Channel assignment for cellular mobile networks with nonuniform cells – an improved heuristic algorithm ! R. Chavez-Santiago, E. Gigi and V. Lyandres Abstract: A heuristic sequential algorithm for fixed channel assignment in cellular mobile networks with nonuniform cells, i.e., cells of different sizes and shapes, is presented. The channel assignment is treated as a nonbinary constraints satisfaction problem. The algorithm uses both the co-channel and adjacent channel interference levels and the number of channel requirements per cell to define the corresponding sequence of assignment. The effects of intercell overlapping and complex propagation mechanisms on the channel assignment results are investigated too.
1
Introduction
The electromagnetic spectrum is one of the most important natural resources and must be administrated as efficiently as possible. Frequency reuse within a wireless communications network can offer considerable reduction of the required spectrum; however, it may also lead to a loss of quality of communication links owing to interference. The solution for such a problem balances the economies of frequency reuse and the loss of quality in the network. The channel assignment problem (CAP) has commonly been represented as a constraint satisfaction problem (CSP) [1], assuming a regular hexagonal network layout and binary constraints between pairs of transmitters (TXs). These constraints have the following form fi fj 4g 8g 0 ð1Þ where fi and fj are the frequencies assigned to transmitters i and j, respectively, and g is a positive integer. Based on this model, the co-channel, adjacent channel, and co-site constraints are defined [2, 3]. However, it has been demonstrated that modelling the CAP with only binary constraints might lose important elements of the original problem [1, 4], which produces solutions that are not necessarily the best ones. The use of nonbinary constraints is an alternative representation of the CAP that consists of finding an assignment of frequencies for all TXs in a mobile network in such a way that an acceptable carrier-tointerference ratio (CIR) over all points of the coverage area is obtained. This approach was used to solve the CAP with the help of simulated annealing (SA) in [3, 5]. Although SA provides very good solutions, it requires very long processing time to reach optimality, which makes SA unpractical for quasidynamic channel assignment schemes, in which the assignment is modified according to the traffic pattern at different hours of the day. r IEE, 2006 IEE Proceedings online no. 20045331 doi:10.1049/ip-com:20045331 Paper first received 12th December 2004 and in final revised form 28th June 2005 The authors are with Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva 84105, Israel E-mail:
[email protected] IEE Proc.-Commun., Vol. 153, No. 1, February 2006
A heuristic sequential algorithm with nonbinary constraints (CIR constraints) that takes into account the co-channel interference level to compute a difficulty degree (DD) that defines the sequence of assignment was presented in [6]; this CIR constraint algorithm is based on the representation of the CAP as a network colouring problem [7], and it solves the CAP fast enough for quasydynamic channel assignment applications. The algorithm solves the CAP in cellular mobile networks in which the coverage area is divided into a number of circular cells with different sizes. Conventional channel assignment (CA) techniques [2, 3, 8] cannot be directly applied to this kind of networks that do not have a regular frequency reuse pattern. Unfortunately, the algorithm considered in [6] is based on several assumptions that simplify the problem but, in most of the cases, do not capture properly the elements involved with real networks, reducing significantly its practical application. In this paper we present an improved algorithm that takes into account every base station’s (BS) transmit power and the adjacent channel interference to compute the CIR constraint. We also incorporate a more accurate urban propagation model. 2
CIR constraint algorithm
Let Iij be the co-channel interference from cell j to cell i in a cellular mobile network, and bi the set of all the cells (excluding cell i) that use the same channel as cell i. The CIR at the input of a mobile station (MS) located at the contour (assuming that the cells have circular shape) of cell i can be expressed in its more general form as Ci CIRi ¼ P Iij
ð2Þ
j2bi
where Ci is the signal power from the BS in cell i as received at the contour of the cell. Using the simplest model that expresses the propagation loss L proportionally to the path length d as L d 4 in an urban environment, and assuming that all the BS antennas in the network radiate the same power, we can 61
3
rewrite (2) as follows CIRi ¼
R4 P i 4 dij j2bi
a
ð3Þ
where Ri is the radius of cell i, a is a specified CIR threshold, and dij is the worst case distance between the interfering cell j and cell i, given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 xi xj þ yi yj Ri ð4Þ dij ¼ where ðxi ; yi Þ and xj ; yj are the Cartesian coordinates of the BSs of cells i and j, respectively. Defining g ¼ 1=a, (3) can be written as follows X dij 4 g ð5Þ Ri j2b i
The last expression defines the co-channel interference constraint (hereinafter referred to as CIR constraint) that is used in the assignment algorithm as it will be explained in the next Section.
