The Batch Markovian Arrival Process Subject to Renewal Generated Geometric Catastrophes

REFERENCES

• Bartoszynski , R. ; Bühler , W.J. ; Chan , W. ; Pearl , D.K. Population processes under the influence of disasters occurring independently of population size . J. Math. Biol. 1989 , 27 , 167 – 178 .
   Web of Science ®Google Scholar
• Bini , D. ; Meini , B. On the solution of a nonlinear matrix equation arising in queueing problems . SIAM J. Math. Anal. 1996 , 17 , 906 – 926 .
   Web of Science ®Google Scholar
• Bini , D. ; Latouche , G. ; Meini , B. Numerical Methods for Structured Markov Chains ; Oxford University Press : Oxford , 2005 .
   Google Scholar
• Brockwell , P.J. ; Gani , J. ; Resnick , S.I. Birth, immigration and catastrophe processes . Adv. Appl. Prob. 1982 , 14 , 709 – 731 .
   Web of Science ®Google Scholar
• Brockwell , P.J. The extinction time of a birth, death and catastrophe process and of a related diffusion model . Adv. Appl. Prob. 1985 , 17 , 42 – 52 .
   Web of Science ®Google Scholar
• Cairns , B. ; Pollett , P.K. Extinction times for a general birth, death and catastrophe process . J. Appl. Prob. 2004 , 41 , 1211 – 1218 .
   Google Scholar
• Chao , X. ; Zheng , Y. Transient analysis of immigration birth-death processes with total catastrophes . Prob. Eng. Inf. Sci. 2003 , 17 , 83 – 106 .
   Web of Science ®Google Scholar
• Chen , A. ; Renshaw , E. The M/M/1 queue with mass exodus and mass arrivals when empty . J. Appl. Prob. 1997 , 34 , 192 – 207 .
   Web of Science ®Google Scholar
• Çinlar , E. Introduction to Stochastic Processes ; Prentice-Hall, Inc. : Englewood Cliffs , NJ , 1975 .
   Google Scholar
• Di Crescenzo , A. ; Giorno , V. ; Nobile , A.G. ; Ricciardi , L.M. On the M/M/1 queue with catastrophes and its continuous approximation . Queueing Systems 2003 , 43 , 329 – 347 .
   Web of Science ®Google Scholar
• Dudin , A.N. ; Nishimura , S. A BMAP/SM/1 queueing system with Markovian arrival input of disasters . J. Appl. Prob. 1999 , 36 , 868 – 881 .
   Web of Science ®Google Scholar
• Economou , A. ; Fakinos , D. A continuous-time Markov chain under the influence of a regulating point process and applications in stochastic models with catastrophes . Europ. J. Operat. Res. 2003 , 149 , 625 – 640 .
   Web of Science ®Google Scholar
• Economou , A. The compound Poisson immigration process subject to binomial catastrophes. J. Appl. Prob. 2004, 41, 508–523.
   Web of Science ®Google Scholar
• Gail , H.R. ; Hantler , S.L. ; Taylor , B.A. Spectral analysis of M/G/1 and G/M/1 type Markov chains . Adv. Appl. Prob. 1996 , 28 , 114 – 165 .
   Web of Science ®Google Scholar
• Gail , H.R. ; Hantler , S.L. ; Taylor , B.A. Non-skip-free M/G/1 and G/M/1 type Markov chains . Adv. Appl. Prob. 1997 , 29 , 733 – 758 .
   Web of Science ®Google Scholar
• Gail , H.R. ; Hantler , S.L. ; Taylor , B.A. Use of characteristic roots for solving infinite state Markov chains . In Computational Probability ; Grassmann , W.K. , Ed.; Kluwer , 2000 ; 205 – 254 .
   Google Scholar
• Gómez-Corral , A. On a tandem G-network with blocking . Adv. Appl. Prob. 2002 , 34 , 626 – 661 .
   Google Scholar
• Grassmann , W.K. ; Stanford , D.A. Matrix analytic methods . In Computational Probability ; Grassmann , W.K. , Ed.; Kluwer , 2000 ; 153 – 202 .
   Google Scholar
• Hanson , F.B. ; Tuckwell , H.C. Population growth with randomly distributed jumps . J. Math. Biol. 1997 , 36 , 169 – 187 .
