Alternative Approaches for the Transient Analysis of Markov Chains with Catastrophes

In this paper we present various approaches for the transient analysis of a Markovian population process with total catastrophes. We discuss the pros and the cons of these methodologies and point out how they lead to different tractable extensions. As an illustrating example, we consider the nonhomogeneous Poisson process with total catastrophes. The extension of the probabilistic methodologies for analyzing models with binomial catastrophes is also discussed.

[1]  Gerardo Rubino,et al.  Dual processes to solve single server systems , 2005 .

[2]  H. Tuckwell,et al.  Population growth with randomly distributed jumps , 1997 .

[3]  J. Gani,et al.  Birth, immigration and catastrophe processes , 1982, Advances in Applied Probability.

[4]  H J Bremermann,et al.  A competitive exclusion principle for pathogen virulence , 1989, Journal of mathematical biology.

[5]  Robert Bartoszyński,et al.  Population processes under the influence of disasters occurring independently of population size , 1989 .

[6]  A. Krinik,et al.  Birth-multiple catastrophe processes , 2007 .

[7]  J R Artalejo,et al.  Evaluating growth measures in an immigration process subject to binomial and geometric catastrophes. , 2007, Mathematical biosciences and engineering : MBE.

[8]  Joseph Gani,et al.  Death and Birth-Death and Immigration Processes with Catastrophes , 2007 .

[9]  Antonis Economou,et al.  The compound Poisson immigration process subject to binomial catastrophes , 2004, Journal of Applied Probability.

[10]  Randall J. Swift,et al.  TRANSIENT PROBABILITIES FOR A SIMPLE BIRTH-DEATH-IMMIGRATION PROCESS UNDER THE INFLUENCE OF TOTAL CATASTROPHES , 2001 .

[11]  Fabio Spizzichino,et al.  Time-lagged point processes with the order-statistics property , 2000, Math. Methods Oper. Res..

[12]  Antonis Economou,et al.  A continuous-time Markov chain under the influence of a regulating point process and applications in stochastic models with catastrophes , 2003, Eur. J. Oper. Res..

[13]  Xiuli Chao,et al.  TRANSIENT ANALYSIS OF IMMIGRATION BIRTH–DEATH PROCESSES WITH TOTAL CATASTROPHES , 2003, Probability in the Engineering and Informational Sciences.

[14]  Paul J. Schweitzer,et al.  Stochastic Models, an Algorithmic Approach , by Henk C. Tijms (Chichester: Wiley, 1994), 375 pages, paperback. , 1996, Probability in the Engineering and Informational Sciences.

[15]  Vaidyanathan Ramaswami,et al.  Introduction to Matrix Analytic Methods in Stochastic Modeling , 1999, ASA-SIAM Series on Statistics and Applied Mathematics.

[16]  Alan C. Krinik,et al.  Transient probability functions of finite birth–death processes with catastrophes , 2007 .

[17]  C. Lee,et al.  The density of the extinction probability of a time homogeneous linear birth and death process under the influence of randomly occurring disasters. , 2000, Mathematical biosciences.

[18]  Prem S. Puri.  ON THE CHARACTERIZATION OF POINT PROCESSES WITH THE ORDER STATISTIC PROPERTY WITHOUT THE MOMENT CONDITION , 1982 .

[19]  B. Krishna Kumar,et al.  Transient solution of an M/M/1 queue with catastrophes , 2000 .

[20]  E. G. Kyriakidis.  Stationary probabilities for a simple immigration-birth-death process under the influence of total catastrophes , 1994 .

[21]  D. Stirzaker.  PROCESSES WITH RANDOM REGULATION , 2006, Probability in the Engineering and Informational Sciences.

[22]  P. J. Brockwell.  The extinction time of a general birth and death process with catastrophes , 1986 .