American Institute of Mathematical Sciences

Advanced
October  2014, 10(4): 1191-1208. doi: 10.3934/jimo.2014.10.1191

On the multi-server machine interference with modified Bernoulli vacation

Tzu-Hsin Liu 1, and  Jau-Chuan Ke 1,

1. 

Department of Applied Statistics, National Taichung University of Science and Technology, Taichung, 404, Taiwan, Taiwan

Received  April 2013 Revised  November 2013 Published  February 2014

We study the multi-server machine interference problem with repair pressure coefficient and a modified Bernoulli vacation. The repair rate depends on the number of failed machines waiting in the system. In congestion, the server may increase the repair rate with pressure coefficient $\theta$ to reduce the queue length. At each repair completion of a server, the server may go for a vacation of random length with probability $p$ or may continue to repair the next failed machine, if any, with probability $1-p$. The entire system is modeled as a finite-state Markov chain and its steady state distribution is obtained by a recursive matrix approach. The major performance measures are evaluated based on this distribution. Under a cost structure, we propose to use the Quasi-Newton method and probabilistic global search Lausanne method to search for the global optimal system parameters. Numerical examples are presented to demonstrate the application of our approach.
Keywords: probabilistic global search Lausanne., repair pressure, Quasi-Newton method, Bernoulli vacation schedule, machine interference problem.
Mathematics Subject Classification: Primary: 90B22; Secondary: 60K2.
Citation: 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
References:
[1]

F. Benson and D. R. Cox, The productivity of machine requiring attention at random intervals,, Journal of the Royal Statistical Society B, 13 (1951), 65.   Google Scholar

[2]

E. K. P. Chong and S. H. Zak, An Introduction of Optimization,, 2nd edition, (2001).  doi: 10.1002/9781118033340.  Google Scholar

[3]

G. Choudhury, A two-stage batch arrival queueing system with a modified Bernoulli schedule vacation under N-policy,, Mathematical and Computer Modeling, 42 (2005), 71.  doi: 10.1016/j.mcm.2005.04.003.  Google Scholar

[4]

G. Choudhury and K. Deka, An $M^X/G/1$ unreliable retrial queue with two phases of service and Bernoulli admission mechanism,, Applied Mathematics and Comutation, 215 (2009), 936.  doi: 10.1016/j.amc.2009.06.015.  Google Scholar

[5]

G. Choudhury and K. C. Madan, A two phase batch arrival queueing system with a vacation time under Bernoulli schedule,, Applied Mathematics and Computation, 149 (2004), 337.  doi: 10.1016/S0096-3003(03)00138-3.  Google Scholar

[6]

G. Choudhury and L. Tadj, The optimal control of an $M^X/G/1$ unreliable server queue with two phases of service and Bernoulli vacation schedule,, Mathematical and Computer Modelling, 54 (2011), 673.  doi: 10.1016/j.mcm.2011.03.010.  Google Scholar

[7]

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

[8]

S. M. Gupta, Machine interference problem with warm spares, server vacations and exhaustive service,, Performance Evaluation, 29 (1997), 195.  doi: 10.1016/S0166-5316(96)00046-6.  Google Scholar

[9]

J. C. Ke, Vacation policies for machine interference problem with an unreliable server and state-dependent service rate,, Journal of the Chineses Institute of Industrial Engineers, 23 (2006), 100.   Google Scholar

[10]

J. C. Ke, S. L. Lee and C. H. Liou, Machine repair problem in production systems with spares and server vacations,, RAIRO-Operations Research, 43 (2009), 35.  doi: 10.1051/ro/2009004.  Google Scholar

[11]

J. C. Ke and C. H. Lin, A Markov repairable system involving an imperfect service station with multiple vacations,, Asia-Pacific Journal of Operational Research, 22 (2005), 555.  doi: 10.1142/S021759590500073X.  Google Scholar

[12]

J. C. Ke and K. H. Wang, Vacation policies for machine repair problem with two type spares,, Applied Mathematical Modelling, 31 (2007), 880.  doi: 10.1016/j.apm.2006.02.009.  Google Scholar

[13]

J. C. Ke, C. H. Wu and W. L. Pearn, Algorithmic analysis of the multi-server system with a modified Bernoulli vacation schedule,, Applied Mathematical Modelling, 35 (2011), 2196.  doi: 10.1016/j.apm.2010.11.019.  Google Scholar

[14]

J. C. Ke, C. H. Wu and Z. Zhang, Recent developments in vacation queueing models: A short survey,, International Journal of Operations Research, 7 (2010), 3.   Google Scholar

[15]

K. C. Madan, W. Abu-Dayyeh and F. Taiyyan, A two server queue with Bernoulli schedules and a single vacation policy,, Applied Mathematics and Computation, 145 (2003), 59.  doi: 10.1016/S0096-3003(02)00469-1.  Google Scholar

[16]

R. Oliva, Tradeoffs in responses to work pressure in the service industry,, California Management Review, 30 (2001), 26.  doi: 10.1109/EMR.2002.1022405.  Google Scholar

[17]

B. Raphael and I. F. C. Smith, A direct stochastic algorithm for global search,, Journal of Applied Mathematics and Computation, 146 (2003), 729.  doi: 10.1016/S0096-3003(02)00629-X.  Google Scholar

[18]

L. Tadj, G. Choudhury and C. Tadj, A quorum queueing system with a random setup time under N-policy and with Bernoulli vacation schedule,, Quality Technology and Quantitative Management, 3 (2006), 145.   Google Scholar

