Base station location in a cellular CDMA system

Telecommunication Systems 14 (2000) 163–173 163

Base station location in a cellular CDMA system

Dong-Wan Tcha a,∗, Young-Soo Myung b,∗∗ and June-hyuk Kwon c

a Graduate School of Management, Korea Advanced Institute of Science and Technology,207-43 Cheongryangri-Dong, Dongdaemun-Gu, Seoul 130-012, Korea

b Department of Business Administration, Dankook University, Cheonan, Chungnam 330-714, Koreac Radio Network Engineering Team, LG TeleCom, 145-6 Doksan-Dong, Kumchun-Gu, Seoul 153-010,

Korea

We consider a cellular CDMA system in which blocking is enforced when the relativeinterference exceeds a certain threshold level. This paper addresses a radio network designproblem in such a CDMA system. Given the data of call-traffic distributed over the servicearea and potential sites of base stations, the objective of the problem is to locate basestations so as to minimize the associated cost for establishing base stations while keepingthe probability of blocking under control. We develop an efficient algorithm for solving thedesign problem. Computational experiments with real-world data are conducted to showboth the efficiency and the practicality of the proposed design method.

1. Introduction

Digital cellular radio systems must incorporate multiple access schemes that makeefficient use of the allocated bandwidth and the radio cell infrastructure with min-imum cost and maximum performance. Code Division Multiple Access (CDMA) isthe multiple access technique, the acceptance of which in the global market placehas significantly grown ever since its first commercial system deployment in 1995.Especially in Korea, two mobile service providers (using 800 MHz band) and threePCS providers (using 1.8 GHz band) are providing national-scale commercial servicesusing the CDMA technology.

The successful deployment of a cellular system depends greatly upon the cellularnetwork planning process which includes such operations as transmission and radiopropagation predictions, geographical and traffic parameters evaluation, optimal radionetwork design, and network resource allocation. From among the operations, we focuson radio network design, in which a planner’s primary concern is on how and whereto locate Base Stations (BSs). Considering the large cost of establishing a BS, the keyto the successful initial deployment is on reducing the cost as much as possible whilekeeping the service quality. To fit the reality, the costs considered for establishing aBS are not confined to the BS equipments, but defined to cover the associated real

∗ To whom correspondences should be addressed.∗∗ The author wishes to acknowledge that this research has been supported by a 1997 Korea Research

Foundation Grant.

J.C. Baltzer AG, Science Publishers

164 D.-W. Tcha et al. / Base station location in a CDMA system

estate. Since various cash flows may differ in their temporal occurrences as can beseen from different contract terms of real estates, each of them is adjusted to a fixedcharge at the base period using the common discount factor. Studies in this directionhave been traditionally very scarce from the days of analog cellular systems, only tobe more acute for the digital CDMA system.

With CDMA, all (active) users share a common spectral frequency, users are beingdifferentiated by uniquely assigned digital codes. Since other user signals interferewith the desired signal, the power level of the desired signal is always below theinterference level. This contrasts with analog and TDMA systems, in which users usedifferent frequencies or time slots so that the level of the desired signal should alwaysbe much stronger than the interference level. This difference in interference-relatedenvironments gives rise to a call blocking mechanism for the CDMA system differentfrom that for analog and TDMA systems. In analog and TDMA systems, blockingoccurs to a call request when there is no frequency or time slot left from amongthose not interfering with the already assigned frequencies. In the CDMA systemhowever, a call request is not admitted when the total interference level exceeds acertain threshold level, leading to the term “soft blocking” [13]. This also implies thatall active calls are guaranteed to meet the quality standard insofar as the interferencelevel is within the threshold value. From here on, we shall simply call the soft-blockingas the blocking, assuming that the call admission policy based on the threshold valueis enforced.

In the CDMA system, a call quality is determined by the bit error rate, whichis known to nonincrease with respect to the relative interference, represented by theratio of bit energy to noise spectral density, Eb/N0. The objective of our CDMA radionetwork design is to locate BSs with the least amount of investment insofar as everyuser’s Eb/N0 does not fall below the given threshold value. Thus in a sense, onemay think that the CDMA network design is easier and simpler than the counterpartin analog and TDMA systems where locating BSs are intertwined with allocatingfrequencies to interfering neighboring cells [6].