2.1
Algorithm description
The heuristic algorithm presented in [6] establishes a sequence of assignment in descending order according to the DD of every BS. Ten different DDs were defined in terms of interference from neighbouring cells and the number of channel requirements per cell, denoted by mi. The first and simplest DD is given by 4 N X dij DDi ð1Þ ¼ mi ð6Þ Ri j¼1 j6¼i
whereas the last and most complex one is given by " 4 # N X dij 4 dji DDi ð10Þ ¼ xij mi þ Ri Rj j¼1
ð7Þ
j6¼i
where N is the total number of cells in the network and xij is an influence factor whose value is 1 if the cells i and j use the same channel simultaneously, and 0 otherwise. The assignment process basically consists of a requirement–exhaustive strategy [6, 8, 9]. First, the algorithm arranges the cells in descending order in terms of their DD. Then, the first channel is taken and assigned to the first cell in the sequence, i.e., the cell that has the largest DD; after this, all the rest cells in the sequence are inspected in descending order according to their corresponding DD to determine whether the channel can be reused or not. Only if the CIR constraint given by (5) is satisfied, the channel reuse is allowed. After all the cells have been inspected, the sequence of assignment is reordered and then the second channel is considered. This process is repeated iteratively until all channel requirements have been satisfied. This algorithm belongs to the so-called exhaustive search techniques, which can prove optimality and are quicker than other heuristics and metaheuristics (e.g., genetic algorithms, simulated annealing, tabu search, etc.) in providing a solution. These facts were corroborated by a channel assignment example in a network with uniform cells [6]; the algorithm reached the theoretical lower bound of 275 channels predicted by [10]. Detailed analyses of the computational complexity of the algorithm can be found in [9, 11], which show that the algorithm is faster than any metaheuristic for problems of the same size. These characteristics justify further development and improvement of this channel assignment technique. 62
Improved CIR constraint algorithm
Although the algorithm described in the previous Section has provided satisfactory results in some practical examples [6], there are a number of inconveniences that limit its application to realistic scenarios: The algorithm was designed for channel assignment in networks where cells have different sizes; therefore, it cannot be assumed that all the BS antennas radiate the same power because the size of a cell is determined, among other things, by its antenna’s transmit power. The algorithm considers only co-channel interference. Although neighbouring cells using the same channel are the main source of interference, adjacent channels also contribute to the deterioration of the CIR and affect the assignment results. Considering only co-channel interference provides assignment plans that economise spectrum, but that in most of the cases maintain a considerable level of harmful interference. The effects of complex radio propagation mechanisms also affect the final assignment results. Thus, a more sophisticated propagation model than L d 4 must be considered in some special cases. It is unrealistic to assume that all the cells have a regular shape, e.g., perfect hexagons or circles. An algorithm for more realistic scenarios must take into account the fact that most cells might be irregular-polygon-shaped.