[19]

H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation, Vacation and Priority Systems. part I, vol. 1,, North-Holland, (1991).   Google Scholar

[20]

N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications,, Springer-Verlag, (2006).   Google Scholar

[21]

K. H. Wang, W. L. Chen and D. Y. Yang, Optimal management of the machine repair problem with working vacation: Newton's method,, Journal of Computational and Applied Mathematics, 233 (2009), 449.  doi: 10.1016/j.cam.2009.07.043.  Google Scholar

show all references

References:
[1]

F. Benson and D. R. Cox, The productivity of machine requiring attention at random intervals,, Journal of the Royal Statistical Society B, 13 (1951), 65.   Google Scholar

[2]

E. K. P. Chong and S. H. Zak, An Introduction of Optimization,, 2nd edition, (2001).  doi: 10.1002/9781118033340.  Google Scholar

[3]

G. Choudhury, A two-stage batch arrival queueing system with a modified Bernoulli schedule vacation under N-policy,, Mathematical and Computer Modeling, 42 (2005), 71.  doi: 10.1016/j.mcm.2005.04.003.  Google Scholar

[4]

G. Choudhury and K. Deka, An $M^X/G/1$ unreliable retrial queue with two phases of service and Bernoulli admission mechanism,, Applied Mathematics and Comutation, 215 (2009), 936.  doi: 10.1016/j.amc.2009.06.015.  Google Scholar

[5]

G. Choudhury and K. C. Madan, A two phase batch arrival queueing system with a vacation time under Bernoulli schedule,, Applied Mathematics and Computation, 149 (2004), 337.  doi: 10.1016/S0096-3003(03)00138-3.  Google Scholar

[6]

G. Choudhury and L. Tadj, The optimal control of an $M^X/G/1$ unreliable server queue with two phases of service and Bernoulli vacation schedule,, Mathematical and Computer Modelling, 54 (2011), 673.  doi: 10.1016/j.mcm.2011.03.010.  Google Scholar

[7]

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

[8]

S. M. Gupta, Machine interference problem with warm spares, server vacations and exhaustive service,, Performance Evaluation, 29 (1997), 195.  doi: 10.1016/S0166-5316(96)00046-6.  Google Scholar

[9]

J. C. Ke, Vacation policies for machine interference problem with an unreliable server and state-dependent service rate,, Journal of the Chineses Institute of Industrial Engineers, 23 (2006), 100.   Google Scholar

[10]

J. C. Ke, S. L. Lee and C. H. Liou, Machine repair problem in production systems with spares and server vacations,, RAIRO-Operations Research, 43 (2009), 35.  doi: 10.1051/ro/2009004.  Google Scholar

[11]

J. C. Ke and C. H. Lin, A Markov repairable system involving an imperfect service station with multiple vacations,, Asia-Pacific Journal of Operational Research, 22 (2005), 555.  doi: 10.1142/S021759590500073X.  Google Scholar

[12]

J. C. Ke and K. H. Wang, Vacation policies for machine repair problem with two type spares,, Applied Mathematical Modelling, 31 (2007), 880.  doi: 10.1016/j.apm.2006.02.009.  Google Scholar

[13]

J. C. Ke, C. H. Wu and W. L. Pearn, Algorithmic analysis of the multi-server system with a modified Bernoulli vacation schedule,, Applied Mathematical Modelling, 35 (2011), 2196.  doi: 10.1016/j.apm.2010.11.019.  Google Scholar

[14]

J. C. Ke, C. H. Wu and Z. Zhang, Recent developments in vacation queueing models: A short survey,, International Journal of Operations Research, 7 (2010), 3.   Google Scholar

[15]

K. C. Madan, W. Abu-Dayyeh and F. Taiyyan, A two server queue with Bernoulli schedules and a single vacation policy,, Applied Mathematics and Computation, 145 (2003), 59.  doi: 10.1016/S0096-3003(02)00469-1.  Google Scholar

[16]

R. Oliva, Tradeoffs in responses to work pressure in the service industry,, California Management Review, 30 (2001), 26.  doi: 10.1109/EMR.2002.1022405.  Google Scholar

[17]

B. Raphael and I. F. C. Smith, A direct stochastic algorithm for global search,, Journal of Applied Mathematics and Computation, 146 (2003), 729.  doi: 10.1016/S0096-3003(02)00629-X.  Google Scholar

[18]

L. Tadj, G. Choudhury and C. Tadj, A quorum queueing system with a random setup time under N-policy and with Bernoulli vacation schedule,, Quality Technology and Quantitative Management, 3 (2006), 145.   Google Scholar

[19]

H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation, Vacation and Priority Systems. part I, vol. 1,, North-Holland, (1991).   Google Scholar

[20]

N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications,, Springer-Verlag, (2006).   Google Scholar

[21]

K. H. Wang, W. L. Chen and D. Y. Yang, Optimal management of the machine repair problem with working vacation: Newton's method,, Journal of Computational and Applied Mathematics, 233 (2009), 449.  doi: 10.1016/j.cam.2009.07.043.  Google Scholar

[1]

Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149
[2]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056
[3]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[4]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094
[5]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212
[6]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021
[7]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397
[8]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617
[9]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463
[10]

Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81
[11]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027
[12]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327
[13]

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
[14]

Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151
[15]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
[16]

Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065
[17]

Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006
[18]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017
[19]

Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018
[20]

Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences