# American Institute of Mathematical Sciences

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

## Equilibrium balking strategies in renewal input queue with Bernoulli-schedule controlled vacation and vacation interruption

 1 School of Basic Scienes, Indian Institute of Technology, Bhubaneswar-751007, India, India, India 2 School of Computer Application, KIIT University, Bhubaneswar-751024, India

Received  September 2014 Revised  March 2015 Published  September 2015

We consider a single server renewal input queueing system under multiple vacation policy. When the system becomes empty, the server commences a vacation of random length, and either begins an ordinary vacation with probability $q\, (0\le q\le 1)$ or takes a working vacation with probability $1-q$. During a working vacation period, customers can be served at a rate lower than the service rate during a normal busy period. If there are customers in the system at a service completion instant, the working vacation can be interrupted and the server will come back to a normal busy period with probability $p\, (0\le p\le 1)$ or continue the working vacation with probability $1-p$. The server leaves for repeated vacations as soon as the system becomes empty. Upon arrival, customers decide for themselves whether to join or to balk, based on the observation of the system-length and/or state of the server. The equilibrium threshold balking strategies of customers under four cases: fully observable, almost observable, almost unobservable and fully unobservable have been studied using embedded Markov chain approach and linear reward-cost structure. The probability distribution of the system-length at pre-arrival epoch is derived using the roots method and then the system-length at an arbitrary epoch is derived with the help of the Markov renewal theory and semi-Markov processes. Various performance measures such as mean system-length, sojourn times, net benefit are derived. Finally, we present several numerical results to demonstrate the effect of the system parameters on the performance measures.
Citation: 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 & Management Optimization, 2016, 12 (3) : 851-878. doi: 10.3934/jimo.2016.12.851
##### 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] A. D. Banik, U. C. Gupta and S. 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. [3] M. A. A. Boon, R. D. van der Mei and E. M. M. Winands, Applications of polling systems,, Surveys in Operations Research and Management Science, 16 (2011), 67. doi: 10.1016/j.sorms.2011.01.001. [4] M. L. Chaudhry and J. G. C. Templeton, A First Course in Bulk Queues,, Wiley, (1983). [5] M. L. Chaudhry, C. M. Harris and W. G. Marchal, Robustness of rootfinding in single-server queueing models,, ORSA Journal on Computing, 2 (1990), 273. [6] H. Chen, J. Li and N. Tian, The GI/M/1 queue with phase-type working vacations and vacation interruption,, Journal of Applied Mathematics and Computing, 30 (2009), 121. doi: 10.1007/s12190-008-0161-1. [7] J. L. Dorsman, O. J. Boxma and R. D. van der Mei, On two-queue Markovian polling systems with exhaustive service,, Queueing Systems, 78 (2014), 287. doi: 10.1007/s11134-014-9413-y. [8] B. T. Doshi, Queueing systems with vacations-A survey,, Queueing Systems, 1 (1986), 29. doi: 10.1007/BF01149327. [9] A. Economou, A. Gómez-Corral and S. Kanta, Optimal balking strategies in single-server queues with general service and vacation times,, Performance Evaluation, 68 (2011), 967. doi: 10.1016/j.peva.2011.07.001. [10] A. Economou and S. Kanta, Equilibrium balking strategies in the observable single-server queue with breakdowns and