On the Properties of a Tree-Structured Server Process

Let X0 be a nonnegative integer-valued random variable and let an independent copy of X0 be assigned to each leaf of a binary tree of depth k. If X0 and X0′ are adjacent leaves, let X1=(X0−1)++(X0′−1)+ be assigned to the parent node. In general, if Xj and Xj′ are assigned to adjacent nodes at level j = 0,⋯, k − 1, then Xj and Xj′ are, in turn, independent and the value assigned to their parent node is then Xj+1=(Xj−1)++(Xj′−1)+. We ask what is the behavior of Xk as k→∞. We give sufficient conditions for Xk→∞ and for Xk→0 and ask whether these are the only nontrivial possibilities. The problem is of interest because it asks for the asymptotics of a nonlinear transform which has an expansive term (the + in the sense of addition) and a contractive term (the + in the sense of positive part).