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

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[6]	E. Gelenbe, Production-form queueing networks with negative and positive customers, Journal of Applied Probability, 28 (1991), 656-663. doi: 10.2307/3214499.
[7]	F. A. Haight, Queueing with balking, Biometrika., 44 (1957), 360-369.
[8]	F. A. Haight, Queueing with balking, Biometrika., 47 (1960), 285-296.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	S. N. Raju and U. N. Bhat, A computationally oriented analysis of the $G$/$M$/$1$ queue, Opsearch, 19 (1982), 67-83.
[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.
[17]	A. M. K. Tarabia, Transient analysis of a non-empty $M$/$M$/$1$/$N$ queue-an alternative approach, Opsearch, 38 (2001), 431-440.
[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.
[19]	K.-H. Wang and Y.-C Chang, Cost analysis of a finite $M$/$M$/$\mathbb{R}^{N}$ queueing system with balking, reneging, and server breakdowns, Mathematical Methods of Operations Research, 56 (2002), 169-180. doi: 10.1007/s001860200206.

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References:

[1]	M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions," New York, Dover, 1970.
[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.
[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.
[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.
[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.
[6]	E. Gelenbe, Production-form queueing networks with negative and positive customers, Journal of Applied Probability, 28 (1991), 656-663. doi: 10.2307/3214499.
[7]	F. A. Haight, Queueing with balking, Biometrika., 44 (1957), 360-369.
[8]	F. A. Haight, Queueing with balking, Biometrika., 47 (1960), 285-296.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	S. N. Raju and U. N. Bhat, A computationally oriented analysis of the $G$/$M$/$1$ queue, Opsearch, 19 (1982), 67-83.
[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.
[17]	A. M. K. Tarabia, Transient analysis of a non-empty $M$/$M$/$1$/$N$ queue-an alternative approach, Opsearch, 38 (2001), 431-440.
[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.
[19]	K.-H. Wang and Y.-C Chang, Cost analysis of a finite $M$/$M$/$\mathbb{R}^{N}$ queueing system with balking, reneging, and server breakdowns, Mathematical Methods of Operations Research, 56 (2002), 169-180. doi: 10.1007/s001860200206.

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