Ann Oper Res DOI 10.1007/s10479-006-0046-3
Optimizing base station location and configuration in UMTS networks∗ Edoardo Amaldi · Pietro Belotti · Antonio Capone · Federico Malucelli
C
Springer Science + Business Media, LLC 2006
Abstract Radio planning and coverage optimization are critical issues when deploying and expanding third generation cellular systems. We investigate mixed integer programming models for locating and configuring base stations in UMTS networks so as to maximize coverage and minimize installation costs. The overall model considers both uplink and downlink directions, that we studied separately in Amaldi et al. (2002, 2003b). The two-stage Tabu Search algorithm we propose exploits solutions of a simplified model for the uplink direction to drastically reduce the computational time required to find good approximate solutions of the overall uplink and downlink model. Computational results obtained for realistic instances are reported and discussed. Keywords Network planning . Location problem . Base station configuration . Mixed integer programming models . Cutting planes . Tabu search 1. Introduction In the last years, third generation cellular systems such as Universal Mobile Telecommunication Systems (UMTS), which are based on the more flexible but also more complex Wideband Code Division Multiple radio Access scheme (W-CDMA) (Holma and Toskala, 2000), have been standardized and network deployment has started. Since operators have to face the huge costs of the service licenses and the critical market situation, there is an acute need for planning tools that help designing and expanding UMTS networks in an efficient way. Given a set of candidate sites where base stations can be installed, a set of possible configurations for each base station (sectors orientation, antenna tilt, antenna height, maximum ∗ Research carried out within the national project “Optimization, simulation and complexity in the design and management of telecommunication networks” funded by the Italian Ministry of Education, University and Scientific Research (MIUR).
E. Amaldi () . P. Belotti . A. Capone . F. Malucelli Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza L. da Vinci 32, 20133 Milano, Italy e-mail:
[email protected] Springer
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power, etc.), the traffic estimates and the signal propagation information, the general problem is to select the location and configuration of the base stations so as to minimize installation costs and to meet the traffic demand as well as possible. Due to the peculiarities of W-CDMA, the radio planning problem cannot be subdivided into a coverage problem (to locate and configure base stations according to traffic distribution) and a frequency allocation problem (to allocate capacity based on traffic requirements and service quality) like for planning second generation cellular systems with a Time Division Multiple Access scheme (TDMA). In W-CDMA the bandwidth is shared by all active connections and no actual frequency assignment is strictly required, but the planning problem is more complex since the coverage and capacity aspects must be simultaneously addressed. In fact, the area actually covered by a base station (BS) also depends on the signal quality constraints, usually expressed in terms of Signal-to-Interference Ratio (SIR), and on the traffic distribution (Berruto et al., 1998). In this paper we investigate mathematical programming models for BS location and configuration, which account for both uplink and downlink directions. Pilot signals, which are emitted by BSs so as to broadcast system information and to allow mobile terminals to select the best serving BSs, are also considered. The power control (PC) mechanism, which dynamically adjusts emission powers so as to reduce interference and guarantee signal quality, is considered at different levels of detail. The paper is organized as follows. In Section 2 we summarize the main features of WCDMA that limit UMTS network capacity and affect coverage and signal quality. In Section 3, we present a general mixed integer programming model for locating and configuring BSs which considers both uplink and downlink directions. Since even simplified instances of this NP-hard problem are beyond the reach of state-of-the-art techniques for mixed integer programming, we propose in Section 4 a two-stage Tabu Search algorithm which exploits the solutions of the uplink model with a simplified PC mechanism to drastically reduce the computing time needed to obtain good approximate solutions of the overall uplink and downlink model with a more accurate PC mechanism. Computational results obtained for realistic instances are reported and discussed in Section 5. Finally, Section 6 contains some concluding remarks.
