations in a Single Variable,", PWN-Polish Scientific Publishers, (1968).
[6]	Y. Levy and U. Yechiali, An M/M/s queue with server's vacations,, In INFOR 14, (1976), 153.
[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.
[8]	H. Takagi, "Analysis of Polling Systems,", MIT Press, (1986).
[9]	H. Takagi, "Queueing Analysis - A Foundation of Performance Evaluation, Vacation and Prority Systems,", North-Holland, (1991).
[10]	N. Tian and G. Zhang, "Vacation Queueing Models: Theory and Applications. Series: International Series in Operations Research & Management Science,", Springer-Verlag, (2006).
[11]	N. Tian and L. Li, The M/M/c queue with PH synchronous vacations,, Journal of Systems Science and Complexity, 13 (2000), 007.
[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.
[13]	R. W. Wolff, Poisson arrivals see times averages,, Operations Research, 30 (1982), 223. doi: 10.1287/opre.30.2.223.
[14]	W. Yue, Y. Takahashi and H. Takagi, "Advances in Queueing Theory and Network Applications,", Springer Science + Business Media, (2010).
[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.
[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.
[17]	R. W. Wolff, "Stochastic Modeling and the Theory of Queues,", Prentice-Hall, (1989).

[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]	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
[8]	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
[9]	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
[10]	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
[11]	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
[12]	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
[13]	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
[14]	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
[15]	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
[16]	Ruiling Tian, Dequan Yue, Wuyi Yue. Optimal balking strategies in an M/G/1 queueing system with a removable server under N-policy. Journal of Industrial & Management Optimization, 2015, 11 (3) : 715-731. doi: 10.3934/jimo.2015.11.715
[17]	Dequan Yue, Wuyi Yue. Block-partitioning matrix solution of M/M/R/N queueing system with balking, reneging and server breakdowns. Journal of Industrial & Management Optimization, 2009, 5 (3) : 417-430. doi: 10.3934/jimo.2009.5.417
[18]	Zsolt Saffer, Miklós Telek. Analysis of BMAP vacation queue and its application to IEEE 802.16e sleep mode. Journal of Industrial & Management Optimization, 2010, 6 (3) : 661-690. doi: 10.3934/jimo.2010.6.661
[19]	Veena Goswami, Pikkala Vijaya Laxmi. Analysis of renewal input bulk arrival queue with single working vacation and partial batch rejection. Journal of Industrial & Management Optimization, 2010, 6 (4) : 911-927. doi: 10.3934/jimo.2010.6.911
[20]	Ahmed M. K. Tarabia. Transient and steady state analysis of an M/M/1 queue with balking, catastrophes, server failures and repairs. Journal of Industrial & Management Optimization, 2011, 7 (4) : 811-823. doi: 10.3934/jimo.2011.7.811

2017 Impact Factor: 0.994

American Institute of Mathematical Sciences