American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Single-machine scheduling with stepwise tardiness costs and release times
  • JIMO Home
  • This Issue
  • Next Article
    Nonsmooth optimization over the (weakly or properly) Pareto set of a linear-quadratic multi-objective control problem: Explicit optimality conditions
October  2011, 7(4): 811-823. doi: 10.3934/jimo.2011.7.811

Transient and steady state analysis of an M/M/1 queue with balking, catastrophes, server failures and repairs

Ahmed M. K. Tarabia 1,

1. 

Mathematics Department, Damietta Faculty of Science, New Damietta, Egypt

Received  June 2010 Revised  May 2011 Published  August 2011

Recently Krishna Kumar and Pavai [10] have obtained the transient distribution for the queue length of the system an M/M/1 queueing system with catastrophes, server failures using a direct technique. In this paper, we consider Krishna Kumar and Pavai [10] model with balking feature. Based on the generating function technique and a direct approach, transient and steady state analysis of the queue length is carried out Krishna Kumar and Pavai [10] model can be deduced from the new model. Moreover, some other special cases are shown as special cases of our solution.
Keywords: catastrophes, server failure, Markovian queue, balking., single server queue.
Mathematics Subject Classification: Primary: 60K25; Secondary: 90B2.
Citation: 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
References:
[1]

M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions," New York, Dover, 1970. Google Scholar

[2]

C. J. Jr. Ancker 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.  Google Scholar

[3]

C. J. Jr. Ancker 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.  Google Scholar

[4]

I. Atencia and P. Moreno, The discrete time $Geo$/$Geo$/$1$ queue with negative customers and disasters, Computers and Operations Research, 9 (2004), 1537-1548. doi: 10.1016/S0305-0548(03)00107-2.  Google Scholar

[5]

X. Chao, A queueing network model with catastrophes and product form solution, Operations Research Letters., 18 (1995), 75-79. doi: 10.1016/0167-6377(95)00029-0.  Google Scholar

[6]

E. Gelenbe, Production-form queueing networks with negative and positive customers, Journal of Applied Probability, 28 (1991), 656-663. doi: 10.2307/3214499.  Google Scholar

[7]

F. A. Haight, Queueing with balking, Biometrika., 44 (1957), 360-369.  Google Scholar

[8]

F. A. Haight, Queueing with balking, Biometrika., 47 (1960), 285-296.  Google Scholar

[9]

B. Krishna Kumar and D. Arivudainambi, Transient solution of an $M$/$M$/$1$ queue with catastrophes, Computers and Mathematics with Applications, 40 (2000), 1233-1240. doi: 10.1016/S0898-1221(00)00234-0.  Google Scholar

[10]

B. Krishna Kumar and S. Pavai Madheswari, Transient analysis of an $M$/$M$/$1$ queue subject to catastrophes and server failures, Stochastic Analysis and Applications, 23 (2005), 329-340. doi: 10.1081/SAP-200050101.  Google Scholar

[11]

B. Krishna Kumar, A. Krishnamoorthy, S. Pavai Madheswari and S. Sadiq Basha, Transient analysis of a single server queue with catastrophes, failures and repairs, Queueing Systems., 56 (2007), 133-141. doi: 10.1007/s11134-007-9014-0.  Google Scholar

[12]

B. Krishna Kumar, P. R. Parthasarathy and M. Sharafali, Transient solution of an $M$/$M$/$1$ queue with balking, Queueing Systems Theory Appl., 13 (1993), 441-448. doi: 10.1007/BF01149265.  Google Scholar

[13]

A. Montazer-Haghighi, J. Medhi and S. G. Mohanty, On a multiserver Markovian queueing system with balking and reneging, Comput. Oper. Res., 13 (1986), 421-425. doi: 10.1016/0305-0548(86)90029-8.  Google Scholar

[14]

P. R. Parthasarathy and M. Sharafali, Transient solution to the many-server Poisson queue: A simple approach, Journal of Applied Probability, 26 (1986), 584-594. doi: 10.2307/3214415.  Google Scholar

[15]

