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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

 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.
Citation: Zsolt Saffer, Wuyi Yue. M/M/c multiple synchronous vacation model with gated discipline. Journal of Industrial and 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. [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. [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. [4] B. T. Doshi, Queueing systems with vacations-a survey, Queueing Systems, 1 (1986), 29-66. doi: 10.1007/BF01149327. [5] M. Kuczma, "Functional Equations in a Single Variable," PWN-Polish Scientific Publishers, Warsaw, 1968. [6] Y. Levy and U. Yechiali, An M/M/s queue with server's vacations, In INFOR 14, (1976), 153-163. [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. [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, New York, 1991. [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. [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. [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. [13] R. W. Wolff, Poisson arrivals see times averages, Operations Research, 30 (1982), 223-231. 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, New York, 2010. [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. [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. [17] R. W. Wolff, "Stochastic Modeling and the Theory of Queues," Prentice-Hall, Englewood Cliffs, NJ, 1989.

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##### References:
 [1] A. Begum and M. Nadarajan, Multiserver markovian queueing system with vacation, Optimization, 41 (1997), 71-78. doi: 10.1080/02331939708844326. [2] S. C. Borst and O. J. Boxma, Polling models with and without switch over times, Operations Research, 45 (1997), 536-543. 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), 392-406. doi: 10.1016/S0377-2217(97)00253-1. [4] B. T. Doshi, Queueing systems with vacations-a survey, Queueing Systems, 1 (1986), 29-66. doi: 10.1007/BF01149327. [5] M. Kuczma, "Functional Equations in a Single Variable," PWN-Polish Scientific Publishers, Warsaw, 1968. [6] Y. Levy and U. Yechiali, An M/M/s queue with server's vacations, In INFOR 14, (1976), 153-163. [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. [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, New York, 1991. [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. [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. [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. [13] R. W. Wolff, Poisson arrivals see times averages, Operations Research, 30 (1982), 223-231. 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, New York, 2010. [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. [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. [17] R. W. Wolff, "Stochastic Modeling and the Theory of Queues," Prentice-Hall, Englewood Cliffs, NJ, 1989.
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