# American Institute of Mathematical Sciences

2014, 10(3): 839-857. doi: 10.3934/jimo.2014.10.839

## Performance analysis of renewal input $(a,c,b)$ policy queue with multiple working vacations and change over times

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

Received  June 2012 Revised  June 2013 Published  November 2013

This paper analyzes a renewal input multiple working vacations queue with change over times under $(a,c,b)$ policy. The service and vacation times are exponentially distributed. The server begins service if there are at least $c$ units in the queue and the service takes place in batches with a minimum of size $a$ and a maximum of size $b~ (a\leq c \leq b)$. The steady state queue length distributions at arbitrary and pre-arrival epochs are obtained along with some special cases of the model. Performance measures and optimal cost policy is presented with numerical experiences for some particular values of the parameters. The genetic algorithm is employed to search the optimal values of some important parameters of the system.
Citation: Pikkala Vijaya Laxmi, Seleshi Demie. Performance analysis of renewal input $(a,c,b)$ policy queue with multiple working vacations and change over times. Journal of Industrial & Management Optimization, 2014, 10 (3) : 839-857. doi: 10.3934/jimo.2014.10.839
##### References:
 [1] Y. Baba, Analysis of a GI/M/1 queue with multiple working vacations,, Operations Research Letters, 33 (2005), 201. doi: 10.1016/j.orl.2004.05.006. [2] C. Baburaj, A discrete time $(a, c, d)$ policy bulk service queue,, International Journal of Information and Management Sciences, 21 (2010), 469. [3] C. Baburaj and T. M. Surendranath, An M/M/1 bulk service queue under the policy $(a, c, d)$,, International Journal of Agricultural and Statistical Sciences, 1 (2005), 27. [4] A. Banik, U. C. Gupta and S. Pathak, On the GI/M/1/N queue with multiple working vacations - Analytic analysis and computation,, Applied Mathematical Modelling, 31 (2007), 1701. doi: 10.1016/j.apm.2006.05.010. [5] G. D. Fatta, F. Hoffmann, G. L. Re and A. Urso, A genetic algorithm for the design of a fuzzy controller for active queue management,, IEEE Transactions on Systems, 33 (2003), 313. [6] B. T. Doshi, Queueing systems with vacations: A survey,, Queueing Systems Theory Appl., 1 (1986), 29. doi: 10.1007/BF01149327. [7] V. Goswami and G. B. Mund, Analysis of discrete-time batch service renewal input queue with multiple working vacations,, Computers $&$ Industrial Engineering, 61 (2011), 629. doi: 10.1016/j.cie.2011.04.018. [8] R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms,, 2nd edition, (2004). [9] J. H. Holland, Adaptation in Natural and Artificial Systems. An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence,, The University of Michigan Press, (1975). [10] H.-I. Huang, P.-C. Hsu and J.-C. Ke, Controlling arrival and service of a two-removable-server system using genetic algorithm,, Expert Systems with Applications, 38 (2011), 10054. doi: 10.1016/j.eswa.2011.02.011. [11] M. Jain and P. Singh, State dependent bulk service queue with delayed vacations,, JKAU Engineering Sciences, 16 (2005), 3. doi: 10.4197/Eng.16-1.1. [12] J.-C. Ke, C.-H. Wu and Z. G. Zhang, Recent developments in vacation queueing Models: A short survey,, International Journal of Operations Research, 7 (2010), 3. [13] P. V. Laxmi, V. Goswami and D. Seleshi, Renewal input (a,c,b) policy queue with multiple vacations and change over times,, International Journal of Mathematics in Operational Research, 5 (2013), 466. [14] H. W. Lee, D. I. Jung and S. S. Lee, Decompositions of Batch Service Queue with Server Vacations: Markovian Case,, Research Report, (1994). [15] J.-H. Li, N.-S. Tian and W.-Y. Liu, Discrete time GI/Geo/1 queue with multiple working vacations,, Queueing Systems, 56 (2007), 53. doi: 10.1007/s11134-007-9030-0. [16] C.-H. Lin and J.-C. Ke, Genetic algorithm for optimal thresholds of an infinite capacity multi-server system with triadic policy,, Expert Systems with Applications, 37 (2010), 4276. [17] C.-H. Lin and J.-C. Ke, Optimization analysis for an infinite capacity queueing system with multiple queue-dependent servers: Genetic algorithm,, International Journal of Computer Mathematics, 88 (2011), 1430. doi: 10.1080/00207160.2010.509791. [18] S. S. Rao, Engineering Optimization: Theory and Practice,, 4th edition, (2009). [19] L. D. Servi and S. G. Finn, M/M/1 queues with working vacations (M/M/1/WV),, Performance Evaluation, 50 (2002), 41. [20] L. Tadj and C. Abid, Optimal management policy for a single and bulk service queue,, International Journal of Advanced Operations Management, 3 (2011), 175. [21] L. Tadj and G. Choudhury, Optimal design and control of queues,, Top, 13 (2005), 359. doi: 10.1007/BF02579061. [22] L. Tadj, G. Choudhury and C. Tadj, A bulk quorum queueing system with a random setup time under $N$- policy and with Bernoulli vacation schedule,, Stochastics: An International Journal of Probability and Stochastic Processes, 78 (2006), 1. doi: 10.1080/17442500500397574. [23] H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation. Vol. 1. Vacation and Priority Systems. Part 1,, North Holland, (1991). [24] N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications,, International Series in Operations Research & Management Science, (2006).

