Abstract
In this paper, we consider a BMAP/G/1 retrial queue with a server subject to breakdowns and repairs, where the life time of the server is exponential and the repair time is general. We use the supplementary variable method, which combines with the matrix-analytic method and the censoring technique, to study the system. We apply the RG-factorization of a level-dependent continuous-time Markov chain of M/G/1 type to provide the stationary performance measures of the system, for example, the stationary availability, failure frequency and queue length. Furthermore, we use the RG-factorization of a level-dependent Markov renewal process of M/G/1 type to express the Laplace transform of the distribution of a first passage time such as the reliability function and the busy period.
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References
Aissani, A.A. (1994). “Retrial Queue with Redundancy and Unreliable Server.” Queueing Systems 17, 431–449.
Aissani, A. and J.R. Artalejo. (1998). “On the Single Server Retrial Queue Subject to Breakdowns.” Queueing Systems 30, 309–321.
Artalejo, J.R. (1994). “New Results in Retrial Queueing Systems with Breakdowns of the Servers.” Statistics Neerlandica 48, 23–36.
Artalejo, J.R. (1999). “A Classified Bibliography of Research on Retrial Queues: Progress in 1990–1999.” Top 7, 187–211.
Artalejo, J.R. and A. Gómez-Corral. (1998). “Unreliable Retrial Queues due to Service Interruptions Arising from Facsimile Networks.” Belg J Oper Res Statist Comput Sci 38, 31–41.
Asmussen, S. (1987). Applied Probability and Queues. John Wiley & Sons.
Breuer, L., A. Dudin, and V. Klimenok. (2002). “A Retrial BMAP/PH/N System.” Queueing Systems 40, 433–457.
Bright, L. and P.G. Taylor. (1995). “Calculating the Equilibrium Distribution in Level Dependent Quasi-birth-and-death Processes.” Stochastic Models 11, 497–525.
Cao, J. and K. Chen. (1982). “Analysis of the M/G/1 Queueing System with Repairable Service Station.” Acta Math. Appl. Sinica 5, 113–127.
Chakravarthy, S.R. (2001). “The Batch Markovian Arrival Processes: A Review and Future Work.” In A. Krishnamoorthy et al. (eds.), Advances in Probability Theory and Stochastic Processes. Notable Publications, Inc., New Jersey, pp. 21–49.
Chakravarthy, S.R. and A. Dudin. (2002). “A Multi-server Retrial Queue with BMAP Arrivals and Group Services.” Queueing Systems 42, 5–31.
Chakravarthy, S.R. and A. Dudin. (2003). “Analysis of a Retrial Queuing Model with MAP Arrivals and Two Types of Customers.” Math. Comput. Modelling 37, 343–363.
Choi, B.D., Y. Yang, and B. Kim, (1999). “MAP1/MAP2/M/c Retrial Queue with Guard Channels and its Applications to Cellular Networks.” Top 7, 231–248.
Choi, B.D., Y.H. Chung, and A. Dudin. (2001). “The BMAP/SM/1 Retrial Queue with Controllable Operation Modes.” European J. Oper. Res. 131, 16–30.
Diamond, J.E. and A.S. Alfa. (1995). “Matrix Analytic Methods for M/PH/1 Retrial Queues with Buffers.” Stochastic Models 11, 447–470.
Diamond, J.E. and A.S. Alfa (1998). “Matrix Analytic Methods for MAP/PH/1 Retrial Queues with Buffers.” Stochastic Models 14, 1151–1187.
Diamond, J.E. and A.S. Alfa. (1999). “Matrix Analytic Methods for Multi-server Retrial Queues with Buffers.” Top 7, 249–266.
Dudin, A. and V. Klimenok (1999). “BMAP/SM/1 Model with Markov Modulated Retrials.” Top 7, 267–278.
Dudin, A. and V. Klimenok. (2000). “A Retrial BMAP/SM/1 System with Linear Requests.” Queueing Systems 34, 47–66.
Falin, G.I. (1990). “A Survey of Retrial Queues.” Queueing Systems 7, 127–167.
Falin, G.I. and J.G.C. Templeton. (1997). Retrial Queues. Chapman & Hall: London.
Ferng, H.W. and J.F. Chang. (2001). “Departure Processes of BMAP/G/1 Queues.” Queueing Systems 39, 109–135.
Gnedenko, B.V. and I.N. Kovalenko. (1989). Introduction to Queueing Theory, Second Edition. Birkhauser: Boston.
Grassmann, W.K. and D.P. Heyman. (1990). “Equilibrium Distribution of Block-structured Markov Chains with Repeating Rows.” J. Appl. Prob. 27, 557–576.
He, Q., H. Li, and Y.Q. Zhao. (2000). “Ergodicity of the BMAP/PH/s/s+k Retrial Queue with PH-retrial Times.” Queueing Systems 35, 323–347.
Horn, R.A and C.R. Johnson. (1985). Matrix Analysis. Cambridge University Press: London.
Hsu, G.H., Yuan, X. and Li, Q.L. (2000). “First Passage Times for Markov Renewal Processes and Application.” Science in China, Series A 43, 1238–1249.
Latouche, G. and V. Ramaswami. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. SIAM, Philadelphia.
Lee, G. and J. Jeon. (1999). “Analysis of an N/G/1 Finite Queue with the Supplementary Variable Method.” J Appl Math Stochastic Anal 12, 429–434.
