  We construct a generic, simple, and efficient scheduling policy for
stochastic processing networks, and provide a general framework to establish
its stability. Our policy is randomized and prioritized: with high probability
it prioritizes jobs which have been least routed through the network. We show
that the network is globally stable under this policy if there exists an
appropriate quadratic local Lyapunov function that provides a negative drift
with respect to nominal loads at servers. Applying this generic framework, we
obtain stability results for our policy in many important examples of
stochastic processing networks: open multiclass queueing networks, parallel
server networks, networks of input-queued switches, and a variety of wireless
network models with interference constraints. Our main novelty is the
construction of an appropriate global Lyapunov function from quadratic local
Lyapunov functions, which we believe to be of broader interest.
