Area Packing Problem

SEA23 decided to re-approach the problem from a perspective of an area-packing problem based on feedback from the second progress review. Using a search theory scheme outlined in Professor Harney’s 2013 Combat Systems Engineering, there are two methods of packing circles: hexagonal and square. Each node can represent a circular search pattern since the antennas are omnidirectional. Figure 28 shows both hexagonal and square tiling.

Figure 28. Hexagonal and Square Tiling (Packing) Patterns. Source: Harney (2013).

A cursory glance shows that hexagonal tiling offers the least amount of “white space” that is the area not covered. Because this is not a search problem, these nodes will be static for the purposes of the model. Therefore, there must be overlap so that the

“white space” is nonexistent in order to ensure that the network can communicate 100 percent of the time. While it is a geometry problem to determine the amount of overlap required to rid the area of uncovered space, it involves calculating several different geometries and subtracting to find the leftover space. Figure 29 shows this problem’s geometry.

Figure 29. Geometry of Circular Overlap Problem. Source: Harney (2013).

In Figure 29, (a) is the radius of the circle and (2d) will represent the distance between the circles. For this study, (2d), the radius of the circle, represents the maximum range of the communications system and the distance between nodes. The parameter that the team is interested in is what the distance between nodes needs to be in relation to the max range of the node. Harney (2013) defines this overlap parameter asξ =d a/ . Through this relationship, the team determines how far apart the nodes can be for each network and how many will fill the area of operations. Consulting Figure 30 shows what this parameter needs to be in order to have complete overlap.

Figure 30. Relationship of Overlap Parameter ( )ξ to the Fractional Coverage ( )η and Coverage Efficiency( )ε . Source: Harney (2013).

Using Figure 30, it is determined that the fractional coverage is 1.00 with an overlap parameter of approximately 0.86. To determine the number of nodes required, the range of the communications node must be multiplied by 0.86. The area of operation is a half-circle with a radius of 500 nautical miles. The area of this shape is 392,699 square nautical miles. By dividing the total coverage area by the area covered by a single node, the total number of nodes required is determined. The area of a single node depends upon its maximum transmission range and is simply the area of a circle:

r2

π

where (r) is the radius, or maximum transmission range of the individual node. Figure 31 shows the results of these calculations.

Figure 31. Minimum Number of Nodes Required to Cover Area Given the Range of the Communications Node. Note: X-Axis Does Not Start at Zero.

The team discovered an error after running several trial models that drastically changed the project team’s calculations for the minimum number of nodes required. The project team had thought of these nodes as sensor nodes as in search theory instead of communications nodes. In a searching problem, too much overlap results in wasted coverage and time that assets could be looking for their target. However, this is a communications problem and overlap is required. This error led to an under-calculation of the minimum number of nodes required to fill the area. The communications signal must be able to reach another node. Therefore, the nodes can only be as far apart as their maximum range. From Figure 29, the parameter (d) must be equal to zero. The project team had incorrectly utilized the radius of the maximum range of a single node as the maximum distance between nodes. This means that there will be much more overlap than 0.86. The overlap must cover the next node as shown in Figure 32.

Figure 32. Diagram Showing Required Overlap of Nodal Coverage to Complete Network.

To model this overlap, the team performed another hexagonal area-packing problem. However, this time the team utilized half of the maximum range for a given TDL and assumed zero overlap. Since the nodes are communications nodes only and not searching for a target, overlap is not required. It is just required that the circles touch so that each circle modeled covers half the distance to the next node. Figure 33 shows how the models circles work in comparison to the physical range of the nodes.

Figure 33. Comparison of Real Node Distance (Blue Circles) to Modelled Circles (Red Circles) for Area Packing Problem.

Since there is no overlap in this updated packing model, the team needed to figure out what was the total area to cover. For hexagonal packing with zero overlap, the area covered is equal to 0.9069 of the total area. (Eagle 2013) Since the total area is 392, 699 NM², the area required to cover is:

2 2

392699NM 0.9069=356139NM

SEA23 found the number of nodes by dividing the area required to cover by the area of a single node. Table 13 shows the results of the required number of nodes for each TDL considered. Additionally, both CEC and the Other network category reach their power limits at or prior to their horizon limits. Therefore, only Link-16 will show a benefit to increasing altitude as discussed earlier.

Table 13. Number of Nodes Required at Various Altitudes for TDLs Considered