  We study an open discrete-time queueing network that models the collection of
data in a multi-hop sensor network. We assume data is generated at the sensor
nodes as a discrete-time Bernoulli process. All nodes in the network maintain a
queue and relay data, which is to be finally collected by a designated sink. We
prove that the resulting multi-dimensional Markov chain representing the queue
size of nodes has two behavior regimes depending on the value of the rate of
data generation. In particular, we show that there is a non-trivial critical
value of data rate below which the chain is ergodic and converges to a
stationary distribution and above which it is non-ergodic, i.e., the queues at
the nodes grow in an unbounded manner. We show that the rate of convergence to
stationarity is geometric in the sub-critical regime. We also show the
connections of this process to a class of Laplacian systems of equations whose
solutions include the important problem of finding the effective resistance
between two nodes, a subroutine that has been widely used to develop efficient
algorithms for a number of computational problems. Hence our work provides the
theoretical basis for a new class of distributed algorithms for these problems.
