This paper presents a new QoS guarantee scheme called R-(m,k)-firm (Relaxed-(m,k)-firm) which provides the guarantee on transmission delay of at least m out of any k consecutive packets (m´k). It has several advantages: (1) during network congestion, packets are dropped according to the (m,k) model rather than uncontrollably as the case of TD and RED, avoiding thus undesirable long consecutive packet drops; (2) it allows to admit more real-time flows than the traditional over-provisioning approach. A new mechanism, called DLB (Double Leaks Bucket) is also proposed for dropping a proportion of packets of a flow or of aggregated-flows in case of network congestion while still guaranteeing the R-(m,k)-firm constraint. The sufficient condition for this guarantee is given for configuring the DLB parameters. It is easy to implement DLB in the actual IntServ and Diffserv architectures (by simply replacing the actual leaky bucket by DLB) for providing respectively per flow and per class (m,k) guarantee, or event per flow R-(m,k)-firm guarantee in Diffserv.