End-to-end QoS requires mapping of QoS between heterogeneous subnetworks of the end-to-end path. For this mapping to be accurate, we argue, in this paper, that it should be based on the performance metrics. We propose a method to map loss and delay between IP, particularly integrated services (intserv), and ATM, using deterministic service curves. First, we consider a single IP flow and show our original method for mapping of backlog and delay between IP and ATM. Second, we extend our mapping to the case of the three intserv classes served by a single ATM switch implementing a priority-based discipline. Third, we focus on the cases of non-conformant traffic and congested systems. For our mapping method to cover those cases, we introduce loss in the non-lossy service curves approach, through two different ways. Extensive numerical simulations yield qualitative as well as quantitative mapping of loss and delay between intserv and ATM.