It is shown that no condition of the form E[g(N)] =∞, where N denotes a typical family size, is sufficient for explosiveness of a Markov branching process.

This paper presents a variation of a state-dependent reliability model first proposed in Lemoine and Wenocur [4], [5], and develops some of its corresponding analytical theory. In particular, we develop a reliability model in which system state is a random process satisfying a stochastic differential equation where the driving process is gamma distributed.

The queueing system considered in this paper consists of r independent arrival channels and s independent service channels, where, as usual, the arrival and service channels are independent. In the queueing system, each server of the system has his own queue and arriving customers join the shortest line in the system. We give functional central limit theorems for the stochastic processes characterizing this system after appropriately scaling and translating the processes in traffic intensity ρ > 1.

We consider a single server first-come-first-served queue with a stationary and ergodic input. The service rate is a general function of the workload in the queue. We provide the necessary and sufficient conditions for the stability of the system and the asymptotic convergence of the workload process to a finite stationary process at large times. Then, we consider acyclic networks of queues in which the service rate of any queue is a function of the workloads of this and of all the preceding queues. The stability problem is again studied. The results are then extended to analogous systems with periodic inputs.

The time to extinction of a subcritical Galton–Watson branching process and the time of last mutation of its infinite-alleles version are maxima of independent random variables having an upper tail of geometric type, and hence they are not attracted to any extreme value distribution. It is shown that Anderson's asymptotic results for maxima of discrete variates are applicable, and this rectifies a false assertion made in respect to the infinite-alleles simple branching process.

We give a probabilistic proof of a result of Shepp, that a symmetric random walk may have jump size distribution with arbitrarily large tails and yet still be persistent.

It is known that if an irreducible and aperiodic Markov chain satisfies a ‘drift' condition in terms of a non-negative measurable function g(x), it is geometrically ergodic. See, e.g. Nummelin (1984), p. 90. We extend the analysis to show that the distance between the nth-step transition probability and the invariant probability measure is bounded above by ρ n(a + bg(x)) for some constants a, b> 0 and ρ < 1. The result is then applied to obtain convergence rates to the invariant probability measures for an autoregressive process and a random walk on a half line.

Structure functions relate the level of operations of a system as a function of the level of the operation of its components. In this paper structure functions are studied which have an intuitive property, called L-superadditive (L-subadditive). Such functions describe whether a system is more series-like or more parallel-like. L-superadditive functions are also known under the names supermodular, quasi-monotone and superadditive and have been studied by many authors. Basic properties of both discrete and continuous (i.e., taking a continuum of values) L-superadditive structure functions are studied. For binary structure functions of binary values, El-Neweihi (1980) showed that L-superadditive structure functions must be series. This continues to hold for binary-valued structure functions even if the component values are continuous (see Proposition 3.1). In the case of non-binary-valued structure functions this is no longer the case. We consider structure functions taking discrete values and obtain results in various cases. A conjecture concerning the general case is made.

We show that the simple characterizations given by Takahashi for the independence and the total dependence of a multivariate extreme value distribution do not hold for the larger class of maximum infinitely divisible (max i.d.) distributions. This holds also for sup self-decomposable distributions.

A conjecture due to Block et al. (1989), concerning the number of critical connection vectors to the various performance levels of a discrete L-superadditive structure function, is proved. When the components of the discrete L-superadditive structure function are further assumed to satisfy a certain relevance condition due to Griffith (1980), it is shown that there is exactly one critical connection vector to each performance level.

By use of a method of dissection, a formula is derived for the variance in the number of lattice points covered by a rectangle of arbitrary size, lying on a square lattice of unit spacing and inclined to the lattice at an angle of which the tangent is a rational fraction.

Power series ΣzXn are studied, where {Xn} is a strictly increasing integer-valued stochastic process.

We correct a theorem of J. E. Littlewood which gives an approximation for the tail of the binomial distribution. We also present several new approximations which are less accurate but have wider scope. One of them gives an estimate with relative error uniformly O(1/σ) over all values of all the parameters, where σ is the standard deviation.

Relations of a Markov chain describing the repeated splitting of polygons to processes known from queueing theory are pointed out.

For an Ehrenfest urn model, the mean transition time from one state to another is represented as a definite integral. Simple explicit expressions are given in some special cases.

A simple proof is given for the best strategy when the game of pure strategy is played.

We shall use three basic properties of Brownian motion to derive in an elegant and non-computational way the probability that standard Brownian motion, starting from 0, will ever cross the halflines t → αt + β or t → γt + δ where γ, δ < 0 < α, β.