On the first-visit-time problem for birth and death processes with catastrophes.pdfVIP

a r X i v : m a t h / 0 3 0 7 2 0 6 v 1 [ m a t h .P R ] 1 5 J u l 2 0 0 3 On the first-visit-time problem for birth and death processes with catastrophes? A. Di Crescenzo(1), V. Giorno(1), A.G. Nobile(1), L.M. Ricciardi(2)? (1) Dipartimento di Matematica e Informatica Universita? di Salerno Via S. Allende, I-84081 Baronissi (SA), Italy. Email address: {adicrescenzo,giorno,nobile}@unisa.it (2) Dipartimento di Matematica e Applicazioni Universita? di Napoli Federico II Via Cintia, I-80126 Napoli, Italy. Email address: luigi.ricciardi@unina.it February 1, 2008 Abstract For a birth-death process subject to catastrophes, defined on the state- space S = {r, r+1, r+2, . . .}, with r a positive integer or zero, the first-visit time to a state k ∈ S is considered and the Laplace transform of its prob- ability density function is determined, use of which is then made to obtain mean and variance. The Laplace transform of the probability density function of the first effective catastrophe occurrence time and its expected value are also obtained. Some extensions to time-non-homogeneous pro- cesses are then provided. Finally, certain additional results concerning the determination of the steady-state distribution and the representation of the transition probabilities are worked out, while some applications to particular birth-death processes are shown in the Appendix. Keywords: birth-death processes; catastrophes; first-visit times. AMS 2000 Subject Classification: Primary 60J27 Secondary 60J80 1 Introduction Great attention has been paid in the literature to the description of the evo- lution of systems modeled via discrete state-space random processes such as ?short title: “Birth-death processes in the presence of catastrophes” ?Corresponding author 1 populations evolving in random environments or queueing and service systems under various operating protocols. However, with a few exceptions that will be indicated below, in these studies the systems under considerations