Abstract: The paper addresses the issue of reserving resources at ATM switches along the path of calls requiring a deterministic bound on end-to-end delay. The switches are assumed to schedule outgoing cells using the Earliest- Deadline -First (EDF) scheduling discipline. EDF is known to be an optimal scheduling discipline when providing delay bounds to a number of calls being served by a single scheduler. We present the algorithm for call admission control (CAC), and propose a number of resource division policies used for mapping the end-to-end delay requirement of a call into local delay deadlines to be reserved at each scheduler.