When multiple applications at two sites connected by a best-effort service network through gateway (GW) equipment communicate simultaneously, the receive rate of a high-priority application communication flow can be smaller than its necessary bandwidth (BW). In this paper, we refer to this problem as the deficit in bandwidth of a high-priority flow (DBHPF) problem. In order to handle this problem, we consider controlling the BW assigned to each flow based on the available bandwidth (ABW) estimated by the GW. The estimated ABW can be larger than the actual ABW due to the error in the estimation. Thus, the receive rate of a high-priority application can be smaller than its necessary BW, even if the actual ABW is larger than the necessary BW. In this paper, we propose a priority-based BW control method that estimates the receive rate of each flow using estimated ABW and related information, and mitigates the effect of the DBHPF problem by controlling the transmission BW of each flow in order to compensate for the difference between the estimated receive rate and the necessary BW according to the priorities of flows. We call the proposed method estimated-receive-rate-based bandwidth control (eR2BC). We also propose a method for ABW estimation with less overhead than existing methods. We conducted experiments using the proposed methods in a virtual network constructed with virtual machines and confirmed that the proposed methods can mitigate the effect of the DBHPF problem better than existing methods.