3.1
Extended CIR constraint
The points listed above motivated the modification of the algorithm considered in [6] in order to make it practical for assignments in more realistic scenarios. The first improvement we introduced to the algorithm is the extension of the CIR constraint to include every BS’s transmit power and the adjacent channel interference. Adjacent channel interference is reduced by filter selectivity and its harmful effects decrease as the spectrum separation between channels increases. A simple model to compute the filter attenuation on an adjacent channel is given by [1] adj factorg ¼ ð1 þ log2 gÞ 8g40
ð8Þ
where g is the spectrum separation (in number of channels) between the adjacent channel frequency and the filter’s central frequency and L is an attenuation constant. A typical value for L is 18 dB; therefore, for g ¼ 1, an adjacent channel is attenuated by a factor equal to 0.015. Let bi;g define the set of all the cells (excluding cell i) that use an adjacent channel with a spectrum separation g from a channel used in cell i; thus (3) can be extended, including transmit power and adjacent channel interference, to the following form CIRi ¼
Pi R4 i P j2bi
Pj dij4
þ
n P g¼1
adj factorg
P k2bi;g
!a Pk dik4 ð9Þ
where Pi is the transmit power of the BS antenna in cell i and n is the maximum number of spectrum separation channels to be considered for the computation of adjacent channel interference. In a straightforward way, (5) can be IEE Proc.-Commun., Vol. 153, No. 1, February 2006
rewritten to define an extended CIR constraint as 0 1 n X Pk d 4 X Pj dij4 X ik Ag þ adj factorg @ P R4 g ¼ 1 Pi R4 i j2b i i k2b i
3.2
ð10Þ
i;g
Propagation model
Propagation loss in urban environments is usually predicted by empirical and semideterministic models. The inclusion of one of these models in (9) and (10) is relatively easy; the propagation loss L computed with an adequate model should replace R4 and d 4 in all the cases. Most propagation models consist of a formula that defines L as a function of the propagation path d and the operation frequency f, i.e., Lðd; f Þ. Thus, the CIR constraint can be written in a general form as follows 0 1 n X X Pk Lðdik ; f Þ X Pj L dij ; f Ag þ adj factorg @ P i Lð R i ; f Þ g ¼ 1 Pi LðRi ; f Þ j2b k2b i
i;g
ð11Þ The Walfisch-Ikegami model (WIM) [12] is a semideterministic propagation model for a network with medium/large size cells in built-up areas, adopted within the framework of the Cooperation in the Field of Scientific and Technical Research (COST 231). It demonstrates a good fit to measured propagation data for frequencies in the range of 800–2000 MHz and propagation paths in the range of 0.02–5 km. The WIM provides formulas for line-of-sight (LOS) and non-line-of-sight (NLOS) propagation situations. In a LOS situation the WIM gives the propagation loss in dB by ð12Þ LLOS ¼ 42:64 þ 26 log10 d½km þ 20 log10 f½MHz For NLOS path situations, the WIM uses the parameters illustrated in Fig. 1. Such parameters are: hb BS antenna height over street level in metres (4–50 m) hm MS antenna height in metres (1–3 m) hB nominal height of building roofs in metres Dhb ¼ hb hB height of BS antenna above rooftops in metres Dhm ¼ hB hm height of MS antenna below rooftops in metres b building separation in metres (use 20–50 m if no data provided)
w width of street (use b/2 if no data provided) f angle of incident wave with respect to the street (use 901 if no data provided). The propagation loss in dB for NLOS conditions is given by Lfs þ Lrts þ Lmds ; Lrts þ Lmds 0 ð13Þ LNLOS ¼ Lfs ; Lrts þ Lmds o0 where Lrts is the roof-to-street diffraction and scatter loss, Lmsd is the multi-screen diffraction loss, and Lfs is the free space loss given by Lfs ¼ 32:45 þ 20 log10 d½km þ 20 log10 f½MHz
Lrts and Lmsd are functions of the NLOS parameters listed above; a comprehensive description of their computation can be found in [12].
3.3
Irregular-polygon-shaped cells
For the case of networks with irregular-polygon-shaped cells, we define a matrix of ‘radiuses’ Ri and a matrix of distances dij for the worst-case interference point between each and every pair of cells. The matrices of ‘radiuses’ and distances have the following form, respectively 3 2 2 3
6 R21 R ¼ 6 4 .. . Ri1
R12 .. . R12
... ... .. . ...
Rij R2j 7 .. 7 5; .
6 d21 6 D ¼ 6 . 4 .. di1
. . . dij . . . d2j 7 7 . 7 .. . .. 5 ...