We now elaborate on our CDMA radio network design problem. Given are dataof call-traffic distributed over the entire service area, potential sites on which BSs canbe located, and the cost for locating a BS on each potential site. Some assumptionsare imposed: all BSs are with omni-antennas, and call qualities on the reverse link areonly considered as usually done in the CDMA literature [7,13]. Note the feeblenessof the reverse link not only from the need for accurate power control but also fromthe nonexistence of code synchronization among users, as compared to the forwardlink [12]. The objective is to locate BSs so as to minimize the total siting costs whilekeeping the Eb/N0 on the reverse link for every users within the threshold limit.Design studies on a general cellular network are so scarce, and we could not find evena single study directly related to ours in the CDMA literature. However the followingfew studies may be listed as related ones in a broader sense: Gamst [4], Lee [6], Becaet al. [1], and the more recent ones by Stamatelos and Ephremides [11] and Sherali etal. [10].

D.-W. Tcha et al. / Base station location in a CDMA system 165

This paper is organized as follows. We first describe the characteristics and thedesign concept of the CDMA system, and formulate a mathematical model in section 2.Section 3 describes an algorithm developed for the model. Computational experimentson data extracted from some service area in Korea and randomly generated data areelaborated on to show the efficiency of the proposed design method in section 4.Finally, section 5 provides concluding remarks.

2. Design concept and model

Assuming mobiles are arbitrarily distributed over the service area, we partitionthe whole service area into identical square subareas with the size of ∆x2 km2. We callthis subarea Control Point (CP), and assume that all mobiles in the same CP are locatedat the center of CP and served by the same BS. As the size of CP becomes smaller,our assumption gets more realistic but the problem becomes more complicated. LetJ = {1, . . . ,m} denote the index set of partitioned CPs. Assume that the call arrivalsat a BS from the mobiles in CP j follow a Poisson process with mean λj , and thecall duration times are exponentially distributed with mean 1/µ. Then, the number ofactive calls in CP j, denoted by Aj , is a Poisson random variable with mean λj/µunder the assumption that there is no hard limit on servers in BSs [13,14].

Let the index set of the potential sites for BS location be denoted by I ={1, . . . ,n}. We also use cell i as referring to the subarea served by BS i. In CDMAsystem, blocking a call occurs when Eb/N0 falls below some threshold value C. Ifwe let (Eb/N0)i be the received relative interference on the reverse link at the BS i,the blocking probability (p) of that link is defined as Pr((Eb/N0)i < C). As stated inthe previous section, the system constraint requires that this value should be below aprespecified value, say β. So, the system constraint can be presented as follows:

Pr((Eb/N0)i < C

)< β, for all i ∈ I∗, (1)

where I∗ ⊆ I denotes the index set of the selected sites on which BSs are located.Assuming the environment that all individual signals on the reverse link are

received at the same power level of S, we then have (Eb/N0)i for a call in cell i as(Eb

N0

)i

=W/R

(Fi/S) + (η/S), (2)

where Fi is the interference from other signals, R is the information bit rate and η isthe background noise due to spurious interference as well as thermal noise containedin the total spread bandwidth, W .

The interference Fi can be classified into two kinds. One is the intra-cell inter-ference which is introduced by other call signals within the same cell and the other isthe inter-cell interference introduced by the signals from other surrounding cells. If Ncalls are active in cell i, the intra-cell interference is not greater than (N − 1)S whichis further reduced by the voice activity factor, α [5]. Let I∗ be the index set of the


selected BS locations and J(I∗, i) for each i ∈ I∗ be the set of CPs which are coveredby BS i. Then the intra-cell interference is

A(I∗,i)−1∑p=1

χpS,

where A(I∗, i) =∑

j∈J(I∗,i)Aj and χp are i.i.d. random variables, representing thevoice activity, with distribution

χp = χ =

{1, with probability α,0, with probability 1− α.