2. Main features of UMTS systems Consider UMTS networks based on W-CDMA and frequency division duplexing, i.e, systems in which two separate frequency wide bands are used for uplink and downlink. Prior to transmission, signals are spread over such a wide band by using special codes. Spreading codes used for signals transmitted in downlink by the same BS are mutually orthogonal, while those used for signals emitted by different stations (base or mobile) can be considered as pseudo-random. In an ideal environment, the de-spreading process performed at the receiving end can completely avoid the interference of orthogonal signals and reduce that of the others by the spreading factor (SF), which is the ratio between the spread signal rate and the user rate. In wireless environments, due to multipath propagation, the interference among orthogonal signals cannot be completely avoided and the SIR is given by: SIR = SF
Springer
Pr eceived α Iin + Iout + η
(1)
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where Pr eceived is the received power of the signal, Iin is the total interference due to the signals transmitted by the same BS (intra-cell interference), Iout that due to signals of the other BSs (inter-cell interference), α is the orthogonality loss factor (0 ≤ α ≤ 1), and η the thermal noise power. In the uplink case, no orthogonality must be accounted for and α = 1, while in the downlink usually α 1. Since the quality of the received signal, usually expressed in terms of bit error rate (BER), mainly depends on the SIR, one usually imposes signal quality constraints requiring that the SIR exceeds a minimum value γ which may vary according to the communication service considered (voice, video, data packets, etc.). For the sake of simplicity, in the sequel we refer to the minimum SIR before de-spreading as SIRmin = γ /SF. The SIR level of each connection depends on the received powers of the relevant signal and of the interfering signals. These in turn depend on the emitted powers and the attenuation of the radio links between the sources and destinations. A power control (PC) mechanism is in charge of dynamically adjusting the emitted power according to the propagation conditions so as to reduce interference and guarantee quality. For UMTS dedicated channels a SIR-based PC mechanism is adopted: each emitted power is adjusted through a closed-loop control procedure so that the SIR of the corresponding connection is equal to a target value SIRtar , with SIRtar ≥ SIRmin (Holma and Toskala, 2000). In Amaldi, Capone, and Malucelli (2003) we have considered a simpler power-based PC model which assumes that each emitted power is adjusted so as to guarantee a target power value Ptar at the receiving end. In this case the emitted power of each connection just depends on the attenuation between the source and destination. Given any set of SIR constraints, the power-based PC mechanism requires higher powers than the more complex SIR-based one, where the power emitted by each station depends on that emitted by all the others. Thus, from the network planning point of view, adopting a power-based PC mechanism instead of a SIR-based one may lead to a more conservative dimensioning of the system. The amount of traffic that can be simultaneously served by each BS is not limited a priori by a fixed channel assignment as in TDMA systems (see e.g. Aardal et al., 2003), but it is limited by the SIR constraints of all active connections (Gilhousen et al., 1991). Since the radio resources are dynamically assigned according to interference levels and traffic distribution, the network planning problem is more complex. Due to SIR constraints and the power control mechanism, the area actually covered by each BS is affected by the traffic distribution and its size can vary when the interference level changes (the so-called cell breathing effect) (Veeravalli and Sendonaris, 1995). In the presence of symmetrical traffic such as voice and video calls, the model considering only the uplink SIR constraints is in general tighter than that considering only the downlink SIR constraints. This is mainly due to the different levels of inter-cell interference (α values), see e.g. (Holma and Toskala, 2000). But, since data services such as web browsing or email yield asymmetrical traffic, network capacity may be also limited by the downlink SIR constraints. Designing a new UMTS network, or expanding an existing one to serve increased traffic, involves deciding where to install new BSs and how to configure them. In particular, the sectors orientation, tilt, height, maximum power as well as the pilot power level of each BS have to be selected. A reasonable objective is to aim at the best trade-off between maximizing the amount of traffic served and minimizing the total installation costs (Amaldi et al., 2003b). Since the number of candidate sites is often limited by authority regulations, operators are also interested in maximizing coverage by modifying the configurations of a given set of BSs, adding new BSs only when they are strictly needed. Springer