These matrices are used for the CIR computation between pairs of TXs. Let us consider an example to illustrate this point. Figure 2 depicts a network with three irregularpolygon-shaped cells, where the worst-case interference points between every pair of cell are indicated by black dots and the ‘radius’ and distance for every case are labelled. Thus, the co-channel CIRin cell 1 causedby 4 4 ¼ P R P2 d12 þ cell 2 and 3 is given by CIR 1 1 12 4 4 P3 d13 , using only the co-channel interference P1 R13 part of (10); in a straightforward way, the co-channel 3 caused by cell 2 is equal to CIR3 ¼ CIR 4incell 4 P2 d32 . P3 R32 cell 1 ( P1)
cell 2 ( P2)
d
d12
R13 cell 3 ( P3)
buildings
d13 P2
hb
d12 .. . d12
ð15Þ
R12
BS antenna
ð14Þ
d32
R32
P3
hB
w b
BS antenna
hm
worst-case interference point MS antenna
Fig. 2 ∆hb = hb − hB MS
incident wave
Fig. 1
Network with irregular-polygon-shaped cells
∆hm = hB − hm
NLOS parameters for the WIM
IEE Proc.-Commun., Vol. 153, No. 1, February 2006
4 direction of travel
Examples and results
We implemented the heuristic algorithm including all our proposed modifications, and investigated the impact that such modifications have on the channel assignment result in networks with both regular and irregular layouts. All the 63
4.1
Effects of adjacent channel interference
We investigated the effects of adjacent channel interference on the assignment results using the example presented in [6], which is based on the Philadelphia benchmark [8]. In this example, the cells are represented by circles of radius equal to 1 km (Fig. 3), contrarily to the original Philadelphia benchmark where the cells are hexagons. In Fig. 3, every cell is labelled with its corresponding number of channel requirements. The resulting assignment’s span and order (the difference between the minimum and the maximum used channel, and the total number of the used channels, respectively) are summarised in Table 1.
changing the BS locations, we varied the radiuses of the cells to form different network configurations ranging from nonoverlapping to highly overlapping, as depicted in Fig. 4. When the radiuses are equal to 1 km, we obtain a transition configuration where cells are adjacent to each other but without overlapping, as shown in Fig. 4b. The assignment results for every configuration are summarised in Table 2.
base station location and coverage area (radius 500 m) 10 9 8 7 Y, Km
results we present were obtained using the DD(10) given by (7), and a CIR threshold a equal to 18 dB, unless otherwise indicated.
6 5 4
1 km 8
25
8
8
3
8
2 15
18
52
77
28
13
15
1 2
31
15
36
57
28
4
6
8 X, Km
8
10
12
14
a 10
13
8
base station location coverage area (radius 1 Km) 10
Fig. 3
Network structure of the Philadelphia-like example
9 8 7
Cochannel interference
Cochannel and adjacent channel interference
n¼0
n¼1
Y, Km
Table 1: Philadelphia-like example results
6 5 4
n¼2
Span
Order
Span
Order
Span
Order
275
275
448
345
448
345
3 2 1 2
4.2
Effects of intercell overlapping
Although intercell overlapping might be desirable for handover processes, an excessive overlapping might increase interference and therefore the values of the assignment’s span and order. This problem does not exist in networks with regular hexagonal layouts, but unfortunately, such ideal layouts are inexistent in real systems. In order to evaluate the impact of intercell overlapping, we selected 21 BS location points to form a regular network structure similar to the Philadelphia example, where every location point is the centre of a circular cell. Without 64
6
8
10
12
14
X, Km b base station location and coverage area (radius 1.25 Km) 10 9 8 7 Y, Km
In the case of only co-channel interference, we obtained an assignment’s span and order equal to 275 channels, the same result reported in [6]. However, interesting results are obtained when we consider also adjacent channel interference. In this case, both span and order increase to 448 and 345 channels, respectively. The same values are obtained whether n ¼ 1 or n ¼ 2, which suggests that only the first adjacent channels’ effects are significant and therefore they must be taken into account, whereas the second (and further) adjacent channels’ can be neglected. The overoptimistic assignment obtained with n ¼ 0 does not guarantee an interference-free network environment.