The calculation of inter-cell interference is made based on the assumptions that allinterfering calls are also power-controlled by their incumbent BSs, and signal strengthsdecrease according to the 4th power of distance. If a call in cell k is made at distance r0

from BS i for some i 6= k, and at r1 from BS k, then an active call in cell k, producesan interference to a call in cell i proportional to (r1/r0)4S [2,8].

Based on the above observations, for a given I∗ ⊆ I , Fi/S for each i ∈ I∗ canbe represented as follows:

FiS

=

A(I∗,i)−1∑p=1

χp +∑k 6=i

∑j∈J(I∗,k)

Ajχ

(rjkrji

)4

, (3)

where rji denotes the distance between CP j and BS i. Note that when I∗ is given,covering CP j by the nearest one among the selected BSs, never increases Fi/S forany i ∈ I∗. Therefore, we always let J(I∗, i) be the set of CPs to which BS i is thenearest.

From (1) and (2), the system constraint can be rewritten as

Pr(Fi/S > δ) < β, for all i ∈ I∗, (4)

where δ = (W/R)(1/C) − η/S. Since Aj and χ are independent random variables,we obtain the mean and variance of Fi/S in the appendix by using the result in [9] asfollows:

E

(FiS

)= α

( ∑j∈J(I∗,i)

λjµ− 1

)+∑k 6=i

∑j∈J(I∗,k)

λjµα

(rjkrji

)4

, (5)

Var

(FiS

)=α

( ∑j∈J(I∗,i)

λjµ− 1 + α

)

+∑k 6=i

∑j∈J(I∗,k)

(rjkrji

)8{(λjµ

)2(α− α2)+

λjµα

}. (6)


If we approximate Fi/S using the central limit theorem, the blocking probability (p)can be obtained by

Pr

(FiS> δ

)≈ Q

(δ −E(Fi/S)√

Var(Fi/S)

), for all i ∈ I∗,

where

Q(v) =

∫ ∞v

1√2π

e−w2/2 dw.

Let Nc be a constant of the standard normal distribution for the blocking probabilityβ, then the system constraint can be expressed as

δ −E(Fi/S)√Var(Fi/S)

> Nc, for all i ∈ I∗. (7)

Recall that if the BS locations are fixed, the coverage of each BS is easilydetermined by assigning CPs to the nearest BS. So, the radio network design problemwe consider reduces to the following Base station Location Problem:

(BLP) min∑i∈I∗

fi,

s.t. I∗ ⊆ I and satisfies (7),

where fi is the fixed cost needed to locate a BS at potential site i for each i ∈ I . Wesay that I∗ ⊆ I is a feasible set of BS locations for the BLP if I∗ satisfies (7).

3. Solution method

In this section, we describe an algorithm for obtaining a low cost feasible setof BS locations for the BLP. It consists of two heuristics: the construction heuristicfor choosing an initial feasible subset of potential sites and the improvement heuristicfor reducing the cost associated with the selected subset by changing some of itsconstituent sites.

The construction heuristic selects a low cost feasible set of BS locations, byrepeatedly solving the Minimum cost Covering Location Problem (MCLP). In theMCLP, BS i is said to cover CP j, if CP j lies within the prespecified distance, say θ,from BS i. A subset I∗ ⊆ I is called a cover of J , if every CP j, j ∈ J , can becovered by at least one BS in I∗. Let Ij = {i ∈ I | rij 6 θ} for each j ∈ J denotethe set of potential sites covering CP j. Then I∗ ∩ Ij 6= ∅, j ∈ J , for any cover I∗

of J . We now list the MCLP of finding a minimum cost cover which is formulatedas follows:


