The online buffer management problem formulates the problem of queuing policies of network switches supporting QoS (Quality of Service) guarantee. In this paper, we consider one of the most standard models, called multi-queue switches model. In this model, Albers et al. gave a lower bound $\frac{e}{e-1}$, and Azar et al. gave an upper bound $\frac{e}{e-1}$ on the competitive ratio when m, the number of input ports, is large. They are tight, but there still remains a gap for small m. In this paper, we consider the case where m = 2, namely, a switch is equipped with two ports, which is called a bicordal buffer model. We propose an online algorithm called Segmental Greedy Algorithm (SG) and show that its competitive ratio is at most $\frac{16}{13} (\simeq 1.231)$, improving the previous upper bound by $\frac{9}{7} (\simeq 1.286)$. This matches the lower bound given by Schmidt.