  Importance sampling (IS) is a variance reduction method for simulating rare
events. A recent paper by Dupuis, Wang and Sezer (Ann. App. Probab. 17(4):1306-
1346, 2007) exploits connections between IS and stochastic games and optimal
control problems to show how to design and analyze simple and efficient IS
algorithms for various overflow events for tandem Jackson networks. The present
paper uses the same approach to build asymptotically optimal IS schemes for
stable open Jackson networks with a tree topology. Customers arrive at the
single root of the tree. The rare overflow event we consider is the following:
given that initially the network is empty, the system experiences a buffer
overflow before returning to the empty state. Two types of buffer structures
are considered: 1) A single system-wide buffer of size $n$ shared by all nodes,
2) each node $i$ has its own buffer of size $\beta_i n$, $\beta_i \in (0,1)$.
