American Institute of Mathematical Sciences

Advanced
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

Gopinath Panda 1, , Veena Goswami 2, , Abhijit Datta Banik 1, and  Dibyajyoti Guha 1,

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.
Keywords: Bernoulli schedule, $GI/M/1$ queue, working vacations, roots., equilibrium balking strategies.
Mathematics Subject Classification: Primary: 60K25, 68M20; Secondary: 90B2.
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), 201-209. doi: 10.1016/j.orl.2004.05.006.  Google Scholar

[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), 1701-1710. doi: 10.1016/j.apm.2006.05.010.  Google Scholar

[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), 67-82. doi: 10.1016/j.sorms.2011.01.001.  Google Scholar

[4]

M. L. Chaudhry and J. G. C. Templeton, A First Course in Bulk Queues, Wiley, New York, 1983.  Google Scholar

[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), 273-286. Google Scholar

[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), 121-141. doi: 10.1007/s12190-008-0161-1.  Google Scholar

[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), 287-311. doi: 10.1007/s11134-014-9413-y.  Google Scholar

[8]

B. T. Doshi, Queueing systems with vacations-A survey, Queueing Systems, 1 (1986), 29-66. doi: 10.1007/BF01149327.  Google Scholar

[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), 967-982. doi: 10.1016/j.peva.2011.07.001.  Google Scholar

[10]

A. Economou and S. Kanta, Equilibrium balking strategies in the observable single-server queue with breakdowns and repairs, Operations Research Letters, 36 (2008), 696-699. doi: 10.1016/j.orl.2008.06.006.  Google Scholar

[11]

N. M. Edelson and D. K. Hilderbrand, Congestion tolls for Poisson queuing processes, Econometrica: Journal of the Econometric Society, 43 (1975), 81-92. doi: 10.2307/1913415.  Google Scholar

[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), 911-927. doi: 10.3934/jimo.2010.6.911.  Google Scholar

[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), 199-209. doi: 10.1016/j.ejor.2008.07.035.  Google Scholar

[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.  Google Scholar

[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), 3-8. Google Scholar

[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), 790-802. doi: 10.2307/3214016.  Google Scholar

[17]

G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, ASA-SIAM Series on Statistics and Applied Probability, SIAM, Philadelphia, PA, 1999. doi: 10.1137/1.9780898719734.  Google Scholar

[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), 121-127. doi: 10.1007/s11518-006-5030-6.  Google Scholar

[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), 2715-2730. doi: 10.1016/j.apm.2007.09.017.  Google Scholar

[20]

P. Naor, The regulation of queue size by levying tolls, Econometrica, 37 (1969), 15-24. doi: 10.2307/1909200.  Google Scholar

[21]

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.  Google Scholar

[22]

L. Takács, Introduction to the Theory of Queues, University Texts in the Mathematical Sciences, Oxford University Press, New York, 1962.  Google Scholar

[23]

H. Takagi, Analysis and application of polling models, in Performance Evaluation: Origins and Directions, Lecture Notes in Computer Science, 1769, Springer, 2000, 423-442. doi: 10.1007/3-540-46506-5_18.  Google Scholar

[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), 4401-4413. doi: 10.1016/j.amc.2011.10.017.  Google Scholar

[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), 3724-3735. doi: 10.1016/j.apm.2012.07.045.  Google Scholar

[26]

N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, International Series in Operations Research & Management Science, 93, Springer, New York, 2006.  Google Scholar

[27]

J. Wang and F. Zhang, Equilibrium analysis of the observable queues with balking and delayed repairs, Applied Mathematics and Computation, 218 (2011), 2716-2729. doi: 10.1016/j.amc.2011.08.012.  Google Scholar

[28]

U. Yechiali, On optimal balking rules and toll charges in the GI/M/1 queuing process, Operations Research, 19 (1971), 349-370. doi: 10.1287/opre.19.2.349.  Google Scholar

[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), 895-908. doi: 10.3934/jimo.2012.8.895.  Google Scholar

[30]

F. Zhang, J. Wang and B. Liu, Equilibrium balking strategies in Markovian queues with working vacations, Applied Mathematical Modelling, 37 (2013), 8264-8282. doi: 10.1016/j.apm.2013.03.049.  Google Scholar

[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), 579-587.  Google Scholar

[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), 351-363.  Google Scholar

show all references

References:
[1]

