## 4 Reader Collision
\[p(x)=\frac{1}{2 CW}\ (CW-x),\ \ \ \ \ \ 0<x\leq CW \tag{17}\]  
Figure 7: Graphical representation of reader transmission flow (**a**) Conventional Pulse Protocol. (**b**) Modified Pulse Protocol. The average of the distribution derives as  
\[CW_{avg}=\frac{\sum_{0}^{CW}x\ p(x)}{\sum_{0}^{CW}x}. \tag{18}\]  
Referring to Figure 7, backoff time is decremented after maximum beacon interval, called Adaptive Beacon Interval (ABI). The maximum adaptive beacon interval of the algorithm is \(T_{success}=T_{r}+T_{max}+1\). Correspondingly, during the collision period, readers wait for collision slot and next empty slot. Hence, the duration of ABI is \(Tc=collision\ period+empty\ period\). Empty period refers to when no collision or successful events take place. Active ABI is defined when at least one reader communicates. The probability of active ABI is given as [20],  
\[P_{active}=1-\left(1-\frac{1}{CW_{avg}}\right)^{N} \tag{19}\]  
The probability of successful transmission in ABI duration is given by [20],  
\[P_{success}=\frac{N}{CW_{avg}}\bigg{(}1-\frac{1}{CW_{avg}}\bigg{)}^{N}. \tag{20}\]  
Therefore, the probability of collision event in ABI duration is given as [20],  
\[P_{collision}=1-\bigg{(}1-\frac{1}{CW_{avg}}\bigg{)}^{N}-\frac{N}{CW_{avg}} \bigg{(}1-\frac{1}{CW_{avg}}\bigg{)}^{N}. \tag{21}\]  
The total probability of empty slot and collision slot and successful slot is 1. Therefore,  
\[P_{empty}+P_{success}+P_{collision}=1 \tag{22}\]  
\[P_{empty}=\bigg{(}1-\frac{1}{CW_{avg}}\bigg{)}^{N}. \tag{23}\]  
Average duration of ABI can be obtained from,  
\[A_{ABI}=P_{empty}T_{empty}+P_{success}T_{success}+P_{collision}T_{collision}. \tag{24}\]  
The probability of collision of \(R_{1}\)'s and \(R_{2}\)'s transmission is \(1/CW_{avg}\). If there are \(N\) number of readers, the probability of collision in \(R_{1}\)'s transmission is,  
\[p(x)=\frac{1}{2 CW}\ (CW-x),\ \ \ \ \ \ 0<x\leq CW \tag{17}\]0  
If collision with probability of P occurs, reader selects another random backoff value from [0, CW] and transmit again at the end the backoff value. Therefore, according to the geometric distribution, the number of transmission trials can be defined with the probability of success (\(1-\mathrm{P}\)). Thus, until the successful transmission happens the expected number of ABI is defined by [20],  
\[p(x)=\frac{1}{2 CW}\ (CW-x),\ \ \ \ \ \ 0<x\leq CW \tag{17}\]1  
\[p(x)=\frac{1}{2 CW}\ (CW-x),\ \ \ \ \ \ 0<x\leq CW \tag{17}\]2The average duration of reader transmission cycle is given by:  
\[p(x)=\frac{1}{2 CW}\ (CW-x),\ \ \ \ \ \ 0<x\leq CW \tag{17}\]3  
The number of queries is determined by the reader over data channel within \(T_{read}\) period. The maximum number of beacon intervals is \"a\". Therefore, \(T_{read}=a\times T_{bi}\). In each slot of the \"a\" number of beacon intervals, one beacon is transmitted on the control channel and multiple number of queries on the data channel. \(T_{read}\) can be calculated for a beacon interval as [20],  
\[p(x)=\frac{1}{2 CW}\ (CW-x),\ \ \ \ \ \ 0<x\leq CW \tag{17}\]4  
\[p(x)=\frac{1}{2 CW}\ (CW-x),\ \ \ \ \ \ 0<x\leq CW \tag{17}\]5  
Throughput and utilization describe the system's merit. The number of successful periods by all readers in one cycle is determined by \(N_{ABI}\times P_{success}\). The number of successful queries sent by reader is \(Q_{T}\times N_{ABI}\times P_{success}\). Therefore, total throughput is defined as [16]  
\[p(x)=\frac{1}{2 CW}\ (CW-x),\ \ \ \ \ \ 0<x\leq CW \tag{17}\]6  
Utilization is obtained from,  
\[p(x)=\frac{1}{2 CW}\ (CW-x),\ \ \ \ \ \ 0<x\leq CW \tag{17}\]7  
The parameters considered in the proposed Algorithm 1 with the corresponding values is tabulated in \[p(x)=\frac{1}{2 CW}\ (CW-x),\ \ \ \ \ \ 0<x\leq CW \tag{17}\]8  
As can be seen in Figures 8 and 9, the simulation graph depicts the system's throughput and utilization. Figures 8 and 9 show the responses of the conventional pulse protocol and the modified pulse protocol.