THE EFFECTIVE COVERAGE IN WIRELESS REGIONAL NETWORKS

P O Z NA N UN I V E R S ITY O F TE C H N O LO GY A C A D E M IC J O U R N AL S

No 54 Electrical Engineering 2007

Mirosław HAJDER

1

Aneta FILIPAK-KARASIŃSKA

1

Paweł DYMORA

1

THE EFFECTIVE COVERAGE IN WIRELESS REGIONAL

NETWORKS

The designing, constructing and exploitation of cellular wireless network is connected with the minimizing of network load. One of most effective ways of wireless network base stations load minimizing is additional division of subscribers nodes between neighboring cells. This work presents a new algorithms of wireless network decomposition into cells with taking into consideration limitations on its diameter and stream intensity in each of them.

Keywords: computer network, network topology, fault tolerance, graph theory.

1. DIVISION TASK DEFINITION

Let's consider the i -ry wireless network cell. We can present it with the help of

coherent finite undirected graph G_i =

(

V E_i, _i

)

, where: V – is the set of graph _i

nodes

(

{

}

)

1, , n , 0

i i i

V = v … v n> ; E _i – is the set of graph edges

{

}

(

Ei = ei1, ,… eim ,m>0

)

, and for each k

(

k= …1, ,n

)

, e is a pair of elements from ik

the set V [4]. The traffic in i -ry cell is described by the internodal flow matrix _i

[ ]

kl n n

L= λ _× , where: λ_kl – is the information stream between nodes k and l ,

, _i

k l V∈ . The position of subscribers nodes is defined by the table W =

[

α ϕ_k, _{k n}

]

_×2 of geographical coordinates, where: α ϕ_k, _k – is a node longitude and latitude, respectively. The length of wireless communication channel d is defined by the _kl

equation of the orthodrome length [5]:

( ) ( )

( ) ( ) (

)

[

]

arccos sin sin cos cos cos

kl k l k l k l

d = ϕ ϕ + ϕ ϕ α −α , (1)

1 _{Rzeszów University of Technology}

2007

Poznańskie Warsztaty Telekomunikacyjne Poznań 6 - 7 grudnia 2007 POZNAN UNIVERSITY OF TECHNOLOGY ACADEMIC JOURNALS

where:

(

α ϕ_k, _k

)

,

(

α ϕ_l, _l

)

– is the nodes k , l geographical coordinates; ,α ϕ – is a node longitude and latitude, respectively. However, the channel length obtained from the equation (1) is characterized by a significant error, which can be minimized by equation transformation to the below form [5]:

(

)

2

_{( ) ( )}

(

)

2

2 arcsin sin cos cos sin

2 2 k l k l kl k l d = ⎢⎡ _⎢⎡ ϕ ϕ− ⎤_⎥ + ϕ ϕ ⎡_⎢ α −α _⎥⎤ ⎥⎤ ⎢ ⎣ ⎦ ⎣ ⎦ ⎥ ⎣ ⎦ . (2) The main designing goal function is the network building cost minimization obtained for minimal number of cells and base stations covering cluster [1],[2],[3]. If the multicriterion designing is acceptable as the additional goal functions some standard function used in network designing process can be used e.g.: communication delays minimization, total traffic capacity maximization, minimization of maximal communication channel load and other. The most important designing limitations are: the maximal acceptable cell diameter as well as acceptable capacity of a base station.

Because the division task of wireless network into cells treats to group of linear programming tasks, so in order to its effective solution the lower and upper bound of acceptable solutions and initial approximate solution should be defined. The lower bound of base stations number is defined for the following input data: S –_p

network area; s – area of i -ary cell; _i r – radius of i -ary cell; _i λ_kl – intensity of information stream between subscribers v and _k v ; _{l ,}

1, n k k l l k l λ = ≠ Λ =

∑

– total information stream of a node v ; _k V – the set of subscribers nodes of i -ary cell. _i

Let’s assume that any i -ary cell is operated by B_i-ary base station, which maximal operate information stream is equal Λ . The total intensity of i -ary cell stream is _app equal

i i

B i

V

Λ =

∑

Λ . In optimal case, the network division into cells works by such selection of area covering by a cell so that each base station as well as each cell had a total cell stream intensity equal to maximal total stream operated by the base station, i.e. |

i i

B app ∀B

Λ = Λ . If the network is presented by an ellipse, the lower bound of base stations number is defined by round up to integer number of area quotient: network and one cell, i.e.: n_{= ⎢}⎡2πab 3 3r⎤_{⎥ . In real conditions traffic in}

cells is heterogeneous and average intensity of inner network stream as well as inner cells is different. Because of that the above estimation may be little precise.