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Given the various issues that affect system performance, UMTS network planning is a complex task and optimization models and algorithms can make a substantial difference. 2.1. Related work In Amaldi, Capone, and Malucelli (2003) we discussed previous work on planning third generation cellular systems and differences with respect to planning second generation systems. Although several papers address network planning problems for CDMA systems, many of them still rely on a classical coverage approach based on set covering formulations and often neglect the interference effect. For instance, in Cal´egari et al. (1997) a simple model based on the minimum dominating set problem is considered, while in Lee and Kang (2000) the traffic capacity is also taken into account and the resulting classical capacitated facility location problem is tackled with a Tabu Search algorithm. One of the first contributions accounting for simplified signal quality constraints is Galota et al. (2001), where a simplified model for locating BSs is proposed and a polynomial time approximation scheme is presented. However, only intra-cell interference is considered and the crucial aspect of inter-cell interference is neglected. See also (Mathar and Schmeink, 2001, 2002). In Eisenblatter et al. (2003) the work carried out within the European project MOMENTUM is described. In particular, various modeling aspects of the UMTS radio planning are discussed and a detailed mathematical program is presented. The focus is on accounting for as many system features as possible, including BS configurations, pilot signals, the number of codes and traffic snapshots, without considering, at least at this stage, the difficulties in tackling the problem. Indeed, the proposed mathematical programming models are huge and beyond the reach of state-of-the-art mixed integer programming (MIP) solvers except for all but very small instances. In our work we have adopted a different approach: we started from simpler models accounting for some important aspects of the problem and extended them progressively. First we have investigated the problem of locating BSs for the uplink direction and compared the models with power-based and SIR-based PC mechanism (Amaldi, Capone, and Malucelli, 2003). Then we have considered more complex configurations (Amaldi et al., 2002) and the downlink direction (Amaldi et al., 2003b). In the process we have been trying to exploit the insight gained from the study of the simplified models to tackle the more detailed ones.
3. Base station location and configuration model Assume that a set of candidate sites (CS) S = {1, . . . , m} where a BS can be installed, is given. For every CS j, j ∈ S, let the set K j index all the possible configurations of the BS that can be installed in j. Since the installation cost may vary with the BS configuration (e.g., its maximum emission power or the antenna height), an installation cost c jk is associated with every pair of CS j and BS configuration k, j ∈ S, k ∈ K j . A set of test points (TPs) I = {1, . . . , n} is also given. Each TP i ∈ I can be considered as a centroid where a given traffic demand u i is requested and where a certain level of service (measured in terms of SIR) must be guaranteed (Tutschku, 1998). The propagation information is also supposed to be known. Let gi jk , 0 ≤ gi jk ≤ 1, be the attenuation factor of the radio link between TP i, i ∈ I , and a BS installed in CS j, j ∈ S, with configuration k ∈ K j . We assume that all characteristics of the radio link contribute to the propagation factor, including propagation from source to destination over multiple paths and radiation pattern of BS antenna. The propagation matrix G = [gi jk ]i∈I, j∈S,k∈K j is either Springer
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estimated by using prediction tools (e.g., Hata’s models or ray tracing (Parsons, 1996)) or obtained by actual measurements. We consider directive BS antennas covering a particular sector, for instance 120 degree sectors. Let the index set I jk ⊆ I denote the set of all TPs that fall within the sector of the BS installed in CS j with configuration k. Note that, for directive BSs with uniform antenna diagram along the horizontal axis, the propagation factors gi jk only depend on the distance between TP i and CS j and not on the sector in which TP i falls. In the UMTS base station location and configuration problem one wishes to select a subset of candidate sites where to install directive BSs as well as their configurations, and to assign the TPs to the available BSs so as to maximize the traffic covered and/or minimize the installation costs. In this paper we do not explicitly consider soft-handover (each TP is assigned to at most one BS) and we assume that the number of connections assigned to any BS does not exceed the number of available spreading codes. Since soft-handover tends to increase SIR values, we may be too conservative, but this can be at least partially compensated for by decreasing the value of SIRmin . As we shall see, this assumption is useful from a computational point of view: 0 − 1 assignment variables allow to express the crucial constraints related to signal quality and emission power in a simpler way. While a very large number of nonorthogonal codes are available in uplink, standard cardinality constraints can be added to the model in downlink to account for at most SF orthogonal codes. To describe the mixed integer programming model, we consider the two following classes of binary variables: y jk = xi jk =
1
if a BS is installed in j with configuration k
0
otherwise
∀ j ∈ S, k ∈ K j
1 if TP i is assigned to BS j with configuration k 0
otherwise
∀i ∈ I, j ∈ S, k ∈ K j .