(P(θ)) min∑i∈I

fiyi,

s.t.∑i∈Ij

yi > 1, j ∈ J , (8)

yi = 0 or 1, i ∈ I , (9)

where yi is equal to 1 if a BS is sited at i and 0 otherwise.Our strategy is to solve P(θ) at different values of θ to obtain a low cost feasible

cover I∗ ⊆ I which satisfies the system constraint (7). Note that the smaller θis, the larger the cardinality of a feasible cover of J becomes. Note also that ifany feasible cover I∗ ⊆ I of P(θ) satisfies the system constraint (7), it is also afeasible set of BS locations for the BLP. P(θ) is the well-known set covering problem,which can be solved efficiently by the existing solution methods such as the dual-based algorithm by Fisher and Kedia [3]. Their dual-based algorithm is composedof the two heuristics, the dual and the primal heuristic. The dual heuristic obtains anear optimal solution to the dual of the linear programming relaxation of P(θ) wherethe integrality condition (9) is replaced by 0 6 yi 6 1 for each i ∈ I , and theprimal heuristic constructs a good-quality feasible cover from the dual solution thusobtained [3].

However, P(θ) is still an NP-hard problem and thus we must reasonably de-termine the θ-values for which P(θ) is solved. Let θt denote the θ-value for whichwe solve P(θ) at iteration t. We first set θ1 as θ1 = maxj∈J mini∈I rij , because forany θ < θ1, P(θ) has no feasible solution. Suppose that θt < θt+1 and It and It+1

are the minimum cost covers of P(θt) and P(θt+1), respectively. Then the cost cor-responding to It+1 is less than or equal to that corresponding to It. On the otherhand, It+1 is more likely to violate the system constraint (7). Based on this obser-vation, at iteration t, if P(θt) provides a feasible cover I∗ ⊆ I satisfying (7), we setθt+1 = θt + ∆x and repeat the process. Otherwise, we stop and update I∗ to be theleast cost feasible set of BS locations among the obtained ones. Recall that ∆x isthe length of one side of each square CP. If P(θ1) does not provide a feasible coversatisfying (7), we set I∗ = I . If even I violates (7), the original BLP has no feasiblesolution.

The second part of our algorithm is an improvement heuristic for reducing thecost of the selected set of BS locations. We first consider whether any BS in the setcan be deleted without violating the system constraint (7). If not possible, we nextcheck whether we can reduce the cost of the selected subset by interchanging a BS inthe set with one not selected. Suppose that I∗ ⊆ I is the obtained set of BS locations,then we arbitrarily choose i ∈ I∗ and also select k ∈ I\I∗ such that fk < fi. IfI∗ ∪ {k}\{i} satisfies (7), we obtain a better solution for the BLP. When selecting apotential BS site for interchanging, only those sites within a certain distance from iare searched to save computation.


4. Computational results

The proposed algorithm for solving the BLP was coded in C language and testedon an HP workstation (9000 series 715/33). We performed computational experimentsusing two classes of data, one from real-world problems and the other from randomlygenerated problems. Most of the input data for the first class of test problems wasgiven by a service provider with geographical data taken from the business districtwith the approximate size of 64 km2 of the city of Taejon, which is located around thecenter of South Korea. With ∆x set at 500 m, the area was partitioned into 256 CPs.Two call arrival rates (λj), one at and the other outside the peak traffic time, wereprovided for each CP j, and the mean service time (1/µ) was also supplied by aservice provider. The fixed cost for establishing a BS (fi) for each potential site i ∈ Iwas set to cover not only equipment costs but also that of real estate, giving rise to thepractical range of 300, 000 6 fi 6 500, 000 ($). The allocated total spread bandwidth(W ), the bit rate (R), and the voice activity factor (α) were assumed to be 1.25 MHz,8 kb/s, and 3/8, respectively. We also assumed that the value of δ was 30 and thethreshold value of Eb/N0, C, was 5 as in [5], and β was set equal to 1 (%).

Computational experiments were conducted with six cases, three different num-bers of potential sites (100, 125 and 150 sites) and two values of λj as mentioned above.For each of the above six cases, three input instances were considered by differingpotential site locations. The results for the first class of test problems are summarizedin table 1, the figures in which represent the average results of the corresponding threeinstances. The “number of steps” in the sixth column means the number of differentθ-values used to obtain an initial feasible solution and the “number of improvements”in the seventh column indicates the number of solution improvements made with ourimprovement heuristic.