S. N. Raju and U. N. Bhat, A computationally oriented analysis of the $G$/$M$/$1$ queue, Opsearch, 19 (1982), 67-83.  Google Scholar

[16]

L. Takács, "The Transient Behaviour of a Single Server Queueing Process with a Poisson Input," Proc. 4th Berkeley Symp. On Mathematical Statistics and Probability, Vol. II, Univ. California Press, Berkeley, Calif., (1961), 535-567.  Google Scholar

[17]

A. M. K. Tarabia, Transient analysis of a non-empty $M$/$M$/$1$/$N$ queue-an alternative approach, Opsearch, 38 (2001), 431-440.  Google Scholar

[18]

A. M. K. Tarabia, A new formula for the transient behaviour of a non-empty $M$/$M$/$1$/$infty$ queue, Applied Mathematics and Computation, 132 (2002), 1-10. doi: 10.1016/S0096-3003(01)00145-X.  Google Scholar

[19]

K.-H. Wang and Y.-C Chang, Cost analysis of a finite $M$/$M$/$R$ queueing system with balking, reneging, and server breakdowns, Mathematical Methods of Operations Research, 56 (2002), 169-180. doi: 10.1007/s001860200206.  Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions," New York, Dover, 1970. Google Scholar

[2]

C. J. Jr. Ancker 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.  Google Scholar

[3]

C. J. Jr. Ancker 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.  Google Scholar

[4]

I. Atencia and P. Moreno, The discrete time $Geo$/$Geo$/$1$ queue with negative customers and disasters, Computers and Operations Research, 9 (2004), 1537-1548. doi: 10.1016/S0305-0548(03)00107-2.  Google Scholar

[5]

X. Chao, A queueing network model with catastrophes and product form solution, Operations Research Letters., 18 (1995), 75-79. doi: 10.1016/0167-6377(95)00029-0.  Google Scholar

[6]

E. Gelenbe, Production-form queueing networks with negative and positive customers, Journal of Applied Probability, 28 (1991), 656-663. doi: 10.2307/3214499.  Google Scholar

[7]

F. A. Haight, Queueing with balking, Biometrika., 44 (1957), 360-369.  Google Scholar

[8]

F. A. Haight, Queueing with balking, Biometrika., 47 (1960), 285-296.  Google Scholar

[9]

B. Krishna Kumar and D. Arivudainambi, Transient solution of an $M$/$M$/$1$ queue with catastrophes, Computers and Mathematics with Applications, 40 (2000), 1233-1240. doi: 10.1016/S0898-1221(00)00234-0.  Google Scholar

[10]

B. Krishna Kumar and S. Pavai Madheswari, Transient analysis of an $M$/$M$/$1$ queue subject to catastrophes and server failures, Stochastic Analysis and Applications, 23 (2005), 329-340. doi: 10.1081/SAP-200050101.  Google Scholar

[11]

B. Krishna Kumar, A. Krishnamoorthy, S. Pavai Madheswari and S. Sadiq Basha, Transient analysis of a single server queue with catastrophes, failures and repairs, Queueing Systems., 56 (2007), 133-141. doi: 10.1007/s11134-007-9014-0.  Google Scholar

[12]

B. Krishna Kumar, P. R. Parthasarathy and M. Sharafali, Transient solution of an $M$/$M$/$1$ queue with balking, Queueing Systems Theory Appl., 13 (1993), 441-448. doi: 10.1007/BF01149265.  Google Scholar

[13]

A. Montazer-Haghighi, J. Medhi and S. G. Mohanty, On a multiserver Markovian queueing system with balking and reneging, Comput. Oper. Res., 13 (1986), 421-425. doi: 10.1016/0305-0548(86)90029-8.  Google Scholar

[14]

P. R. Parthasarathy and M. Sharafali, Transient solution to the many-server Poisson queue: A simple approach, Journal of Applied Probability, 26 (1986), 584-594. doi: 10.2307/3214415.  Google Scholar

[15]

S. N. Raju and U. N. Bhat, A computationally oriented analysis of the $G$/$M$/$1$ queue, Opsearch, 19 (1982), 67-83.  Google Scholar

