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Unlike conventional deterministic bounding curve characterization, quality-of-service (QoS) dependent (e.g., the portion of buffer-overflow time intervals) curve characterization allows bound-violation for a certain portion of time and results in flexible, pointwisely smaller constraint functions to utilize valuable network resources. Our derivations show that the portion of bound-violation for an L/sup P/-sense (1/spl les/p/spl les//spl infin/) constraint function has a normalized tail bound decaying with x/sup -p/, and L/sup p/-sense constraint functions reduce to deterministic ones when p=/spl infin/. Therefore, our QoS-dependent constraint functions are obvious generalizations of conventional deterministic constraint functions. We further apply L/sup P/-sense constraint functions to reduce buffer requirements for multi-hop connections according to the required bound on the portion of time prone to buffer overflows.