Ant colony optimization for optimum service times in a Bernoulli schedule vacation interruption queue with balking and reneging

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October 2016, 12(4): 1199-1214. doi: 10.3934/jimo.2016.12.1199

Ant colony optimization for optimum service times in a Bernoulli schedule vacation interruption queue with balking and reneging

Pikkala Vijaya Laxmi ^1, , Singuluri Indira ^1, and Kanithi Jyothsna ^1,

1.	Department of Applied Mathematics, Andhra University, Visakhapatnam - 530 003, India, India, India

Received October 2014 Revised April 2015 Published January 2016

Abstract
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This paper analyzes a finite buffer multiple working vacations queue with balking, reneging and Bernoulli schedule vacation interruption. Arriving customers decide either to join the system or to balk. After joining the queue the customers may renege based on their desire for service or their unwillingness for waiting. At a service completion instant during working vacations, the server can decide either to continue the vacation with probability $q$ or interrupt the vacation and resume regular service period with probability $1-q$. The inter-arrival times of customers are assumed to be arbitrarily distributed. Service times during a regular service period, during a working vacation period and vacation times are assumed to be exponentially distributed. Using recursive technique, the steady state system length distributions at various epochs are obtained. Some performance measures of the model and cost analysis using ant colony optimization are presented. Finally, numerical results showing the effect of model parameters on key performance measures are presented.

Keywords: Bernoulli schedule vacation interruption, balking, ant colony optimization., reneging, multiple working vacations.

Mathematics Subject Classification: Primary: 60K25; Secondary: 90B22, 68M2.

Citation: 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

References:

[1]	M. O. Abou-El-Ata and A. I. Shawky, The single-server Markovian overflow queue with balking, reneging and an additional server for longer queues, Microelectronics and Reliability, 32 (1992), 1389-1394. doi: 10.1016/0026-2714(92)90008-9.
[2]	R. O. Al-Seedy, A. A. El-Sherbiny, S. A. El-Shehawy and S. I. Ammar, Transient solution of the M/M/c queue with balking and reneging, Computer and Mathematics with Applications, 57 (2009), 1280-1285. doi: 10.1016/j.camwa.2009.01.017.
[3]	S. I. Ammar, A. A. El-Sherbiny, S. A. El-Shehawy and R. O. Al-Seedy, A matrix approach for the transient solution of an M/M/1/N queue with discouraged arrivals and reneging, International Journal of Computer Mathematics, 89 (2012), 482-491. doi: 10.1080/00207160.2011.637553.
[4]	C. J. Ancker Jr. and A. V. Gafarian, Some queueing problems with balking and reneging: I, Operations Research, 11 (1963), 88-100. doi: 10.1287/opre.11.1.88.
[5]	C. J. Ancker Jr. and A. V. Gafarian, Some queueing problems with balking and reneging: II, Operations Research, 11 (1963), 928-937. doi: 10.1287/opre.11.6.928.
[6]	A. D. Banik, U. C. Gupta and S. S. Pathak, On the GI/M/1/N queue with multiple working vacations-analytical analysis and computation, Applied Mathematical Modelling, 31 (2007), 1701-1710.
[7]	A. Colorni, M. Dorigo and V. Maniezzo, Distributed optimization by ant colonies, in Proceedings of ECAL'91 European Conference on Artificial Life, Elsevier Publishing, 1991, 134-142.
[8]	M. Dorigo, V. Maniezzo and A. Colorni, The ant system: Optimization by a colony of cooperating agents, IEEE Transactions on Systems, Man, and Cybernetics-Part B, 26 (1996), 29-41. doi: 10.1109/3477.484436.
[9]	V. Goswami, Study of customers' impatience in a GI/M/1/N queue with working vacations, International Journal of Management Science and Engineering Management, 2015.
[10]	F. A. Haight, Queueing with balking, Biometrika, 44 (1957), 360-369. doi: 10.1093/biomet/44.3-4.360.
[11]	F. A. Haight, Queueing with reneging, Metrika, 2 (1959), 186-197. doi: 10.1007/BF02613734.
[12]	J. Li and N. Tian, The M/M/1 queue with working vacations and vacation interruption, Journal of Systems Science and Systems Engineering, 16 (2007), 121-127. doi: 10.1007/s11518-006-5030-6.
[13]	J. Li, N. Tian and Z. Ma, Performance analysis of GI/M/1 queue with working vacations and vacation interruption, Applied Mathematical Modelling, 32 (2008), 2715-2730. doi: 10.1016/j.apm.2007.09.017.
[14]	J. Li, Z. Wang and Z. Liu, The GI/M/1 queue with Bernoulli-schedule-controlled vacation and vacation interruption, Applied Mathematical Modelling, 37 (2013), 3724-3735. doi: 10.1016/j.apm.2012.07.045.
[15]	L. D. Servi and S. G. Finn, M/M/1 queues with working vacations (M/M/1/WV), Performance Evaluation, 50 (2002), 41-52. doi: 10.1016/S0166-5316(02)00057-3.
[16]	P. Vijaya Laxmi, V. Goswami and K. Jyothsna, Optimization of balking and reneging queue with vacation interruption under $N$-Policy, Journal of Optimization, 2013 (2013), Article ID 683708, 9pp. doi: 10.1155/2013/683708.
[17]	P. Vijaya Laxmi and K. Jyothsna, Impatient customer queue with Bernoulli schedule vacation interruption, Computers & Operations Research, 56 (2015), 1-7. doi: 10.1016/j.cor.2014.08.018.
[18]	D. Yue, W. Yue and G. Xu, Analysis of customers' impatience in an M/M/1 queue with working vacations, Journal of Industrial and Management Optimization, 8 (2012), 895-908. doi: 10.3934/jimo.2012.8.895.
[19]	H. Zhang and D. Shi, The M/M/1 queue with Bernoulli-schedule-controlled vacation and vacation interruption, International Journal of Information and Management Sciences, 20 (2009), 579-587.

