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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

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 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 $M$/$M$/$R$ queueing system with balking, reneging, and server breakdowns, Mathematical Methods of Operations Research, 56 (2002), 169-180. doi: 10.1007/s001860200206.

show all references

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 $M$/$M$/$R$ 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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