The Batch Markovian Arrival Process Subject to Renewal Generated Geometric Catastrophes

Abstract

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/G/1-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.

Keywords:

• Batch Markovian arrival process
• Extinction time
• Geometric catastrophes
• Markov chain of GI/G/1-type
• Markov chain of M/G/1-type
• Population processes
• RG-factorization
• Stationary distribution

Mathematics Subject Classification:

• Primary 60K20
• Secondary 60J10, 60K15

ACKNOWLEDGMENTS

The authors thank the associate editor and two anonymous referees for valuable comments and remarks, and acknowledge that this work was supported by the DGINV through project no. MTM2005-01248. A. Economou was also supported by the University of Athens grant ELKE/70/4/6415 and by the European Union and the Greek Ministry of Education program PYTHA-GORAS/2004.