  Routing control is one of important traffic management strategies against
urban congestion. However, it could be compromised by heterogeneous driver
non-compliance with routing instructions. In this article we model the
compliance in a stochastic manner and investigate its impacts on routing
control. We consider traffic routing for two parallel links. Particularly, we
consider two scenarios: one ignores congestion spillback while the other one
considers it. We formulate the problem as a non-linear Markov chain, given
random compliance rates. Then we propose the stability and instability
conditions to reveal when the routing is able or unable to stabilize the
traffic. We show that for links without congestion spillback there exists a
sufficient and necessary stability criterion. For links admiting congestion
propagation, we present one stability condition and one instability condition.
This stability conditions allow us to quantify the impacts of driver
non-compliance on the two-link network in terms of throughput. Finally, we
illustrate the results with a set of numerical examples.
