# Energy Efficient Access Point Placement for Distributed Massive MIMO
###### Abstract
In the literature, the radiation pattern of the arrays in the APs is often ignored or assumed to be isotropic, which is not realistic: even the radiation pattern of antenna arrays with dipole elements is typically very directional in practice [31]. We have observed in our earlier work [5] that the radiation pattern has a significant impact on the optimal AP placement. To incorporate the radiation pattern, we need to take into account the broadside direction of the arrays at different locations, which critically depends on the environment of the AP locations. When deploying an AP on a wall, the AP can be installed in a less visible way [32] and catch less wind, for aesthetics and deployment simplicity. The broadside direction of the array will be perpendicular to the wall and towards the street. In this paper, we consider different antenna types for the array and we deploy the APs on the walls of buildings. For the second aspect, the obstacles in the environment, e.g., buildings, have a significant impact on the channels. The channel models applied in the current literature for optimizing the AP placement take a limited amount of environment information into account [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. Specifically, large-scale fading is usually modeled based on certain statistical distributions and the Euclidean distance between user locations and APs. Those models work well for the traditional cellular systems where the coverage region is large. In that case, the propagation distance between a base station and a user location is large and can be well approximated by a statistical distribution parametrized by the Euclidean distance. However, as coverage regions become small, the discrepancy between the propagation distance and Euclidean distance becomes significant, and empirical propagation conditions are highly location-specific. Therefore, those statistical models are no longer appropriate. As suggested by [33], a propagation model which is based on real-world geographic information is indispensable for realistic optimization of the AP placement. In [33], the authors model the propagation environment using a ray-tracing simulator (RTS) together with an iterative procedure for the AP placement optimization. An RTS considers multipath radio propagation as rays, taking into account different radio propagation mechanisms,namely reflection, scattering, and diffraction [34]. Therefore, an RTS is able to accurately simulate the channel gain considering the impact from the environment. However, using an RTS for channel evaluation is computationally intensive [34], especially if this has to be done for many potential AP locations and mobile users that may be anywhere in the coverage zone. Therefore, it is not scalable to apply an RTS for the AP placement optimization. The impact of buildings on the channel gain is also considered by [35, 36] where different pathloss exponent values for different scenarios together with the Euclidean distance are utilized. However, estimating the pathloss exponent values for a large number of scenarios is a cumbersome task. In this paper, we introduce a graph-based approach to effectively capture environment information. Given that planning the AP placement is a strategic level decision, elements in the environment that move or change within short time intervals, e.g., cars, are not considered. As illustrated in Figure 1, this study focuses on the modeling part of the AP placement optimization problem. The contributions of this paper are as follows:  
* We formulate the AP placement optimization as a combinatorial optimization problem. This opens a new door for tackling the AP placement problem. Many combinatorial optimization techniques such as integer programming [37] or meta-heuristics [38] can be utilized for solving the actual problem. * We propose a graph-based approach for modeling the physical propagation.