## 5 Splitting Strategy  
Let us recall from previous section that the differential sojourn time required to achieve a given ICL level increases with the size \(N\) of the set of tags. This observation suggests that if items could be rearranged in smaller subsets the overall sojourn time (to identify all the subsets) may decrease, achieving the same global ICL confidence. In this section the influence of this effect is studied for mixed scenarios. Specifically, the aim is at deciding if splitting the set of tags reduces the overall identification time, and computing the optimal configuration of the subsets.  
In the splitting model, the \(N\) tags of a group are assumed to be uniformly redistributed into \(J\) smaller subsets of size \(N^{\prime}=\frac{N}{J}\) tags (for simplicity we consider a perfect partition, _i.e._, \(N^{\prime}\) is integer). Let \(V^{\prime}_{N}\) denote the (random) identification time of the \(J\) subsets. It can be expressed as,  
\[V^{\prime}_{N}=\sum_{i=1}^{J}V_{N^{\prime}} \tag{9}\]  
Since the reading time of one subset is independent from the reading time of any other subset the mass probability function of \(V^{\prime}_{N}\) is just,  
\[P[V^{\prime}_{N}=s]=P[V_{N^{\prime}}=s]*\stackrel{{ J}}{{\dots}}*P[V_{N^{\prime}}=s] \tag{10}\]  
where the \(*\) operator denotes convolution. From the previous equation the cumulative distribution function of \(V^{\prime}_{N}\) is obtained. As an example of the results, Figure 7 shows the CDF of the identification time (measured in frames or in slots) achieved when a population of \(N=400\) tags is split into \(J\) = 2, 4 and 8 subsets (plus the original set of 400 tags), for a frame length of \(K\) = 128. In this scenario, for the same ICL level as in the previous numerical examples, the configuration of two sets of 200 tags provides the best results, being the first configuration able to surpass the ICL = 0.99 at frame 13. Therefore, these results confirm our previous discussion, that is, the fact that the derivative of the sojourn time increases with \(N\) effectively indicates that reducing the size of the set increases the reading efficiency. Hence, splitting is useful and the next step is to find the optimal size of subsets.  
Besides, due to the partition of the sets a time lapse between each subset should be reckoned on in a complete model of the mixed scenario. Let \(\tau\) denote this time lapse considered between two subsets. Parameter \(\tau\) can be regarded either as a safeguard time for the identification of tags (e.g., a temporal spacing in a conveyor belt) or an operational parameter that takes into account an overhead for splitting the sets (e.g., modelling a robotic arm that removes a subset from the reading area and introduces a new one). With this restriction into account, the overall identification time (\(T\)) is given in next equation. For simplicity, \(\tau\) is considered as an integer number of frames (conversely, slots).  
\[T=V^{\prime}_{N}+(J-1)\tau \tag{11}\]  
And its mass probability function is,  
\[P[T=s]=P[V^{\prime}_{N}=s-(J-1)\tau] \tag{12}\]From the former equation the CDF of \(T\) is extracted and the minimum sojourn time is directly determined with the same procedure as Section 3.2. The minimum sojourn time has been evaluated for different subset sizes and \(\tau\) settings. Figures 8, 9 and 10 show the results for \(N\) = 400 tags, \(J\) = 1, 2, 3 and 4, \(\tau\) = 0, 100, and 1,000 slots, ICL = 0.99, and different frame lengths \(K\) =