Strong Masting Conjecture for Multiple Size Hexagonal Tessellation in GSM Network Design

One way to improve cellular network performance is to use efficient handover method and design pattern among other factors. The efficient design pattern has been proven geometrically to be hexagonal (Hales, 2001, pp. 122) due to its maximum tessellable area coverage. But uneven geographical distribution of subscribers requires tessellable hexagons of different radii due to variation of costs of GSM masts. This will call for an overlap difference. The constraint of minimum overlap difference for multiple cell range is a new area that is untapped in cell planning. This paper addresses such multiple size hexagonal tessellation problem using a conjecture. Data from MTN River State-Nigeria, was collected. Multiple Size Hexagonal Tessellation Model (MSHTM) conjecture for masting three (3) different size MTN GSM masts in River State, accounted for least overlap difference with area of 148.3km using 36 GSM masts instead of the original 21.48 km for 50 GSM masts. Our conjecture generally holds for k-different (kk ≥ 2) cell range.


Introduction
Cell planning is the most significant operations in GSM design network.It includes the choice of design pattern (triangular, square or hexagon), geographic, environmental and network parameters such as terrain and artificial structures, base station location and transmission power among others.But the hexagonal design has least overlap and hence has a strength higher than both a square and an equi-triangular polygon.The hexagon motivated circular shaped cells are produced when two or more sector signal radiated antenna are used.These circular cells overlap significantly and is crucial for subscriber movement-handover.This overlap removes signal loss due to no coverage or ensures soft handover due to better overlapping regions.How much overlap difference to be permitted in the uniform design network has been studied extensively by Donkoh E. K. et.al (2015a).Uneven geographical distribution of subscribers require varied cell ranges.This will result in variation in overlap difference.Both the design pattern and geometry of overlap is complex but the result is more economical as fewer GSM masts will be used.Realistically, how much overlap has been a thriving challenge in recent times due to variation in subscriber geographical distribution, affordability and significance of mobile telephone in this age of technology.

Related Literature
Antenna signal radiation in GSM network design takes many form.But the most profitable radiation pattern is known to be the sector motivated circular shape (Azad, 2012.).Due to it convenience and usefulness manufacturers of GSM antenna like Mobile Mark, Multiband Technologies, Global Source, Asian Creation among others have designed GSM antenna's with variety of sector angles including 30 0 , 60 0 , 90 0 , 120 0 as the common ones.Nonetheless, designing GSM network due to uneven geographical distribution of subscribers requires geometry of 2-D hexagonal tessellation .Unfortunately, multiple size hexagonal design has since not been generally practiced due to the complexity in the position of the base station that offers minimum overlap difference.This paper uses the least tessellable polygonal overlap difference to conjecture a multiple size design formula for hexagons of several dimensions.Donkoh & Opoku (2016, pp.33) emphasize with geometric proof that the hexagon has the least overlap difference of 13.4% and hence has the strongest tessellable area coverage.Donkoh E. K et.al (2015a, pp. 5) investigated the hexagonal overlap difference for a uniform 0.6km cell range of 50 GSM masts design and obtain an overlap difference of 5.788km using 35 GSM masts instead of the original 26.884km.Christaller (1933) classic theory, hexagonal tessellation has been advocated for thematic cartography by Carr et.al (1992), and has been used to study cluster perception in animated maps (Griffin, 2006), as well as color perception (Brewer, 1996).

Beyond applications of
However, geometry of multiple size hexagonal tessellation in covering bounded areas, optimizing the overlap difference for k-different ( ≥ 2) cell ranges have not been studied.We therefore proposed the strong multiple size masting conjecture for solving the bounded area coverage problem in GSM network design.

Computational Experience
We consider intersecting circular cells superimposed on non-hexagonal polygon with cell range  1 and  2 and 1 apothems r 1 and  2 respectively.In Figure 1, XZ is one side of the non-hexagonal polygon and the cells overlap for covering purposes with an overlap difference(d) of  2 −  2 +  1 −  1 .(1) where h is a side of the right triangle.
Equating (4) and ( 5), 1 Apothem is a line drawn from the center of a regular polygon to an edge and perpendicular to that edge.It is the perpendicular bisector of that edge and also the radius of the inscribed circle to that polygon.
where h is a side of the right triangle.Also,  1 =  1  1 Equating (2) and ( 5) and extending it to radius Similarly, in triangle ′, A single overlap difference for Figure 2 is (10) Generally n non-uniform overlaps, (11) Equation ( 9), (10) or ( 11) is used to calculate the overlap difference for non-uniform disks as shown in Figure 1.We established a formula for calculating the area of a pair of overlap for non-hexagonal polygon inscribed disks.From Figure 1 we have: Area of single overlap difference (Area of sector ′ − Area of triangle ′ ) + (Area of sector  − Area of triangle  ) Substituting ( 7) and ( 8) into ( 12) Generally for n different overlaps, Equation ( 13) is the area of each non-uniform cell range in terms of overlap difference that is not created by hexagon.The value   can be calculated from Figure 1 in Autocad environment or using equation (13) as shown in column 4 and 8 in Table 2 where  1 and  2 are the cell ranges of the GSM masts.The equation  =   −   is the overlap difference between disks with centres m and n.    2( 1 −  1 )

