## 1 Introduction  
The existence of idle resources and people's willingness to well use them to promote the sharing economy has brought several lifestyle changes, including various traffic modes. The sharing economy means people can share resources (e.g., services, skills, assets, etc.) through a network of private individuals and businesses, which may often but not always be at lower costs. Shared mobility, a typical resource sharing pattern, has become more popular and common, such as bike-sharing [1], ride-sharing [2], car-sharing [2] and electric vehicle sharing [3]. Apparently, the shared transport emergence contributes to protecting the environment, conserving energy, reducing traffic congestion and improving transportation resource utilization and availability [4]. However, there remain some limitations of current shared transport systems. For instance, users cannot find an available shared vehicle or bicycle nearby sometimes, or they have to spend too much time seeking idle ones when they are in a hurry. Sometimes, the idle cars or bikes are parked in low demand areas, causing potential profit loss for operators [3]. An essential reason is _tidal commuter flows_[1]; more specifically, citizens usually travel from residential areas (or home) to public transit locations (subway stations) or popular areas (e.g., commercial zones) during peak hours, thereby resulting in rare bikes in residential zones, whereas the overwhelming number of shared bikes near popular zones [5]. An optional solution for improving this is to rebalance or reposition shared resources, e.g., reposition shared-bikes by trucks in bike-sharing systems (see [6]) and reposition shared cars in vehicle sharing systems (see [3]). This paper focuses on bike-sharing and aims at solving rebalance problems existing in bike-sharing systems. Generally, there are two typical types of bike-sharing systems: the traditional bike-sharing system with docking stations (e.g., Citi Bike, Divvy and Ford GoBike in America, see [1]) and the dockless bike-sharing system (e.g., Mobike in China, see [5]). The dockless bike-sharing system, the focus of this work and the latest type of bike-sharing system, also termed the free-floating bike-sharing system, implies users are allowed to park their bikes everywhere they want rather than fixed docking stations. As the dockless bike-sharing system emerges, the traditional bike-sharing system tends to lose competitiveness and gradually vanish in some countries, especially China. Numerous previous studies concentrate on optimizing rebalance (or relocation) decisions, which includes two main streams: vehicle-based (also terms operator-based) approach and user-based approach [7]. For vehicle-based approach, the rebalance decision involves pick-up decisions (from which station to pick up how many idle bikes), drop off decisions (to which station to drop off bikes) and the routing decisions, and then the rebalance strategy is executed by a fleet of trucks (see, e.g., [6; 8; 9; 10]). Such a rebalance approach requires accurate demand prediction and cannot handle dynamic settings in real-time well. The user-based approach attempts to guide customers to rebalance the shared-bikes by providing monetary incentives, e.g., recommend pick-up or drop-off areas for users (see [5]), design incentive programs (see, e.g., the Bike Angels program in [1]). Instead of determining the optimal practical level decisions (i.e., pick-up and delivery decisions) and incentive policies (for user-based rebalance), this work focuses on analyzing how the key factors affect system performance based on the Markovian queueing networks formulation and optimizing the rebalance frequency. The rest of this paper is organized as follows. The relevant literature on resource-sharing system and rebalance in bike-sharing system is reviewed in Section 2. In Section 3, we first present a complete problem description and then introduce the Markovian queueing networks formulation in Section 3.1.