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 packets using the Packet-by-Packet Generalized Processor Sharing (PGPS) scheduling discipline. We propose an algorithm for call admission control (CAC), and present the simulation results used to evaluate the performance of the resource division policies used for mapping the end-to-end delay requirement into service rates to be reserved at each switch. The simulation results also show the performance gain when a simple resource-based routing algorithm is used.