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In this article we provide new results for the asymptotic behavior of a time-fractional birth and death process $N_{\alpha}(t)$ , whose transition probabilities $\mathbb{P}[N_{\alpha}(t)=\,j\mid N_{\alpha}(0)=i]$ are governed by a time-fractional system of differential equations, under the condition that it is not killed. More specifically, we prove that the concepts of quasi-limiting distribution and quasi-stationary distribution do not coincide, which is a consequence of the long-memory nature of the process. In addition, exact formulas for the quasi-limiting distribution and its rate convergence are presented. In the first sections, we revisit the two equivalent characterizations for this process: the first one is a time-changed classic birth and death process, whereas the second one is a Markov renewal process. Finally, we apply our main theorems to the linear model originally introduced by Orsingher and Polito [23]. [ABSTRACT FROM AUTHOR]

In this paper we introduce new birth-and-death processes with partial catastrophe and study some of their properties. In particular, we obtain some estimates for the mean catastrophe time, and the first and second moments of the distribution of the process at a fixed time t. This is completed by some asymptotic results. [ABSTRACT FROM AUTHOR]

We consider the distribution of the age of an individual picked uniformly at random at some fixed time in a linear birth-and-death process. By exploiting a bijection between the birth-and-death tree and a contour process, we derive the cumulative distribution function for this distribution. In the critical and supercritical cases, we also give rates for the convergence in terms of the total variation and other metrics towards the appropriate exponential distribution. [ABSTRACT FROM AUTHOR]

In this paper we consider time-changed models of population evolution Xf (t) = X(Hf (t)), where Xf is a counting process and Hf is a subordinator with Laplace exponent f. In the case where X is a pure birth process, we study the form of the distribution, the intertimes between successive jumps, and the condition of explosion (also in the case of killed subordinators). We also investigate the case whereXrepresents a death process (linear or sublinear) and study the extinction probabilities as a function of the initial population size n0. Finally, the subordinated linear birth-death process is considered. Special attention is devoted to the case where birth and death rates coincide; the sojourn times are also analysed. [ABSTRACT FROM AUTHOR]

We study the conditions for positive recurrence and transience of multi-dimensional birth-and-death processes describing the evolution of a large class of stochastic systems, a typical example being the randomly varying number of flow-level transfers in a telecommunication wire-line or wireless network. First, using an associated deterministic dynamical system, we provide a generic method to construct a Lyapunov function when the drift is a smooth function on RN. This approach gives an elementary and direct proof of ergodicity. We also provide instability conditions. Our main contribution consists of showing how discontinuous drifts change the nature of the stability conditions and of providing generic sufficient stability conditions having a simple geometric interpretation. These conditions turn out to be necessary (outside a negligible set of the parameter space) for piecewise constant drifts in dimension two. [ABSTRACT FROM AUTHOR]

Markovian systems with multiple interacting subsystems under the influence of a control unit are considered. The state spaces of the subsystems are countably infinite, whereas that of the control unit is finite. A recent infinite level-dependent quasi-birth-and-death model for such systems is extended by facilitating the automatic representation and generation of the nonzero blocks in its underlying infinitesimal generator matrix with sums of Kronecker products. Experiments are performed on systems of stochastic chemical kinetics having two or more countably infinite state space subsystems. Results indicate that, even though more memory is consumed, there are many cases where a matrix-analytic solution coupled with Lyapunov theory yields a faster and more accurate steady-state measure compared to that obtained with simulation. [ABSTRACT FROM AUTHOR]

For a sequence of finite state space birth--death processes, each having a single absorbing state, we show that, under certain conditions, as the size of the state space tends to infinity, the quasistationary distributions converge to the stationary distribution of a limiting infinite state space birth-death process. This generalizes a result of Keilson and Ramaswamy by allowing birth and death rates to depend upon the size of the state space. We give sufficient conditions under which the convergence result of Keilson and Ramaswamy remains valid. The generalization allows us to apply our convergence result to examples from population biology: a Pearl-Verhulst logistic population growth model and the susceptible-infective-susceptible (SIS) model for infectious spread. The limit distributions obtained suggest new finite-population approximations to the quasistationary distributions of these models, obtained by the method of cumulant closure. The new approximations are found to be both simple in form and accurate. [ABSTRACT FROM AUTHOR]

