## 1 Introduction  
With the rapid development of electric technology and the increase in environmental awareness, electric vehicles (EVs), as an advanced transportation tool with environmental and energy benefits, have been adopted worldwide. Recognizing these advantages, many local governments have strongly encouraged the adoption of electric taxis [1]. To further accelerate the adoption of EVs, they have implemented them in bus transit and taxi areas. According to Shanghai Municipal Development & Reform Commission, the taxi industry cumulatively updated over 22,000 electric taxis by the end of 2022 [2]. Similar adjustments and the optimization of the transportation energy structure have also been adopted in cities across China and Europe [3, 4]. Numerous constraints have hindered the widespread adoption of EVs, one of which is the inability to meet charging requirements [5]. Despite advancements in battery efficiency, it is unlikely that electric taxis will only require a single recharge outside of working hours in the foreseeable future due to their long travel distances and limited battery capacities [6]. In contrast to traditional taxis, which can be refueled within minutes, the recharging process for electric taxis can extend to as long as an hour. Although a battery exchange takes approximately 20 min, this is not widely adopted [7, 8]. This prolonged cycle underscores the importance of prudent charging station deployment. To address spatial and temporal charging demands, the deployment of a charging infrastructure at an appropriate scale and with an appropriate spatial distribution has garnered attention [9, 10, 11]. A large number of studies have focused on the deployment of charging stations for EVs in general, not specifically related to electric taxis [12; 13]. Acknowledging both the similarities and differences in various EV uses, the core research processes employed in the field of charging station deployment can indeed be borrowed and adapted. Researchers have dedicated efforts to developing methodologies for EV charging station deployment through the optimization or data-driven approaches [10; 14]. In recent decades, optimization has been the predominant approach to EV charging station deployment [11; 15; 16; 17]. Given that optimizing the location and scale of EV charging stations inherently poses a non-convex problem formulation, the Genetic Algorithm (GA) has emerged as a prevalent tool in optimizing charging station deployment with the aim of minimizing total costs or maximizing the covered demand [18; 19; 20; 21; 22]. [PERSON] and [PERSON] [23] incorporated factors such as charging price elasticity, travel times, and on-site charging queues into their GA framework to maximize profit. Furthermore, [PERSON] et al. [24] optimized expressway charging stations with a bilevel model and GA, minimizing costs and fostering the adoption of battery electric vehicles. [PERSON] et al. [25] developed heuristic algorithms using two types of greedy algorithm for multi-period deployment optimization. In addition to GA, other optimization algorithms have also been explored and various Machine Learning (ML) algorithms have also been widely used in the charging station deployment problem [26]. [PERSON] et al. [27] formulated this problem as a finite Markov decision process in a reinforcement learning framework, employing a recurrent neural network. Several studies have applied the Particle Swarm Optimization (PSO) algorithm to minimize total costs while considering both energy grid loss and charging service coverage [15; 16; 28; 29]. Moreover, [PERSON] et al. [30], [PERSON] et al. [31], and [PERSON] and [PERSON] [32] developed Integer Linear Program (ILP)-solvers to address various optimization objectives, including minimizing energy and customer losses, and maximizing path flow coverage. Numerous studies have focused on utilizing data-driven approaches to find the optimal deployment of charging stations. To maximize realism and streamline research, the queuing theory has been introduced into such investigations.