A continuous-time Markov chain under the influence of a regulating point process and applications in stochastic models with catastrophes
A continuous time Markov process with discrete state space \(S\) is modified by introducing additional jumps. They occur at time instants \(t_i\) and result in state \(A(t_i)\), \(i=1,2,\dots\), where \(A : [0,\infty)\to S\) is a given piecewise constant function. The time instants \(t_i\) may be random, i.e. be given by a point process \(X\) on \([0,\infty)\). It seems that the authors presume that the sequence has a.s. no finite accumulation points. They discuss several special cases, where the transition probabilities of the modified process can be determined more or less explicitly, for instance the case where \(X\) is a renewal process with special life time distributions, the case where the original Markov process is time homogeneous, is a birth-death process, or is the queueing process in a Markovian setup. Numerical results on the transient behavior of the modified process in some special examples are added.
scientific article; zbMATH DE number 1957063
scientific article