repairs,, Operations Research Letters, 36 (2008), 696. doi: 10.1016/j.orl.2008.06.006. [11] N. M. Edelson and D. K. Hilderbrand, Congestion tolls for Poisson queuing processes,, Econometrica: Journal of the Econometric Society, 43 (1975), 81. doi: 10.2307/1913415. [12] V. Goswami and P. V. Laxmi, Analysis of renewal input bulk arrival queue with single working vacation and partial batch rejection,, Journal of Industrial and Management Optimization, 6 (2010), 911. doi: 10.3934/jimo.2010.6.911. [13] P. Guo and P. Zipkin, The effects of the availability of waiting-time information on a balking queue,, European Journal of Operational Research, 198 (2009), 199. doi: 10.1016/j.ejor.2008.07.035. [14] R. Hassin and M. Haviv, To Queue or not to Queue: Equilibrium Behavior in Queueing Systems,, Springer, (2003). doi: 10.1007/978-1-4615-0359-0. [15] J. Ke, C. Wu and Z. G. Zhang, Recent developments in vacation queueing models: A short survey,, International Journal of Operations Research, 7 (2010), 3. [16] J. Keilson and L. D. Servi, Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedules,, Journal of Applied Probability, 23 (1986), 790. doi: 10.2307/3214016. [17] G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling,, ASA-SIAM Series on Statistics and Applied Probability, (1999). doi: 10.1137/1.9780898719734. [18] J. Li and N. Tian, The M/M/1 queue with working vacations and vacation interruptions,, Journal of Systems Science and Systems Engineering, 16 (2007), 121. doi: 10.1007/s11518-006-5030-6. [19] 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. doi: 10.1016/j.apm.2007.09.017. [20] P. Naor, The regulation of queue size by levying tolls,, Econometrica, 37 (1969), 15. doi: 10.2307/1909200. [21] L. D. Servi and S. G. Finn, M/M/1 queues with working vacations (M/M/1/WV),, Performance Evaluation, 50 (2002), 41. doi: 10.1016/S0166-5316(02)00057-3. [22] L. Takács, Introduction to the Theory of Queues,, University Texts in the Mathematical Sciences, (1962). [23] H. Takagi, Analysis and application of polling models,, in Performance Evaluation: Origins and Directions, (1769), 423. doi: 10.1007/3-540-46506-5_18. [24] L. Tao, Z. Liu and Z. Wang, The GI/M/1 queue with start-up period and single working vacation and Bernoulli vacation interruption,, Applied Mathematics and Computation, 218 (2011), 4401. doi: 10.1016/j.amc.2011.10.017. [25] L. Tao, Z. Wang and Z. Liu, The GI/M/1 queue with Bernoulli-schedule-controlled vacation and vacation interruption,, Applied Mathematical Modelling, 37 (2013), 3724. doi: 10.1016/j.apm.2012.07.045. [26] N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications,, International Series in Operations Research & Management Science, (2006). [27] J. Wang and F. Zhang, Equilibrium analysis of the observable queues with balking and delayed repairs,, Applied Mathematics and Computation, 218 (2011), 2716. doi: 10.1016/j.amc.2011.08.012. [28] U. Yechiali, On optimal balking rules and toll charges in the GI/M/1 queuing process,, Operations Research, 19 (1971), 349. doi: 10.1287/opre.19.2.349. [29] 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. doi: 10.3934/jimo.2012.8.895. [30] F. Zhang, J. Wang and B. Liu, Equilibrium balking strategies in Markovian queues with working vacations,, Applied Mathematical Modelling, 37 (2013), 8264. doi: 10.1016/j.apm.2013.03.049. [31] H. Zhang and D. Shi, The M/M/1 queue with Bernoulli-schedule-controlled vacation and vacation interruption,, Int. J. Inform. Manage. Sci, 20 (2009), 579. [32] G. Zhao, X. Du and N. Tian, GI/M/1 queue with set-up period and working vacation and vacation interruption,, Int. J. Inform. Manage. Sci, 20 (2009), 351.