4
6 5 4 3 2 1 2
4
6
8
10
12
14
16
X, Km c
Fig. 4
Various network configurations
a Nonoverlapping network configuration b Nonoverlapping configuration with adjacent cells; this is the transition configuration c Overlapping configuration IEE Proc.-Commun., Vol. 153, No. 1, February 2006
Table 2: Intercell overlapping results
Table 3: Urban propagation results with DD(1)
Radius, m
Model
Cochannel interference n¼0 Span
Cochannel and adjacent channel Interference n¼1
Order
Span
Radius, m
n¼2 Order
Span
Order
Cochannel interference
Cochannel and adjacent channel interference
n¼0
n¼1
Span LBd–4
Order
Span
n¼2 Order
Span
Order
200
229
77
229
77
229
77
100
229
86
229
86
229
86
500
229
145
229
146
260
189
200
229
188
229
208
229
208
1000
240
240
330
330
338
337
250
237
237
325
325
457
276
1250
280
280
449
344
449
345
300
279
279
457
329
457
329
1500
331
331
439
365
481
374
400
360
360
439
388
601
375
2000
409
397
825
438
865
449 100
229
91
229
91
229
91
200
229
189
229
211
229
214
250
241
241
342
342
474
318
300
281
281
457
329
457
329
400
360
360
435
397
599
394
WIM
The results show that the radiuses of the cells have no significant impact on the assignment’ span when there is no overlapping, and only the assignment’s order increases significantly as the radiuses of the cells increase. This can be seen in the configurations with radiuses of 200 m and 500 m for n ¼ 0, in which the same assignment’s span of 229 channels was obtained in both cases, contrasting with an assignment’s order of 77 and 145 channels, respectively. Moreover, adjacent channel interference can be neglected in nonoverlapping configurations. These facts change drastically in overlapping network configurations, where adjacent channel interference must be taken into account to guarantee an interference-free environment. In highly overlapping configurations (radiuses of 1.5 km and 2 km), even second adjacent channels (n ¼ 2) affect the assignment results. However, in moderate overlapping networks (radius of 1.25 km), the assignment results are practically the same if we take into account the first and second adjacent channels (n ¼ 2) or only the first ones (n ¼ 1). Since the vast majority of networks have a moderate intercell overlapping because nonoverlapping and highly overlapping configurations are practically inexistent in real systems, the results obtained reaffirm the conclusion of the previous Section: for practical purposes, co-channel and first adjacent channel interference should be taken into account for CIR computation, whereas further adjacent channels’ effects can be neglected.
4.3
Table 4: Urban propagation results with DD(10) Model
LBd–4
WIM
Radius, m
Cochannel interference
Cochannel and adjacent channel Interference
n¼0
n¼1
n¼2
Span
Order
Span
Order
Span
Order
100
230
105
230
105
230
105
200
229
188
229
209
229
209
250
238
238
328
328
459
278
300
278
278
457
329
457
329
400
360
360
439
388
601
375
100
231
113
231
113
231
113
200
229
189
229
216
229
216
250
242
242
341
341
467
327
300
281
281
457
329
457
329
400
360
360
435
397
599
394
Urban propagation
The propagation model L d 4 is a fair approximation in urban macrocellular environments. Nevertheless, more precise estimation of interference might be necessary when the scarcity of available channels is critical and overestimation of interference leads to waste of spectrum. Alternatively, underestimation of interference might produce assignments in which harmful interference levels still remain; also in this case, more accurate estimation of interference might be necessary. We incorporated the WIM [12] to our heuristic algorithm and obtained the assignment plans for different network configurations using the structure of the previous Section, but reduced to a scale 1 : 4; e.g., the same configuration of Fig. 4b will be obtained here with a radius equal to 250 m. We varied the radiuses to obtain several configurations ranging from nonoverlapping to highly overlapping, and obtained the assignment plan for each case using both, the L d 4 and the WIM models. The input data assumed for the WIM are: f ¼ 850 MHz, hb ¼ 40 m, hm ¼ 2 m, hB ¼ 20 m, b ¼ 30 m, w ¼ 15 m, and f ¼ 901. The results are summarised in Tables 3 and 4 for DD(1) and DD(10), respectively. IEE Proc.-Commun., Vol. 153, No. 1, February 2006
In this example, the L d 4 model underestimated the interference with respect to the WIM’s predictions in most of the cases, which resulted in a slight difference of assignment’s span and a more significant difference of assignment’s order for the co-channel interference case in the overlapping configurations. This difference is more significant when adjacent channel interference is considered. For nonoverlapping configurations (radiuses of 100 m and 200 m), the same assignment results are obtained with both propagation models. Though the assignment’s span and order obtained with the L d 4 model are smaller than those obtained with the WIM, it does not necessarily mean that they are better solutions. It is important to remember that the optimal solution of the CAP consists of the assignment that best balances the usage of spectrum and interference in the network to satisfy the traffic demand. Nevertheless, the use of a complex propagation model should be limited to the special cases mentioned above, as for practical cases the results obtained with the conventional power law model are very similar to those obtained with the WIM. 65
Regarding the performance of the algorithm, we can notice that the DD(1) provides better results than the DD(10) in this particular example. This suggests that in some cases it is not necessary to use complex difficulty degrees, since simpler ones can produce similar or even better results.
