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).

show all references

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).

[1]

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

[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]

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

[4]

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

[5]

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

[6]

Abdel-Rahman Hedar, Alaa Fahim. Filter-based genetic algorithm for mixed variable programming. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 99-116. doi: 10.3934/naco.2011.1.99

[7]

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

[8]

Bara Kim, Jeongsim Kim. Explicit solution for the stationary distribution of a discrete-time finite buffer queue. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1121-1133. doi: 10.3934/jimo.2016.12.1121

[9]

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

[10]

Sheng Zhu, Jinting Wang. Strategic behavior and optimal strategies in an M/G/1 queue with Bernoulli vacations. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-26. doi: 10.3934/jimo.2018008

[11]

Pikkala Vijaya Laxmi, Obsie Mussa Yesuf. Analysis of a finite buffer general input queue with Markovian service process and accessible and non-accessible batch service. Journal of Industrial & Management Optimization, 2010, 6 (4) : 929-944. doi: 10.3934/jimo.2010.6.929

[12]

Veena Goswami, Pikkala Vijaya Laxmi. Analysis of renewal input bulk arrival queue with single working vacation and partial batch rejection. Journal of Industrial & Management Optimization, 2010, 6 (4) : 911-927. doi: 10.3934/jimo.2010.6.911

[13]

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

[14]

Didem Cinar, José António Oliveira, Y. Ilker Topcu, Panos M. Pardalos. A priority-based genetic algorithm for a flexible job shop scheduling problem. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1391-1415. doi: 10.3934/jimo.2016.12.1391

[15]

Yaw Chang, Lin Chen. Solve the vehicle routing problem with time windows via a genetic algorithm. Conference Publications, 2007, 2007 (Special) : 240-249. doi: 10.3934/proc.2007.2007.240

[16]

Qingzhi Yang. The revisit of a projection algorithm with variable steps for variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (2) : 211-217. doi: 10.3934/jimo.2005.1.211

[17]

Yoshiaki Inoue, Tetsuya Takine. The FIFO single-server queue with disasters and multiple Markovian arrival streams. Journal of Industrial & Management Optimization, 2014, 10 (1) : 57-87. doi: 10.3934/jimo.2014.10.57

[18]

Tomás Caraballo, Gábor Kiss. Attractors for differential equations with multiple variable delays. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1365-1374. doi: 10.3934/dcds.2013.33.1365

[19]

I-Lin Wang, Shiou-Jie Lin. A network simplex algorithm for solving the minimum distribution cost problem. Journal of Industrial & Management Optimization, 2009, 5 (4) : 929-950. doi: 10.3934/jimo.2009.5.929

[20]

Ping-Chen Lin. Portfolio optimization and risk measurement based on non-dominated sorting genetic algorithm. Journal of Industrial & Management Optimization, 2012, 8 (3) : 549-564. doi: 10.3934/jimo.2012.8.549

2016 Impact Factor: 0.994

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]