show all references

References:

[1]	M. O. Abou-El-Ata and A. I. Shawky, The single-server Markovian overflow queue with balking, reneging and an additional server for longer queues, Microelectronics and Reliability, 32 (1992), 1389-1394. doi: 10.1016/0026-2714(92)90008-9.
[2]	R. O. Al-Seedy, A. A. El-Sherbiny, S. A. El-Shehawy and S. I. Ammar, Transient solution of the M/M/c queue with balking and reneging, Computer and Mathematics with Applications, 57 (2009), 1280-1285. doi: 10.1016/j.camwa.2009.01.017.
[3]	S. I. Ammar, A. A. El-Sherbiny, S. A. El-Shehawy and R. O. Al-Seedy, A matrix approach for the transient solution of an M/M/1/N queue with discouraged arrivals and reneging, International Journal of Computer Mathematics, 89 (2012), 482-491. doi: 10.1080/00207160.2011.637553.
[4]	C. J. Ancker Jr. and A. V. Gafarian, Some queueing problems with balking and reneging: I, Operations Research, 11 (1963), 88-100. doi: 10.1287/opre.11.1.88.
[5]	C. J. Ancker Jr. and A. V. Gafarian, Some queueing problems with balking and reneging: II, Operations Research, 11 (1963), 928-937. doi: 10.1287/opre.11.6.928.
[6]	A. D. Banik, U. C. Gupta and S. S. Pathak, On the GI/M/1/N queue with multiple working vacations-analytical analysis and computation, Applied Mathematical Modelling, 31 (2007), 1701-1710.
[7]	A. Colorni, M. Dorigo and V. Maniezzo, Distributed optimization by ant colonies, in Proceedings of ECAL'91 European Conference on Artificial Life, Elsevier Publishing, 1991, 134-142.
[8]	M. Dorigo, V. Maniezzo and A. Colorni, The ant system: Optimization by a colony of cooperating agents, IEEE Transactions on Systems, Man, and Cybernetics-Part B, 26 (1996), 29-41. doi: 10.1109/3477.484436.
[9]	V. Goswami, Study of customers' impatience in a GI/M/1/N queue with working vacations, International Journal of Management Science and Engineering Management, 2015.
[10]	F. A. Haight, Queueing with balking, Biometrika, 44 (1957), 360-369. doi: 10.1093/biomet/44.3-4.360.
[11]	F. A. Haight, Queueing with reneging, Metrika, 2 (1959), 186-197. doi: 10.1007/BF02613734.
[12]	J. Li and N. Tian, The M/M/1 queue with working vacations and vacation interruption, Journal of Systems Science and Systems Engineering, 16 (2007), 121-127. doi: 10.1007/s11518-006-5030-6.
[13]	J. Li, N. Tian and Z. Ma, Performance analysis of GI/M/1 queue with working vacations and vacation interruption, Applied Mathematical Modelling, 32 (2008), 2715-2730. doi: 10.1016/j.apm.2007.09.017.
[14]	J. Li, Z. Wang and Z. Liu, The GI/M/1 queue with Bernoulli-schedule-controlled vacation and vacation interruption, Applied Mathematical Modelling, 37 (2013), 3724-3735. doi: 10.1016/j.apm.2012.07.045.
[15]	L. D. Servi and S. G. Finn, M/M/1 queues with working vacations (M/M/1/WV), Performance Evaluation, 50 (2002), 41-52. doi: 10.1016/S0166-5316(02)00057-3.
[16]	P. Vijaya Laxmi, V. Goswami and K. Jyothsna, Optimization of balking and reneging queue with vacation interruption under $N$-Policy, Journal of Optimization, 2013 (2013), Article ID 683708, 9pp. doi: 10.1155/2013/683708.
[17]	P. Vijaya Laxmi and K. Jyothsna, Impatient customer queue with Bernoulli schedule vacation interruption, Computers & Operations Research, 56 (2015), 1-7. doi: 10.1016/j.cor.2014.08.018.
[18]	D. Yue, W. Yue and G. Xu, Analysis of customers' impatience in an M/M/1 queue with working vacations, Journal of Industrial and Management Optimization, 8 (2012), 895-908. doi: 10.3934/jimo.2012.8.895.
[19]	H. Zhang and D. Shi, The M/M/1 queue with Bernoulli-schedule-controlled vacation and vacation interruption, International Journal of Information and Management Sciences, 20 (2009), 579-587.

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