Cell loss, which occurs when buffers overflow in ATM switches, is a key measure for the performance of an ATM network .An ATM network must be designed for very low and predictable cell loss probabilities to provide adequate quality of service for all users. Many methods have been proposed for allocating bandwidth and determining buffer size requirements to provide quality of service guarantees. These methods are subject to overly optimistic allocations or overly pessimistic allocations. The authors present an exact analysis of cell for on off traffic source which can be used in concert with probabilistic methods to ensure that buffer and bandwidth allocation are mode between the best case and worst case requirements.