We also tested our algorithm on the second class of test problems which weregenerated with varying both size (|J | × |I|) and β, and by randomly selecting thevalue of λj from three different ranges. Two kinds of cost settings were considered:constant fi’s commonly fixed at $100,000, and variable fi’s randomly selected from

Table 1Computational results for the case of Taejon city.

Size λj Obj.a Number of BSs Avg. number of Number Number of CPUvalue selected calls (per BS) of steps improvements (s)

256× 100 peakb 45.0 14.0 27.1 2.3 11.0 114.0normalc 32.0 10.3 30.4 2.7 8.0 181.3

256× 125 peak 45.7 14.3 26.4 3.0 4.0 139.3normal 33.7 10.7 29.5 3.3 6.0 219.3

256× 150 peak 41.3 13.3 28.2 3.0 6.4 175.7normal 32.3 10.3 30.4 3.0 9.0 235.0

a $100,000 per unit cost.b Range of λj for peak time: 1.12–2.29.c Range of λj for normal time: 0.94–1.92.


Table 2Computational results for randomly generated problems.

Size Fixed β Obj. Number of Avg. number of Number of Number of CPUcost (%) value BSs selected calls (per BS) steps improvements (s)

100× 70 var. 1 30.6 9.6 30.2 3.0 6.4 14.02 29.4 9.4 30.8 3.0 6.6 14.03 27.4 8.8 32.8 3.0 7.2 14.4

const. 1 10.2 10.2 29.2 2.8 4.0 11.42 9.4 9.4 31.6 2.8 4.8 11.43 9.2 9.2 32.2 2.8 5.0 10.8

150× 100 var. 1 51.2 15.4 28.2 2.4 6.6 80.42 47.4 14.2 30.6 2.4 7.4 92.83 45.6 13.8 31.6 2.4 8.0 72.2

const. 1 15.0 15.0 28.6 2.0 21.8 54.42 14.6 14.6 29.6 2.2 6.4 52.03 14.0 14.0 30.6 2.2 7.0 50.8

200× 150 var. 1 74.2 22.2 26.8 2.4 13.2 261.02 68.6 20.4 29.4 2.4 15.0 279.83 64.4 19.2 31.2 2.4 16.2 256.2

const. 1 21.0 21.0 28.2 2.6 18.6 253.22 20.4 20.4 29.0 3.0 6.4 222.23 19.6 19.6 30.2 3.0 7.2 171.6

∗ Range of λj : 1–5.

$300,000 to $500,000 for each i. The other system parameters were set the same asin the first set of data. Both the test problems and the results of our runs are given intables 2 and 3.

Each number in the table 2 indicates the average result over the test runs on thefive generated instances. Our algorithm solved most of the problems within reasonabletime and showed consistent performance over varying types of data instances. Anotherpurpose of dealing with small sized problems in the second experiment is to assess theperformance of the proposed algorithm by comparing our solutions with exact onesobtained by complete enumeration. As the listed computation times hint, we could notafford to obtain the exact solutions for larger problems. The comparative results arepresented in table 3, the figures in which show the average results over ten runs foreach case. In general, the gap between our solution and the optimal one increases withthe problem size. However, the results show that the quality of the obtained solutionsdoes not depend on the input parameters.

5. Conclusion

We have considered the CDMA radio network design problem, the objectiveof which is to locate BSs so as to minimize the associated total siting costs whilemaintaining the quality on every active call guaranteed. The design problem was suc-


Table 3Comparison of our solutions with exact ones.

