Current System-on-Chips (SoCs) execute applications with task dependency that compete for shared resources such as buses, memories, and accelerators. In such a structure, the arbitration policy becomes a critical part of the system to guarantee access and bandwidth suitable for the competing applications. Some strategies proposed in the literature to cope with these issues are Round-Robin, Weighted Round-Robin, Lottery, Time Division Access Multiplexing (TDMA), and combinations. However, a fine-grained bandwidth control arbitration policy is missing from the literature. We propose an innovative arbitration policy based on opportunistic access and a supervised utilization of the bus in terms of transmitted flits (transmission units) that settle the access and fine-grained control. In our proposal, every competing element has a budget. Opportunistic access grants the bus to request even if the component has spent all its flits. Supervised debt accounts a record for every transmitted flit when it has no flits to spend. Our proposal applies to interconnection systems such as buses, switches, and routers. The presented approach achieves deadlock-free behavior even with task dependency applications in the scenarios analyzed through cycle-accurate simulation models. The synergy between opportunistic and supervised debt techniques outperforms Lottery, TDMA, and Weighted Round-Robin in terms of bandwidth control in the experimental studies performed.