# Decision System Based on Markov Chains for Sizing the Rebalancing Fleet of Bike Sharing Stations
## 3 Methodology
### Practical Evaluation for the Proposed Model  
In the case of BSSs, the practical evaluations regarding constraints (e.g., reduction of unworking time, lost customer rate) are related to the on-time arrival of the rebalancing agents [54; 55; 56; 57; 58]. Therefore, the proposed model must have all the values for the state transition rates correlated to the time intervals requested to the trucks, in order to provide good performance. This quality-related constraint of the model can be formulated as in the equation below:  
\[\widehat{\lambda}_{ij}=\frac{1}{\gamma(m_{ij})}=\frac{1}{\Gamma\cdot m_{ij}}, \tag{21}\]  
where \(\widehat{\lambda}_{ij}\) is the ideal transition rate; \(\gamma(m_{ij})\) is the maximum time allowed for a truck to travel to \(m_{ij}\) bike stations for rebalancing; \(\Gamma\) is a constant value equal to the amount of time required to travel to each bike station (as part of the trip formed by the \(m_{ij}\) stations). In practice, the ideal transition rate does not always happen, because the high dynamism of the bike stations, with their unpredictable and ever-changing nature, cannot be captured by the model without a degree of tolerance. Therefore, the acceptable transition rate, which also incorporates a degree of noise, is defined below:  
\[ \begin{split}\widetilde{\lambda}_{ij}\approx\widehat{\lambda}_{ ij}\iff\alpha_{LB}\cdot\widehat{\lambda}_{ij}\leq\widetilde{\lambda}_{ij} \leq\alpha_{UB}\cdot\widehat{\lambda}_{ij},\\ \forall i,j\in\{1,\ldots,N\},i\
eq j,\end{split}  \tag{22}\]  
where \(\widetilde{\lambda}_{ij}\) is the acceptable transition rate from state \(S_{i}\) to \(S_{j}\); \(\widehat{\lambda}_{ij}\), with the formula in Equation (21), is the ideal transition rate relevant for supporting a truck that serves \(m_{ij}\) bike stations; \(\alpha_{LB}=\frac{1}{2},\alpha_{UB}=2\) are constant values corresponding to factors that are applied to \(\widehat{\lambda}_{ij}\), in order to obtain the tolerated lower and, respectively, upper bound for \(\widetilde{\lambda}_{ij}\). For the particular case of Citi Bike, according to [7; 59], the truck velocity during rebalancing is expected to reach up to 25.5 km/h, and the average of the distances in between two bike stations to be handled during rebalancing trips is a maximum of 2 km. This leads to the following equality:  
\[\Gamma(\mathrm{Citi\ Bike})=4\;\mathrm{min\ and\ 42\;s}, \tag{23}\]  
where \(\Gamma(BSS)\) is the constant value corresponding to the given BSS equal to the amount of time required to travel to each bike station during the rebalancing trip. The worst-case-scenario number \(m_{ij}\) of bike stations to be served when transitioning from \(S_{i}\) to \(S_{j}\) is the maximum difference between any number from \((v_{i-1},v_{i})\) and any number from \((v_{j-1},v_{j})\). This can be formulated as follows:  
\[m_{ij}=\max(|v_{j}-v_{i-1}|,|v_{i}-v_{j-1}|) \tag{24}\]In addition to the maximum arrival-time constraint, defined in Equation (23), another common operational goal for BSSs is the reduction of the unworking time [54]. The unworking time refers to the cumulative duration during a rebalancing chain of orders when a station has critical load.