## 3 System Model
### Intra-Cluster Communication Session
The energy used in the sleep state is given by:

\[P_{S_{1}S_{2}}=1/u_{S_{1}}*\Pr_{sch} \tag{16}\]4

where \(Pw_{s_{1}}\) is the unit time energy spend in the sleep state and \(T_{s_{1}}\) is the mean time the node takes in the sleep state. The energy dissipated by a node in the back-off state is given by:

\[P_{S_{1}S_{2}}=1/u_{S_{1}}*\Pr_{sch} \tag{16}\]5

where \(Pw_{s_{3}}\) is the unit time energy spend in the back-off state and \(T_{s_{3}}\) is the mean time a node spends in back-off state. The energy \(E_{s_{2}}\) in the active communication state comprises of the transmitting and the receiving energy and the time it takes to service the packet at the node depending on the length of the packet:

\[ \begin{array}{c}E_{s_{2}}=(Pw_{tx}+Pw_{rx})\\ *\left(\overline{T_{s_{2}}}*\frac{L_{pac_{bits}}}{R_{bits/sec}}+\frac{L_{ack_{ bits}}}{R_{bits/sec}}+\frac{L_{rts_{bits}}}{R_{bits/sec}}\right.\\ +\left.\frac{L_{schbits}}{R_{bits/sec}}+\frac{L_{rsubits}}{R_{bits/sec}}+\frac{ L_{cts_{bits}}}{R_{bits/sec}}\right)\end{array}  \tag{30}\]

where \(Pw_{tx}\) and \(Pw_{rx}\) are the unit energy to transmit and receive a bit of information respectively and \(\overline{T_{s_{2}}}\) is the average time a node is active. The energy for each cluster is given as the sum of energy used for transmission of every cluster member, the total energy utilized in the cluster is given by:

\[E_{tot-intra}=\sum_{n=1}^{CN}E_{n} \tag{31}\]

The energy consumed in inter-cluster communication depends on the communication of the cluster head nodes. The communication is either direct or cooperative; the energy consumed by each type of node depends on the amount of time it spends in each of the states based on individual role in the network. This also depends on the traffic of each node as detailed in Equation (7). #### 4.5.2 Lifetime

The estimated lifetime of the node is determined from its initial energy and the total energy it uses for transmission [28; 29]:

\[E|L|=\frac{E_{0}-E_{u}}{E_{tot}} \tag{32}\]where \(E_{u}\) is the energy level at which the sensor is considered unusable, which is decided on implementation, \(E_{0}\) is the node's initial deployment energy level and \(E_{tot}\) is the sum of the energy for intra-cluster and inter-cluster communication. ### Delay

The delay performance result in this work is based on a simulation model. The average end to end delay is considered as the time from packet inception to successful receipt at the destination. With the effect of fading the delay is dependent on the quality of the channel as it impacts the number of retransmissions. For direct the average delay is given by:

\[D_{Avrg_{direct}}=T_{rts}+T_{cts}+T_{dat}+T_{ack} \tag{33}\]

where \(T_{rts}\) is time duration to send RTS by source node \(T_{cts}\) is time to send CTS by destination node, \(T_{dat}\) is time duration for data transmission for a single packet, and \(T_{ack}\) is the delay to send the acknowledgement packet.The transmit delay of the each packet is the packet length in bits over the average data transmittance of the network given by:

\[T_{pac_{type}}=\frac{L_{pac_{type_{bits}}}}{R_{bits/s}} \tag{34}\]

where \(T_{pac_{type}}\) is the delay per packet type, \(L_{pac_{type_{bits}}}\) is the length of the packets in bits and \(R_{bits/s}\) is the data rate. The delay incurred for cooperative transmission of packets between source, relay and destination is given by:

\[D_{Avrg_{coop}}=T_{rts}+T_{cts}+T_{hcts}*RL+T_{rsu}+T_{dat_{sr}}+T_{dat_{rd}}+T _{ack} \tag{35}\]

where \(RL\) is the number of possible relay nodes, \(T_{rsu}\) is the amount of time incurred to transmit the relay selection update and \(T_{dat_{sr}}\) and \(T_{dat_{rd}}\) is the time to transmit the data from source to relay and from relay to destination in a cooperative transmission. ## 5 Performance Results

This section presents the simulation and analytical results for the proposed energy-efficient MAC protocol for cluster-based wireless sensor networks (EEMACCSN) MAC framework.