In this paper, we consider the online buffer management problem, which formulates the problem of managing network switches supporting Quality of Service guarantee. We improve competitive ratios of the 2-value multi-queue switch model, where the value of a packet is restricted to 1 or alpha(>= 1). We use a similar approach as Azar and Richter (STOC 2003 and Algorithmica 43(1-2), 2005) did for the multi-value multi-queue switch model. Namely, we show that the competitive ratio of "the relaxed model" of the 2-value multi-queue switch model is at most x = min{c + 2-c/alpha(2-c)+(c-1), c alpha}, if the competitive ratio of an online algorithm for the unit-value multi-queue switch model is at most c. Azar and Richter's technique implies that if the competitive ratio of the 2-value single-queue switch model is x', then the competitive ratio of the 2-value multi-queue switch model is at most xx'. We obtain several results using known c and x'. (C) 2017 Elsevier B.V. All rights reserved.