On the Properties of a Tree-Structured Server Process

Let X 0 be a nonnegative integer-valued random variable and let an independent copy of X 0 be assigned to each leaf of a binary tree of depth k. If X 0 and X0′ are adjacent leaves let X 1 = (X 0 − 1 ) + + (X0′ − 1 ) + be assigned to the parent node. In general, if X j and Xj′ , are assigned to adjacent nodes at level j = 0 , ... , k − 1, then X j and Xj′ are, in turn, independent and the value assigned to their parent node is then X j + 1 = (X j − 1 ) + + (Xj′ − 1 ) + . We ask what is the behavior of X k as k → ∞. We give sufficient conditions for X k → ∞ and for X k → 0 and ask whether these are the only nontrivial possibilities. The problem is of interest because it asks for the asymptotics of a non-linear transform which has an expansive term (the + in the sense of addition) and a contractive term (the +’s in the sense of positive part).