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We introduce a class of branching processes in which the reproduction or lifetime distribution at a given time depends on the total cumulative number of individuals who have been born in the population until that time. We focus on a continuous-time version of these processes, called total-progeny-dependent birth-and-death processes, and study some of their properties through the analysis of their deterministic (fluid) approximation. These properties include the maximum population size, the total progeny size at extinction, the time to reach the maximum population size, and the time until extinction. As the fluid approximation does not allow us to determine the time until extinction directly, we propose several methods to complement this approach. We also use the deterministic approach to study the behavior of the processes as we increase the magnitude of the individual's birth rate. [ABSTRACT FROM AUTHOR]

The setting considered in this paper concerns a discrete-time multivariate population process under Markov modulation. Our objective is to estimate the model parameters, based on periodic observations of the network population vector. These parameters relate to the arrival, routing and departure processes, but also to the (unobservable) Markovian background process. When opting for the classical likelihood-based approach, the evaluation of the likelihood is problematic. We show however, how an accurate saddlepoint approximation can be used. Numerical experiments illustrate our method and show that even under relatively complicated conditions the parameters are estimated relatively precisely. [ABSTRACT FROM AUTHOR]

The theory of matrix-analytic methods, pioneered by Professor Marcel Neuts (1935-2014), the founding editor of I Stochastic Models i , is an area of applied probability that focuses on constructing methods of analysis that can be implemented using efficient algorithms and allow for practical numerical analysis of many real-life systems. The seven papers contained in this issue cover a broad range of research in the areas of quasi-birth-and-death processes, phase-type and matrix exponential distributions, continuous-state Markov processes, and applications. Gábor Horváth, Illés Horváth and Miklós Telek in "High order concentrated matrix-exponential distributions" construct explicitly a family of matrix-exponential distributions that are highly concentrated, i.e., their coefficient of variation is as small as possible. [Extracted from the article]

Algorithmic methods developed for structured Markov chains are extended to solve questions about a class of continuous-time branching processes calledMarkovian binary trees(MBTs). This approach allows us to compute the extinction probability of an MBT, the distribution of its maximal size given extinction, the expected time until extinction, and the mean time until the tree reaches a given size. The resulting algorithms are efficient in practice for processes of relatively small dimension. The numerical methods are illustrated on MBTs evolving in a Markovian random environment. [ABSTRACT FROM AUTHOR]

We study transient features and the extinction probability of a particular class of multi-type Markovian branching processes called Markovian binary trees, subject at random epochs to catastrophes, controlled by a Markovian arrival process, and killing random numbers of living individuals. We provide two types of probabilistic methods to numerically compute the extinction probability: the first one is based on an integral equation for the probability generating function of the population size, and the second one is based on the correspondence between Markovian branching processes with catastrophes and structured Markov chains of G/M/1-type. [ABSTRACT FROM AUTHOR]