On the Properties of a Tree-Structured Server Process

Let X0be a nonnegative integer-valued random variable and let an independent copy of X0be assigned to each leaf of a binary tree of depth k. If X0and X'0are adjacent leaves, let X1= (X0- 1)++ (X'0- 1)+be assigned to the parent node. In general, if Xjand X'jare assigned to adjacent nodes at level j = 0, ⋯, k - 1, then Xjand X'jare, 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 Xkas 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).