This is a discussion of the place of empirical methods in the study of probability models. It is argued that, for models of great complexity, careful experimentation may be a legitimate source of factual information on the behavior of such models. The need for rigorous standards of empirical methodology and of appropriate reporting is stressed. A simple, but interesting question on the distribution of the maximal eigenvalue of positive matrices with random elements is used as an example of an experimental study. In the course of the discussion of that example, several operational rules of sound computer experimentation are stated.

A generalization of the Bernoulli–Laplace diffusion model is proposed. We consider the case where the number of balls exchanged is greater than one. We show that the stationary distribution is the same as in the classical scheme and we give the mean and the variance of the process. In a second stage, we study the asymptotic approximation based on the diffusion process. A solution of transition density is given using Legendre polynomials.

We consider an infinite server queueing system. An examination of sample path dynamics allows a straightforward development of integral equations having solutions that give time-dependent occupancy (number of customers) and backlog (unfinished work) distributions (conditioned on the time of the first arrival) for the GI/G/∞ queue. These integral equations are amenable to numerical evaluation and can be generalized to characterize GIX/G/∞ queue. Two examples are given to illustrate the results.

Known results on the moments of the distribution generated by the two-locus Wright–Fisher diffusion model, and the duality between the diffusion process and the ancestral process with recombination are briefly summarized. A numerical method for computing moments using a Markov chain Monte Carlo simulation and a method to compute closed-form expressions of the moments are presented. By applying the duality argument, the properties of the ancestral recombination graph are studied in terms of the moments.

A random vibration model is investigated in this paper. The model is formulated as a cosine function with a constant frequency and a random walk phase. We show that this model is second-order stationary and can be rewritten as a vector-valued AR(1) model as well as a scalar ARMA(2, 1) model. The linear innovation sequence of the AR(1) model is shown to be a martingale difference sequence while the linear innovation sequence of the ARMA(2, 1) model is only an uncorrelated sequence. A non-linear predictor is derived from the AR(1) model while a linear predictor is derived from the ARMA(2, 1) model. We deduce that the non-linear predictor of this model has less mean square error than that of the linear predictor. This has significance, for example, for predicting seasonal phenomena with this model. In addition, the limit distributions of the sample mean, the finite Fourier transforms and the autocovariance functions are derived using a martingale approach. The limit distribution of autocovariance functions differs from the classical result given by Bartlett's formula.

It is proved that, in a non-homogeneous Poisson cluster process whose cluster centre rate is λ(t), the long-term behaviour of several associated quantities is equivalent to their behaviour in a homogeneous process with centre rate Λ and the same cluster structure if λ(t)→ λ as t→∞.

We present a stand-alone simple proof of a probabilistic interpretation of the Gaussian binomial coefficients by conditioning a random walk to hit a given lattice point at a given time.

For indivisible strong work-conserving queueing models with a Poisson arrival process, each of the following two properties is a sufficient condition for insensitivity. (1) The completed workload of a job receiving service is independent of the number of jobs in the system. (2) Independently of the completed workloads of the jobs in the system, they all are equally likely to be in service. For models which additionally belong to the class described by two families of parameters each of these properties is also necessary for insensitivity.

It is well known that a simple closed-form formula exists for the stationary distribution of the workload in M/GI/1 queues. In this paper, we extend this to the general stationary framework. Namely, we consider a work-conserving single-server queueing system, where the sequence of customers’ arrival epochs and their service times is described as a general stationary marked point process, and we derive a closed-form formula for the stationary workload distribution. The key to our proof is two-fold: one is the Palm-martingale calculus, that is, the connection between the notion of Palm probability and that of stochastic intensity. The other is the preemptive-resume last-come, first-served discipline.

We propose an estimator for the number of sources, N, which transmit signals according to a Poisson process with a common rate μ. This estimation problem was originated by Starr and pursued by Vardi. Our procedure is a ‘double sampling’ one and has the advantage that it can be used when μ is unknown.

A stationary Gaussian process is exhibited with the following property: the covariance function of the process is not differentiable at the origin and yet almost all the sample paths of the process are differentiable in a set of points of the power of the continuum. The process provides a counter example to a statement of Slepian.

