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Strategies for connection admission control in asynchronous transfer mode (ATM) networks are considered. Without any Poisson or renewal assumptions, two easily computable upper bounds on the time congestion in a finite buffer are derived. The first upper bound is valid for arbitrary peak and mean-rate-policed sources, whereas the second (and, in principle, tighter) bound is valid for sources of the on/off type. The tightnesses of the bounds are evaluated by a new periodic queuing model taking into account the maximum allowed burst duration. It is concluded that the bounds form a basis for a realization of a simple admission control algorithm. Furthermore, it is pointed out that the derivation of the on/off bound induces a decomposition of the queuing process into a cell-scale contribution and a burst-scale contribution, a decomposition which is superior to traditional Markov modulated approaches both in accuracy, and in offering insight into the queuing process