## 3 Identification Process in Mixed Scenarios
### Computation of the Minimum Sojourn Time for a Given ICL  
As stated in the introduction, when tags enter the reading area in mixed scenarios new groups of tags do not enter until the previous one has left it. Therefore, the bulk of tags remains in the reading area only for a bounded sojourn time and some tags may be still unidentified after leaving. Figure 2 depicts this operation mode. Hence, for these type of scenarios the goal is _to minimize the number of times this event of losing tags can occur_. Let us remark that the identification time is an unbounded random variable for FSA systems (_i.e._, the identification time can be arbitrarily high with probability greater than zero). That is, it is not possible to establish a sojourn time which guarantees in the 100% of cases the identification of all the tags. Instead, a minimum sojourn time can be selected to guarantee that in all the cases the probability of identifying all the tags is higher than a given probability threshold, termed as Identification Confidence Level (ICL). From a practical point of view, the design process would be as follows: first, it starts with a quality requirement in the form of a given ICL parameter, then the minimum sojourn time needed to achieve the ICL is computed, and finally, the tunable system parameters (speed, tag population or others) are set to appropriate values to get the desired sojourn time. In the following sections we provide a method to compute the sojourn time for an ICL level.  
The analysis and computation of the minimum sojourn time needed to achieve an ICL level uses the DTMC built in the previous section. The Markovian analysis to compute the transition probabilities is the same as in the static scenario. Hence, we start the analysis from the transition probability matrix \(P\) given in Equation (2). Let us denote \(\pi^{(s)}=[\pi^{(s)}_{0},\ldots,\pi^{(s)}_{N}]\) the state probability vector of our DMTC at frame \(s\), that is, a row vector where the \(i\)-th element (\(\pi^{(s)}_{i}\)) represents the probability that \(i\) tags have been identified in \(s\) frames. From another point of view, \(\pi^{(s)}_{i}\) is the probability of being at a particular system state \(N-i\) in the frame \(s\). Besides, let \(\pi^{(0)}=[1,0,\ldots,0]\) be the initial distribution (since none of the \(N\) tags is identified before identification process starts). Then:  
\[\pi^{(s)}=\pi^{(0)}\cdot P^{s} \tag{6}\]  
That is, the state probability vector at frame \(s\) is given by the product of the initial distribution and the \(s\) power of the transition matrix \(P\). Let \(V_{N}\) denote the random variable that indicates the number of frames required to identify \(N\) tags. Its _cumulative distribution function_ (CDF) can be computed from Equation (6) as:  
\[P[V_{N}\leq s]=\pi^{(s)}_{N} \tag{7}\]  
From the CDF of \(V_{N}\), the minimum number of frames \(s\) required to achieve a given ICL level can be computed. The time elapsed in these frames is the minimum sojourn time needed.  
Figure 3 illustrates this method for ICL set to 0.99. The \(s\)-th column represents the probability of having identified the whole set of \(N\) tags at end of frame \(s\), that is \(\pi^{(s)}_{N}\). In this example, after six frames, the probability of identifying all tags is higher than the ICL selected. That is, each group of tags should  
Figure 2: Typical identification procedure in semi-static scenario. A set of tags enter the coverage area and stay for a sojourn time (during which tags are being identified) before a new set enters.  
remain in coverage for at least 6 identification frames in order to guarantee that, in the 99% of the cases, all tags are successfully identified. Let us remark that this is not the same as guaranteeing that 99% of tags are successfully identified in each group, being the former condition stronger that the latter one.