If we assume a SIR-based PC mechanism, we also need for each TP i ∈ I the continuous up variables pi to indicate the power emitted by the mobile in TP i towards the BS it is assigned to (uplink direction) and pidw to indicate the power received at each TP i from the BS it is assigned to (downlink direction). Although a budget could be imposed on the total installation costs, if we aim at a trade-off between maximizing the total traffic covered and minimizing the total installation costs, we maximize:
u i xi jk − λ
i∈I j∈J k∈K j
c jk y jk
(2)
j∈J k∈K j
where λ ≥ 0 is a trade-off parameter between the two contrasting objectives. The first three groups of constraints ensure the coherence of the location and assignment variables. Each TP i can be assigned to at most one BS:
xi jk ≤ 1
∀i ∈ I
(3)
j∈S k∈K j
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and at most one BS configuration can be selected for CS j:
y jk ≤ 1
∀ j ∈ S.
(4)
k∈K j
Due to the binary variables xi jk , all served TPs i are assigned to a single BS, that is no soft-handover is considered. Note that, since more than one BS (with different sectors) can be installed in the same location, additional technology driven constraints may be needed to avoid co-location of BSs with conflicting configurations (e.g., with overlapping sectors). If for a given CS j we have y jk = 0 for all k ∈ K j , no BS is installed in CS j. A TP i can be assigned to a CS j only if a BS with some configuration k, k ∈ K j , has been installed in j:
xi jk ≤
k∈K j
y jk
∀i ∈ I, j ∈ S.
(5)
k∈K j
In uplink, the power emitted by any mobile terminal at TP i cannot exceed a maximum max−up power Pi : up
max−up
0 ≤ pi ≤ Pi
∀i ∈ I.
(6)
In downlink, besides the limit on the total power emitted by each BS j, we also consider an upper bound on the power used for each connection: 0 ≤ pidw ≤
P max−dw gi jk xi jk
∀i ∈ I,
(7)
j∈S k∈K j
where P max−dw denotes the maximum power per connection. According to constraints (3) at most one of the xi jk variables in (7) is equal to 1 and the right-hand side amounts to the maximum power available in downlink. Thus BSs cannot use too much of their power for transmission towards mobiles with bad propagation conditions (Holma and Toskala, 2000). For the downlink direction, we also introduce for every CS j, j ∈ S, the power pˆ j of the pilot signal emitted by the BS installed in j. These pilot powers can either be considered as variables or, as a first approximation, be taken as equal to a fraction of the total power P jtot−dw emitted by the BS: pˆ j = δk P jtot−dw
∀j ∈ S
(8)
where δk is a characteristic of the BS configuration k (for instance δk = 0.15). Then the limit on the total power emitted by each BS j can be expressed as: p dw i xi jk + pˆ j ≤ P jtot−dw y jk g i∈I k∈K j i jk k∈K j
∀ j ∈ S.
