# Decision System Based on Markov Chains for Sizing the Rebalancing Fleet of Bike Sharing Stations
## 4 Results  
In Figures 12 and 13, two MCs are presented, corresponding to the subsystems with the parameters documented in Tables 3 and 4, respectively. Figure 11: Negligible error of probabilistic functions for model equivalence. The parameters from Table 3 can be interpreted as follows:  
* \(m\), having the value of 20 bike stations, aligns with the usual size of 10-30 bike stations of clusters in the context of rebalancing operations [60; 61; 62];
* \(\Delta t_{REF}\) corresponds to early morning to mid-morning, correlated to the time of departure of workers to their activities, which might be a hint for increased predictability--hence, the subsystem discovery;
* \(T_{I}=T_{E}\) values represent compact chunks of circa 3 months that were used for identification and evaluation;
* \(v_{1}=25\%\), \(v_{2}=50\%\), and \(v_{3}=75\%\) are interval delimiters corresponding to the \(N=4\) states; so, the states \(S_{1}\), \(S_{2}\), \(S_{3}\), and \(S_{4}\) are associated with overall criticality levels inside intervals of [0,25%), [25,50%), [50,75%), and [75,100%], respectively, in accordance with the procedure derived from Equation (8). The content of Figure 12 is an instance derived from the general representation shown in Figure 5. In this instance, \({}^{k}\lambda_{ij}\) is the transition rate from state \(S_{i}\) to state \(S_{j}\), where \(1\leq i\leq 4\) and \(1\leq j\leq 4\); \(k=1\) specifies the context of the first subsystem. Another important aspect, which is particular to the first subsystem, is that each state \(S_{i}\) in the MC is connected to either itself or to the state with neighboring indexes \(S_{i-1}\) and \(S_{i+1}\). This is the most common pattern observed throughout Citi Bike data and implies the fact that it is quite rare to observe sudden jumps in overall criticality, i.e., it requires too many bike stations to become simultaneously critical. The parameters from Table 4 can be interpreted as follows:  
* \(m\), having the value of 16 bike stations, is also usual for clusters in the context of rebalancing operations [60; 61; 62]; \begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameter** & **Property of** & **Value** \\ \hline \(m\) & Subsystem & 16 [bike stations] \\ \(\Delta t_{REF}\) & Identification/Evaluation & Tuesday 6:35 PM–9:16 PM \\ \(T_{I}\) & Identification & 87 [days] \\ \(T_{E}\) & Evaluation & 87 [days] \\ \(v_{1}\) & Quantization & 25\% \\ \(v_{2}\) & Quantization & 50\% \\ \(v_{3}\) & Quantization & 75\% \\ \(N\) & MC & 4 [states] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Subsystem 2 parameters. Figure 12: MC of the first subsystem identified. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameter** & **Property of** & **Value** \\ \hline \(m\) & Subsystem & 20 [bike stations] \\ \(\Delta t_{REF}\) & Identification/Evaluation & Wednesday 5:47 AM–10:08 AM \\ \(T_{I}\) & Identification & 103 [days] \\ \(T_{E}\) & Evaluation & 103 [days] \\ \(v_{1}\) & Quantization & 25\% \\ \(v_{2}\) & Quantization & 50\% \\ \(v_{3}\) & Quantization & 75\% \\ \(N\) & MC & 4 [states] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Subsystem 1 parameters. * \(\Delta t_{REF}\) overlaps with the usual time of checking out of work, which might explain the subsystem predictability;
* \(T_{I}=T_{E}\) have values for the time interval of identification and evaluation of almost 3 months;
* \(v_{1}=25\%\), \(\
u_{2}=50\%\), and \(v_{3}=75\%\) are interval delimiters corresponding to the \(N=4\) states; so, the states \(S_{1}\), \(S_{2}\), \(S_{3}\), and \(S_{4}\) are associated with overall criticality levels inside intervals of [0,25%), [25,50%), [50,75%) and [75,100%], respectively, in accordance with the procedure derived from Equation (8). Compared to Table 3, the number of bike stations that could be captured was reduced by 20%, and this was to be expected because the second subsystem is showing additional transition rates, which would be restricted if too many bike stations were introduced.