   Web of Science ®Google Scholar
• Krinik , A. ; Rubino , G. ; Marcus , D. ; Swift , R.J. ; Kasfy , H. ; Lam , H. Dual processes to solve single server systems . J. Statist. Plann. Inference 2005 , 135 , 121 – 147 .
   Web of Science ®Google Scholar
• Krishna Kumar , B. ; Arivudainambi , D. Transient solution of an M/M/1 queue with catastrophes . Comp. Math. Appl. 2000 , 40 , 1233 – 1240 .
   Web of Science ®Google Scholar
• Kulkarni , V.G. Modeling and Analysis of Stochastic Systems ; Chapman and Hall : London , 1995 .
   Google Scholar
• Kyriakidis , E.G. The transient probabilities of the simple immigration-catastrophe process . Math. Scientist 2001 , 26 , 56 – 58 .
   Google Scholar
• Latouche , G. Algorithms for infinite Markov chains with repeating columns . In Linear Algebra, Markov Chains, and Queueing Models ; Meyer , C.D. , Plemmons , R.J. , Eds.; Springer , 1993 ; 231 – 265 .
   Google Scholar
• Latouche , G. ; Ramaswami , V. Introduction to Matrix Analytic Methods in Stochastic Modeling ; ASA-SIAM : Philadelphia , 1999 .
   Google Scholar
• Lee , C. The density of the extinction probability of a time homogeneous linear birth and death process under the influence of randomly occurring disasters . Math. Biosci. 2000 , 164 , 93 – 102 .
   PubMed Web of Science ®Google Scholar
• Li , Q.-L. ; Zhao , Y.Q. Light-tailed asymptotics of stationary probability vectors of Markov chains of GI/G/1-type . Adv. Appl. Prob. 2005 , 37 , 1075 – 1093 .
   Web of Science ®Google Scholar
• Lucantoni , D.M. ; Meier-Hellstern , K.S. ; Neuts , M.F. A single-server queue with server vacations and a class of non-renewal arrival processes . Adv. Appl. Prob. 1990 , 22 , 676 – 705 .
   Web of Science ®Google Scholar
• Lucantoni , D.M. New results on the single server queue with a batch Markovian arrival process . Stochastic Models 1991 , 7 , 1 – 46 .
   Google Scholar
• Neuts , M.F. A versatile Markovian point process . J. Appl. Prob. 1979 , 16 , 764 – 779 .
   Web of Science ®Google Scholar
• Neuts , M. F. Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach ; The Johns Hopkins University Press : Baltimore , 1981 . (Reprinted: Dover Publications: New York, 1994) .
   Google Scholar
• Neuts , M.F. Structured Stochastic Matrices of M/G/1 Type and Their Applications ; Marcel Dekker, Inc. : New York , 1989 .
   Google Scholar
• Neuts , M.F. An interesting random walk on the non-negative integers . J. Appl. Prob. 1994 , 31 , 48 – 58 .
   Web of Science ®Google Scholar
• Ramaswami , V. A stable recursion for the steady state vector in Markov chains of M/G/1 type . Stochastic Models 1988 , 4 , 183 – 188 .
   Google Scholar
• Shafer , C.L. Inter-reserve distance . Biol. Conservation 2001 , 100 , 215 – 227 .
   Web of Science ®Google Scholar
• Shaffer , M.L. Minimum population sizes for species conservation . BioScience 1981 , 31 , 131 – 134 .
   Web of Science ®Google Scholar
• Swift , R.J. A simple immigration-catastrophe process . Math. Scientist 2000 , 25 , 32 – 36 .
   Google Scholar
• Van Doorn , E.A. ; Zeifman , A.I. Extinction probability in a birth–death process with killing . J. Appl. Prob. 2005 , 42 , 185 – 198 .
   Google Scholar
• Van Doorn , E.A. ; Zeifman , A.I. Birth–death processes with killing . Statistics and Probability Letters 2005 , 72 , 33 – 42 .
   Google Scholar
• Zhao , Y.Q. Censoring technique in studying block-structured Markov chains . In Advances in Algorithmic Methods for Stochastic Models ; Latouche , G. , Taylor , P. , Eds.; Notable Publications, Inc. : NJ , 2000 ; 417 – 433 .
   Google Scholar
• Zhao , Y.Q. ; Li , W. ; Braun , W.J. Censoring, factorizations, and spectral analysis for transition matrices with block-repeating entries . Methodology and Computing in Applied Probability 2003 , 5 , 35 – 58 .
   Web of Science ®Google Scholar