We supply some relations that establish intertwining from duality and give a probabilistic interpretation. This is carried out in the context of discrete Markov chains, fixing up the background of previous relations established for monotone chains and their Siegmund duals. We revisit the duality for birth-and-death chains and the nonneutral Moran model, and we also explore the duality relations in an ultrametric-type dual that extends the Siegmund kernel. Finally, we discuss the sharp dual, following closely the Diaconis-Fill study. [ABSTRACT FROM AUTHOR]

In this paper we describe methods for randomly thinning certain classes of spatial point processes. In the case of a Markov point process, the proposed method involves a dependent thinning of a spatial birth-and-death process, where clans of ancestors associated with the original points are identified, and where we simulate backwards and forwards in order to obtain the thinned process. In the case of a Cox process, a simple independent thinning technique is proposed. In both cases, the thinning results in a Poisson process if and only if the true Papangelou conditional intensity is used, and, thus, can be used as a graphical exploratory tool for inspecting the goodness-of-fit of a spatial point process model. Several examples, including clustered and inhibitive point processes, are considered. [ABSTRACT FROM AUTHOR]

We propose the following simple stochastic model for phylogenetic trees. New types are born and die according to a birth and death chain. At each birth we associate a fitness to the new type sampled from a fixed distribution. At each death the type with the smallest fitness is killed. We show that if the birth (i.e. mutation) rate is subcritical, we obtain a phylogenetic tree consistent with an influenza tree (few types at any given time and one dominating type lasting a long time). When the birth rate is supercritical, we obtain a phylogenetic tree consistent with an HIV tree (many types at any given time, none lasting very long). [ABSTRACT FROM AUTHOR]

In this paper we consider a class of quasi-birth-and-death processes for which explicit solutions can be obtained for the rate matrix R and the associated matrix G. The probabilistic interpretations of these matrices allow us to describe their elements in terms of paths on the two-dimensional lattice. Then determining explicit expressions for the matrices becomes equivalent to solving a lattice path counting problem, the solution of which is derived using path decomposition, Bernoulli excursions, and hypergeometric functions. A few applications are provided, including classical models for which we obtain some new results. [ABSTRACT FROM AUTHOR]

Consider a continuous nonnegative random variable Xwith mean ? and hazard function h. Assume further that a?h(t)?bfor all t?0. Under these constraints, we obtain sharp two-sided bounds for $\bar{F}(t) = \hbox{Pr}(X \gt t)$. An application to birth and death processes is discussed. [ABSTRACT FROM AUTHOR]

We present bounds on the decay parameter for absorbing birth-death processes adapted from results of Chen (2000), (2001). We address numerical issues associated with computing these bounds, and assess their accuracy for several models, including the stochastic logistic model, for which estimates of the decay parameter have been obtained previously by Nåsell (2001). [ABSTRACT FROM AUTHOR]

We complete a paper written by Edward Pollak in 1974 on a multitype branching process the generating functions of whose birth law are fractional linear functions with the same denominator. The main tool is a parameterization of these functions adapted using the mean matrix M and an element w of the first quadrant. We use this opportunity to give a self-contained presentation of Pollak's theory. [ABSTRACT FROM AUTHOR]

We study the genealogy of so-called immortal branching processes, i.e. branching processes where each individual upon death is replaced by at least one new individual, and conclude that their marginal distributions are compound geometric. The result also implies that the limiting distributions of properly scaled supercritical branching processes are compound geometric. We exemplify our results with an expression for the marginal distribution for a class of branching processes that have recently appeared in the theory of coalescent processes and continuous stable random trees. The limiting distribution can be expressed in terms of the Fox H-function, and in special cases by the Meijer G-function. [ABSTRACT FROM AUTHOR]

In this paper we introduce the concepts of instantaneous reversibility and instantaneous entropy production rate for inhomogeneous Markov chains with denumerable state spaces. The following statements are proved to be equivalent: the inhomogeneous Markov chain is instantaneously reversible; it is in detailed balance; its entropy production rate vanishes. In particular, for a time-periodic birth-death chain, which can be regarded as a simple version of a physical model (Brownian motors), we prove that its rotation number is 0 when it is instantaneously reversible or periodically reversible. Hence, in our model of Markov chains, the directed transport phenomenon of Brownian motors can occur only in nonequilibrium and irreversible systems. [ABSTRACT FROM AUTHOR]

We consider the class of level-independent quasi-birth-and-death (QBD) processes that have countably many phases and generator matrices with tridiagonal blocks that are themselves tridiagonal and phase independent. We derive simple conditions for possible decay rates of the stationary distribution of the ‘level’ process. It may be possible to obtain decay rates satisfying these conditions by varying only the transition structure at level 0. Our results generalize those of Kroese, Scheinhardt, and Taylor, who studied in detail a particular example, the tandem Jackson network, from the class of QBD processes studied here. The conditions derived here are applied to three practical examples. [ABSTRACT FROM AUTHOR]