(9)
We now turn to the SIR constraints, which express the signal quality requirements, and consider first the uplink direction. Given the SIR-based PC mechanism, for every triple of Springer
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BS j ∈ S with configuration k ∈ K j and TP i ∈ I we have the uplink SIR constraint: up
h∈I ijk
up
u h ph gh jk
p gi jk xi jk i t∈S
l∈K t
up
x htl − pi gi jk + η j
= SIRtar xi jk ,
(10)
i where η j denotes the thermal noise at BS j and the index set I jk denotes the set of all TPs in I that are covered together with TP i by a BS located in CS j with configuration k. For any single connection between a TP i and a BS installed in CS j with configuration k, the numerator of the left-hand-side term corresponds to the power of the relevant signal arriving from TP i at CS j with BS configuration k while the denominator amounts to the total i interference due to all other active connections in the system. According the definition of I jk , the triple summation term expresses the total power received at the BS in j with configuration up k from all TPs h that are served. Indeed, ph gh jk indicates the power received at the BS j from TP h and, according to (3), t∈S l∈K t x htl is equal to 1 if and only if TP h is assigned to a BS, namely is served. The total interference, is then obtained by just subtracting the received power of the relevant signal. In the downlink direction, the SIR constraints must also take into account the pilot powers. For each triple of TP i ∈ I and BS j ∈ S with configuration k ∈ K j , we have the downlink SIR constraint:
α Iin +
l∈S l = j
z∈K l
pidw xi jk = SIRtar xi jk phdw ˆ h∈Ilzi u h gilz ghlz x hlz + Pi + ηi
(11)
where
Iin =
u h gi jk
h∈I ijk
phdw x h jk − pidw gh jk
(12)
is the intra-cell interference and Pˆi = αˆpj gijk +
l∈S l = j
pˆl gilz ylz
(13)
z∈K l
amounts to the interference at TP i due to all pilot signals. ηi denotes the thermal noise of mobile terminal at TP i. For any single connection between a BS located in CS j with configuration k and a TP i covered by this BS, the numerator of the left-hand-side term corresponds to the power of the relevant signal received at TP i from the BS j (definition of pidw ) and the denominator amounts to the total interference due to all other active connections in the system. The interpretation of (11) is similar to that of (10) except for the pilot powers and for the orthogonality loss factor α in the SIR formula (1), which is strictly smaller than 1 in downlink. Thus, constraints (10) and (11) ensure that if a connection is active between a TP i and a BS j with configuration k (i.e., xi jk = 1) then the corresponding SIR value is equal to SIRtar . Springer
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Finally, for every triple BS j with configuration k and TP i, we have to consider an additional downlink SIR constraint regarding the pilot signal:
α
h∈I ijk
u h gi jk gphhjk
pˆ j gi jk xi jk x h jk + l∈S z∈Kl pˆl gilz ylz + l∈S z∈Kl h∈Ilzi u h gilz gphlzh x hlz +ηi l = j
l = j
= SIRtar xi jk
(14)
The numerator of the left-hand-side term corresponds to the pilot power received at TP i from the BS j and the denominator amounts to the total interference due to all other signal including all other pilot powers. The resulting model is a mixed integer program with nonlinear SIR constraints since they up contain products of assignment variables (xi jk and y jk ) and power variables ( pi and pidw ). 3.1. Model assumptions and extensions We now comment on the model assumptions and on the possible extensions. By restricting the assignment variables xi jk to take binary values, we require that each active connection is assigned to a single BS. Therefore, we do not explicitly consider softhandover which allows a TP to be assigned to a set of BSs (active set). In uplink, the signal emitted by a mobile terminal is received by all BSs in the active set and then combined at the network level. In downlink, each BS of the active set must transmit towards the mobile terminal, which then combines all received signals. Although power control clearly becomes more involved, the global effect is that of increasing SIR values. According to the