Y. Baba, Analysis of a GI/M/1 queue with multiple working vacations, Operations Research Letters, 33 (2005), 201-209. doi: 10.1016/j.orl.2004.05.006.  Google Scholar

[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), 1701-1710. doi: 10.1016/j.apm.2006.05.010.  Google Scholar

[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), 67-82. doi: 10.1016/j.sorms.2011.01.001.  Google Scholar

[4]

M. L. Chaudhry and J. G. C. Templeton, A First Course in Bulk Queues, Wiley, New York, 1983.  Google Scholar

[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), 273-286. Google Scholar

[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), 121-141. doi: 10.1007/s12190-008-0161-1.  Google Scholar

[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), 287-311. doi: 10.1007/s11134-014-9413-y.  Google Scholar

[8]

B. T. Doshi, Queueing systems with vacations-A survey, Queueing Systems, 1 (1986), 29-66. doi: 10.1007/BF01149327.  Google Scholar

[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), 967-982. doi: 10.1016/j.peva.2011.07.001.  Google Scholar

[10]

A. Economou and S. Kanta, Equilibrium balking strategies in the observable single-server queue with breakdowns and repairs, Operations Research Letters, 36 (2008), 696-699. doi: 10.1016/j.orl.2008.06.006.  Google Scholar

[11]

N. M. Edelson and D. K. Hilderbrand, Congestion tolls for Poisson queuing processes, Econometrica: Journal of the Econometric Society, 43 (1975), 81-92. doi: 10.2307/1913415.  Google Scholar

[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), 911-927. doi: 10.3934/jimo.2010.6.911.  Google Scholar

[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), 199-209. doi: 10.1016/j.ejor.2008.07.035.  Google Scholar

[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.  Google Scholar

[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), 3-8. Google Scholar

[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), 790-802. doi: 10.2307/3214016.  Google Scholar

[17]

G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, ASA-SIAM Series on Statistics and Applied Probability, SIAM, Philadelphia, PA, 1999. doi: 10.1137/1.9780898719734.  Google Scholar

[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), 121-127. doi: 10.1007/s11518-006-5030-6.  Google Scholar

[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), 2715-2730. doi: 10.1016/j.apm.2007.09.017.  Google Scholar

[20]

P. Naor, The regulation of queue size by levying tolls, Econometrica, 37 (1969), 15-24. doi: 10.2307/1909200.  Google Scholar

[21]

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.  Google Scholar

[22]

L. Takács, Introduction to the Theory of Queues, University Texts in the Mathematical Sciences, Oxford University Press, New York, 1962.  Google Scholar

[23]

H. Takagi, Analysis and application of polling models, in Performance Evaluation: Origins and Directions, Lecture Notes in Computer Science, 1769, Springer, 2000, 423-442. doi: 10.1007/3-540-46506-5_18.  Google Scholar

[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), 4401-4413. doi: 10.1016/j.amc.2011.10.017.  Google Scholar

[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), 3724-3735. doi: 10.1016/j.apm.2012.07.045.  Google Scholar

[26]

N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, International Series in Operations Research & Management Science, 93, Springer, New York, 2006.  Google Scholar

[27]

J. Wang and F. Zhang, Equilibrium analysis of the observable queues with balking and delayed repairs, Applied Mathematics and Computation, 218 (2011), 2716-2729. doi: 10.1016/j.amc.2011.08.012.  Google Scholar

[28]

U. Yechiali, On optimal balking rules and toll charges in the GI/M/1 queuing process, Operations Research, 19 (1971), 349-370. doi: 10.1287/opre.19.2.349.  Google Scholar

[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), 895-908. doi: 10.3934/jimo.2012.8.895.  Google Scholar

[30]

F. Zhang, J. Wang and B. Liu, Equilibrium balking strategies in Markovian queues with working vacations, Applied Mathematical Modelling, 37 (2013), 8264-8282. doi: 10.1016/j.apm.2013.03.049.  Google Scholar

[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), 579-587.  Google Scholar

[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), 351-363.  Google Scholar

[1]

Sheng Zhu, Jinting Wang. Strategic behavior and optimal strategies in an M/G/1 queue with Bernoulli vacations. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1297-1322. doi: 10.3934/jimo.2018008
[2]

Dequan Yue, Wuyi Yue, Gang Xu. Analysis of customers' impatience in an M/M/1 queue with working vacations. Journal of Industrial & Management Optimization, 2012, 8 (4) : 895-908. doi: 10.3934/jimo.2012.8.895
[3]

