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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

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, (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.  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.  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.  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.  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.  doi: 10.2307/3214499.  Google Scholar

[7]

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

[8]

F. A. Haight, Queueing with balking,, Biometrika., 47 (1960), 285.   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.  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.  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.  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.  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.  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.  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.   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, (1961), 535.   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.   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.  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.  doi: 10.1007/s001860200206.  Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions,", New York, (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.  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.  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.  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.  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.  doi: 10.2307/3214499.  Google Scholar

[7]

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

[8]

F. A. Haight, Queueing with balking,, Biometrika., 47 (1960), 285.   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.  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.  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.  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.  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.  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.  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.   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, (1961), 535.   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.   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.  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.  doi: 10.1007/s001860200206.  Google Scholar

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