A Markovian growth-collapse model

Abstract

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, (Xt)t≥0, and the distributions of the hitting times Ta = inf{t ≥ 0 : Xt = a}, a > 0. After presenting the general theory of these GCPs, several important special cases are studied. We also take a brief look at the Markov-modulated case. In particular, we present a method of computing the distribution of min[Ta, 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.

TCP and iso-stationary transformations

“…The processes in this paper belong to the special class of growth-collapse processes for which we refer to [6].…”

Hitting Times and the Running Maximum of Markovian Growth-Collapse Processes

“…We have b ≥ 0 because ν(x) is nondecreasing. By Proposition 3 in [8], X t is ergodic if lim sup x→∞ λ(x)…”

“…MGCPs can be encountered in a large variety of applications, of which we mention growth population models, risk processes, neuron firing, and window sizes in transmission control protocols, and have been studied in [14,8,7,28]. They form a special class of piecewise deterministic Markov processes [11,10,6].…”

“…Propositions 2 and 3 in [8] imply that for an MGCP Eτ n y < ∞ for all y ≥ 0 and all n ∈ N and that X t is ergodic if lim sup x→∞ λ(x) x 0 µ x (y) dy > 1. Moreover, it can be shown (see e.g.…”

On Growth-Collapse Processes with Stationary Structure and Their Shot-Noise Counterparts

“…For recent papers on growth-collapse models and their applications, see [1], [2], [3], [5], [6], [8], [9], and the references therein.…”