  We consider optimal control of a stochastic network,where service is
controlled to prevent buffer overflow. We use a risk-sensitive escape time
criterion, which in comparison to the ordinary escape time criteria heavily
penalizes exits which occur on short time intervals. A limit as the buffer
sizes tend to infinity is considered. In [2] we showed that, for a large class
of networks, the limit of the normalized cost agrees with the value function of
a differential game. The game's value is characterized in [2] as the unique
solution to a Hamilton-Jacobi-Bellman Partial Differential Equation (PDE). In
the current paper we apply this general theory to the important case of a
network of queues in tandem. Our main results are: (i) the construction of an
explicit solution to the corresponding PDE, and (ii) drawing out the
implications for optimal risk-sensitive and robust regulation of the network.
In particular, the following general principle can be extracted. To avoid
buffer overflow there is a natural competition between two tendencies. One may
choose to serve a particular queue, since that will help prevent its own buffer
from overflowing, or one may prefer to stop service, with the goal of
preventing overflow of buffers further down the line. The solution to the PDE
indicates the optimal choice between these two, specifying the parts of the
state space where each queue must be served (so as not to lose optimality), and
where it can idle.
