# Decision System Based on Markov Chains for Sizing the Rebalancing Fleet of Bike Sharing Stations
## 1 Introduction  
The imperative to shift travel modes from vehicles to eco-friendly bikes has become evident, due to traffic congestion and pollution. Motor vehicles contribute significantly to emissions, primarily from fossil fuel consumption. To address these challenges and promote sustainable cities, integrating Bike Share Programs (BSPs) into transportation systems has gained traction. BSPs mitigate pollution, conserve resources, and improve air quality. They offer environmental benefits, promote physical activity, and provide cost-effective and efficient transportation. Implementing BSPs fosters greener urban environments, economically viable transportation, and healthier societies, facilitating the transition to sustainable modes of travel [1]. Bike sharing primarily replaces sustainable modes of transportation, although the rate at which it replaces cars is relatively low. However, bike sharing can indirectly reduce car travel by creating synergies with public transport, such as expanding catchment areas, reducing overcrowding, and improving overall travel times. Different types of modal shift result in different benefits. Transitioning public transport users to bike sharing can alleviate overcrowding in saturated public transport networks. It has been observed that sociallydisadvantaged groups are underrepresented among bike sharing users, but implementing equity measures and awareness-raising campaigns could help increase their usage rates [2]. The availability of bike sharing services presents both benefits and challenges in traffic management. One of the challenges is ensuring efficient bike sharing dispatch. Polarized traffic patterns within a bike sharing network can give rise to hotspots, which are locations characterized by an imbalance of bikes, either in excess or deficiency. Insufficient bike supply can result in lower user acceptance, while increasing the supply can lead to higher dispatch costs and potential underutilization of resources [3; 4]. The operators responsible for bike dispatching play a crucial role in balancing the demand for bicycles. They ensure that bikes are replenished at the most popular starting stations and that docking bases are available at the most popular destination stations [5]. It is important to note that how effectively the operators handle this issue directly impacts the satisfaction level of the service users. This presents another significant challenge, as it requires precise control on the part of the operators, and determining the optimal solution remains a complex and unresolved problem. Sub-optimal solutions for the control of bike sharing rebalancing operators can lead to several pitfalls. Inadequate control measures may result in inefficient redistribution of bikes, leading to imbalanced bike availability at different stations. This can result in a shortage of bikes at popular starting stations, causing frustration and inconvenience for users. Conversely, an excess of bikes at popular destination stations can lead to overcrowding and limited docking space, making it difficult for users to return their bikes. Sub-optimal control strategies may fail to consider dynamic changes in demand patterns, resulting in a mismatch between bike supply and user demand. This can further exacerbate the imbalances and hinder the overall effectiveness of the Bike Sharing System (BSS). Sub-optimal control of operators may lead to increased operational costs, as resources are not efficiently utilized. Therefore, it is crucial to develop robust control strategies that consider real-time demand fluctuations and optimize the redistribution of bikes, to ensure a seamless and satisfactory BSS user experience [6]. The present study introduces an approach based on capacity scaling in the BSS by utilizing bike counts as a measure of load. This allows for the identification of highly dynamic bike stations, which are then analyzed as a separate subsystem. A state vector is associated with this subsystem, capturing the frequency of critical events at the bike stations.