Queueing problems are investigated for very wide classes of input traffic and service time models to obtain good loss probability and waiting time probability approximation. The proposed approximation is based on the fundamental recursion formula and the Chernoff bound technique, both of which requires no particular assumption for the stochastic nature of input traffic and service time, such as renewal or markovian properties. The only essential assumption is stationarity. We see that the accuracy of the obtained approximation is confirmed by comparison with computer simulation. There are a number of advantages of the proposed method of approximation when we apply it to network capacity design or path accommodation design problems. First, the proposed method has the advantage of applying to multi-media traffic. In the ATM network, a variety of bursty or non-bursty cell traffic exist and are superposed, so some unified analysis methodology is required without depending each traffic's characteristics. Since our method assumes only the stationarity of input and service process, it is applicable to arbitrary types of cell streams. Further, this approach can be used for the unexpected future traffic models. The second advantage in application is that the proposed probability approximation requires only small amount of computational complexity. Because of the use of the Chernoff bound technique, the convolution of every traffic's probability density fnuction is replaced by the product of probability generating functions. Hence, the proposed method provides a fast algorithm for, say, the call admission control problem. Third, it has the advantage of accuracy. In this paper, we applied the approxmation to the cases of homogeneous CBR traffic, non-homogeneous CBR traffic, M/D/1, AR(1)/D/1, M/M/1 and D/M/1. In all cases, the approximating values have enough accuracy for the exact values or computer simulation results from low traffic load to high load. Moreover, in all cases of the numerical comparison, our approximations are upper bounds of the real values. This is very important for the sake of conservative network design.