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Loss of the routing protocol messages due to network congestion can cause peering session failures in routers, leading to route flaps and routing instabilities. We study the effects of traffic overload on routing protocols by quantifying the stability and robustness properties of two common Internet routing protocols, OSPF and BGP, when the routing control traffic is not isolated from data traffic. We develop analytical models to quantify the effect of congestion on the robustness of OSPF and BGP as a function of the traffic overload factor, queueing delays, and packet sizes. We perform extensive measurements in an experimental network of routers to validate the analytical results. Subsequently we use the analytical framework to investigate the effect of factors that are difficult to incorporate into an experimental setup, such as a wide range of link propagation delays and packet dropping policies. Our results show that increased queueing and propagation delays adversely affect BGP's resilience to congestion, in spite of its use of a reliable transport protocol. Our findings demonstrate the importance of selective treatment of routing protocol messages from other traffic, by using scheduling and utilizing buffer management policies in the routers, to achieve stable and robust network operation.