# LBTM: Listen-before-Talk Protocol for Multiclass UHF RFID Networks
## 6 Hospital Example
### Results  
Figure 8 shows the number of inventory cycles allocated for each type of reader (\(\phi_{1},\phi_{2},\phi_{3}\)) for a fixed \(\Delta_{1}=1\) (that is, no back-off time after the first LT for the patient monitoring readers) and either fixing \(\Delta_{3}=100\) or \(1000\) (for the inventorying readers inside the storage rooms) or fixing \(\Delta_{2}=50\) or \(200\). The \(x\)-axis indicates the free \(\Delta\) parameter.  
As with the grocery shop example, the variation of the \(\Delta\) parameters produces a noticeable change in the system performance, showing a clear trade-off between different traffic types. For example, when \(\Delta_{2}\) grows, readers of the first and third class increment their cycles/s (a significant amount in the case of first-class readers). The same occurs when \(\Delta_{2}\) is fixed but \(\Delta_{3}\) grows, reducing the priority of those readers and increasing the inventory frequency for the rest.  
Finally, we have also developed an expression to measure the _utility_ of a given configuration, in order to ease the interpretation of the results. This utility function is given by  
\[u(\phi_{1},\phi_{2},\phi_{3})=\text{sigm}(a_{1}\phi_{1}-b_{1})\text{sigm}(a_{ 2}\phi_{2}-b_{2})\text{sigm}(a_{3}\phi_{1}-b_{3}), \tag{1}\]  
where sigm denotes the _sigmoid_ function and \(a_{j},b_{j}\) for \(j=1,2,3\) are class parameters selected such that the sigmoid has values 0.1 and 0.9 respectively for a given minimum and a maximum number of cycles per second. The idea behind this expression is that for a configuration of \(\Delta\)'s being \"good\", it is required that it achieves a given minimum frequency of operation for each traffic. On the other hand, if the frequencies are too high, no additional benefit is obtained. The sigmoid product allows for capturing this behavior.  
The frequencies used for each traffic (see Table 2) as references are the following:  
* For patient monitoring readers, the minimum and maximum frequencies are respectively 1 and 1.5 cycles/s;
* For door readers, the minimum and maximum frequencies are respectively 0.2 and 0.5 cycles/s;
* For room-inventorying readers, the minimum and maximum frequencies are respectively 0.1 and 0.2 cycles/s.  
Figure 7: Hospital layout.  
For this setup, Figure 9 shows a color map with the resulting utility. Note that once again, \(\Delta_{1}\) has been set fixed to 1, and value modifications are allowed for \(\Delta_{2}\) and \(\Delta_{3}\). Results show that the optimal utility is above 0.76 and this is achieved for \(\Delta_{2}=220\) and \(\Delta_{3}=750\).  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Traffic Class** & **a** & **b** & \begin{tabular}{c} **\(\boldsymbol{\phi_{\min}}\)** \\ **(cycles)** \\ \end{tabular} & \begin{tabular}{c} **\(\boldsymbol{\phi_{\max}}\)** \\ **(cycles)** \\ \end{tabular} & \(\boldsymbol{u(\phi^{\star})}\) & \(\boldsymbol{\Delta^{\star}}\) & **Notes** \\ \hline Patient monitoring & 8.78 & 10.98 & 1 & 1.5 & 0.82 & 1 & \begin{tabular}{c} Reference frequencies are \\ selected to achieved a number \\ of at least 1 cycle/s \\ \end{tabular} \\ \hline Doors & 14.64 & 5.12 & 0.2 & 0.5 & 0.95 & 220 & \
\begin{tabular}{c} Reference frequencies aim at \\ getting at least 1 interrogation \\ opportunity for an maximum \\ walking speed of 6 km/h \\ considering a 2-m interrogation \\ radius around door \\ \end{tabular} \\ \hline Room inventorying & 43.94 & 6.59 & 0.1 & 0.2 & 0.97 & 750 &
\begin{tabular}{c} Reference frequencies aim at \\ detecting critical equipment \\ transfers in less than 1 min \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 2: Utility parameters. \(\Delta^{\star}\) are the \(\Delta^{\star}\)s of the optimal utility.  
Figure 8: Inventory frequencies for different \(\Delta\) setups.  
In summary, this utility function is a powerful _offline_ tool for a network designer to select a proper balance for the network configuration. An _online_ alternative, allowing the readers to autoselect their \(\Delta\) parameter, is described in the next section.