• 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

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 & 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]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311

[2]

Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228

[3]

W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349

[4]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024

[5]

Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225

[6]

Johannes Kellendonk, Lorenzo Sadun. Conjugacies of model sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3805-3830. doi: 10.3934/dcds.2017161

[7]

Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, Olivier Saut. A viscoelastic model for avascular tumor growth. Conference Publications, 2009, 2009 (Special) : 101-108. doi: 10.3934/proc.2009.2009.101

[8]

Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427

[9]

Martin Bohner, Sabrina Streipert. Optimal harvesting policy for the Beverton--Holt model. Mathematical Biosciences & Engineering, 2016, 13 (4) : 673-695. doi: 10.3934/mbe.2016014

[10]

Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks. Networks & Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53

[11]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973

[12]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089

[13]

Alba Málaga Sabogal, Serge Troubetzkoy. Minimality of the Ehrenfest wind-tree model. Journal of Modern Dynamics, 2016, 10: 209-228. doi: 10.3934/jmd.2016.10.209

[14]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183

[15]

Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1

[16]

Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623

[17]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367

[18]

Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021035

[19]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301

[20]

Michel Chipot, Mingmin Zhang. On some model problem for the propagation of interacting species in a special environment. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020401

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (31)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]