Skeletons of branching processes are defined as trees of lineages characterized by an appropriate signature of future reproduction success. In the supercritical case a natural choice is to look for the lineages that survive forever (O'Connell (1993)). In the critical case it was suggested that the particles with the total number of descendants exceeding a certain threshold could be distinguished (see Sagitov (1997)). These two definitions lead to asymptotic representations of the skeletons as either pure birth process (in the slightly supercritical case) or critical birth-death processes (in the critical case conditioned on the total number of particles exceeding a high threshold value). The limit skeletons reveal typical survival scenarios for the underlying branching processes. In this paper we consider near-critical Bienaymé-Galton-Watson processes and define their skeletons using marking of particles. If marking is rare, such skeletons are approximated by birth and death processes, which can be subcritical, critical, or supercritical. We obtain the limit skeleton for a sequential mutation model (Sagitov and Serra (2009)) and compute the density distribution function for the time to escape from extinction.

In this paper, we investigate computable lower bounds for the best strongly ergodic rate of convergence of the transient probability distribution to the stationary distribution for stochastically monotone continuous-time Markov chains and reversible continuous-time Markov chains, using a drift function and the expectation of the first hitting time on some state. We apply these results to birth–death processes, branching processes and population processes.

On the basis of the Karlin and McGregor result, which states that the transition probability functions of a birth and death process can be expressed via the introduction of an orthogonal polynomial system and a spectral measure, we investigate in this paper how the Laplace transforms and the distributions of different transient characteristics related to excursions of a birth and death process can be expressed by means of the basic orthogonal polynomial system and the spectral measure. This allows us in particular to give a probabilistic interpretation of the series introduced by Stieltjes to study the convergence of the fundamental continued fraction associated with the system. Throughout the paper, we pay special attention to the case when the birth and death process is ergodic. Under the assumption that the spectrum of the spectral measure is discrete, we show how the distributions of different random variables associated with excursions depend on the fundamental continued fraction, the orthogonal polynomial system and the spectral measure.

We provide a probabilistic proof of the Stein's factors based on properties of birth and death Markov chains, solving a tantalizing puzzle in using Markov chain knowledge to view the celebrated Stein–Chen method for Poisson approximations. This work complements the work of Barbour (1988) for the case of Poisson random variable approximation.

The time until extinction for the closed SIS stochastic logistic epidemic model is investigated. We derive the asymptotic distribution for the extinction time as the population grows to infinity, under different initial conditions and for different values of the infection rate.

This paper considers the asymptotic distribution of the maximum number of infectives in an epidemic model by showing that, as the initial number of susceptibles converges to infinity, the process of infectives converges almost surely to a birth and death process. The model studied here is more general than usual (see e.g. Bailey (1975), Bharucha-Reid (1960), Keilson (1979)) in that it incorporates immigration and the limiting birth and death process is non-linear. The main novelty of the present paper is the martingale approach used to prove the above-mentioned convergence.

We study conditions for the existence of non-trivial quasi-stationary distributions for the birth-and-death chain with 0 as absorbing state. We reduce our problem to a continued fractions one that can be solved by using extensions of classical results of this theory. We also prove that there exist normalized quasi-stationary distributions if and only if 0 is geometrically absorbing.

Asymptotic approximations and numerical computations are used to estimate the accuracy of the diffusion approximation for the expected time to extinction for some stochastic processes. The results differ for processes with a continuant transition matrix (e.g. a birth and death process), and those with a noncontinuant transition matrix (e.g. a non-linear branching process). In the latter case, the diffusion equation does not hold near the point of exit. Consequently, high-order corrections do not result in substantial improvement over the diffusion approximation.

This is a study of simple random walks, birth and death processes, and M/M/s queues that have transition probabilities and rates that are sequentially controlled at jump times of the processes. Each control action yields a one-step reward depending on the chosen probabilities or transition rates and the state of the process. The aim is to find control policies that maximize the total discounted or average reward. Conditions are given for these processes to have certain natural monotone optimal policies. Under such a policy for the M/M/s queue, for example, the service and arrival rates are non-decreasing and non-increasing functions, respectively, of the queue length. Properties of these policies and a linear program for computing them are also discussed.

A generalization to continuous time is given for a discrete-time model of a birth and death process in a random environment. Some important properties of this process in the continuous-time setting are stated and proved including instability and extinction conditions, and when suitable absorbing barriers have been defined, methods are given for the calculation of extinction probabilities and the expected duration of the process.

This article deals with the generalized Smoluchowski process, {n(t), t ≧ 0}, defined by the temporal fluctuating of the numbers of randomly moving particles contained in m < ∞ disjoint regions of space. The relationship of the Smoluchowski process {n(t), t ≧ 0} to the emigration–immigration process is discussed and conditions for their equivalence are presented.

We prove and extend Burke's output theorem for queues in equilibrium with the help of a filtering formula giving the time evolution of the least square estimate of the state with respect to the output.

We consider the problem of product pricing when the firm's market share is changing stochastically according to a birth and death process. The current market share together with the price prevailing determine the current rate of profit made as well as the birth and death rates. The optimal pricing policy must balance the immediate advantage of setting a high price in terms of increased current profit against the disadvantage in terms of a possible erosion of the future market share. We formulate a continuous-time Markov decision model and analyse it using a recent technique developed by Lippman [6] for optimization of exponential queueing systems. The optimal pricing policy is characterized as having a sort of monotonicity property. We also analyse the dependence of the optimal policy on the problem parameters and indicate further extensions of the model.

Let {XN(t)} be a sequence of continuous time Markov population processes on an n-dimensional integer lattice, such that XN has initial state Nx(0) and has a finite number of possible transitions J from any state X: let the transition X → X + J have rate NgJ(N–1X), and let gJ(x) and x (0) be fixed as N varies. The rate of convergence of √N(N–1XN(t) — ζ (t)) to a Gaussian diffusion is investigated, where ζ(t) is the deterministic approximation to N–1XN(t), and a method of deriving higher order asymptotic expansions for its distribution is justified. The methods are applied to two birth and death processes, and to the closed stochastic epidemic.