10 Y, km
4.4
15
Irregular network layouts
One of the important advantages of modelling the CAP with nonbinary constraints is the possibility of finding assignments in networks in which the cells have different sizes and shapes, and where no regular frequency reuse pattern exists. Conventional CA techniques cannot be applied directly to this kind of irregular network layouts. The work in [13] suggests the possibility of finding an optimal regular reuse pattern in irregular network layouts, which might enable the application of conventional CA algorithms; however, to the best of our knowledge, no further results in this direction were reported. We applied our heuristic algorithm to a practical example based on the structure of a real network in the city of W.urzburg, Germany [14–16], depicted in Fig. 5. In this example, the spatial traffic distribution is represented by discrete points called demand nodes. Such points are the centre of an area containing a quantum of demand from a traffic point of view, accounted in a fixed number of call requests per time unit. Demand nodes are generated by a partition clustering algorithm [14] using available population data; thus, demand nodes are densely distributed in traffic hotspots and sparse in areas of low demand, as shown in Fig. 6. The so-called set cover positioning algorithm (SCBPA) [15] selects the optimal positions of BSs that maximise the number of covered demand nodes. The lines indicating the convex hull around the set of demand nodes covered by each BS delimit the size and shape of the cells (Fig. 5). The transmit power necessary for every BS’s antenna to cover its corresponding cell area is also indicated in Fig. 5. We assumed three cases of traffic load conditions where each demand node takes a value of 1, 3, and 5 calls/h [17], respectively, and computed the number of channel requirements per cell in all the cases (Table 5). The network operates with the global system for mobile communications at 1800 MHz (GSM1800) standard, which means that a maximum of 374 frequency channels (200 kHz wide-carrier per channel) are available [18]. We defined the matrices R
5
0
0
5
10
15
X, km
Fig. 6
Spatial distribution of traffic demand nodes
. Table 5: Channel requirements of the Wurzburg-like example Cell
Number of channel requirements 1 call/h
3 call/h
5 call/h
1
10
22
33
2
9
19
29 13
3
5
9
4
25
61
97
5
5
9
13
6
6
11
16
7
14
31
47
8
7
15
22
9
5
9
13
and D as given by (16), and obtained the assignment plans for each case using the DD(1) and DD(10). The results are summarised in Table 6. This practical example once again reaffirms the conclusions elaborated upon in the previous Section regarding the adjacent channel interference. Moreover, we notice a slight better performance of the DD(10) for the case of only cochannel interference, but it is outperformed by the DD(1) when adjacent channel interference is also taken into
15 cell 1 (54 dBm)
. Table 6: Wurzburg-like example results
cell 3 (55 dBm)
cell 2 (56 dBm)
Cochannel interference n¼0
10 cell 5 (47 dBm)
Span
Y, km
cell 4 (61 dBm)
n¼1 Order
Span
n¼2 Order
Span
Order
DD(1) 1 call/h
cell 7 (61 dBm)
cell 6 (60 dBm)
5
cell 8 (59 dBm)
74
64
101
69
101
72
3 calls/h 182
144
241
157
241
159
5 calls/h 290
222
380
237
380
245
cell 9 (51 dBm)
DD(10) 1 call/h
0 0
5
10 X, km
Fig. 5 66
Cochannel and adjacent channel Interference
Network structure of the W.urzburg-like example
15
73
64
113
72
113
72
3 calls/h 181
144
270
157
270
157
5 calls/h 289
222
424
242
424
242
IEE Proc.-Commun., Vol. 153, No. 1, February 2006
account. However, the practical importance of the assignment plan of Table 6 for the heavy traffic-loaded case, i.e., assuming 5 calls/h, is very poor as the assignment span exceeds the maximum number of available channels. This situation generates a problem referred to as the fixed spectrum problem [3], which consists in finding a feasible assignment subjected to a limited number of available channels. In the case that an assignment without constraints violations is unfeasible, the solution of the fixed spectrum problem must admit some violations in order to satisfy the channel requirements of the network. For our particular way of modelling the CAP, the violation of the CIR constraint in order to obtain a lower assignment span consists in tolerating higher levels of interference. This is done by decreasing the CIR threshold a until a valid assignment is found. Thus, we reduced the CIR threshold a in steps of 1 dB and computed the assignment plan in each case until the span was equal or lower than 374 channels. The results using the DD(1) are summarised in Table 7. Notice that for a ¼ 17 dB the assignment plan is exactly the same as in Table 6, and only with a ¼ 16 dB the span in all the cases is below the