# Formal Algebraic Model of an Edge Data Center with a Redundant Ring Topology
## 3 Basics of the System Proposed  
Let us suppose a square field with a given side length aimed at growing crops, which may easily be divided into four quadrants with their internal boundaries being the coordinate axes and their crossing point being the origin of coordinates, thus allowing the assignment of some bidimensional co-ordinates to the location of any given bush or plant. This way, such a field could be assimilated to the real plane with Euclidean distance, where each quadrant would have a unique combination of signs for both axes, the same way as it happens in trigonometry. In this context, let us suppose four hosts ubicated in each of the corners therein, where every one of those may house diverse remote computing units, such as virtual machines (VMs), which may act as edge computing servers or fog computing servers, depending on their roles, so as to deal with the big data being generated by this whole ecosystem. Each host, ubicated on each corner, is attached to a pole furnished with a 90-degree sectorial antenna emitting with enough power to completely cover its own quadrant, thus the signal radiated by one of them overlaps around the coordinate axes with those coming from the other ones, as depicted in Figure 1.  
This way, communications between any given pair of antennas may be undertaken by means of fiber optical runs, those being either monomode or multimode, depending on the distance among them or the requirements of quality of service, although other middle-range wireless technologies might also be used, such as WiMAX or LoRaWAN, even though diverse long-range wireless solutions may also work, such as cellular deployments or satellite networks. This type of traffic among hosts, or even among VMs, may be identified as control traffic, whereas the sort of traffic among sensors and VMs, or VMs and actuators, may be seen as data traffic. Focusing on the VMs allocated within each host, which is associated to its particular antenna, there are two types of VMs to be considered herein, such as brokers and orchestrators. On the one hand, the former are dealing with data coming from end devices, being those collecting data from sensors, which could be moving ones, such as drones collecting pictures from diverse locations of the field, or fixed ones, such as environmental posts picking up information about temperature or humidity out of polling stations located around the terrain. On the other hand, the latter are working with data furnished by the former, which could be seen as a higher hierarchical level entities, whose role is to influence the decisions made by the former. Attending to the origin of the data they operate with and also to their position in the remote computing hierarchy, the former might be considered as edge servers and the latter, as fog servers. In addition, those fog devices might make use of cloud servers as their backup for foraging data or offloading their processing tasks. Hence, three levels within remote computing hierarchy are being established herein with different roles each, such as edge servers, also known as brokers, occupying the lower level, then the fog servers are above the edge ones, taking the middle level, and in turn, the cloud servers are above the fog ones, thus located on some ISP facilities and being placed on the top level. Therefore, the four hosts located on each of the corners of the square field may allocate both the VMs acting as brokers and those playing the part of orchestrators. Within the brokers, the ones dealing with environmental data will be always standing in the same  
Figure 1: Draft of the square field and the coverage area of the sectorial antennas located in each corner. host, where basically there will be one situated on each host at all times so as to process the data being received from fixed sensors situated on its own quadrant.