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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] A. D. Banik, U. C. Gupta and S. 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. [3] M. A. A. Boon, R. D. van der Mei and E. M. M. Winands, Applications of polling systems,, Surveys in Operations Research and Management Science, 16 (2011), 67. doi: 10.1016/j.sorms.2011.01.001. [4] M. L. Chaudhry and J. G. C. Templeton, A First Course in Bulk Queues,, Wiley, (1983). [5] M. L. Chaudhry, C. M. Harris and W. G. Marchal, Robustness of rootfinding in single-server queueing models,, ORSA Journal on Computing, 2 (1990), 273. [6] H. Chen, J. Li and N. Tian, The GI/M/1 queue with phase-type working vacations and vacation interruption,, Journal of Applied Mathematics and Computing, 30 (2009), 121. doi: 10.1007/s12190-008-0161-1. [7] J. L. Dorsman, O. J. Boxma and R. D. van der Mei, On two-queue Markovian polling systems with exhaustive service,, Queueing Systems, 78 (2014), 287. doi: 10.1007/s11134-014-9413-y. [8] B. T. Doshi, Queueing systems with vacations-A survey,, Queueing Systems, 1 (1986), 29. doi: 10.1007/BF01149327. [9] A. Economou, A. Gómez-Corral and S. Kanta, Optimal balking strategies in single-server queues with general service and vacation times,, Performance Evaluation, 68 (2011), 967. doi: 10.1016/j.peva.2011.07.001. [10] A. Economou and S. Kanta, Equilibrium balking strategies in the observable single-server queue with breakdowns and repairs,, Operations Research Letters, 36 (2008), 696. doi: 10.1016/j.orl.2008.06.006. [11] N. M. Edelson and D. K. Hilderbrand, Congestion tolls for Poisson queuing processes,, Econometrica: Journal of the Econometric Society, 43 (1975), 81. doi: 10.2307/1913415. [12] V. Goswami and P. V. Laxmi, Analysis of renewal input bulk arrival queue with single working vacation and partial batch rejection,, Journal of Industrial and Management Optimization, 6 (2010), 911. doi: 10.3934/jimo.2010.6.911. [13] P. Guo and P. Zipkin, The effects of the availability of waiting-time information on a balking queue,, European Journal of Operational Research, 198 (2009), 199. doi: 10.1016/j.ejor.2008.07.035. [14] R. Hassin and M. Haviv, To Queue or not to Queue: Equilibrium Behavior in Queueing Systems,, Springer, (2003). doi: 10.1007/978-1-4615-0359-0. [15] J. Ke, C. Wu and Z. G. Zhang, Recent developments in vacation queueing models: A short survey,, International Journal of Operations Research, 7 (2010), 3. [16] J. Keilson and L. D. Servi, Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedules,, Journal of Applied Probability, 23 (1986), 790. doi: 10.2307/3214016. [17] G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling,, ASA-SIAM Series on Statistics and Applied Probability, (1999). doi: 10.1137/1.9780898719734. [18] J. Li and N. Tian, The M/M/1 queue with working vacations and vacation interruptions,, Journal of Systems Science and Systems Engineering, 16 (2007), 121. doi: 10.1007/s11518-006-5030-6. [19] 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. doi: 10.1016/j.apm.2007.09.017. [20] P. Naor, The regulation of queue size by levying tolls,, Econometrica, 37 (1969), 15. doi: 10.2307/1909200. [21] L. D. Servi and S. G. Finn, M/M/1 queues with working vacations (M/M/1/WV),, Performance Evaluation, 50 (2002), 41. doi: 10.1016/S0166-5316(02)00057-3. [22] L. Takács, Introduction to the Theory of Queues,, University Texts in the Mathematical Sciences, (1962). [23] H. Takagi, Analysis and application of polling models,, in Performance Evaluation: Origins and Directions, (1769), 423. doi: 10.1007/3-540-46506-5_18. [24] L. Tao, Z. Liu and Z. Wang, The GI/M/1 queue with start-up period and single working vacation and Bernoulli vacation interruption,, Applied Mathematics and Computation, 218 (2011), 4401. doi: 10.1016/j.amc.2011.10.017. [25] L. Tao, Z. Wang and Z. Liu, The GI/M/1 queue with Bernoulli-schedule-controlled vacation and vacation interruption,, Applied Mathematical Modelling, 37 (2013), 3724. doi: 10.1016/j.apm.2012.07.045. [26] N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications,, International Series in Operations Research & Management Science, (2006). [27] J. Wang and F. Zhang, Equilibrium analysis of the observable queues with balking and delayed repairs,, Applied Mathematics and Computation, 218 (2011), 2716. doi: 10.1016/j.amc.2011.08.012. [28] U. Yechiali, On optimal balking rules and toll charges in the GI/M/1 queuing process,, Operations Research, 19 (1971), 349. doi: 10.1287/opre.19.2.349. [29] 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. doi: 10.3934/jimo.2012.8.895. [30] F. Zhang, J. Wang and B. Liu, Equilibrium balking strategies in Markovian queues with working vacations,, Applied Mathematical Modelling, 37 (2013), 8264. doi: 10.1016/j.apm.2013.03.049. [31] H. Zhang and D. Shi, The M/M/1 queue with Bernoulli-schedule-controlled vacation and vacation interruption,, Int. J. Inform. Manage. Sci, 20 (2009), 579. [32] G. Zhao, X. Du and N. Tian, GI/M/1 queue with set-up period and working vacation and vacation interruption,, Int. J. Inform. Manage. Sci, 20 (2009), 351.
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