Lee, G. and Jeon, J. (2000). “A New Approach to an N/G/1 Queue.” Queueing Systems 35, 317–322.
Li, Q.L. (1996). “Queue System M/SM(PH/SM)/1 with Repairable Service Station.” Mathematical Applicata 9, 422–428.
Li, Q.L. and J. Cao. (2000). “The Repairable Queueing System MAP/PH(M/PH)/2 with Dependent Repairs.” Systems Sciences and Mathematical Science 20, 78–86.
Li, Q.L., M. Tan, and Y. Sun. (1999). “The SM/PH/1 Queue with Repairable Server of PH Lifetime.” In IFAC 14th Triennial World Congress,Vol. A. Beijing, P.R. China, 297–305.
Li, Q.L., D.J. Xu, and J. Cao. (1997). “Reliability Approximation of a Markov Queueing System with Server Breakdown and Repair.” Microelectron. Reliab. 37, 1203–1212.
Li, Q.L. and Y.Q. Zhao. (2002). “A Constructive Method for Finding β-invariant Measures for Transition Matrices of M/G/1 type.” In G. Latouche and P.G. Taylor (eds.), Matrix Analytic Methods Theory and Applications. World Scientific, pp. 237–264.
Li, Q.L. and Y.Q. Zhao. (2004). “The RG-factorization in block-structured Markov renewal process with applications.” In X. Zhu (ed.), Observation, Theory and Modeling of Atmospheric Variability. World Scientific, pp. 545–568.
Li, Q.L. and Y.Q. Zhao. (2003). “β-invariant Measures for Transition Matrices of GI/M/1 type.” Stochastic Models 19, 201–233.
Li, W., D.H. Shi, and X.L. Chao. (1997). “Reliability Analysis of M/G/1 Queueing Systems with Server Breakdowns and Vacations.” J. Appl. Prob. 34, 546–555.
Liang, H.M. and V.G. Kulkarni. (1993). “Stability Condition for a Single Retrial Queue.” Adv. in Appl. Prob. 25, 690–701.
Lucantoni, D.M. (1991). “New Results on the Single Server Queue with a Batch Markovian Arrival Process.” Stochastic Models 7, 1–46.
Lucantoni, D.M. (1993). “The BMAP/G/1 queue: A tutorial.” In L. Donatiello and R. Nelson (eds.), Models and Techniques for Performance Evaluation of Computer and Communication Systems Springer-Verlag: New York.
Lucantoni, D.M., G.L. Choudhury, and W. Whitt. (1994). “The Transient BMAP/G/1 Queue.” Stochastic Models 10, 145–182.
Lucantoni, D.M. and M.F. Neuts. (1994). “Some Steady-state Distributions for the MAP/SM/1 Queue.” Stochastic Models 10, 575–598.
Kulkarni, V.G. and B.D. Choi. (1990). “Retrial Queues with Server Subject to Breakdowns and Repairs.” Queueing Systems 7, 191–208.
Kulkarni, V.G. and H.M. Liang. (1997). “Retrial Queues Revisited.” In J.H. Dshalalow (ed.), Frontiers in Queueing: Models and Applications in Science and Engineering. CRC Press: Boca Raton, FL, pp. 19–34.
Mitrany, I.L. and B. Avi-Ttzhak. (1968). “A Many Server Queue with Service Interruptions.” Operations Research 16, 628–638.
Neuts, M.F. (1989). Structured Stochastic Matrices of M/G/1 Type and Their Applications. Marcel Decker Inc.: New York.
Neuts, M.F. (1995). “Matrix-analytic methods in the Theory of queues.” In J.H. Dshalalow (ed.), Advances in Queueing: Theory, Methods and Open Problems. CRC Press: Boca Raton, FL, pp. 265–292.
Neuts, M.F. and D.M. Lucantoni. (1979). “A Markovian Queue with N Servers Subject to Breakdowns and Repairs.” Managm. Sci. 25, 849–861.
Neuts, M.F. and B.M. Rao. (1990). “Numerical Investigation of a Multiserver Retrial Model.” Queueing Systems 7, 169–190.
Ramaswami, V. (1980). “The N/G/1 Queue and its Detailed Analysis.” Adv. Appl. Prob. 12, 222–261.
Wang, J., J. Cao, and Q.L. Li. (2001). “Reliability Analysis of the Retrial Queue with Server Breakdowns and Repairs.” Queueing Systems 38, 363–380.
Yang, T. and H. Li. (1994). “The M/G/1 Retrial Queue with the Server Subject to Starting Failures.” Queueing Systems 16, 83–96.
Yang, T. and J.G.C. Templeton. (1987). “A Survey on Retrial Queues.” Queueing Systems 2, 201–233.
Zhao, Y.Q. (2000). “Censoring technique in studying block-structured Markov chains.” In G. Latouche and P.G. Taylor (eds.), Advances in Algorithmic Methods for Stochastic Models. Notable Publications Inc.: New Jersey, pp. 417–433.
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Li, QL., Ying, Y. & Zhao, Y.Q. A BMAP/G/1 Retrial Queue with a Server Subject to Breakdowns and Repairs. Ann Oper Res 141, 233–270 (2006). https://doi.org/10.1007/s10479-006-5301-0
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DOI: https://doi.org/10.1007/s10479-006-5301-0