We consider a class of branching processes with countably many types which we refer to as Lower Hessenberg branching processes. These are multitype Galton-Watson processes with typeset X = {0, 1, 2, . . . }, in which individuals of type i may give birth to offspring of type j ≤ i + 1 only. For this class of processes, we study the set S of fixed points of the progeny generating function. In particular, we highlight the existence of a continuum of fixed points whose minimum is the global extinction probability vector q and whose maximum is the partial extinction probability vectorq. In the case whereq = 1, we derive a global extinction criterion which holds under second moment conditions, and whenq < 1 we develop necessary and sufficient conditions for q =q.

We consider a class of multitype Galton–Watson branching processes with a countably infinite type set $\mathcal{X}_d$ whose mean progeny matrices have a block lower Hessenberg form. For these processes, we study the probabilities $\textbf{\textit{q}}(A)$ of extinction in sets of types $A\subseteq \mathcal{X}_d$ . We compare $\textbf{\textit{q}}(A)$ with the global extinction probability $\textbf{\textit{q}} = \textbf{\textit{q}}(\mathcal{X}_d)$ , that is, the probability that the population eventually becomes empty, and with the partial extinction probability $\tilde{\textbf{\textit{q}}}$ , that is, the probability that all types eventually disappear from the population. After deriving partial and global extinction criteria, we develop conditions for $\textbf{\textit{q}} < \textbf{\textit{q}}(A) < \tilde{\textbf{\textit{q}}}$ . We then present an iterative method to compute the vector $\textbf{\textit{q}}(A)$ for any set A. Finally, we investigate the location of the vectors $\textbf{\textit{q}}(A)$ in the set of fixed points of the progeny generating vector.

We consider the extinction events of Galton-Watson processes with countably infinitely many types. In particular, we construct truncated and augmented Galton-Watson processes with finite but increasing sets of types. A pathwise approach is then used to show that, under some sufficient conditions, the corresponding sequence of extinction probability vectors converges to the global extinction probability vector of the Galton-Watson process with countably infinitely many types. Besides giving rise to a family of new iterative methods for computing the global extinction probability vector, our approach paves the way to new global extinction criteria for branching processes with countably infinitely many types.

In an M/M/1/C queue, customers are lost when they arrive to find C customers already present. Assuming that each arriving customer brings a certain amount of revenue, we are interested in calculating the value of an extra waiting place in terms of the expected amount of extra revenue that the queue will earn over a finite time horizon [0, t].There are different ways of approaching this problem. One involves the derivation of Markov renewal equations, conditioning on the first instance at which the state of the queue changes, a second involves expressing the value of capacity in terms of the entries of a transient analogue of the deviation matrix, discussed by Coolen-Schrijner and van Doorn in [6], and a third involves an elegant coupling argument.In this paper, we shall compare and contrast these approaches and, in particular, use the coupling analysis to explain why the value of an extra unit of capacity remains the same when the arrival and service rates are interchanged when the queue starts at full capacity.

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