Publication: The batch Markovian arrival process subject to renewal generated geometric catastrophes
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2007
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Taylor & Francis
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We deal with a population of individuals that grows stochastically according to a batch Markovian arrival process and is subject to renewal generated geometric catastrophes. Our interest is in the semi-regenerative process that describes the population size at arbitrary times. The main feature of the underlying Markov renewal process is the block structure of its embedded Markov chain. Specifically, the embedded Markov chain at post-catastrophe epochs may be thought of as a Markov chain of GI/G1-type, which is indeed amenable to be studied through its R- and G-measures, and a suitably defined Markov chain of M/G/1-type. We present tractable formulae for a variety of probabilistic descriptors of the population, including the equilibrium distribution of the population size and the distribution of the time to extinction for present units at post-catastrophe epochs.
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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.
Bini, D.; Meini, B. On the solution of a nonlinear matrix equation arising in queueing problems.SIAM J. Math. Anal. 1996, 17, 906–926.
Bini, D.; Latouche, G.; Meini, B. Numerical Methods for Structured Markov Chains; Oxford University Press: Oxford, 2005.
Brockwell, P.J.; Gani, J.; Resnick, S.I. Birth, immigration and catastrophe processes. Adv. Appl.Prob. 1982, 14, 709–731.
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.
Cairns, B.; Pollett, P.K. Extinction times for a general birth, death and catastrophe process.J. Appl. Prob. 2004, 41, 1211–1218.
Chao, X.; Zheng, Y. Transient analysis of immigration birth-death processes with total catastrophes. Prob. Eng. Inf. Sci. 2003, 17, 83–106.
Chen, A.; Renshaw, E. The M/M/1 queue with mass exodus and mass arrivals when empty.J. Appl. Prob. 1997, 34, 192–207.
Çinlar, E. Introduction to Stochastic Processes; Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1975.
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.
Dudin, A.N.; Nishimura, S. A BMAP/SM/1 queueing system with Markovian arrival input of disasters. J. Appl. Prob. 1999, 36, 868–881.
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.
Economou, A. The compound Poisson immigration process subject to binomial catastrophes. J.Appl. Prob. 2004, 41, 508–523.
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.
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.
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.
Gómez-Corral, A. On a tandem G-network with blocking. Adv. Appl. Prob. 2002, 34, 626–661.
Grassmann, W.K.; Stanford, D.A. Matrix analytic methods. In Computational Probability;Grassmann, W.K., Ed.; Kluwer, 2000; 153–202.
Hanson, F.B.; Tuckwell, H.C. Population growth with randomly distributed jumps. J. Math. Biol.1997, 36, 169–187.
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.
Krishna Kumar, B.; Arivudainambi, D. Transient solution of an M/M/1 queue with catastrophes.Comp. Math. Appl. 2000, 40, 1233–1240.
Kulkarni, V.G. Modeling and Analysis of Stochastic Systems; Chapman and Hall: London, 1995.
Kyriakidis, E.G. The transient probabilities of the simple immigration-catastrophe process. Math.Scientist 2001, 26, 56–58.
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.
Latouche, G.; Ramaswami, V. Introduction to Matrix Analytic Methods in Stochastic Modeling; ASASIAM: Philadelphia, 1999.
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.
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.
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.
Lucantoni, D.M. New results on the single server queue with a batch Markovian arrival process.Stochastic Models 1991, 7, 1–46.
Neuts, M.F. A versatile Markovian point process. J. Appl. Prob. 1979, 16, 764–779.
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).
Neuts, M.F. Structured Stochastic Matrices of M/G/1 Type and Their Applications; Marcel Dekker, Inc.: New York, 1989.
Neuts, M.F. An interesting random walk on the non-negative integers. J. Appl. Prob. 1994, 31,48–58.
Ramaswami, V. A stable recursion for the steady state vector in Markov chains of M/G/1 type. Stochastic Models 1988, 4, 183–188.
Shafer, C.L. Inter-reserve distance. Biol. Conservation 2001, 100, 215–227.
Shaffer, M.L. Minimum population sizes for species conservation. BioScience 1981, 31, 131–134.
Swift, R.J. A simple immigration-catastrophe process. Math. Scientist 2000, 25, 32–36.
Van Doorn, E.A.; Zeifman, A.I. Extinction probability in a birth–death process with killing. J.Appl. Prob. 2005, 42, 185–198.
Van Doorn, E.A.; Zeifman, A.I. Birth–death processes with killing. Statistics and Probability Letters 2005, 72, 33–42.
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.
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.