number of available channels. A further reduction to a ¼ 15 dB leaves the previous results practically unchanged. This assignment meets perfectly the requirements of the GSM1800 standard, which recommends a CIR threshold of 15 dB for planning purposes [19]. This example showed the effectiveness of our algorithm at generating appropriate assignments with realistic constraints regarding number of available channels, cells size and shape, and transmit power. In this case, the transmit power was not the same for all BSs and this fact was taken into account by our algorithm. The impact of modelling different transmit powers into the channel assignment process can be shown by obtaining the channel assignment assuming the same transmit power for all BSs and comparing it to the results in Table 7. We obtained such an assignment for a ¼ 15 dB using the DD(1) (Table 8). We observe that assuming the same transmit power for all BSs produces significant lower assignment spans than those shown in Table 7 when both co-channel and adjacent channel interference are taken into account. For the case of only co-channel interference the results are practically the same. This means that the assignments in Table 8 might not
Table 7: Results for different CIR thresholds Cochannel interference
Cochannel and adjacent channel Interference
n¼0
n¼1
n¼2
Span
Order Span
Order Span
Order
a ¼ 17 dB 1 call/h
74
64
101
69
101
72
3 calls/h
182
144
241
157
241
159
5 calls/h
290
222
380
237
380
245
a ¼ 16 dB 1 call/h
74
64
83
65
83
69
3 calls/h
182
144
200
143
200
152
5 calls/h
290
221
317
221
317
233
a ¼ 15 dB 1 call/h
74
56
83
62
83
65
3 calls/h
181
124
200
138
200
145
5 calls/h
289
189
317
213
317
223
IEE Proc.-Commun., Vol. 153, No. 1, February 2006
Table 8: Results for equal transmit power in all BSs Cochannel interference
Cochannel and adjacent channel Interference
n¼0 Span
n¼1 Order
Span
n¼2 Order
Span
Order
1 call/h
75
63
83
64
83
64
3 calls/h
182
142
194
144
194
144
5 calls/h
290
219
307
222
307
222
guarantee an interference-free network environment. The difference between both results is due to a less inaccurate estimation of CIR values during the assignment process when the assumption of equal transmit power is made. 5
Conclusions
We have presented an improved heuristic sequential algorithm for fixed channel assignment that uses nonbinary constraints. The algorithm provides an acceptable carrierto-interference value at the worst case interference point of each and every cell using the same channel. We demonstrated that it is necessary to take into account adjacent channel interference for the computation of carrier-tointerference ratio, but for practical purposes we recommend to consider only the first adjacent channels since that is enough to obtain interference-free assignments. This algorithm solves the minimum span problem for a fixed carrier-to-interference ratio, i.e., the problem of finding an assignment to meet a given carrier-to-interference ratio threshold criteria. In the case that the algorithm produces an assignment with more channels than the limit assigned to a network operator, the given threshold should be reduced to tolerate more interference. We recommend reducing such a threshold in steps of 2 dB; lower values might not produce significant changes. The channel assignment must be computed using every new threshold value until a valid assignment is found. If the achieved carrier-to-interference ratio is below the standard value for a specific system, the network might not work properly and other techniques such as cell splitting should be applied. Further research in sequential algorithms for fixed channel assignment should not be devoted to the development of more elaborated difficulty degrees or different sequence arrange criteria, since we have demonstrated that simple difficulty degrees provide similar and even better results than complex ones in practical examples. With the development of metaheuristic algorithms that can solve assignment problems with binary constraints even to optimality in some cases, it seems that the only way to improve the usage of spectrum is by modelling the channel assignment problem using nonbinary constraints. However, the key to obtain maximum profit of the nonbinary constraints approach resides in the proper estimation of interference. Thus, research efforts must be concentrated in this direction. 6
Acknowledgments
The authors are grateful to the referees who reviewed the manuscript. Their comments were extremely helpful and improved the paper significantly. R. Cha! vez-Santiago thanks the National Council of Science and Technology (Conacyt–Mexico) for the financial support given under Graduate Scholarship number 130737. 67
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