# Decision System Based on Markov Chains for Sizing the Rebalancing Fleet of Bike Sharing Stations
## 4 Results
The content of Figure 13 is another instance derived from the general representation shown in Figure 5, where \({}^{k}\lambda_{ij}\) is the transition rate from the state \(S_{i}\) to the state \(S_{j};1\leq i\leq 4\) and \(1\leq j\leq 4;k=2\) specifies the context of the second subsystem. Compared to Figure 12, the MC features two extra transition rates (\({}^{2}\lambda_{13}\) and \({}^{2}\lambda_{24}\)). This is a good opportunity to test if the theoretical and practical evaluations passed, given the fact that the data points available for deriving these two rates were fewer, so the risk was higher because of the potentially lower signal-to-noise ratio. The MCs had the following state transition matrices:  
\[Q_{1}= \begin{bmatrix}^{1}\lambda_{11}&^{1}\lambda_{12}&0&0\\ ^{1}\lambda_{21}&^{1}\lambda_{22}&^{1}\lambda_{23}&0\\ 0&^{1}\lambda_{32}&^{1}\lambda_{33}&^{1}\lambda_{34}\\ 0&0&^{1}\lambda_{43}&^{1}\lambda_{44}\end{bmatrix} , \tag{31}\]  
\[Q_{2}= \begin{bmatrix}^{2}\lambda_{11}&^{2}\lambda_{12}&^{2}\lambda_{13}&0\\ ^{2}\lambda_{21}&^{2}\lambda_{22}&^{2}\lambda_{23}&^{2}\lambda_{24}\\ 0&^{2}\lambda_{32}&^{2}\lambda_{33}&^{2}\lambda_{34}\\ 0&0&^{2}\lambda_{43}&^{2}\lambda_{44}\end{bmatrix} , \tag{32}\]  
and the following values for the transition rates (Tables 5 and 6): \begin{table}
\begin{tabular}{r r r r r} \hline \hline \({}^{1}\lambda_{ij}\) & & & & & \\ \([10^{-4}\text{s}^{-1}]\) & \(\mathbf{j=1}\) & \(\mathbf{2}\) & \(\mathbf{3}\) & \(\mathbf{4}\) \\ \hline i = 1 & \(-7.48\) & \(7.48\) & \(0\) & \(0\) \\
2 & 6.71 & \(-12.52\) & \(5.81\) & \(0\) \\
3 & 0 & 6.39 & \(-10.66\) & \(4.27\) \\
4 & 0 & 0 & 9.10 & \(-9.10\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Transition rate values of Subsystem 1. Figure 13: MC of the second subsystem identified. Given that no transition rate reached a deviation of 3%, according to Equation (13), the theoretical evaluation passed. As expected and formulated in Equation (11), the minor values of \({}^{2}\lambda_{13}\) and \({}^{2}\lambda_{24}\) are clearly visible through the two isolated curves inside Figure 15.  
The practical evaluation of the (relevant) transition rates, in accordance with Equation (22), is presented below in Figures 16 and 17. The blue and red delineations correspond to the lower bound and upper bound, respectively, of the acceptable domain of the transition rate, while the green line marks the actual value identified. Therefore, an acceptable green line position is always in between blue and red, never above red and never below blue. For irrelevant transition rates evaluation (value \"0\" or \(\lambda_{ii}\) in Tables 5 and 6), \"N/A\" (not applicable) text was introduced. Due to the additional transitions present in the second subsystem, the identified values of all rates were closer to the bounds, due to the lower signal-to-noise ratio. Figure 15: Graphical representation of Subsystem 2 model identification and theoretical evaluation. Nevertheless, in both subsystems, all rates are within the bounds, meaning that the practical evaluation also passed. In Figure 18, the subsystems' overall unworking time is shown. The term \"assisted\" refers to the rebalancing that used the presented model; \"unsassisted\" means that the rebalancing was performed without the aid of the model. The average, worst, and best cases are defined in Equations (25)-(27), respectively. Based on the customer feedback mentioned in published studies [54], the improvement caused by decreasing the unworking times by 15-20% from values around 100 s was noticeable by the end users. So, the results of the \"assisted\" cases justified the usage of the model. Figure 16: Graphical representation of Subsystem 1 practical evaluation. Figure 17: Graphical representation of Subsystem 2 practical evaluation. In Figure 19, the subsystems' overall lost customer rate is presented. Based on the obtained results, the model caused a reduction in the lost customer rate, proving its positive effect. Nevertheless, it is desired that no customer is lost [58] and the model cannot guarantee that.