## 3 The Modified BA Protocol (MBA)
Whilst, in phase II, the arriving tags are identified without colliding with the staying tags through multiple rounds (denoted by cycles in BT and GBT), in which the first round is executed by ABS protocol (for \(Tx~{}-~{}T\), initial allocated cycles determined by tags exponential average estimation method and equal to \(0.88\gamma\), in the steady state, in which \(\gamma\) is the number of arriving tags in the \((i+1)\)-th frame \(F_{i+1}\), and the number of staying tags is zero), and each round consists of multiple cycles. Now, let the number of the tags in the \((i)\)-th frame (\(F_{i}\)) be \(n_{r}\), and the number of arriving and leaving tags in the \((i+1)\)-th frame (\(F_{i+1}\)) be \(\gamma\) and \(\beta\), respectively. Since ABS does not estimate the number of tags, its initial allocated cycles in the \((i+1)\)-th frame are equal to \(n_{r}\) when the number of tags in the \((i)\)-th frame is \(n_{r}\). Thus, according to the identification delay analysis of ABS [16], the first round of MBA in phase II can be formulated as follows:  
\[D_{MBA}(F_{i+1}|F_{i})\Bigg{|}~{}~{}j=1~{}~{}~{}=0.88\gamma~{}~{}1+~{}D_{BT}( \gamma)=0.88\gamma-1+1+2C_{BT}~{}(rj)~{}=0.88\gamma+~{}2C_{BT}(rj) \tag{13}\]  
\(nr\) and \(D_{BT}(\gamma+1)\) in \(D_{ABS}(F_{i+1}|F_{i})\) were replaced by \(0.88\gamma\) and \(D_{BT}(\gamma)\) in \(D_{MBA}(F_{i+1}|F_{i})\Bigg{|}~{}~{}j=1~{}~{}\), respectively, since the number of allocated cycles of \(D_{ABS}(F_{i+1}|F_{i})\) was replaced by \(0.88\gamma\), and the number of staying tags in \(D_{MBA}(F_{i+1}|F_{i})\Bigg{|}~{}~{}j=1~{}~{}~{}\) is zero. For GBT, from Equation (10), we obtain  
\[D_{GBT}(F_{i+1}|F_{i})=2C_{BT}(n_{r}-~{}\beta+~{}\gamma)~{}+~{}r \tag{14}\]  
while for MBA in \(F_{i+1}\) frame in phase I, the number of cycles equals the sum of the number of staying and leaving tags (blocking protocol). In phase II, only the arriving tags collide with each other according to Equation (13) in the first round and with BT protocol in the remaining rounds; Therefore,  
\[r_{j}=D_{BT}^{1}(u_{j}) \tag{6}\]0  
When \(\alpha\) is equal to zero, i.e., \(r=2\), the third term in Equation (15) will be eliminated, and \(r_{j}\) is equal to \(\gamma\). The first term represents one idle cycle consumed in the last round, i.e., \(j=r\), to ensure that all the tags have been identified. We notice, from Equations (14) and (15), when \(\gamma=~{}\beta\), MBA outperforms GBT and TBT (since \(D_{TBT}\) = \(D_{GBT}+EC\), according to [5]) significantly in consuming a lower total number of cycles when \(\gamma\) is equal to or less than \(n_{r}/2\). MBA also enhances the identification efficiency when \(\beta\) decreases, but the identification delay is not affected. As explained previously, the former protocols, GQT1, GQT2, GBT, and TBT, do not use blocking technique, and thus there is no ability to distinguish between collision and interference signals (responses of tags) for the entire identification delay period, while MBA recognizes between collision and interference responses for the period of staying tags. Thus, the identification delay of GBT is much more affected than MBA when interference signals exist according to the following equations:  
\[r_{j}=D_{BT}^{1}(u_{j}) \tag{6}\]1\[D_{MBA}(F_{i+1}|F_{i})=\left.n_{r}+\left[0.88\gamma\right.+\left.2C_{BT}(r\beta) \right]\right.+\left.\sum_{j=2}^{r-1}\left[1+2C_{BT}(r_{j})\right]\right.+\left.1 \right|\Bigg{|}_{ \begin{array}{c}j=r\end{array} }+\left[\frac{(n_{r}-\beta)}{ 1-p\mu}-(n_{r}-\beta)\right]+2\left[\frac{\gamma}{1-p\mu}-\gamma\right] \tag{17}\]  
The last term in Equation (16) and last two terms in Equation (17) denote the number of the additional cycles (\(\mathbf{\xi}\)) when interference signals exist. The last term in both equations is equal to the number of interference responses multiplied by 2, since GBT and MBA consume two additional cycles (one collision and one idle) for each interference response without the distinguishing from collision response (the entire period for GBT, and the arriving tags period for MBA).