A single-flux-quantum (SFQ) device provides fast operation with low power consumption. As SFQ devices can be applied in network switches requiring high throughput, we have been developing an SFQ network switch system. Not only the switch fabric but also the switch scheduler is a key component for a high-throughput network switch. The switch scheduler orders multiple inputs to a single output at the same time to prevent collisions from occurring in the switch fabric. The scheduling time becomes longer as the number of switch ports increases, because the scheduler must scan requests from all input ports. We designed two types of 4-port arbiters (i.e. switch scheduler sub-circuits) with structures based on serial or parallel processing. We chose to implement and demonstrate the serial arbiter because it has a smaller number of gates. It consisted of about 1800 Josephson junctions, including on-chip test circuitry. We tested a fabricated circuit at high speed and confirmed its correct operation at 40 GHz.