propagation conditions and soft-handover procedures, this effect can thus be taken into account by decreasing the value of SIRmin . In our model, we assume that the number of connections assigned to any BS does not exceed the number of available spreading codes. In uplink, there is a very large number of nonorthogonal codes, while in downlink, where there are at most SF orthogonal codes, standard cardinality constraints can be easily added to the model. Finally, if we assume a power-based PC mechanism instead of a SIR-based one, the up model can be substantially simplified. Indeed, the powers pi emitted from any TP i in dw uplink and the powers pi received at any TP i from the BS they are assigned to, are no longer variable. Since all emitted powers are adjusted so as to guarantee a received power up of Ptar , pi and pidw , for all i ∈ I , just depend on the value of Ptar and on the propagation factor of the corresponding radio links. To obtain the simplified model, we take into account that: up
pi =
Ptar xi jk g j∈S k∈K j i jk
pidw = Ptar
(15)
and in the SIR constraints we require that the left-hand-side terms of Eqs. (10), (11), and (14) are greater or equal to SIRmin xi jk . The resulting SIR constraints, which have the general form Ptar ≥ SIRmin xi jk (α Iin + Iout + η)
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(16)
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where Ptar is a constant, and Iin and Iout are linear functions in the x and y variables, can be linearized as follows: (α Iin + Iout + η) ≤
1 + Mi jk (1 − xi jk ) (17) SIRmin
for large enough values of the constants Mi jk .
4. Computational approaches Since the simplified version of the BS location problem neglecting inter-cell interference is NP-hard (Galota et al., 2001), the overall UMTS BS location and configuration problem is also NP-hard. In practice, even small instances of the overall problem are often beyond the reach of state-of-the-art mixed integer programming techniques. Also simplified versions, such as for instance the model for locating BSs with fixed configurations considering only the uplink direction, are challenging computationally. 4.1. Exact solution Assuming a power-based PC mechanism and fixed configurations, the uplink location model amounts to the following binary program: max
u i xi j − λ
j∈J i∈I j
s.t.
cj yj
j∈S
xi j ≤ 1 i ∈ I j
(18)
j ∈ S, i ∈ I j
(19)
j∈S
xi j ≤ y j h∈I
u h gh j
Ptar ≥ SIRmin y j Ptar x ht − Ptar t∈S ght
xi j , y j ∈ {0, 1}
j ∈ S, i ∈ I j ,
j∈S
(20)
(21)
where, for each CS j ∈ S, I j ⊆ I denotes the set of all TPs i that can be assigned to j while max−up respecting the power limit, that is such that (Ptar /gi j ) ≤ Pi . Since the power received by any given BS j from all TPs assigned to it is by definition equal to Ptar , the nonlinear SIR constraints corresponding to all these TPs are identical and reduce to a single constraint (20), which must be enforced whenever y j = 1. For each pair of TP i ∈ I and CS j ∈ S, (19) is the coherence inequality. Note that the upper bound on the power emitted by the mobile max−up terminal in any TP i, namely (Ptar /gi j ) xi j ≤ Pi y j , is implicit due to the definition of I j . See (Amaldi, Capone, and Malucelli, 2003) for more details. To obtain a compact formulation, we consider the packing constraints (18) and the aggregated coherence constraints:
xi j ≤ |I j |y j
j ∈ S.
(22)
i∈I j
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The SIR constraints can be linearized as follows: h∈I t∈J
u h gh j
x ht 1 −1≤ + M j (1 − y j ) ght SIRmin
j∈S
(23)
where M j = h∈I u h gh j γh and γh = 1/ mint∈Ih ght . To verify that h∈I t∈J u h gh j gxhtht is bounded from above by M j , it suffices to notice that due to the packing constraint (18) at most one of the variables x ht is nonzero in any feasible solution, hence t∈J x ht /ght is not greater than γh = 1/ mint∈Ih ght . Since the linear relaxation of this compact formulation is efficiently solved even for large instances, we adopt a cutting plane approach. Several classes of valid inequalities can be used, including the coherence constraints xi j ≤ y j and the obnoxious facility location constraints: yj +
xik ≤ 1. (24)
k∈J :gik
