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 & 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

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