## 4 The Proposed Routing Protocol
### The Network Model  
We consider a sensor network that consists of a base station and some sensor nodes that are energy-constrained immobile nodes and are deployed randomly in a complex region. It is considered that sensor nodes are not grouped into clusters or trees. Furthermore, most nodes cannot send the packet to base station directly, e.g., there exist obstacles or weak signals of sensor nodes. In this region, the sensors communicate with each other by short-haul radio communication, and most of the nodes cannot transmit their data packets to the base station directly. Thus, they are required to transmit the packets using hop-to-hop delivery. Moreover, each node knows its own location by the Global Positioning System (GPS) technology [42]. Sensor nodes are known as three-dimensional positions to determine their neighbors correctly. Furthermore, all sensor nodes know the position of the immobile base station. After deploying the sensor nodes in a certain network, they will detect their neighbors by broadcasting a Neighbors Discovery (ND) message. Each node that received this message, responds to it by sending a Neighbors Acceptance (NA) message. When the sensor node receives the NA message, it searches the sender node in its neighbors list; if it does not exist, its sender ID is appended to the neighbors list. The sensor nodes perform two main functions: sensing and relaying. The sensing element probes its environment to track an event. Then, the sensed data are relayed to the base station trough selected nodes. It is necessary to express that data is generated by the Poisson distribution [36]. The process of data generation is done in rounds, which are determined by this distribution. Note that the packets have no defined finite life time. In fact, time-to-live of packets are considered infinite until they are delivered to the base station. The network model for transmitting the data packets by the proposed protocol is shown in Figure 3. It consistsof some sensor nodes deployed throughout the network and a base station located within the radio range of a small number of nodes. Furthermore, any node that has a new sensed data to transmit to the base station is called _sender node_.  
We apply the method Presented in [31, 32] to estimate the communication energy consumption of the sensor nodes. In this model, the main energy parameters for communication are the energy/bit consumed by the transmitter electronics (\(\alpha_{I}\)), dissipated energy in the transmit OPAMP (\(\alpha_{2}\)), and energy/bit consumed by the receiver electronics (\(\alpha_{3}\)). We assume a 1/\(d^{\prime}_{n}\) path loss, and the energy consumed is:  
\[\begin{cases}\dot{E}_{t}=(\alpha_{I}+\alpha d^{\prime}_{n})r_{b}\\ \dot{E}_{r}=\alpha_{STb}\end{cases} \tag{1}\]  
Where, \(E_{t}\) is the required energy for sending \(r_{b}\) bits, and \(E_{r}\) is the desired energy consumed for receiving \(r_{b}\) bits; \(\alpha_{I}\) is the dissipated energy in the transmitter electronics (taken to be 50 nJ/b). In addition, \(\alpha_{3}\) is the dissipated energy in the receiver electronics; \(\alpha_{2}\) is the dissipated energy in the transmitter amplifier (taken as equal to 100 PJ/(b.m\({}^{2}\))); \(r_{b}\) is the number of bits in the transmitted packet; and \(d^{\prime}_{n}\) is the distance that should be traversed by packet.