This study introduces a Markov network process called a string-net.
Its state is the vector of quantities of customers or units that move among the
nodes, and a transition of the network consists of a string of instantaneous
vector increments in the state. The rate of such a string transition is a
product of a transition-initiation rate and a string-generation rate. The main
result characterizes the stationary distribution of a string-net. Key
parameters in this distribution satisfy certain "polynomial traffic
equations" involving the string-generation rates. We identify sufficient
conditions for the existence of a solution of the polynomial equations, and we
relate these equations to a partial balance property and throughputs of the
network. Other results describe the stationary behavior of a large class of
string-nets in which the vectors in the strings are unit vectors and a
string-generation rate is a product of Markov routing probabilities. This class
includes recently studied open networks with Jackson-type transitions augmented
by transitions in which a signal (or negative customer) deletes units at nodes
in one or two stages. The family of string-nets contains essentially all Markov
queueing network processes, aside from reversible networks, that have known
formulas for their stationary distributions. We discuss old and new variations
of Jackson networks with batch services, concurrent or multiple-unit movements
of units, state-dependent routings and multiple types of units and routes.