  We consider a basic model of a dynamical distribution network, modeled as a
directed graph with storage variables corresponding to every vertex and flow
inputs corresponding to every edge, subject to unknown but constant inflows and
outflows. In [1] we showed how a distributed proportionalintegral controller
structure, associating with every edge of the graph a controller state,
regulates the state variables of the vertices, irrespective of the unknown
constant inflows and outflows, in the sense that the storage variables converge
to the same value (load balancing or consensus). In many practical cases, the
flows on the edges are constrained. The main result of [1] is a sufficient and
necessary condition, which only depend on the structure of the network, for
load balancing for arbitrary constraint intervals of which the intersection has
nonempty interior. In this paper, we will consider the question about how to
decide the steady states of the same model as in [1] with given network
structure and constraint intervals. We will derive a graphic condition, which
is sufficient and necessary, for load balancing. This will be proved by a
Lyapunov function and the analysis the kernel of incidence matrix of the
network. Furthermore, we will show that by modified PI controller, the storage
variable on the nodes can be driven to an arbitrary point of admissible set.
