A Study on the Stochastic Approximation Method during the Single Server Queueing System with Catastrophe
DOI:
https://doi.org/10.51983/ajsat-2016.5.1.2540Keywords:
Queueing system, Steady state, catastrophe, approximation condition, stability condition, Mean queue lengthAbstract
We consider an M/G/1 queueing system where the customer arrivals occur according to a Poisson process with mean arrival rate λ. The server is subject to catastrophe and repairs while in operation. At failure times, the server still works at a lower service rate rather than completely stopping service.The service time follow general laws with probability distribution function and Laplace Stieltjes transform. Using the approximation conditions in the classical M/G/1 system, we obtain stability inequalities with exact computation of the constants.
References
J.R. Artalejo and A. Gomez-Corral, "Analysis of a stochastic clearing system with repeated attempts," Stochastic Models, vol. 14, pp. 623–645, 1988.
O.J. Boxma, D. Perry, and W. Stadje, "Clearing Models for M/G/1 Queues," Queueing Systems, vol. 38, pp. 287–306, 2001.
P.J. Brockwell, J. Gani, and S.I. Resnick, "Birth, Immigration and Catastrophe Processes," Adv. Appl. Prob., vol. 14, no. 4, pp. 709-731, 1982.
X. Chao, "A queueing network model with catastrophes and product form solution," Oper. Res. Lett., vol. 18, pp. 75–79, 1995.
A. Di Crescenzo, V. Giorno, A.G. Nobile, and L.M. Ricciardi, "On the M/M/1 Queue with Catastrophes and Its Continuous Approximation," Queueing Systems, vol. 43, pp. 329–347, 2003.
A.N. Dudin and S. Nishimura, "A B MAP/SM/1 queueing system with Markovian arrival of catastrophes," J. Appl. Probab., vol. 36, pp. 868–881, 1999.
A.N. Dudin and A.V. Karolik, "BMAP/SM/1 queue with Markovian input of catastrophes and no instantaneous recovery," Performance Eval., vol. 45, pp. 19–32, 2001.
A. Economou and S. Kapodistria, "Synchronized abandonments in a single server unreliable queue," Eur. J. Oper. Res., vol. 203, no. 1, pp. 143–155, 2009.
E. Gelenbe, P. Glynn, and K. Sigman, "Queues with Negative Arrivals," J. Appl. Prob., vol. 28, pp. 245–250, 1991.
D. Gross and C. M. Harris, "Fundamentals of queueing theory," John Wiley & Sons, 1974.
B.K. Kumar, A. Krihnamoorthy, S.P. Madheswari, and S.S. Basha, "Transient analysis of a single server queue with catastrophes, failures, and repairs," Queueing Systems, vol. 56, pp. 133–141, 2007.
M.R. Sagayaraj, S. Anand Gnana Selvam, and R. Reynald Susainathan, "A Study on Stochastic Integral Associated with Catastrophes," International Scholarly and Scientific Research and Innovation, vol. 3, no. 6, 2015.
M.R. Sagayaraj, S. Anand Gnana Selvam, and R. Reynald Susainathan, "Analysis of Transient behavior M/M/1 Queueing model with Catastrophical events," International Journal of Mathematical Sciences & Engineering Applications (IJMSEA), vol. 9, 2015, pp. 149-158.
Y.W. Shin, "Multi-server retrial queue with negative customers and catastrophes," Queueing Systems, vol. 56, pp. 223–237, 2007.
U. Yechiali, "Queues with system catastrophes and impatient customers when system is down," Queueing Systems, vol. 56, pp. 195-202, 2007.
W.S. Yang, J.D. Kim, and K.C. Chae, "Analysis of M/G/1 stochastic clearing systems," Stochastic Analysis and Applications, vol. 20, no. 5, pp. 1083–1100, 2002.
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