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A general model of a branching Markov process on ℝ is considered. Sufficient and necessary conditions are given for the random variable M=supt≥0max1≤k≤N(t)Ξk(t) to be finite. Here Ξk(t) is the position of the kth particle, and N(t) is the size of the population at time t. For some classes of processes (smooth branching diffusions with Feller-type boundary points), this results in a criterion stated in terms of the linear ODEσ2(x)2f″(x)+a(x)f′(x)=λ(x)(1−k(x))f(x). Here σ(x) and a(x) are the diffusion coefficient and the drift of the one-particle diffusion, respectively, and λ(x) and k(x) the intensity of branching and the expected number of offspring at point x, respectively. Similarly, for branching jump Markov processes the conditions are expressed in terms of the relations between the integral μ(x)∫π(x,dy)(f(y)−f(x)) and the product λ(x)(1−k(x))f(x), where λ(x) and k(x) are as before, μ(x) is the intensity of jumping at point x, and π(x,dy) is the distribution of the jump from x to y.

A model of a queueing network is proposed which leads to a stochastic equation generalizing a standard Lindley equation for a single FCFS server. We study the problem of the existence and uniqueness of a stationary solution to this equation and its connection with random processes on a Cayley tree.

R.L. Dobrushin (1929-1995) made substantial contributions to Queueing Network Theory (QNT). A review of results from QNT which arose from his ideas or were connected to him in other ways is given. We also comment on various related open problems.