Overlap for
Let   2,  1 =  +  1 +  2 represents the global least overlap difference for two different size disks superimposed on tessellable hexagon in the ratio 1:k, where Theorem: The apothem   created by  sided tessellable regular polygon inscribed in a disk of radius  1 is Proof.
Donkoh et al (2016, pp.33-34) gave a formal proof of this theorem.Equation ( 18) then becomes Since the polygon is a hexagon  = 6 sided but with different radii.Thus Equation ( 20) is the least overlap difference for GSM network design using two different radii since the 1:  size hexagon tile completely.

Masting Conjecture
Generally for  different tessellable regular polygons  1 ,  2 , …   inscribed in disks with respective radii   ,  −1 , … ,  1 (ℎ   >  −1 ) the least overlap difference is Since we are tilling with hexagon For three different tessellable regular hexagon the least overlap difference can be obtain from equation ( 22) to be This is the one sided least overlap difference of triple non-uniform hexagonal tessellation for masting in GSM network.Equation ( 23) significantly informs cell planners how to design the GSM network for least overlap difference.Overlap difference will be For  overlaps, the difference will be Case II: Hexagons with radii  2 and  1 6(c): Section of hexagon tilling with radii  2 , 1 6(d): Section of hexagon tiling with radii  2 ,  1 Overlap difference will be For  such overlaps, the difference will be Case III: Hexagons with Radii  2 and  1 Overlap difference is For  such overlaps, the difference will be Case IV: Hexagons with Radius  1

Figure 6(e). Section of hexagonal tilling with radius 𝑅𝑅 1
Overlap difference will be For  such overlaps, the difference will be Generally, the various cases put together gives us total overlap difference Where the overlap area is calculated using the inclusive exclusive formula where the first sum is over all  the second sum is over all pairs . with  < , the third sum is over all triples , ,  with  <  <  and so fourth.Table 3. Comparative Analysis of the MSHT model to the Original Layout method for MTN River State, Nigeria.

Discussion
Designing of multiple size hexagonal tessellation with minimum overlap difference is a new area in cell planning in telecommunication network design.Our study conjectures an algorithm for efficient masting with least overlap difference for multiple cell range.Application of this formula to MTN River State GSM network solution resulted in an overlap difference of 12.368km which is a 53.2% reduction over the original overlap difference of 26.413km.The formula also uses 36GSM masts, covering an area of 148.715km 2 compared to the cell engineers original design of 50 masts covering an area of 21.48km 2 .This is equivalent to using 1GSM masts to cover 4.13km 2 in the multiple size hexagonal tessellation model instead of 1GSM masts for 0.43km 2 using the original design.Table 3 shows the results of the computation.

Conclusion
Our study provide an optimal multiple size hexagonal tessellation design with least overlap difference of 12.368km and total coverage area of 148.71km 2 .The number of GSM masts obtained from the MSHTM is 36 as compared to the original design of 50 GSM masts.This gives a 28% reduction over the original number of GSM masts.We used geometry of hexagonal tessellation approach to geometric disks covering for multiple cell range to reach optimality and it is the first study that uses multiple size hexagonal tessellation for covering of point sets to arrive at minimum overlap difference and overlap area.

Figure 2 (Figure 2 .
Figure2(a) illustrates the hexagonal cell layout with inradius and circumradius of the hexagonal cell as  1 and  1 , respectively.In Figure2(b), cells partially overlapped because  1 equals to the hexagon's circumradius.

Figure 4 .
Figure 4. Overlap difference for non-uniform disks, represented by , and GSM cell by .

Figure 6 .
Figure 6.Triple Size Hexagonal Tessellation Continuous upward tiling will lead us to the following overlap differences .Case I: Hexagon with Radius  2 as in 6(b)

Table 1 .
WGS-84 coordinates of 50 MTN GSM masts in River State, Nigeria.Let   be disks with radius ,    be number of cells with radius  then Area of cells = sum of area of all disks− sum of all overlap areas of cells