  Adaptive control is a classical control method for complex cyber-physical
systems, including transportation networks. In this work, we analyze the
convergence properties of such methods on exemplar graphs, both theoretically
and numerically. We first illustrate a limitation of the standard backpressure
algorithm for scheduling optimization, and prove that a re-scaling of the model
state can lead to an improvement in the overall system optimality by a factor
of at most $\mathcal{O}(k)$ depending on the network parameters, where $k$
characterizes the network heterogeneity. We exhaustively describe the
associated transient and steady-state regimes, and derive convergence
properties within this generalized class of backpressure algorithms. Extensive
simulations are conducted on both a synthetic network and on a more realistic
large-scale network modeled on the Manhattan grid on which theoretical results
are verified.