We introduce an idealised model for overland flow generated by rain falling on a hillslope. Our prime motivation is to show how the coalescence of runoff streams promotes the total generation of runoff. We show that, for our model, as the rate of rainfall increases in relation to the soil infiltration rate there is a distinct phase change. For low rainfall (the subcritical case) only the bottom of the hillslope contributes to the total overland runoff, while for high rainfall (the supercritical case) the whole slope contributes and the total runoff increases dramatically. We identify the critical point at which the phase change occurs, and show how it depends on the degree of coalescence. When there is no stream coalescence the critical point occurs when the rainfall rate equals the average infiltration rate, but when we allow coalescence the critical point occurs when the rainfall rate is less than the average infiltration rate, and increasing the amount of coalescence increases the total expected runoff.

We consider the asymmetric simple exclusion process which starts from a product measure such that all the sites to the left of zero (including zero) are occupied and the right of 0 (excluding 0) are empty. We label the particle initially at 0 as the leading particle. We study the long-term behaviour of this process near large sites when the leading particle's holding time is different from that of the other particles. In particular, we assume that the leading particle moves at a slower rate than the other particles. We call this modified asymmetric simple exclusion process the road-hog process. Coupling and stochastic ordering techniques are used to derive the density profile of this process. Road-hog processes are useful in modelling series of exponential queues with Poisson and non-Poisson input process. The density profiles dramatically illustrate the flow of customers through the queues.

Independent observations X0, X1…, XN+1 are drawn from each of N populations whose distribution functions F(x – θi) have means θ i, 0 ≦ i < N, and we wish to calculate the probability Pk;N that X0 is the k th largest observation. For normal populations an approximation is given for PK;N based on a Taylor series expansion in the θ 's. If F(x) has an increasing failure rate, as is the case for the normal, an upper bound can be given for the ‘win' probability P1;N Some moment relations for normal order statistics are also given.

Let {X n , n = 0, 1,…} be the sequence of the lower records for an arbitrary underlying distribution μ on [0, ∞). We show that is equal in distribution to where {τ i , i = 1, 2,…} is a Poisson flow of unit intensity and g is a right-continuous and nonincreasing function defined by μ. This observation allows us to extend results of Bose et al. and simplify their proofs.

The mean and the variance of the time S(t) spent by a system below a random threshold until t are obtained when the system level is modelled by the current value of a sequence of independent and identically distributed random variables appearing at the epochs of a nonhomogeneous Poisson process. In the case of the homogeneous Poisson process, the asymptotic distribution of S(t)/t as t → ∞ is derived.

This note is motivated by Blom's work in 1989. We consider a generalized Ehrenfest urn model in which a randomly-chosen ball has a positive probability of moving from one urn to the other urn. We use recursion relations between the mean transition times to derive formulas in terms of finite sums, which are shown to be equivalent to the definite integrals obtained by Blom.

We consider an insurance model, where the underlying point process is a Cox process. Using a martingale approach applied to diffusion processes, finite-time Lundberg inequalities are obtained. By change-of-measure techniques, Cramér–Lundberg approximations are derived.

We analyse features of the patterns formed from a simple model for a martensitic phase transition that fragments the unit square into rectangles. This is a fragmentation model that can be encoded by a general branching random walk. An important quantity is the distribution of the lengths of the interfaces in the pattern, and we establish limit theorems for some of the asymptotics of the interface profile. In particular, we are able to use a general branching process to show almost sure power law decay of the number of interfaces of at least a certain size and a general branching random walk to examine the numbers of rectangles of a certain aspect ratio. In doing so we extend a theorem on the growth of the general branching random walk as well as developing results on the tail behaviour of the limiting random variable in our general branching process.

Suppose a physical process is modelled by a Markov chain with transition probability on S1 ∪ S2, S1 denoting the transient states and S2 a set of absorbing states.

If v denotes the output distribution on S2, the question arises as to what input distributions (of raw materials) on S1 produce v. In this note we give an alternative to the formulation of Ray and Margo [2] and reduce the problem to one system of linear inequalities. An application to random walk is given and the equiprobability case examined in detail.