Problem Our solutions Exact solutions

Size λj Fixed Obj. Number of CPU Obj. Number of CPU % gapcost value (a) BSs selected (s) value (b) BSs selected (s) ( a−bb × 100)

16× 8 6–10 var. 16.2 4.5 0.70 15.5 4.3 2.03 3.9const. 5.0 5.0 0.67 4.6 4.6 2.03 8.3

1–10 var. 9.1 2.7 0.67 8.8 2.6 0.92 5.0const. 2.6 2.6 0.67 2.4 2.4 1.57 10.4

1–5 var. 3.5 1.1 0.67 3.5 1.1 1.47 0.0const. 1.3 1.3 0.67 1.3 1.3 1.48 0.0

20× 10 6–10 var. 24.8 6.3 0.71 23.8 5.8 2.47 5.1const. 5.8 5.8 0.71 5.0 5.0 2.47 16.4

1–10 var. 13.5 3.7 0.47 12.6 3.3 2.48 7.5const. 3.4 3.4 0.67 3.3 3.3 3.20 4.2

1–5 var. 6.4 2.0 0.67 6.1 1.9 2.47 4.3const. 1.9 1.9 0.67 1.9 1.9 2.47 0.0

25× 12 6–10 var. 30.4 7.8 0.89 24.6 6.2 10.0 24.5const. 7.7 7.7 0.52 7.3 7.3 12.7 5.6

1–10 var. 18.1 4.9 0.77 15.4 4.1 17.9 15.2const. 5.3 5.3 0.75 4.6 4.6 12.8 17.6

1–5 var. 7.1 2.2 0.67 6.8 2.1 17.8 3.8const. 2.5 2.5 0.72 2.2 2.2 17.9 13.3

30× 15 6–10 var. 37.3 9.5 0.65 30.8 7.7 156.5 21.0const. 9.5 9.5 0.82 7.5 7.5 124.0 27.7

1–10 var. 21.2 6.0 0.93 18.4 5.2 178.2 14.6const. 6.6 6.6 0.91 5.6 5.6 162.3 16.7

1–5 var. 9.0 2.9 0.91 8.1 2.6 110.4 13.6const. 3.0 3.0 0.89 2.6 2.6 189.3 20.0

∗ β = 1 (%).

cessfully formulated as a kind of the so-called minimum cost covering location model,for which an efficient algorithm was developed. To show the real-world applicabilityof our design method, we first conducted a computational experiment with the inputdata given by a Korean service provider. In addition, we extensively tested the pro-posed algorithm using randomly generated problems. Computational results showedthat the proposed algorithm consistently performed so well both in solution qualityand in speed as to be practicable. Although a fairly extensive scope was covered inthis study, one may further enhance its practicality by incorporating such factors aslog-normal and multi-path fading, cell sectorization, etc.

Appendix

By [9], the expectation and variance of a sum of a random number of randomvariables are obtained as


E

[N∑i=1

Xi

]= E[N ]E[X]

and

Var

(N∑i=1

Xi

)= E[N ]Var(X) +

(E[X]

)2Var(N ).

Since the random variables χ and Aj for each j ∈ J are independent of eachother, the mean and variance of intra-cell interference of Fi/S become

E

[A(I∗,i)−1∑p=1

χp

]= α

( ∑j∈J(I∗,i)

λjµ− 1

)and

Var

[A(I∗,i)−1∑p=1

χp

]=

( ∑j∈J(I∗,i)

λjµ− 1

)α(1− α) + α2

∑j∈J(I∗,i)

λjµ

=α

( ∑j∈J(I∗,i)

λjµ− 1 + α

).

Note that E[Ajχ] = E[Aj ]E[χ] and that

Var[Ajχ] = E[A2jχ

2]− (E[Aj ]E[χ])2

=

{λjµ

+

(λjµ

)2}α−

(λjµα

)2

.

The independence of random variables gives rise to the following mean andvariance of Fi/S:

E

(FiS

)= α

( ∑j∈J(I∗,i)

λjµ− 1

)+∑k 6=i

∑j∈J(I∗,k)

λjµα

(rjkrji

)4

and

Var

(FiS

)=α

( ∑j∈J(I∗,i)

λjµ− 1 + α

)

+∑k 6=i

∑j∈J(I∗,k)

(rjkrji

)8{(λjµ

)2(α− α2)+

λjµα

}.

References

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Base station location in a cellular CDMA system