## 2 Related Work
First, the sender determines a candidate set, which comes from the Forwarding Nodes Set (FNS). The forwarding nodes set is a collection of such nodes: the nodes are in the sending range of the sender node, but are closer to the sink than the sender. The candidate set contains the nodes in FNS that are close to the sink and have high reliability. When the sender transmits data, it sends the data to all the nodes in the candidate set using the broadcast transmission. As long as any node in the candidate set receives the data, the transmission is successful. Because data will be retransmitted only if multiple candidate nodes do not receive it and the probability that multiple candidate nodes do not receive data is low, the probability of retransmission is minimal. In other words, it can effectively reduce delay [59]. It can be seen that in the broadcast transmission method of low reliability network, one sender corresponds to multiple receivers. The retransmission is caused only when multiple receivers have not received the data packet. Therefore, it can effectively reduce the number of retransmission. This method has also been studied in Ref. [60]. [PERSON] et al. [60] studied this mode of broadcast transmission for data fusion networks, which consists of N packets into a packet. In their research, the network adopts the broadcast way to carry on the data transmission. After broadcasting, the data is received by multiple receiving nodes. The packets received by these receiving nodes continue to broadcast transmission. Therefore, using this method, retransmission is rarely needed in a low reliability network, which reduces the delay on the premise of ensuring reliability. The delay minimizing scheme described above is based on such a hypothesis: nodes in the network are always in the awake state. In this case, the receiver is always able to receive the signal when the sender has a packet to send. But in practice, in order to save energy, many sensor networks use the duty cycle mechanism [42; 61]. In this mechanism, the node converts between two states of activity and sleep. As a result, when the sender has a packet to transmit, the nodes in the forwarding nodes set are not necessarily in the active state. Thus, the sender needs to wait for a period of time until the forwarding node wakes up, then the data can be transmitted. This sleep delay is often much larger than the time used for data transmission [42; 61]. So, in the duty cycle based network, it is more important to reduce the sleep delay due to the duty cycle mechanism, in addition to the earlier methods mentioned above. There are also many factors affecting the sleep delay, in which the node density has a certain effect on the delay [42; 61]. The greater the density of the nodes the smaller the sleep delay is when the duty ratio is fixed. This is because the greater the node density is, the larger the forwarding nodes set is when the sender sends data. The work pattern of the node is asynchronous, so the expected value of waking of the candidate forwarding nodes is smaller. For a wireless sensor network, once the nodes are deployed, the density of nodes is fixed. Therefore, another important factor affecting the sleep delay is the duty cycle [42; 61]. If the duty ratio is large, for example, its value of 1 means that in one cycle the node is always in the active state without sleep. Therefore, the sleep delay is 0, which is the minimum. But the large duty cycle consumes more energy of the node, which seriously reduces the lifetime of the node. For example, the lifetime of a node with a duty cycle of 1 is only about 1/3 of the lifetime of a node with a duty cycle of 0.3. Therefore, it is not a good method to simply adopt the method of increasing the duty cycle. As a result, researchers are seeking to reduce the delay without changing the duty cycle. The improvements in routing algorithms are also conducive to reducing delays in wireless sensor networks based on the duty cycle.