A Markovian growth-collapse model

Onno Boxma, David Perry, Wolfgang Stadje, Shelemyahu Zacks
2006 Advances in Applied Probability  
We consider growth-collapse processes (GCPs) that grow linearly between random partial collapse times, at which they jump down according to some distribution depending on their current level. The jump occurrences are governed by a state-dependent rate function r(x). We deal with the stationary distribution of such a GCP, (X t ) t≥0, and the distributions of the hitting times T a = inf{t ≥ 0 : X t = a}, a > 0. After presenting the general theory of these GCPs, several important special cases
more » ... re studied. We also take a brief look at the Markov-modulated case. In particular, we present a method of computing the distribution of min[T a , σ] in this case (where σ is the time of the first jump), and apply it to determine the long-run average cost of running a certain Markov-modulated disaster-ridden system.
doi:10.1017/s0001867800000884 fatcat:n3pigxjq3nb65pfbkd7mmtjkki