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, making (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, R-(m,k)-firm provides the guarantee on the transmission delay of at least m out of any k consecutive packets (m´k). Instead of imposing a transmission delay constraint per packet (i.e. deadline), R-(m,k)-firm only considers a global transmission 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 network congestion while still guaranteeing the R-(m,k)-firm constraint. The sufficient condition for this guarantee is given for configuring the DLB parameters. Finally a comprehensive discussion on the existing constraint relaxation strategies is developed showing the generality of the R-(m,k)-firm scheme.