Disclosed is a method and apparatus for arbitration between multiple pipelines over shared resources for an SMP computer system. The computer includes logic to defer arbitration until later in the pipeline to help reduce latency to each pipeline. Also, introduced is the concept of retry tags for better priority to avoid lock-out. The system also includes round-robin tokens to manage rejected requests to allow better fairness on conflicts. While the processing logic employed specifically applies to cross-interrogation, the logic can be extended to other common resources. The illustrated SMP computer system also has self-correcting logic to maintain good round-robin tokens.