## 1 Introduction  
The high-speed train (HST) has become the most prominent mode of transport for passengers, especially in densely populated countries. Additionally, wireless channel modeling for HSTs has become a growing study area [1]. Apart from handling critical signaling, the HST communication system in 5G and beyond is expected to provide a variety of stable onboard broadband services such as the Internet of Things for the railway, real-time train dispatch, onboard internet access, etc. However, present technologies allow data rate transmission of up to a maximum of 100 Mbps [2; 3]. This can only be accommodated in the interconnection required for the future HST communication infrastructure [4]. Implementing millimeter wave (mmWave) technologies for future rail communication services is a viable strategy for achieving the required data-rates for 5G transmission and beyond [2; 5; 6]. This has been seen in the development of a mmWave-based mobile hotspot network (MHN) structure in [7]. In this system, the field trials and simulations have shown that a throughput of 7 Gbps can be achieved in the downlink when the train speed is 500 km/h. The single-frequency multi-flow (SFMF) transmission technique was used to achieve this peak throughput. In addition, mmWave band measurements between the train and truck were carried out by [8] at 41 GHz. The channel characteristics like the LCR and average fading duration were investigated thoroughly when the velocity between the receiver and transmitter was 170 km/h. In addition the research on measurement done in [7; 8], simulations of wireless channel models have also been done in [9; 10; 11]. A GBSM approach for mmWave bands was introduced in [9], although this was based on ray tracer simulation. In the work of [11], the channel characteristics of HST environments, such as tunnels and urban environments, were analyzed using ray tracing at 30 GHz. Most of the channel statistics in these environments have been studied using ray tracing, which also has a shortcoming of computational complexity when compared to geometry-based stochastic models (GBSM). Although these studies have been carried out, the implementation of the mmWave band for HST communications in environments like cutting and viaducts continues to face different challenges. One of the challenges is the lack of channel models, especially geometry-based stochastic models, which can mimic the channel environments. The available models have focused on the lower sub-band of 5G, such as in [12; 13]. Results from these models cannot be just extended to mmWave bands without addressing the shortcoming of scatterers, which could be captured using 3D geometry. In addition, some channel models like the IMT-A channel model have not introduced the other high-speed train channel environments, like viaducts and cutting, whereas the impact of breaking distance has not been considered in these channel environments [14]. Thus, with this motivation, a non-stationary 3D GBSM MIMO for viaduct and cutting environments for the 5G channel was proposed. Since coverage is important for mmWave band channels, when modeling a channel, it is important to consider the 3D (elevation angles) and the MIMO. Incorporating the two enables the proposed channel models to be extended to massive MIMO with different planes, such as cylindrical or spherical, to enhance coverage for mmWave channels. Elevation angles are also important in the cutting environment when directional antennas are used and the mobile radio station (MRS) is very close to the base station (BS). In this paper, a generic model is proposed that is an extension of the one in [15], which was considered for this open-case scenario. A GBSM consists of one sphere and many confocal elliptic cylinders representing the stationary and uniformly distributed scatterers and the non-stationary environment situations, respectively. A time-variant Rician K-factor was introduced in the viaduct and cutting scenarios. This is important due to the effect of dimension structure in these two environments.