Bayesian networks are convenient graphical expressions for high-dimensional probability distributions which represent complex relationships between a large number of random variables. They have been employed extensively in areas such as bioinformatics, artificial intelligence, diagnosis, and risk management. The recovery of the structure of a network from data is of prime importance for the purposes of modeling, analysis, and prediction. There has been a great deal of interest in recent years in the NP-hard problem of learning the structure of a Bayesian network from observed data. Typically, one assigns a score to various structures and the search becomes an optimization problem that can be approached with either deterministic or stochastic methods. In this paper, we introduce a new search strategy where one walks through the space of graphs by modeling the appearance and disappearance of edges as a birth and death process. We compare our novel approach with the popular Metropolis–Hastings search strategy and give empirical evidence that the birth and death process has superior mixing properties.

This article provides entropic inequalities for binomial-Poisson distributions, derived from the two point space. They appear as local inequalities of the M/M/∞ queue. They describe in particular the exponential dissipation of Φ-entropies along this process. This simple queueing process appears as a model of “constant curvature”, and plays for the simple Poisson process the role played by the Ornstein-Uhlenbeck process for Brownian Motion. Some of the inequalities are recovered by semi-group interpolation. Additionally, we explore the behaviour of these entropic inequalities under a particular scaling, which sees the Ornstein-Uhlenbeck process as a fluid limit of M/M/∞ queues.Proofs are elementary and rely essentially on the development of a “Φ-calculus”.

In this paper we obtain the rate of convergence to equilibrium of a class of interacting marked point processes, introduced by Kerstan, in two different situations. Indeed, we prove the exponential and subexponential ergodicity of such a class of stochastic processes. Our results are an extension of the corresponding results of Brémaud, Nappo and Torrisi. The generality of the dynamics which we take into account allows the application to the so-called loss networks, and multivariate birth and death processes.

A probabilistic proof of a Stein factor bound is given, extending the Poisson case in Xia (1999).

This paper is concerned with a model for the spread of an epidemic in a closed, homogeneously mixing population in which new infections occur at rate f(x, y) and removals occur at rate g(x, y), where x and y are the numbers of susceptible and infective individuals, respectively, and f and g are arbitrary but specified positive real-valued functions. Sequences of such epidemics, indexed by the initial number of susceptibles n, are considered and conditions are derived under which the epidemic processes converge almost surely to a birth and death process as n tends to infinity. Thus a threshold theorem for such an epidemic model is obtained. The results are extended to models which incorporate immigration and emigration of susceptibles. The theory is illustrated by several examples of models taken from the epidemic literature. Generalizations to multipopulation epidemics are discussed briefly.

Two families of associated Wilson polynomials are introduced. Both families are birth and death process polynomials, satisfying the same recurrence relation but having different initial conditions. Contiguous relations for generalized hypergeometric functions of the type 7F6 are derived and used to find explicit representations for the polynomials and to compute the corresponding continued fractions. The absolutely continuous components of the orthogonality measures of both families are computed. Generating functions are also given.

We discuss the quasi-stationary distribution obtained when a simple birth and death process is conditioned on never exceeding K. An application of this model to one-dimensional circuit-switched communication networks is described, and some special cases examined.

Consider a network of queues in equilibrium. When are the flows observed on two given links of that network equivalent, i.e., when do they have the same law?

The verification of the equivalence of two such point processes is first reduced to an algebraic problem by a technique based on the filtering theory.

This method is then used to show that the arrival and departure processes at an M/M/1 node in a Jackson network are not always equivalent, thereby contradicting a conjecture made in [8]. An example where that equivalence holds is also given; it provides new results on the time reversibility of some familiar processes.

Previous work on the concept of a limiting conditional age distribution of a discrete-state continuous-time Markov process with one absorbing state is generalised. The generalisation allows this process to have a finite number of absorbing states and the associated return process to have an arbitrary initial distribution on the transient states of the absorbing process. If the return process is ρ-recurrent, possesses the strong ratio limit property and satisfies some further requirements then the limiting age distribution exists. The proof of this result requires a new representation of the ρ-invariant measure of the return process.

The following examples are treated, (a) finite state space birth-death processes, (b) Markov branching processes and the linear death process, and (c) the linear birth and death process with killing.

Two-dimensional bounded birth and death processes are considered. State probabilities are given for the generalized case in which the transition rates are arbitrary time-independent functions of the population sizes. Some applications in epidemic theory and species interaction processes are briefly described.

