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 X.0 are adjacent leaves, let X1=(X0.1)++(X.0.1)+ be assigned to the parent node. In general, if Xj and X.j are assigned to adjacent nodes at level j=0,.,k.1, then Xj and X.j are, in turn, independent and the value assigned to their parent node is then Xj+1=(Xj.1)++(X.j.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).