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In this paper, we consider the problem of optimum selection of quality-of-service (QoS) classes in interworking between the networks in a next-generation-network (NGN) environment. After introducing the delay-cost and loss-cost characteristics, we discuss the time-invariant (TI) and time-variant (TV) scenarios. For the TI case, we show that under nearly lossless transmission condition, each network can make its own optimization regardless of other networks. For the TV case, we present sufficient conditions under which the optimum QoS class of each network can be considered fixed with respect to time without considerable degradation in the optimization target. Therefore, under the conditions presented in this paper, the QoS of a flow in each network can be determined solely by considering the characteristics of that network and this QoS class can be held fixed during the flow period.