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

On the multi-server machine interference with modified Bernoulli vacation

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.
Citation: Tzu-Hsin Liu, Jau-Chuan Ke. On the multi-server machine interference with modified Bernoulli vacation. Journal of Industrial and 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), 65-82.

[2]

E. K. P. Chong and S. H. Zak, An Introduction of Optimization, 2nd edition, John Wiley and Sons, New Jersey, 2001. doi: 10.1002/9781118033340.

[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), 71-85. doi: 10.1016/j.mcm.2005.04.003.

[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), 936-949. doi: 10.1016/j.amc.2009.06.015.

[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), 337-349. doi: 10.1016/S0096-3003(03)00138-3.

[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), 673-688. doi: 10.1016/j.mcm.2011.03.010.

[7]

B. T. Doshi, Queueing systems with vacations - a survey, Queueing Systems, 1 (1986), 29-66. doi: 10.1007/BF01149327.

[8]

S. M. Gupta, Machine interference problem with warm spares, server vacations and exhaustive service, Performance Evaluation, 29 (1997), 195-211. doi: 10.1016/S0166-5316(96)00046-6.

[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), 100-114.

[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), 35-54. doi: 10.1051/ro/2009004.

[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), 555-582. doi: 10.1142/S021759590500073X.

[12]

J. C. Ke and K. H. Wang, Vacation policies for machine repair problem with two type spares, Applied Mathematical Modelling, 31 (2007), 880-894. doi: 10.1016/j.apm.2006.02.009.

[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), 2196-2208. doi: 10.1016/j.apm.2010.11.019.

[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), 3-8.

[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), 59-71. doi: 10.1016/S0096-3003(02)00469-1.

[16]

R. Oliva, Tradeoffs in responses to work pressure in the service industry, California Management Review, 30 (2001), 26-43. doi: 10.1109/EMR.2002.1022405.

[17]

B. Raphael and I. F. C. Smith, A direct stochastic algorithm for global search, Journal of Applied Mathematics and Computation, 146 (2003), 729-758. doi: 10.1016/S0096-3003(02)00629-X.

[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), 145-160.

[19]

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

[20]

N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, Springer-Verlag, New York, 2006.

[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), 449-458. doi: 10.1016/j.cam.2009.07.043.

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), 65-82.

[2]

E. K. P. Chong and S. H. Zak, An Introduction of Optimization, 2nd edition, John Wiley and Sons, New Jersey, 2001. doi: 10.1002/9781118033340.

[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), 71-85. doi: 10.1016/j.mcm.2005.04.003.

[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), 936-949. doi: 10.1016/j.amc.2009.06.015.

[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), 337-349. doi: 10.1016/S0096-3003(03)00138-3.

[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), 673-688. doi: 10.1016/j.mcm.2011.03.010.

[7]

B. T. Doshi, Queueing systems with vacations - a survey, Queueing Systems, 1 (1986), 29-66. doi: 10.1007/BF01149327.

[8]

S. M. Gupta, Machine interference problem with warm spares, server vacations and exhaustive service, Performance Evaluation, 29 (1997), 195-211. doi: 10.1016/S0166-5316(96)00046-6.

[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), 100-114.

[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), 35-54. doi: 10.1051/ro/2009004.

[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), 555-582. doi: 10.1142/S021759590500073X.

[12]

J. C. Ke and K. H. Wang, Vacation policies for machine repair problem with two type spares, Applied Mathematical Modelling, 31 (2007), 880-894. doi: 10.1016/j.apm.2006.02.009.

[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), 2196-2208. doi: 10.1016/j.apm.2010.11.019.

[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), 3-8.

[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), 59-71. doi: 10.1016/S0096-3003(02)00469-1.

[16]

R. Oliva, Tradeoffs in responses to work pressure in the service industry, California Management Review, 30 (2001), 26-43. doi: 10.1109/EMR.2002.1022405.

[17]

B. Raphael and I. F. C. Smith, A direct stochastic algorithm for global search, Journal of Applied Mathematics and Computation, 146 (2003), 729-758. doi: 10.1016/S0096-3003(02)00629-X.

[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), 145-160.

[19]

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

[20]

N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, Springer-Verlag, New York, 2006.

[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), 449-458. doi: 10.1016/j.cam.2009.07.043.