[16]

L. Takács, "The Transient Behaviour of a Single Server Queueing Process with a Poisson Input," Proc. 4th Berkeley Symp. On Mathematical Statistics and Probability, Vol. II, Univ. California Press, Berkeley, Calif., (1961), 535-567.  Google Scholar

[17]

A. M. K. Tarabia, Transient analysis of a non-empty $M$/$M$/$1$/$N$ queue-an alternative approach, Opsearch, 38 (2001), 431-440.  Google Scholar

[18]

A. M. K. Tarabia, A new formula for the transient behaviour of a non-empty $M$/$M$/$1$/$infty$ queue, Applied Mathematics and Computation, 132 (2002), 1-10. doi: 10.1016/S0096-3003(01)00145-X.  Google Scholar

[19]

K.-H. Wang and Y.-C Chang, Cost analysis of a finite $M$/$M$/$R$ queueing system with balking, reneging, and server breakdowns, Mathematical Methods of Operations Research, 56 (2002), 169-180. doi: 10.1007/s001860200206.  Google Scholar

[1]

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
[2]

Dhanya Shajin, A. N. Dudin, Olga Dudina, A. Krishnamoorthy. A two-priority single server retrial queue with additional items. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2891-2912. doi: 10.3934/jimo.2019085
[3]

Ke Sun, Jinting Wang, Zhe George Zhang. Strategic joining in a single-server retrial queue with batch service. Journal of Industrial & Management Optimization, 2021, 17 (6) : 3309-3332. doi: 10.3934/jimo.2020120
[4]

Naoto Miyoshi. On the stationary LCFS-PR single-server queue: A characterization via stochastic intensity. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 713-725. doi: 10.3934/naco.2011.1.713
[5]

Yi Peng, Jinbiao Wu. Analysis of a batch arrival retrial queue with impatient customers subject to the server disasters. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2243-2264. doi: 10.3934/jimo.2020067
[6]

Shaojun Lan, Yinghui Tang, Miaomiao Yu. System capacity optimization design and optimal threshold $N^{*}$ for a $GEO/G/1$ discrete-time queue with single server vacation and under the control of Min($N, V$)-policy. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1435-1464. doi: 10.3934/jimo.2016.12.1435
[7]

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
[8]

Ali Delavarkhalafi. On optimal stochastic jumps in multi server queue with impatient customers via stochastic control. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021030
[9]

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
[10]

Tuan Phung-Duc. Single server retrial queues with setup time. Journal of Industrial & Management Optimization, 2017, (3) : 1329-1345. doi: 10.3934/jimo.2016075
[11]

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
[12]

Sherif I. Ammar, Alexander Zeifman, Yacov Satin, Ksenia Kiseleva, Victor Korolev. On limiting characteristics for a non-stationary two-processor heterogeneous system with catastrophes, server failures and repairs. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1057-1068. doi: 10.3934/jimo.2020011
[13]

Shahede Omidi, Jafar Fathali. Inverse single facility location problem on a tree with balancing on the distance of server to clients. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021017
[14]

Tuan Phung-Duc, Wouter Rogiest, Sabine Wittevrongel. Single server retrial queues with speed scaling: Analysis and performance evaluation. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1927-1943. doi: 10.3934/jimo.2017025
[15]

Saroja Kumar Singh. Moderate deviation for maximum likelihood estimators from single server queues. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 2-. doi: 10.1186/s41546-020-00044-z
[16]

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
[17]

Dequan Yue, Wuyi Yue. Block-partitioning matrix solution of M/M/R/N queueing system with balking, reneging and server breakdowns. Journal of Industrial & Management Optimization, 2009, 5 (3) : 417-430. doi: 10.3934/jimo.2009.5.417
[18]

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
[19]

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
[20]

A. Azhagappan, T. Deepa. Transient analysis of N-policy queue with system disaster repair preventive maintenance re-service balking closedown and setup times. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2843-2856. doi: 10.3934/jimo.2019083

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (79)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences