Probabilistic arbitration is combined with distance-based weights to achieve equality of service in interconnection networks, such as those used with chip multiprocessors. This arbitration desirably used incorporates nonlinear weights that are assigned to requests. The nonlinear weights incorporate different arbitration weight metrics, namely fixed weight, constantly increasing weight, and variably increasing weight. Probabilistic arbitration for an on-chip router avoids the need for additional buffers or virtual channels, creating a simple, low-cost mechanism for achieving equality of service. The nonlinearly weighted probabilistic arbitration includes additional benefits such as providing quality-of-service features and fairness in terms of both throughput and latency that approaches the global fairness achieved with age-base arbitration. This provides a more stable network by achieving high sustained throughput beyond saturation. Each router or switch in the network may include an arbiter to apply the weighted probabilistic arbitration.