[1]

Cheng-Dar Liou. Note on "Cost analysis of the M/M/R machine repair problem with second optional repair: Newton-Quasi method". Journal of Industrial and Management Optimization, 2012, 8 (3) : 727-732. doi: 10.3934/jimo.2012.8.727

[2]

Kuo-Hsiung Wang, Chuen-Wen Liao, Tseng-Chang Yen. Cost analysis of the M/M/R machine repair problem with second optional repair: Newton-Quasi method. Journal of Industrial and Management Optimization, 2010, 6 (1) : 197-207. doi: 10.3934/jimo.2010.6.197

[3]

Honglan Zhu, Qin Ni, Meilan Zeng. A quasi-Newton trust region method based on a new fractional model. Numerical Algebra, Control and Optimization, 2015, 5 (3) : 237-249. doi: 10.3934/naco.2015.5.237

[4]

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

[5]

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 and Management Optimization, 2016, 12 (3) : 851-878. doi: 10.3934/jimo.2016.12.851

[6]

Basim A. Hassan. A new type of quasi-newton updating formulas based on the new quasi-newton equation. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 227-235. doi: 10.3934/naco.2019049

[7]

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

[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 and Management Optimization, 2016, 12 (4) : 1199-1214. doi: 10.3934/jimo.2016.12.1199

[9]

Tzu-Hsin Liu, Jau-Chuan Ke, Ching-Chang Kuo. Machine interference problem involving unsuccessful switchover for a group of repairable servers with vacations. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1411-1422. doi: 10.3934/jimo.2020027

[10]

Cheng-Dar Liou. Optimization analysis of the machine repair problem with multiple vacations and working breakdowns. Journal of Industrial and Management Optimization, 2015, 11 (1) : 83-104. doi: 10.3934/jimo.2015.11.83

[11]

Yuhong Dai, Nobuo Yamashita. Convergence analysis of sparse quasi-Newton updates with positive definite matrix completion for two-dimensional functions. Numerical Algebra, Control and Optimization, 2011, 1 (1) : 61-69. doi: 10.3934/naco.2011.1.61

[12]

Shummin Nakayama, Yasushi Narushima, Hiroshi Yabe. Memoryless quasi-Newton methods based on spectral-scaling Broyden family for unconstrained optimization. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1773-1793. doi: 10.3934/jimo.2018122

[13]

Matthias Gerdts, Stefan Horn, Sven-Joachim Kimmerle. Line search globalization of a semismooth Newton method for operator equations in Hilbert spaces with applications in optimal control. Journal of Industrial and Management Optimization, 2017, 13 (1) : 47-62. doi: 10.3934/jimo.2016003

[14]

Peng Guo, Wenming Cheng, Yi Wang. A general variable neighborhood search for single-machine total tardiness scheduling problem with step-deteriorating jobs. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1071-1090. doi: 10.3934/jimo.2014.10.1071

[15]

Xin Li, Ziguan Cui, Linhui Sun, Guanming Lu, Debnath Narayan. Research on iterative repair algorithm of Hyperchaotic image based on support vector machine. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1199-1218. doi: 10.3934/dcdss.2019083

[16]

Liqun Qi, Zheng yan, Hongxia Yin. Semismooth reformulation and Newton's method for the security region problem of power systems. Journal of Industrial and Management Optimization, 2008, 4 (1) : 143-153. doi: 10.3934/jimo.2008.4.143

[17]

Hanchun Yang, Meimei Zhang, Qin Wang. Global solutions of shock reflection problem for the pressure gradient system. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3387-3428. doi: 10.3934/cpaa.2020150

[18]

Gui-Qiang G. Chen, Qin Wang, Shengguo Zhu. Global solutions of a two-dimensional Riemann problem for the pressure gradient system. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2475-2503. doi: 10.3934/cpaa.2021014

[19]

Mingyong Lai, Xiaojiao Tong. A metaheuristic method for vehicle routing problem based on improved ant colony optimization and Tabu search. Journal of Industrial and Management Optimization, 2012, 8 (2) : 469-484. doi: 10.3934/jimo.2012.8.469

[20]

Petteri Piiroinen, Martin Simon. Probabilistic interpretation of the Calderón problem. Inverse Problems and Imaging, 2017, 11 (3) : 553-575. doi: 10.3934/ipi.2017026

2020 Impact Factor: 1.801

Metrics

  • PDF downloads (82)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]