This paper presents a new real-time Quality of Service (QoS) guarantee scheme called Relaxed (m,k)-firm denoted by R-(m,k)-firm. The problem we deal with is that for a set of streams (e.g. sources which generate packets) sharing a common resource, deterministic (m,k)-firm guarantee of each stream can lead to an arbitrarily low resource utilization, which makes (m,k)-firm useless for QoS guarantee in a network. The goal of the proposed constraint relaxation is to achieve a higher resource utilization while still deterministically guarantee the (m,k)-firm constraint. As with (m,k)-firm guarantee, it provides the guarantee on the transmis-sion delay of at least m out of any k consecutive packets (m´k). Instead of imposing a transmis-sion delay constraint on per packet (i.e. deadline), R-(m,k)-firm only considers a global trans-mission delay constraint on a group of any k consecutive packets. This constraint relaxation may be acceptable for a large class of soft real-time applications such as multimedia flow transmissions in the networks for which occasional packet drops can be tolerated. One of the possible implementations of the R-(m,k)-firm scheme is also provided with the development of a new traffic control mechanism, called Double-Leaks Bucket (DLB). DLB selectively drops a proportion of packets of a flow or an aggregated-flows in case of the net-work congestion while still guaranteeing the R-(m,k)-firm constraint. The sufficient condition for this guarantee is given for configuring the DLB parameters. Simulation results show the advantage in terms of avoidance of consecutive dropping and queue length. Additionally, a comprehensive discussion on the existing constraint relaxation strategies is developed showing the generality of the R-(m,k)-firm scheme.