A two-dimensional linear birth and death process is a continuous-time Markov chain Y(·) with state space (Z+)2 which can jump from the point (n, m) to one of its four neighbors, with rates that are linear functions of n and m. Criteria are extended for determining whether such a process has a positive probability or zero probability of escaping to infinity. In the transient case considered, the projections of the imbedded Markov chain {Xn} of the successive states visited by Y(·) on a suitable pair of orthonormal vectors v and w are shown to be regularly varying sequences with index 1. Specifically, (Xn, v)∽δn and (Xn, w)∽ kn/log n for positive constants δ and k.

The purpose of this note is to show how well the coupling device is fitted for use in the study of birth and death process asymptotics: known achievements get new proofs, and new results on rate of ‘forgetfulness of initial state' and stochastic domination are established.

A recent paper in Advances in Applied Probability (Siegel (1976)) considered the duration of the time Tmn for a linear birth and death process to grow from a (large) initial size m to a larger size n. The main aim was to show that, when the birth rate exceeds the death rate, Tmn is close to its mean value, log n/m, with high probability. This paper establishes this result using much simpler techniques.

A limiting distribution for the age of a class of Markov processes is found if the present state of the process is known. We use this distribution to find the age of branching processes. Using the fact that the moments of the age of birth and death processes and of diffusion processes satisfy difference equations and differential equations respectively, we find simple formulas for these moments. For the Wright–Fisher genetic model we find the probability that a given allele is the oldest in the population if all the gene frequencies are known. The proofs of the main results are based on methods from renewal theory.

We consider a birth and death process with Q-matrix of rates qm,m + 1 = mβ, qm,m − 1 = mδ, qm,m = – m(β + δ) and qm,n = 0 otherwise. We assume that 0 < δ < β and β – δ = 1. The asymptotic behavior of first-arrival time at state n given that the process is at state m at time zero is expressed in terms of polynomials and it is shown that if m < n and m is large that the first-arrival time is close to log n/m with high probability.

Let {X1 (t), X2 (t), t ≧ 0} be a bivariate birth and death (Markov) process taking non-negative integer values, such that the process {X2(t), t ≧ 0} may influence the growth of the process {X1(t), t ≧ 0}, while the process X2 (·) itself grows without any influence whatsoever of the first process. The process X2 (·) is taken to be a simple linear birth and death process with λ2 and µ2 as its birth and death rates respectively. The process X1 (·) is also assumed to be a linear birth and death process but with its birth and death rates depending on X2 (·) in the following manner: λ (t) = λ1 (θ + X2 (t)); µ(t) = µ1 (θ + X2 (t)). Here λ i, µi and θ are all non-negative constants. By studying the process X1 (·), first conditionally given a realization of the process {X2 (t), t ≧ 0} and then by unconditioning it later on by taking expectation over the process {X2 (t), t ≧ 0} we obtain explicit solution for G in closed form. Again, it is shown that a proper limit distribution of X1 (t) always exists as t→∞, except only when both λ1 > µ1 and λ2 > µ2. Also, certain problems concerning moments of the process, regression of X1 (t) on X2 (t); time to extinction, and the duration of the interaction between the two processes, etc., are studied in some detail.

The model considered here consists of n operating units which are subject to stochastic failure according to an exponential failure time distribution. These operating units are backed up by mn spare units. Failures can be of two types. With probability p (q) a failure is of type 1(2) and is sent to repair facility 1(2) for repair. Repair facility 1(2) operates as a -server queue with exponential repair times having parameter μ1 (μ2). The number of units waiting for or undergoing repair at each of the two facilities is a continuous parameter Markov chain with finite state space. This paper derives limit theorems for the stationary distribution of this Markov chain as n becomes large under the assumption that and mn grow linearly with n. These limit theorems give very useful approximations, in terms of the seven parameters characterizing the model, to a distribution that would be difficult to calculate in practice.

The model considered here consists of n operating units which are subject to stochastic failure according to an exponential failure time distribution. Failures can be of two types. With probability p(q) a failure is of type 1(2) and is sent to repair facility 1(2) for repair. Repair facility 1(2) operates as a -server queue with exponential repair times having parameter μ1 (μ2). The number of units waiting for or undergoing repair at each of the two facilities is a continuous-parameter Markov chain with finite state space. This paper derives limit theorems for the stationary distribution of this Markov chain as n becomes large under the assumption that both and grow linearly with n. These limit theorems give very useful approximations, in terms of the six parameters characterizing the model, to a distribution that would be difficult to use in practice.