In this study, we characterize the equilibrium behavior of spatial migration processes that represent population migrations, or birth-death processes, in general spaces. These processes are reversible Markov jump processes on measure spaces. As a precursor, we present fundamental properties of reversible Markov jump processes on general spaces. A major result is a canonical formula for the stationary distribution of a reversible process. This involves the characterization of two-way communication in transitions, using certain Radon-Nikodým derivatives. Other results concern a Kolmogorov criterion for reversibility, time reversibility, and several methods of constructing or identifying reversible processes. [ABSTRACT FROM AUTHOR]

We study birth-death processes on the nonnegative integers, where {1, 2,.. } is an irreducible class and 0 an absorbing state, with the additional feature that a transition to state 0 may occur from any state. We give a condition for absorption (extinction) to be certain and obtain the eventual absorption probabilities when absorption is not certain. We also study the rate of convergence, as t → ∞, of the probability of absorption at time t, and relate it to the common rate of convergence of the transition probabilities that do not involve state 0. Finally, we derive upper and lower bounds for the probability of absorption at time it by applying a technique that involves the logarithmic norm of an appropriately defined operator. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of bisexual Gallon-Watson branching processes in which the law of offspring distribution is dependent on the population size. Under a suitable condition on the offspring distribution, we prove that the limit of mean growth-rate per mating unit exists. Based on this limit, we give a criterion to identify whether the process admits ultimate extinction with probability one. [ABSTRACT FROM AUTHOR]

As proposed by Irle and Gani in 2001, a process X is said to be slower in level crossing than a process Y if it takes X stochastically longer to exceed any given level than it does Y. In this paper, we extend a result of file (2003), relative to the level crossing ordering of uniformizable skip-free-to-the-right continuous-time Markov chains, to derive a new set of sufficient conditions for the level crossing ordering of these processes. We apply our findings to birth-death processes with and without catastrophes, and MItvIIsIc systems. [ABSTRACT FROM AUTHOR]

An explicit, computable, and sufficient condition for exponential ergodicity of single- birth processes is presented. The corresponding criterion for birth-death processes is proved using a new method. As an application, some sufficient conditions are obtained for exponential ergodicity of an extended class of continuous-time branching processes and of multidimensional Q-processes, by comparison methods. [ABSTRACT FROM AUTHOR]

The eigentime identity is proved for continuous-time reversible Markov chains with Markov generator L. When the essential spectrum is empty, let {0 = λ0 < λ1 > λ2 > …} be the whole spectrum of L in L2. Then Ξn&les;1 λn-1 < ∞ implies the existence of the spectral gap a of L in L∞. Explicit formulae are presented in the case of birth-death processes and from these formulae it is proved that Ξn&les;1 λn-1 < ∞ if and only if α >0. [ABSTRACT FROM AUTHOR]

We consider an infinite-capacity second-order fluid queue governed by a continuous- time Markov chain and with linear service rate. The variability of the traffic is modeled by a Brownian motion and a local variance function modulated by the Markov chain and proportional to the fluid level in the queue. The behavior of this second-order fluid-flow model is described by a linear stochastic differential equation, satisfied by the transient queue level. We study the transient level's convergence in distribution under weak assumptions and we obtain an expression for the stationary queue level. For the first-order case, we give a simple expression of all its moments as well as of its Laplace transform. For the second-order model we compute its first two moments. [ABSTRACT FROM AUTHOR]

In this article, we analyse the error induced by the Euler scheme combined with a symmetry procedure near the boundary for the simulation of diffusion processes with an oblique reflection on a smooth boundary. This procedure is easy to implement and, in addition, accurate: indeed, we prove that it yields a weak rate of convergence of order 1 with respect to the time-discretization step. [ABSTRACT FROM AUTHOR]

We consider a fluid queue controlled by a semi-Markov process and we apply the Markov- renewal approach developed earlier in the context of quasi-birth-and-death processes and of Markovian fluid queues. We analyze two subfamilies of semi-Markov processes. In the first family, we assume that the intervals during which the input rate is negative have an exponential distribution. In the second family, we take the complementary case and assume that the intervals during which the input rate is positive have an exponential distribution. We thoroughly characterize the structure of the stationary distribution in both cases. [ABSTRACT FROM AUTHOR]

