## 3 Rait
### Theoretical Analysis  
For the data transmissions from a child sensor node to a parent node, CSMA is used to reserve the channel. We assume that each node knows the number of contending neighbor nodes (m) and contends the channel with the optimal probability p = 1/m. The probability that one contending node wins the channel is p\({}_{\text{succ}}=(1-1/\text{m})^{\text{m}-1}\). Since the number of slots needed until the successful reservation is a geometric random variable, the average number of contending slots (ACS) is given by  
\[ACS=\frac{1}{\left(1-\frac{1}{M}\right)^{M-1}} \tag{1}\]  
where M is the Average number of Neighbor nodes.  
Assuming that the time length of the contention slot is T\({}_{\text{s}}\), we have the energy consumption (CE) for channel contention:  
\[CE=T_{s}\times P_{t}\times ACS=T_{s}\times P_{t}\times\frac{1}{\left(1-\frac{1 }{M}\right)^{M-1}} \tag{2}\]  
where P\({}_{\text{t}}\) is the transmission power.  
Assuming that the average number of packets of an image is NPI, the average total ACS to send an image one hop (IACS\({}_{\text{one}}\)) without RAIT is given by  
\[IACS_{one}=NPI\times ACS=NPI\times\frac{1}{\left(1-\frac{1}{M}\right)^{M-1}} \tag{3}\]  
and the average CE to send an image one hop (ICE\({}_{\text{one}}\)) without RAIT is given by \[ICE_{one}=NPI\times CE=NPI\times T_{s}\times P_{t}\times\frac{1}{(1-\frac{1}{M})^{M -1}} \tag{4}\]  
The average total ACS to send an image to the sink (IACS\({}_{sink}\)) without RAIT is given by  
\[IACS_{sink}=\sum_{i=1}^{H}(NPI\times ACS)=\sum_{i=1}^{H}(NPI\times\frac{1}{(1- \frac{1}{M})^{M-1}}) \tag{5}\]  
and the average CE to send an image to the sink (ICE\({}_{sink}\)) without RAIT is given by  
\[ICE_{sink}=\sum_{i=1}^{H}(NPI\times CE)=\sum_{i=1}^{H}(NPI\times T_{s}\times P_ {t}\times\frac{1}{(1-\frac{1}{M})^{M-1}}) \tag{6}\]  
where H is the hop count from the source node to the sink node.  
Assuming that the number of sensor nodes in the sensor network is N, the average total ACS to send whole images to the sink from N sensor nodes (IACS\({}_{total}\)) without RAIT is given by  
\[IACS_{total}=\sum_{i=1}^{N}IACS_{sink}=\sum_{i=1}^{N}\sum_{i=1}^{H}(NPI\times \frac{1}{(1-\frac{1}{M})^{M-1}}) \tag{7}\]  
and the average CE to send whole images from N sensor nodes to the sink (ICE\({}_{total}\)) without RAIT is given by  
\[ICE_{total}=\sum_{i=1}^{N}ICE_{sink}=\sum_{i=1}^{N}\sum_{i=1}^{H}(NPI\times T_{ s}\times P_{t}\times\frac{1}{(1-\frac{1}{M})^{M-1}}) \tag{8}\]  
On the other hand, IACS\({}_{one}\) with RAIT is same with the ACS and ICE\({}_{one}\) is also same with CE because a sensor node with RAIT has to contend just one time and could send whole packets of an image to the parent node without interference.  
The IACS\({}^{RAIT}_{sink}\) with RAIT is given by  
\[IACS_{sink}^{RAIT}=\sum_{i=1}^{H}ACS=\sum_{i=1}^{H}\frac{1}{(1-\frac{1}{M})^{ M-1}} \tag{9}\]  
and ICE\({}^{RAIT}_{sink}\) with RAIT is given by  
\[ICE_{sink}^{RAIT}=\sum_{i=1}^{H}CE=\sum_{i=1}^{H}(T_{s}\times P_{t}\times\frac {1}{(1-\frac{1}{M})^{M-1}}) \tag{10}\]  
IACS\({}^{RAIT}_{total}\) with RAIT is given by  
\[IACS_{total}^{RAIT}=\sum_{i=1}^{N}IACS_{sink}^{RAIT}=\sum_{i=1}^{N}\sum_{i=1}^ {H}(\frac{1}{(1-\frac{1}{M})^{M-1}}) \tag{11}\]  
and ICE\({}^{RAIT}_{total}\) with RAIT is given by Sensors_**2010**, _10_  
\[ICE_{total}^{RAH}=\sum_{i=1}^{N}ICE_{i\equiv k}^{RATT}=\sum_{i=1}^{N}\sum_{i=1}^{H} \left(NPI\times T_{s}\times P_{i}\times\frac{1}{\left(1-\frac{1}{M}\right)^{M-1}}\right) \tag{12}\]