3. THE ESSENCE OF PROPOSED METHOD

fig. 1.a. Circle radius r (cell radius), on which hexagon representing cell is _i

circumscribed, describes maximal acceptable area (region) of subscribers nodes localization. Let’s assume, that sets of nodes for analyzed cells are denoted as V ₁

and V . If, as shown in fig. 1.a, cells are presented in a form of adjustment to each ₂

other polygons, a small fragment of cells is common and subscribers nodes may be operated by two base stations. The set of nodes with access to both base stations is denoted as B_1,2, which consists of elements fulfilling the condition B_1,2= ∪ . It V₁ V₂

may be assumed, that the set B_1,2is always not empty, i.e. in neighboring cells always are nodes which may be operated by two base stations. If B_1,2= ∅ it is a unique network case, which can’t be optimized with the presented below method. Let’s define the necessary and enough condition of subscribers node v _k

membership to the i -ary cell (c_i). The distance between nodes v_k and v_l is denoted as d_kl. As a diameter D_i of a cell c_i we will denote the quantity

( )

max{ | }

k l i

i kl v v c

D = l _{∀ ∈} . Let v and _k v are the nodes of communication channel _l

which determines the cell diameter. In such a case, the sufficient condition of subscribers node v_q membership to the cell c (i.e._i v_q∈ ) if fulfilling the c_i

following condition: 2 _{( )}2 _{( )}2

i kq ql

D ≥ d + d [5]. This condition can be used for determining the subscribers nodes set belonging to the cell c_k:

2 2 2 ( ( ) ( ) ) { | } i kq ql i q D l l V = v _{≥ +} . Fig. 1.

From fig. 1.a we can see that he necessary condition of subset B existence is _1,2

fulfilling of condition (r r₁+ ₂)> , where L₀ L – is a base station range.₀ From the other hand the condition B_1,2≠ ∅ is fulfilled if

1 2 | i i v V v V i _{∈ ∪ ∈} ∃ [6], [7]. Let’s introduce the limitation on a stream intensity as 0,7

i

B app

Λ ≤ Λ , where: app

Λ – the maximal stream intensity operated by the base station [2]. Than with the regard to the above limitation condition the subscribers node v assignment to the _q

set V may be defined as: _i

(

2

( ) ( )

2 2

)

| 0,7 i i q i kq ql i app V V =⎧_⎨v D ≥ d + d ∨⎛_⎜ Λ ≤ Λ ⎞_⎟⎫_⎬ ⎝ ⎠ ⎩

∑

⎭.

Proposed method is based on relocation of subscribers nodes with generated by them streams between neighboring cells. Let’s consider an example of additional division of streams between cells c_1,2, ,… c_3,3 (see fig. 1.b). Let’s assume that maximal stream Λ operated by the base station is equal 60. Than streams _app additional division is realized in reference to the cell c_2,3 in direction of cells with minimal intensity, i.e. c , _1,2 c , _2,4 c . _3,2

4. ALGORITHM OF STREAMS ADDITIONAL DIVISION

Algorithm (see fig. 2) searches for a cell with a maximal intensity, which stream is in a next step divided between neighboring cells.

( ) 0 , 1, , 1, , 1 1, , 1, , 1 1, , 1 rs rs B B rs rs− rs rs+ rs r− −s rs r+s rs r+ −s rs r+s Λ = Λ − Λ Λ Λ Λ Λ Λ 0 rs B app Λ < Λ (λjk−) rs B app Λ ≤ Λ rs B app Λ ≥ Λ (λjk) − rs B app Λ ≤ Λ rs B app Λ ≥ Λ Fig. 2.

In order to do so, among them a cell with minimal total stream intensity is selected. To this cell a subscriber node with minimal value of inner stream is moved. The algorithm work is ended in the moment when in a network there is no more cells for which total inner stream intensity exceeds acceptable base station stream, i.e.

rs

B app

Λ ≥ Λ .Its input data are: type of stream distribution; maximal and minimal streams intensity; acceptable streams intensity value; maximal cell radius; maximal cluster diameter; maximal subscribers nodes number. Algorithm works in two steps. In the first step the designing of division without taking into consideration of streams intensity is realized. In the second the streams intensity is considered in order to create cells with smaller or bigger diameter. The algorithm is presented in fig.3.

Start

Insert initialize data

Determine the upper and the lower bound of cells number in cluster on the basis of equation: 2

3 3 ab n r π ⎡ ⎤ = ⎢_⎢ ⎥_⎥

Determine the intensity of internal and external streams in reference to each cell.

Divide the cluster into minimal number of cells without regard of streams intensity.

Determine the set of subscribers systems incoming to each cell. avg app Λ ≤ Λ Yes No Stop avg app Λ > Λ Yes

No Determine the cells formation with smaller

diameter.

Fig. 3.

