## 4 Reading Process Evaluation  
In this section the reading process is evaluated for mixed scenarios. The goal is to point out the differences in the reading performance if the ICL criterion is used to set the sojourn time, rather than using the simpler approximation of setting the average identification time as the sojourn time. From Equation (7) the sojourn time is computed for an ICL level set to 0.99 assuming a frame length \(K\) = 16, and for a range \(N\) from 10 to 100 tags. Figure 4 shows the results. Let us remark that the results are exactly computed from Equation (7), no simulation has been done. Note that sojourn time is expressed as frames, and since frame length is fixed, the conversion to slots is provided as well. Besides, Figure 4 also shows the average number of frames (slots) required for identification, computed from Equation (5). Let us remark that this second curve corresponds to the criterion of selecting the average identification time as the sojourn time. Therefore, it has been referred to as \"Criterion average\" in the figure. Figure 4 also shows that achieving an ICL level of 0.99 becomes harder as the number of tags increases. In fact, it is the expected result when the frame length \(K\) is not high enough to cope with them. In that case, the frame length must be increased properly. To illustrate it, Table 1 provides the number of frames needed to achieve an ICL level of 0.99 when \(K\) = 16, 32 and 64 slots and slots are \(N\) = 50, 100, 200 and 300 tags. As can be seen, by increasing the frame length according to the expected number of tags it is possible to achieve the given ICL in a reasonable amount of time. Note from Figure 4 that the derivative of the sojourn time seems to increase with \(N\). This has an important implication. Namely, that _the reading efficiency decreases for large sets_. This effect can also be seen in Table 1. In fact, as a result of this observation we propose a identification strategy that improves the reading efficiency by splitting the set of tags in subset of lower size. Section 5 deals with this property and its consequences. Figure 3: Minimum sojourn time for a given ICL. The above-mentioned results indicate that in a mixed scenario the sojourn time of the tags must be much larger than the average time required for reading, specially for large groups of tags. This might result counterintuitive. A justification is provided in Figure 5. It shows the CDF of \(V_{N}\) for a configuration with \(N\) = 100 and \(K\) = 16. For a given sojourn time, it depicts the probability that all tags in the group have been identified. In this figure, the average identification time is marked, and its corresponding probability. The result is notable, since it means that when setting this value as the sojourn time more than 50% of the sets would leave with some unidentified tags. This is not acceptable in most RFID facilities. From another point of view, Figure 6 shows a comparative of the average number of lost tags (\(\bar{L}\)) using both criteria for the sojourn coverage. \(\bar{L}\) is the expectation of the number of unidentified tags at frame \(s\), corresponding to the probability distribution \(\pi^{(s)}\). Namely,  
\[\bar{L}=\sum_{i=0}^{N}(N-i)\pi_{i}^{(s)} \tag{8}\] \begin{table}
\begin{tabular}{|c|c|c|c|c|} \hline \multirow{2}{*}{Frame length (slots)} & \multicolumn{4}{c|}{Number of tags} \\ \cline{2-5}  & 50 & 100 & 200 & 300 \\ \hline
16 & 17 & 169 & \(\infty\) & \(\infty\) \\
32 & 7 & 16 & 144 & \(\infty\) \\
64 & 5 & 7 & 15 & 41 \\ \hline \end{tabular}
\end{table} 
Table 1: Number of frames needed to achieve \(ICL=0.99\) for frame lengths \(K=16,32,64\) and number of tags \(N=50,100,200,300\). Figure 4: Identification time (frames and slots) needed for different tag populations and \(K\) = 16. Criterion \(ICL=0.99\) shows the minimum number of frames computed from Equation (7), whereas Criterion average shows the number of frames needed when computed using Equation (5). The number of slots is given by the number of frames times the frame length.