The on-chip network of emerging many-core CMPs enables the sharing of numerous on-chip components. This on-chip network needs to ensure fairness when accessing the shared resources. In this work, we propose providing equality of service (EoS) in future many-core CMPs on-chip networks by leveraging distance, or hop count, to approximate the age of packets in the network. We propose probabilistic arbitration combined with distance-based weights to achieve EoS and overcome the limitation of conventional round-robin arbiter. We describe how nonlinear weights need to be used with probabilistic arbiters and propose three different arbitration weight metrics - fixed weight, constantly increasing weight, and variably increasing weight. By only modifying the arbitration of an on-chip router, we do not require any additional buffers or virtual channels and create a complexity-effective mechanism for achieving EoS.