5. SIMULATION RESULTS

In order to estimate the efficiency of proposed algorithm let’s compare it with the most common division algorithm based on the method of local minimum. To improve the efficiency of cells division with equal stream intensity an algorithm modification is proposed which further is called as a simple cells algorithm. In this algorithm as opposed to the base one, the initial radius is not always equal to the base station range radius. Before making an initial division an initial coverage radius is obtained. To do this a network area SNet is calculated which is considered

net

P – is a total network traffic;P – is a maximal traffic operated by the base _srv

station. The area SBC of base station coverage in radius function of its range is

expressed as: ₃ 2 ₃

BC

S = r , where: r – is a range radius of a base station. Next, using the quotient: n S= _Net S_Srv, a radius for initial base station localization is determined: r= S P PNet Net Srv 3 3. If obtained value is greater than base station range radius, than initial coverage radius is selected as equal to range radius, if it’s smaller than it’s equal to calculated value. The other algorithm steps are the same as in the base algorithm.

In sequential maximum algorithm, in coverage area the point with maximal subscribers nodes stream intensity is searched, in which the base station is located in the next step. Around this station a new cell is created. Nest, another area with maximal subscribers nodes stream intensity is searched. This process is continued until all subscribers nodes are not allocated to cells.

Next, the following symbols were used: µ λ/ – the ratio of stream intensity operations by subscribers node to incoming stream intensity; P_used/P – the level _app

of network load expressed in percents of maximal acceptable load; D – distance _dist

dispersion between subscribers nodes and a base station; D – traffic dispersion _traf

between network cells; t – time of creating cluster task solution.

As we can see from carried out surveys the proposed algorithm is characterized by a minimal time of task solution, which in some cases is even 2 scale rows smaller than a working time of local maximum algorithm. However, the proposed algorithm a little less accurately realizes the division of streams intensity (i.e. network is characterized by higher traffic intensity dispersion).

2,0 1,8 1,6 1,4 1,2 1,0 0,8 0,6 0,4 0,2 0 20 40 60 80 100 Simple cells Division algorithm Local maximum µ/λ P_used/P_max Fig. 4. 2,0 1,8 1,6 1,4 1,2 1,0 0,8 0,6 0,4 0,2 0 20 40 60 80 100 Simple cells Division algorithm Local maximum µ/λ P_used/P_max Fig. 5.

In fig. 4 the dependence of relative load to the coefficient µ λ/ value is presented by an even subscribers nodes division between cells. The fig. 5 presents similar relation in case of unequal subscribers nodes distribution. From both figures follows that the local maximum method ensures better level of existing

communication sources use.

In fig. 6 a designing task solution time is presented for different values of / µ λ. 2,0 1,8 1,6 1,4 1,2 1,0 0,8 0,6 0,4 0,2 0 500 1000 1500 2000 Simple cells Division algorithm Local maximum µ/λ t, ms Fig. 6.

As it follows from the above figure the solution time of cells division task with the method of local maximum is much greater than a solution time of other methods. It may be explained by the fact that in proposed algorithms an additional subscribers nodes division is not executed in the whole cluster area but only in this cells in which the exceeding of acceptable stream values is observed.

2,0 1,8 1,6 1,4 1,2 1,0 0,8 0,6 0,4 0,2 0 200 400 600 800 µ/λ D_dist , m Simple cells Division algorithm Local maximum Fig. 7. 2,0 1,8 1,6 1,4 1,2 1,0 0,8 0,6 0,4 0,2 0 500 1000 1500 2000 2500 3000 3500 4000 Simple cells Division algorithm Local maximum µ/λ D_traf Fig. 8.

In fig. 7 the value of distance dispersion between subscribers nodes and base stations by an unequal base station location is presented. In fig. 8 the value of traffic dispersion between subscribers nodes and base stations by an unequal base station location is presented.

6. CONCLUSION

Authors of this work concentrates theirs efforts on creating methods and means of corporation network with regional organization designing. It’s assumed, that this class networks will have a three-level hierarchical organization which at the lowest level are wireless cell networks. In this work a thesis was stated that one of

effective ways of wireless network base station load minimizing is additional division of subscribers nodes between neighboring cells. The necessary and enough conditions of additional streams division were defined as also it was presented that presented initial division significantly minimizes the complexity of structural wireless network synthesis task. On the basis of known division algorithms analysis the new algorithm of wireless network decomposition into cells was proposed with taking into consideration limitations on its diameter and stream intensity in each of them.

Also, division algorithm with variable cells radius was proposed, ensuring minimal value of inner cells streams density dispersion. In this work the fullness legitimacy of presented at the beginning thesis, was admitted.

Further authors works will be concentrated on preparing methods and means of topology selection and routes organization in wireless cell networks.

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