It is shown that the method of deriving bounds on the rate of convergence for birth-death processes developed by Zeifman can be effectively applied to stochastic models of chemical reactions. [ABSTRACT FROM AUTHOR]

A new structure with the special property that instantaneous resurrection and mass disaster are imposed on an ordinary birth-death process is considered. Under the condition that the underlying birth-death process is exit or bilateral, we are able to give easily checked existence criteria for such Markov processes. A very simple uniqueness criterion is also established. All honest processes are explicitly constmcted. Ergodicity properties for these processes are investigated. Surprisingly, it can be proved that all the honest processes are not only recurrent but also ergodic without imposing any extra conditions. Equilibrium distributions are then established. Symmetry and reversibility of such processes are also investigated. Several examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]

We study the λ-classification of absorbing birth-and-death processes, giving necessary and sufficient conditions for such processes to be λ-transient, λ-null recurrent and λ-positive recurrent. [ABSTRACT FROM AUTHOR]

For Markov processes on the positive integers with the origin as an absorbing state, Ferrari, Kesten, Martínez and Picco studied the existence of quasi-stationary and limiting conditional distributions by characterizing quasi-stationary distributions as fixed points of a transformation Φ on the space of probability distributions on {1, 2, ...}. In the case of a birth-death process, the components of Φ (v) can be written down explicitly for any given distribution v. Using this explicit representation, we will show that Φ preserves likelihood ratio ordering between distributions. A conjecture of Kryscio and Lefèvre concerning the quasi-stationary distribution of the SIS logistic epidemic follows as a corollary. [ABSTRACT FROM AUTHOR]

Applies coupling methods to study strong ergodicity for Markov processes. Expectations of coupling times; Criteria for one-dimensional diffusions and birth-death processes; Diffusions on compact manifolds.

Discusses general and unifying approaches for computing fundamental characteristics of both continuous-time and discrete-time birth-death processes. Overview of related literature; Discussion on the theory of birth-death processes; Application of a simple conditioning agreement to both continuous- and discrete-time birth-death process.

Presents a study which focused on the convergence to stationarity of birth-death processes. Preliminary results of birth-death processes; Main results; Discussion on birth-death processes in discrete time.

Presents a study which investigated the convergence and limits of certain matrix sequences that arise in a quasi-birth-and-death Markov chains. Results from nonnegative matrix theory; Probabilistic interpretations and equations for matrix sequence.

Discusses the level-phase independence for GI/M/1-type Markov chains. Statement of the problem; Construction of the matrix for a quasi-birth-and-death process; Uniqueness of GI/M1-type Markov chains.

Determines the conditions on the birth and death rates of a birth-death process for the process to be a member of a family of similar processes. Families of similar birth-death processes; Birth-death processes with finite state space.

Examines the application of the perfect simulation Coupling from the Past method to locally stable point processes. Equilibrium distributions of spatial birth-and-death processes; Accounts of the discrete-time Metropolis-Hastings samplers for point processes; Incorporation of perfect simulation into Strauss point process.

Examines the relationship between birth-and-death processes, lattice path combinatorics and continued fractions. Creation of a correspondence between birth-and-death processes and Stieltjes-Jacobi continued fraction; Correlation of weighted lattice paths to continued fractions; Accounts of obtaining Laplace transforms of transient characteristics.

Presents information on a study which investigated the Laplace theory. Transient characteristics associated with the birth and death processes; Methodology of the study; Discussion on the results; Conclusion.

Provides a probabilistic proof of the Stein's factors based on Markov birth and death Markov chains. Why the Stein's factors are important in applying the Stein-Chem method in Poisson approximation; Estimation of the Stein's factors.

Discusses the standard bivariate death process and other Markovian modification and extensions that are of interest in the field of epidemic modeling. Development of an approach that permits the derivation of the exact distribution of the population state at any point in time; Derivation of a central system of relations with two indices for the temporal state distribution.

Describes a spatial symmetry for first-passage-time (FPT) problem for birth-and-death (BD) processes. Details on BD processes; Analysis of truncated BD processes; Symmetry properties for truncated BD processes.

Discusses regularity and reversibility for birth-death-migration processes. Application of Markov chains; Conditions for recurrence; Correlation of geographic distribution and dynamics of population growth.

Analyzes limiting conditional distributions for birth-death processes. Quasi-stationary distributions for a non-explosive; Evanescent processes; Certainty of absorption; Transition rates; Kolmogorov differential equations.