# American Institute of Mathematical Sciences

• 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

 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.
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