Shan Gao, Jinting Wang. On a discrete-time GI$^X$/Geo/1/N-G queue with randomized working vacations and at most $J$ vacations. Journal of Industrial & Management Optimization, 2015, 11 (3) : 779-806. doi: 10.3934/jimo.2015.11.779
[4]

Biao Xu, Xiuli Xu, Zhong Yao. Equilibrium and optimal balking strategies for low-priority customers in the M/G/1 queue with two classes of customers and preemptive priority. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1599-1615. doi: 10.3934/jimo.2018113
[5]

Chia-Huang Wu, Kuo-Hsiung Wang, Jau-Chuan Ke, Jyh-Bin Ke. A heuristic algorithm for the optimization of M/M/$s$ queue with multiple working vacations. Journal of Industrial & Management Optimization, 2012, 8 (1) : 1-17. doi: 10.3934/jimo.2012.8.1
[6]

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 & Management Optimization, 2016, 12 (4) : 1199-1214. doi: 10.3934/jimo.2016.12.1199
[7]

Ahmed M. K. Tarabia. Transient and steady state analysis of an M/M/1 queue with balking, catastrophes, server failures and repairs. Journal of Industrial & Management Optimization, 2011, 7 (4) : 811-823. doi: 10.3934/jimo.2011.7.811
[8]

Dequan Yue, Wuyi Yue. A heterogeneous two-server network system with balking and a Bernoulli vacation schedule. Journal of Industrial & Management Optimization, 2010, 6 (3) : 501-516. doi: 10.3934/jimo.2010.6.501
[9]

Dequan Yue, Wuyi Yue, Guoxi Zhao. Analysis of an M/M/1 queue with vacations and impatience timers which depend on the server's states. Journal of Industrial & Management Optimization, 2016, 12 (2) : 653-666. doi: 10.3934/jimo.2016.12.653
[10]

Ruiling Tian, Dequan Yue, Wuyi Yue. Optimal balking strategies in an M/G/1 queueing system with a removable server under N-policy. Journal of Industrial & Management Optimization, 2015, 11 (3) : 715-731. doi: 10.3934/jimo.2015.11.715
[11]

Shaojun Lan, Yinghui Tang. Performance analysis of a discrete-time $ Geo/G/1$ retrial queue with non-preemptive priority, working vacations and vacation interruption. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1421-1446. doi: 10.3934/jimo.2018102
[12]

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
[13]

Zhanyou Ma, Wenbo Wang, Linmin Hu. Performance evaluation and analysis of a discrete queue system with multiple working vacations and non-preemptive priority. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1135-1148. doi: 10.3934/jimo.2018196
[14]

Feng Zhang, Jinting Wang, Bin Liu. On the optimal and equilibrium retrial rates in an unreliable retrial queue with vacations. Journal of Industrial & Management Optimization, 2012, 8 (4) : 861-875. doi: 10.3934/jimo.2012.8.861
[15]

Sujit Kumar Samanta, Rakesh Nandi. Analysis of $GI^{[X]}/D$-$MSP/1/\infty$ queue using $RG$-factorization. Journal of Industrial & Management Optimization, 2021, 17 (2) : 549-573. doi: 10.3934/jimo.2019123
[16]

Zhanyou Ma, Pengcheng Wang, Wuyi Yue. Performance analysis and optimization of a pseudo-fault Geo/Geo/1 repairable queueing system with N-policy, setup time and multiple working vacations. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1467-1481. doi: 10.3934/jimo.2017002
[17]

Qingqing Ye. Algorithmic computation of MAP/PH/1 queue with finite system capacity and two-stage vacations. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2459-2477. doi: 10.3934/jimo.2019063
[18]

Cheng-Dar Liou. Optimization analysis of the machine repair problem with multiple vacations and working breakdowns. Journal of Industrial & Management Optimization, 2015, 11 (1) : 83-104. doi: 10.3934/jimo.2015.11.83
[19]

Dequan Yue, Jun Yu, Wuyi Yue. A Markovian queue with two heterogeneous servers and multiple vacations. Journal of Industrial & Management Optimization, 2009, 5 (3) : 453-465. doi: 10.3934/jimo.2009.5.453
[20]

Zsolt Saffer, Wuyi Yue. A dual tandem queueing system with GI service time at the first queue. Journal of Industrial & Management Optimization, 2014, 10 (1) : 167-192. doi: 10.3934/jimo.2014.10.167

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (92)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences