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We consider Galton-Watson branching processes with countable typeset X. We study the vectors q(A) = (qx(A))x∈X recording the conditional probabilities of extinction in subsets of types A ⊆ X, given that the type of the initial individual is x. We first investigate the location of the vectors q(A) in the set of fixed points of the progeny generating vector and prove that qx({x}) is larger than or equal to the xth entry of any fixed point, whenever it is different from 1. Next, we present equivalent conditions for qx(A) < qx(B) for any initial type x and A, B ⊆ X. Finally, we develop a general framework to characterise all distinct extinction probability vectors, and thereby to determine whether there are finitely many, countably many, or uncountably many distinct vectors. We illustrate our results with examples, and conclude with open questions. [ABSTRACT FROM AUTHOR]

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 an elegant coupling argument; and a third involves expressing the value of capacity in terms of the entries of a transient analogue of the deviation matrix. In this paper, we shall compare and contrast these approaches and, in particular, use the coupling analysis to explain why the selling price of an extra unit of capacity remains the same when the arrival and service rates are interchanged when the queue starts at full capacity. [ABSTRACT FROM AUTHOR]