We introduce open stochastic fluid networks that can be regarded as continuous analogues or fluid limits of open networks of infinite-server queues. Random exogenous input may come to any of the queues. At each queue, a c.d.f.-valued stochastic process governs the proportion of the input processed by a given time after arrival. The routeing may be deterministic (a specified sequence of successive queue visits) or proportional, i.e. a stochastic transition matrix may govern the proportion of the output routed from one queue to another. This stochastic fluid network with deterministic c.d.f.s governing processing at the queues arises as the limit of normalized networks of infinite-server queues with batch arrival processes where the batch sizes grow. In this limit, one can think of each particle having an evolution through the network, depending on its time and place of arrival, but otherwise independent of all other particles. A key property associated with this independence is the linearity: the workload associated with a superposition of inputs, each possibly having its own pattern of flow through the network, is simply the sum of the component workloads. As with infinite-server queueing models, the tractability makes the linear stochastic fluid network a natural candidate for approximations.

The mathematical model we consider here is the classical Bienaymé–Galton–Watson branching process modified with immigration in the state zero.

We study properties of the waiting time to explosion of the supercritical modified process, i.e. that time until all beginning cycles which die out have disappeared. We then derive the expected total progeny of a cycle and show how higher moments can be computed. With a view to applications the main goal is to show that any statistical inference from observed cycle lengths or estimates of total progeny on the fertility rate of the process must be treated with care. As an example we discuss population experiments with trout.

Recently, Elmes et al. (see [2]) proposed a definition of a quasistationary distribution to accommodate absorbing Markov chains for which absorption occurs with probability less than 1. We will show that the probabilistic interpretation pertaining to cases where absorption is certain (see [13]) does not hold in the present context. We prove that the state probabilities at time t conditional on absorption taking place after t, generally depend on t. Conditions are derived under which there is no initial distribution such that the conditional state probabilities are stationary.

In an attempt to investigate the adequacy of the normal approximation for the number of nuclei in certain growth/coverage models, we consider a Markov chain which has properties in common with related continuous-time Markov processes (as well as being of interest in its own right). We establish that the rate of convergence to normality for the number of ‘drops’ during times 1,2,…n is of the optimal ‘Berry–Esséen’ form, as n → ∞. We also establish a law of the iterated logarithm and a functional central limit theorem.

For (μ,σ2) ≠ (0,1), and 0 < z < ∞, we prove that where φ and Φ are, respectively, the p.d.f. and the c.d.f. of a standard normal random variable. This inequality is sharp in the sense that the right-hand side cannot be replaced by a larger quantity which depends only on μ and σ. In other words, for any given (μ,σ) ≠ (0,1), the infimum, over 0 < z < ∞, of the left-hand side of the inequality is equal to the right-hand side. We also point out how this inequality arises in the context of defining individual bioequivalence.

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.

We consider a classical population flow model in which individuals pass through n strata with certain state-dependent probabilities and at every time t = 0,1,2,…, there is a stochastic inflow of new individuals to every stratum. For a stationary inflow process we prove the convergence of the joint distribution of group sizes and derive the limiting Laplace transform.