\begin{abstract}
{\it Diffracting trees} are an effective and highly scalable
dis\-tri\-bu\-ted-pa\-rallel 
technique for shared counting and load balancing.
This paper presents the first steady-state combinatorial mo\-del and
analysis for diffracting trees, and uses it to answer several critical
algorithmic design questions.  Our model is simple and sufficiently
high level to overcome many implementation specific details, and yet
as we will show it is rich enough to accurately predict empirically
observed behaviors. As a result of our analysis we were able to
identify starvation problems in the original diffracting tree
algorithm and modify it to a create a more stable version.  We are
also able to identify the range in which the diffracting tree performs
most efficiently, and the ranges in which its performance degrades.
We believe our model and modeling approach open the way to
steady-state analysis of other distributed-parallel structures such as
counting networks and elimination trees.
\end{abstract}
