## 1 Introduction  
In 5 th generation (5G) mobile communication wireless networks [1], a maximum transmission speed of 20 Gbps or more, a user transmission speed of 100 Mbps or more, and a transmission delay of less than 1 ms are provided by adopting a millimeter band (mmWave) of 30-100 GHz in addition to the existing sub-6 GHz band. The 6 th generation (6G) mobile communication system is expected to be deployed in 2030. A prospective vision of 6G network as well as its key enabling technologies are being studied to support future networks, such as the fully digital and connected world exploiting a new spectrum, the Internet of Everything, and the native artificial intelligence service of mobile broadband applications [2, 3, 4, 5]. However, due to support for transmission bandwidth under 20 GHz, it is difficult to support the maximum transmission speed of 1 Tbps or more and the user reduction speed of 10 Gbps or more required in the 6G mobile communication wireless networks [2]. Therefore, the terahertz band is being considered as an additional service band to meet the requirements of the 6 th generation mobile communication wireless system [4, 5]. It is also necessary to design a multi-band environment with ultra-wide frequency aggregation technology, provide ultra-high-precision massive beam-based multi-access and mobile connectivity, and provide multi-connectivity in a multi-band environment that simultaneously provides sub-6 GHz band, mmWave band, and Sub-THz band (hereafter, we identify sub-6 GHz band, mmWave band, and Sub-THz band by Frequency-Range 1 (FR1), Frequency-Range 2 (FR2), and Frequency-Range 3 (FR3), unless notified, respectively [3; 4; 5]. In particular, multi-band is required as one of the technologies to overcome unstable radio access in the mmWave band and the Sub-THz band [4; 5]; for instance, the UE association algorithm [5] which selects a base station operated in FR2 and FR3, i.e., multi-band mobile communication networks. However, it considers a single connectivity operation in multi-band mobile communication. In contrast to selecting base station in [5], it is necessary to discover radio access technology to improve system performance through collaboration between bands or access nodes and to design a multi-connectivity operation in mobile communication networks that allows mobile stations to access simultaneously at each access node (or base station) via a transmission point. In addition to this, in order to provide large-capacity transmission wireless access networks, performance analysis considering multi-band radio environments in mobile communication networks should be preceded. Thus, we extend transmission point selection (TPS) problem, also known as user association problem (see [6; 7; 8; 9; 10; 11; 12; 13; 14; 15] and references therein). For example, a cell is selected as the service cell for a user according to the expected service rate based on the received signal strength (RSS), i.e., max-RSS-based TPS [8]. For the spectrum efficiency, particularly in multi-tier networks, biased and load balancing, TPS are provided [9; 10], e.g., conventionally according to received power-based rule [8]. Also, the TP is selected according to at least one of the widely used metrics: outage/coverage probability [8], fairness [11], spectrum and/or energy efficiency [9], and latency [12]. In general, several association algorithms have been studied with these metrics. Meanwhile, some user association algorithms have also been studied recently for AI-embedded wireless networks [13; 14; 15]. To minimize the average age of the information of mobile Internet of Things (IoT) nodes by optimizing the selection of the hybrid access point (HAP) or station IoT node, we consider energy consumption of wireless-powered IoT [13]. In [14], a hierarchical double auction model for a collaborative computing-based system is designed to improve the long-term network capacity.