||In future Broadband ISDN (B-ISDN), Asynchronous Transfer Mode (ATM) networks is designed not only to support a wide range of traffic classes with diverse flow characteristics (e.g., burstiness, bit rate and burst length), but also to guarantee the different Quality of Service (QOS) requirements as well. The QOS are measured in terms of cell loss probability and maximum cell delay. In this thesis, the ATM multiplexer is considered as a single-server queueing system with the implementation of the virtual path (VP) concept. By applying the Markov Modulated Deterministic Process (MMDP) method, an accurate and efficient algorithm is developed to compute the sufficient and minimum capacity required to satisfy all the QOS requirements when multiple classes of on-off sources are multiplexed onto a virtual path. Also, the problem of assigning traffic classes to VPs so as to minimize the total capacity required is addressed. A simple algorithm is proposed to determine the VP combination that achieves the near optimum of total capacity required for satisfying the individual QOS requirements. Numerical results are presented to demonstrate the performance of the algorithm when compared to the optimal total capacity and buffer required. Since the VP bandwidth are normally not shared among the VPs, such approach may not have a good utilization of the network bandwidth and is referred to as static bandwidth allocation scheme. Another more flexible approach is to provide some priority handling mechanism to achieve different QOS while maintaining efficient network use. This priority scheme is a dynamic bandwidth allocation scheme that whenever one queue is empty, its dedicated capacity would be utilized by other non-empty queues in the order according to their priorities. Numerical results of the dynamic scheme will be compared to that of the static scheme to evaluate their performance (i.e., the total capacity and the buffer required). The implication of performance of both schemes to bandwidth allocation of virtual path and call admission control in ATM networks will also be discussed.