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##### References:
 [1] Y. Baba, Analysis of a GI/M/1 queue with multiple working vacations,, Operations Research Letters, 33 (2005), 201. doi: 10.1016/j.orl.2004.05.006. [2] C. Baburaj, A discrete time $(a, c, d)$ policy bulk service queue,, International Journal of Information and Management Sciences, 21 (2010), 469. [3] C. Baburaj and T. M. Surendranath, An M/M/1 bulk service queue under the policy $(a, c, d)$,, International Journal of Agricultural and Statistical Sciences, 1 (2005), 27. [4] A. Banik, U. C. Gupta and S. Pathak, On the GI/M/1/N queue with multiple working vacations - Analytic analysis and computation,, Applied Mathematical Modelling, 31 (2007), 1701. doi: 10.1016/j.apm.2006.05.010. [5] G. D. Fatta, F. Hoffmann, G. L. Re and A. Urso, A genetic algorithm for the design of a fuzzy controller for active queue management,, IEEE Transactions on Systems, 33 (2003), 313. [6] B. T. Doshi, Queueing systems with vacations: A survey,, Queueing Systems Theory Appl., 1 (1986), 29. doi: 10.1007/BF01149327. [7] V. Goswami and G. B. Mund, Analysis of discrete-time batch service renewal input queue with multiple working vacations,, Computers $&$ Industrial Engineering, 61 (2011), 629. doi: 10.1016/j.cie.2011.04.018. [8] R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms,, 2nd edition, (2004). [9] J. H. Holland, Adaptation in Natural and Artificial Systems. An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence,, The University of Michigan Press, (1975). [10] H.-I. Huang, P.-C. Hsu and J.-C. Ke, Controlling arrival and service of a two-removable-server system using genetic algorithm,, Expert Systems with Applications, 38 (2011), 10054. doi: 10.1016/j.eswa.2011.02.011. [11] M. Jain and P. Singh, State dependent bulk service queue with delayed vacations,, JKAU Engineering Sciences, 16 (2005), 3. doi: 10.4197/Eng.16-1.1. [12] J.-C. Ke, C.-H. Wu and Z. G. Zhang, Recent developments in vacation queueing Models: A short survey,, International Journal of Operations Research, 7 (2010), 3. [13] P. V. Laxmi, V. Goswami and D. Seleshi, Renewal input (a,c,b) policy queue with multiple vacations and change over times,, International Journal of Mathematics in Operational Research, 5 (2013), 466. [14] H. W. Lee, D. I. Jung and S. S. Lee, Decompositions of Batch Service Queue with Server Vacations: Markovian Case,, Research Report, (1994). [15] J.-H. Li, N.-S. Tian and W.-Y. Liu, Discrete time GI/Geo/1 queue with multiple working vacations,, Queueing Systems, 56 (2007), 53. doi: 10.1007/s11134-007-9030-0. [16] C.-H. Lin and J.-C. Ke, Genetic algorithm for optimal thresholds of an infinite capacity multi-server system with triadic policy,, Expert Systems with Applications, 37 (2010), 4276. [17] C.-H. Lin and J.-C. Ke, Optimization analysis for an infinite capacity queueing system with multiple queue-dependent servers: Genetic algorithm,, International Journal of Computer Mathematics, 88 (2011), 1430. doi: 10.1080/00207160.2010.509791. [18] S. S. Rao, Engineering Optimization: Theory and Practice,, 4th edition, (2009). [19] L. D. Servi and S. G. Finn, M/M/1 queues with working vacations (M/M/1/WV),, Performance Evaluation, 50 (2002), 41. [20] L. Tadj and C. Abid, Optimal management policy for a single and bulk service queue,, International Journal of Advanced Operations Management, 3 (2011), 175. [21] L. Tadj and G. Choudhury, Optimal design and control of queues,, Top, 13 (2005), 359. doi: 10.1007/BF02579061. [22] L. Tadj, G. Choudhury and C. Tadj, A bulk quorum queueing system with a random setup time under $N$- policy and with Bernoulli vacation schedule,, Stochastics: An International Journal of Probability and Stochastic Processes, 78 (2006), 1. doi: 10.1080/17442500500397574. [23] H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation. Vol. 1. Vacation and Priority Systems. Part 1,, North Holland, (1991). [24] N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications,, International Series in Operations Research & Management Science, (2006).
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