On the multi-server machine interference with modified Bernoulli vacation

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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

Abstract
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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 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.

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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.

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