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In this paper, we consider a feedback-based video transmission system over the ATM ABR service under the constraints of fixed end-to-end delay. The lower bound of the future available rates is obtained by the states of leaky-buckets for high-priority connections and the occupancy of the switch buffer. Based on this bound, an upper bound of feedback rates are derived, that guarantees the end-to-end delay constraints. Numerical results are given for some test sequences and operation parameters.