## 1 Introduction  
Wireless sensor networks (WSNs) have been revolutionizing the way that people interact with the physical world by their diverse applications in different areas [1]. In an application-specific network like WSNs, applications are likely to need specific routing protocols according to the unique features of generated data [2]. For instance, emergency-detecting applications need real-time routing protocols, whereas state-monitoring applications require reliable routing protocols. Moreover, various types of communications such as unicast, multicast and broadcast also need to be supported according to the demands. In the case of fire alert application, for example, the detection of the fire may be delivered to a base station by the real-time unicast protocol, transmitted to fire fighters by the real-time multicast protocol and disseminated to all people in the local area for warning purposes by real-time broadcast protocols. In these scenarios, real-time communication is one of the areas highly focused on to support diverse and novel emerging sensor applications such as urgent data reporting and multimedia traffic management. However, supporting real-time communication is also a challenging issue in WSNs since the distinguishing features of WSNs as compared to traditional wired networks [3; 4] lead to the building of harsh network environments that are composed of a large number of resource-constrained sensor nodes where their links are lossy and unstable. Therefore, WSNs naturally have dynamicenvironments such that states of nodes and links are easily changed. Moreover, the energy efficiency problem is another challenge to increase the network lifetime. In terms of real-time communications, in general, hard real-time is impractical in WSNs due to the constrained resources and the dynamic network environment [5]. Thus, many studies propose soft real-time protocols that try to improve the probability of successful real-time delivery through probabilistic guarantee or heuristic algorithms. Soft real-time protocols, however, cannot guarantee the consistent Quality of Service (QoS) related to real time. Unlike typical approaches based on the hard and soft real-time model, recently, [PERSON] and [PERSON] [6] suggest the (\(m\), \(k\))-firm-based Real-time Unicast Protocol (FRUP), which sets the criterion between hard real-time and soft real-time. The concept of (\(m\), \(k\))-firm real time is defined as at least \(m\) messages among any \(k\) consecutive messages have to meet their real-time constraint from a real-time data stream. This (\(m\), \(k\))-firm real-time model provides a more credible performance evaluation indicator than the simple probabilistic indicator, e.g., 80% real-time success ratio. Technically, FRUP tries to satisfy the (\(m\), \(k\))-firm real-time requirement by adjusting the data path in a way that reselects forwarding nodes by using two state indicators, Distance-Based Priority (DBP) and Local State Indicator (LSI). In addition to FRUP, since all other protocols for the (\(m\), \(k\))-firm real-time work are focused on one-to-one unicast, there is no existing multicast and broadcast protocol to support the (\(m\), \(k\))-firm model yet. Thus, (\(m\), \(k\))-firm based real-time unicast is required to be extended to multicast and network broadcast to apply this idea to various real-time applications. To accommodate the above demands, in this paper, we deal with the (\(m\), \(k\))-firm-based real-time broadcast as the first step of the extension. From the point of view of efficiency, the current FRUP cannot be simply applied to network broadcast for following reasons. If data packets are delivered to each destination by FRUP in the case of network broadcasting (to all sensor nodes), many duplicated packet transmissions may lead to wasting of energy, increasing of transmission delay and congestion. Moreover, the algorithm to decide to rebroadcast in the original flooding and its variants need to be changed to meet the (\(m\), \(k\))-firm requirement.