The probability that two randomly selected phylogenetic trees of the same size are isomorphic is found to be asymptotic to a decreasing exponential modulated by a polynomial factor. The number of symmetrical nodes in a random phylogenetic tree of large size obeys a limiting Gaussian distribution, in the sense of both central and local limits. The probability that two random phylogenetic trees have the same number of symmetries asymptotically obeys an inverse square-root law. Precise estimates for these problems are obtained by methods of analytic combinatorics, involving bivariate generating functions, singularity analysis, and quasi-powers approximations.

We investigate the number of iterations needed by an addition algorithm due to Burks et al. if the input is random. Several authors have obtained results on the average case behaviour, mainly using analytic techniques based on generating functions. Here we take a more probabilistic view which leads to a limit theorem for the distribution of the random number of steps required by the algorithm and also helps to explain the limiting logarithmic periodicity as a simple discretization phenomenon.

In a real n-1 dimensional affine space E, consider a tetrahedron T0, i.e. the convex hull of n points α1, α2, …, αn of E. Choose n independent points β1, β2, …, βn randomly and uniformly in T0, thus obtaining a new tetrahedron T1 contained in T0. Repeat the operation with T1 instead of T0, obtaining T2, and so on. The sequence of the Tk shrinks to a point Y of T0 and this note computes the distribution of the barycentric coordinates of Y with respect to (α1, α2, …, αn) (Corollary 2.3). We also obtain the explicit distribution of Y in more general cases. The technique used is to reduce the problem to the study of a random walk on the semigroup of stochastic (n,n) matrices, and this note is a geometrical application of a former result of Chamayou and Letac (1994).

We investigate critical Bellman-Harris processes which allow immigration of new particles whenever the population size is 0. Under some special conditions on the immigration component the asymptotic behaviour of the probability of extinction is obtained and limit theorems are also proved.

Some classes of population-size-dependent Galton-Watson processes are considered where extinction occurs with probability 1. Results on the asymptotic behaviour of the probability of survival up to time t, mean population size and conditioned limit distributions are found to hold. They correspond to those obtained in the study of Galton-Watson processes with immigration stopped at 0.

A class of two-node queueing networks with general stationary ergodic governing sequence is considered. This means that, in particular, a non-Poissonian arrival process and dependent service times, as well as a non-Bernoulli feedback mechanism are admitted. A mixing condition ensures that the limiting distributions of the number of customers in the nodes observed in continuous time as well as at certain embedded epochs can be expressed by the Palm distributions of appropriately chosen marked point processes. This gives the possibility of connecting the classical concept of embedding with a general point-process approach. Furthermore, it leads to simple proofs of relationships between the limiting distributions. An example is given to illustrate how these relationships can be used to derive explicit formulas for various stationary queueing characteristics.

Some classes of population-size-dependent Galton–Watson processes {Z(t)}t=0,1, …, whose transition probability generating functions allow for certain upper or lower bounds, can be treated by means of functional iteration methods. Criteria for almost certain extinction are obtained as well as gammatype limit distributions for Z(t)/t as t → ∞ the results can be stated under conditions on moments of the reproduction distributions.

This paper deals with continuous-time branching processes which allow a temporally-decreasing immigration whenever the population size is 0. In the critical case the asymptotic behaviour of the probability of non-extinction and of the first two moments is investigated and different types of limit theorems are also proved.

This paper deals with the Foster–Pakes model for Galton–Watson branching processes allowing immigration whenever the number of particles is 0. In the critical case we investigate the asymptotic behaviour of the probability of non-extinction, of the expectation and of the variance, and obtain different types of limit theorems depending on the temporally-decreasing sizes of the immigrants.

A GI/G/r(x) store is considered with independently and identically distributed inputs occurring in a renewal process, with a general release rate r(·) depending on the content. The (pseudo) extinction time, or the content, just before inputs is a Markov process which can be represented by a random walk on and below a bent line; this results in an integral equation of the form gn+1(y) = ∫ l(y, w)gn(w) dw with l(y, w) a known conditional density function. An approximating solution is found using Hermite or modified Hermite polynomial expansions resulting in a Gram–Charlier or generalized Gram–Charlier representation, with the coefficients being determined by a matrix equation. Evaluation of the elements of the matrix involves two-dimensional numerical integration for which Gauss–Hermite–Laguerre integration is effective. A number of examples illustrate the quality of the approximating procedure against exact and simulated results.

This paper considers a semi-infinite storage model, of the type studied by Senturia and Puri [13] and Balagopal [2], defined on a Markov renewal process, {(Xn, Tn), n = 0, 1, ·· ·}, with 0 = T0 < T1 < · ··, almost surely, where Xn takes values in the set {1, 2, ·· ·}. If at Tn, Xn = j, then there is a random ‘input' Vn (j) (a negative input implying a demand) of ‘type' j, having distribution function Fj(·). We assume that {Vn (j)} is an i.i.d. sequence of random variables, taken to be independent of {(Xn, Tn)} and of {Vn (k)}, for k ≠ j, and that Vn (j) has first and second moments. Here the random variables Vn (j) represent instantaneous ‘inputs' (a negative value implying a demand) of type j for our storage model. Under these assumptions, we establish certain limit distributions for the joint process (Z(t), L(t)), where Z(t) (defined in (2)) is the level of storage at time t and L(t) (defined in (3)) is the demand lost due to shortage of supply during [0, t]. Different limit distributions are obtained for the cases when the ‘average stationary input' ρ, as defined in (5), is positive, zero or negative.

In 1968 Lampard determined some limit distributions for a counter process in which the input and output are independent Poisson processes. In 1974 Phatarfod dealt with the generalizations of Lampard's formulas for the case when the input and output form a bivariate Poisson process; however, his reasoning is erroneous. The object of this paper is to determine the correct limit distributions for the generalized process.

This paper gives a survey of the historical development of the solutions of various fluctuation problems in the theory of queues from the point of view of the mathematical methods used. These methods include Markov chains, Markov processes, integral equations, complex functions, generating functions, Laplace transforms, factorization of functions, operator calculus, Banach algebras and some particular methods, such as calculus of finite differences and combinatorics. In addition, the paper contains several recent results of the author for semi-Markov queuing processes.