# Decision System Based on Markov Chains for Sizing the Rebalancing Fleet of Bike Sharing Stations
## 2 Literature Review
An MC provides a powerful framework for capturing the dynamic nature of a BSS, where the availability of bikes at different stations changes over time, due to user demand and system operations. Several studies have explored the use of MCs to analyze and improve various aspects of BSSs, such as station capacity planning [31], rebalancing strategies [32], and user behavior prediction [30]. In contrast, the current work focused on analyzing highly dynamic stations in a BSS. Such stations are characterized by frequent fluctuations in their bike availability, often being either completely full or empty. This enables the tailoring of rebalancing control strategies more effectively, ensuring compliance with specific system time constraints. In other words, it addresses, specifically, the problematic areas within the system in a flexible manner, instead of imposing a system universal solution that is not always requested and not always feasible. Researchers have developed models that incorporate factors like weather conditions, temporal patterns, and station characteristics, to accurately predict bike availability and demand [33; 34; 35]. These models have been used to optimize the allocation of bikes across stations, minimize rebalancing costs, and enhance the overall efficiency and user experience of BSSs. Furthermore, in their work, [PERSON] et al. [36] demonstrated that the use of MCs allows for the evaluation of different operational policies and the assessment of their impact on system performance. Overall, the application of MCs in the context of BSSs has shown great potential in improving the sustainability, reliability, and usability of these systems. However, several studies have concluded that there are certain drawbacks associated with the use of Markov models as modeling tools [37]. Firstly, Markov models do not have the capability to represent confliction, parallelism, preemption, and synchronization [37]. Secondly, it is difficult to extend a system that is modeled using a Markov model. Even a small change in the system design can result in significant changes to the MC structure [38]. Cluster-based approaches offer a promising solution to addressing these two key limitations of MCs [37]. With respect to the first issue mentioned, the inability to convey different forms of concurrency, study [39] showed that clustering allows a more granular representation of system dynamics, which enables the modeling of complex behaviors that are not easily captured by traditional Markov models. For the second issue mentioned, the model extension, by organizing the system into clusters, each representing a subset of the overall system, [PERSON] et al. [40] demonstrated that it becomes easier to incorporate new components or modify existing ones without disrupting the entire model. Obtaining a model based on a BSS using an MC opens up a wide range of applications in the field of transportation and urban planning. One potential application highlighted by [35] is predicting the availability of bikes at different stations throughout the day. By analyzing the transition probabilities between states, [PERSON] et al. [33] used their proposed model to provide insights into the patterns of bike usage and to help optimize the distribution of bikes across stations. This information can be valuable for bike sharing operators, to ensure a sufficient supply of bikes at popular locations and avoid overcrowding or shortages. Another application showcased in [41] is evaluating the impact of different policies or interventions on the system. By simulating the MC model in various scenarios, decision makers can assess the effectiveness of strategies, such as adding new stations, adjusting pricing schemes, or implementing incentives to encourage bike usage. Additionally, the model was used in [42] to analyze the spatial and temporal dynamics of BSSs, identifying hotspots in demand and understanding the factors influencing bike usage patterns.