Green Base Station Placeent for Microwave Backhaul Links Alonso Silva, Antonia Masucci To cite this version: Alonso Silva, Antonia Masucci. Green Base Station Placeent for Microwave Backhaul Links. Proceedings of the International Syposiu on Ubiquitous Networking (UNet 17), May 2017, Casablanca, Morocco. Proceedings of the International Syposiu on Ubiquitous Networking (UNet 17), <http://www.unet-conf.org/>. <hal-01563371> HAL Id: hal-01563371 https://hal.archives-ouvertes.fr/hal-01563371 Subitted on 17 Jul 2017 HAL is a ulti-disciplinary open access archive for the deposit and disseination of scientific research docuents, whether they are published or not. The docuents ay coe fro teaching and research institutions in France or abroad, or fro public or private research centers. Copyright L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de docuents scientifiques de niveau recherche, publiés ou non, éanant des établisseents d enseigneent et de recherche français ou étrangers, des laboratoires publics ou privés.
Green Base Station Placeent for Microwave Backhaul Links Alonso Silva 1 and Antonia Maria Masucci 2 1 Nokia Bell Labs, Centre de Villarceaux, Route de Villejust, 91620 Nozay, France 2 Orange Labs, 44 Avenue de la République, 92320 Châtillon, France Abstract Wireless obile backhaul networks have been proposed as a substitute in cases in which wired alternatives are not available due to econoical or geographical reasons. In this work, we study the location proble of base stations in a given region where obile terinals are distributed according to a certain probability density function and the base stations counicate through icrowave backhaul links. Using results of optial transport theory, we provide the optial asyptotic distribution of base stations in the considered setting by iniizing the total power over the whole network. 1 Introduction There are several scenarios in which wired alternatives arenot the best solution to satisfy the traffic deand of users due to econoical or geographical reasons. Wireless obile backhaul networks have been proposed as a solution to these types of situations. Since these networks do not require costly cable constructions, they reduce total investent costs. However, achieving high speed and long range in wireless backhaul networks reains a significant technical challenge. In this work, we consider a clique network of base Eail: alonso.silva@nokia-bell-labs.co To who correspondence should be addressed. Eail: antoniaaria.asucci@orange.co stations and we assue that data is transitted independently in different radio frequency channels. We use optial transport theory, also known as theory of ass transportation, to deterine in this siplified scenario the optial asyptotic placeent of base stations counicating through backhaul links. Optial transport theory has its origins in planning probles, where a central planner needs to find a transport plan between two non-negative probability easures which iniizes the average transport cost. Resource allocation probles and/or assignent probles coing fro engineering or econoics are coon applications of this theory. In the present work, we study the proble of iniizingthetotalpowerusedbythenetworktoachieve a certain throughput and we use recent results of optial transport theory to find the optial asyptotic base stations locations. 1.1 Related Works Location gaes have been introduced by Hotelling [1], who odeled the spatial copetition along a street between two firs for persuading the largest nuber of custoers which are uniforly distributed. Probles siilar to location gaes, as for exaple the axiu capture proble, have been analyzed by [2, 3] and references therein. Within the counication networks counity, Altan et al. [4, 5] studied the duopoly situation in the uplink scenario of a cellular network where users are placed on a line segent. Considering the par-
ticular cost structure that arises in the cellular context, the authors observe that coplex cell shapes are obtained at equilibriu. Silva et al. [6, 7, 8] analyzed the proble of obile terinals association to base stations using optial transport theory and considering the data traffic congestion produced by this obile terinals to base stations association. 1.2 Energy efficiency Our objective is to conceive wireless backhaul networks able to guarantee quality of service while iniizing the energy consuption of the syste. We follow the free space path loss odel in which the signal strength drops in proportion to the square of the distance between transitter and receiver since it is a good approxiation for outdoor scenarios. The works on stochastic geoetry are related to our study (see e.g. the books of Baccelli and Blaszczyszyn [9], [10]) but we do not consider any particular deployent distribution function such as e.g. Poisson point processes. The reaining of this work is organized as follows. In Section 2 we provide the odel forulation of the considered proble where we redefine the probability density function of obile terinals to incorporate their throughput requireents and deterine the power cost function of inter-cell and intracell counications. In Section 3 we provide the ain results of our work by considering the free-path loss approxiation and asyptotic results fro optial transportation theory. In Section 4 we provide illustrative siulations for the asyptotic results obtained in the previous section, and in Section 5 we conclude our work. 2 Model forulation A suary of the notation used on this work can be found in Table 1. We are interested on the analysis of a icrowave backhaul network deployed over a bounded region, which we denote by D, over the two-diensional plane. Mobile terinals are distributed according to N K f Table 1: Notation Total nuber of obile terinals in the network Total nuber of base stations Deployent distribution function of obile terinals (x k,y k ) Position of the k-th base station N i h i h ij i Cell deterined by the i-th base station Nuber of obile terinals associated to the i-th BS Channel gain function in the i-th cell Channel gain between base station i and base station j Traffic requireent satisfied by base station i Total traffic requireent satisfied by the network a given probability density function f(x, y). The proportion of obile terinals in a sub-region A D is A f(x,y)dxdy. The nuber of obile terinals in sub-region A D, denoted by N(A), can be approxiated by ( ) N(A) = N f(x,y)dxdy, A where N denotes the total nuber of obile terinals in the network. We consider K base stations in the network, denoted by BS 1,BS 2,...,BS K, at positions (x 1,y 1 ),(x 2,y 2 ),...,(x K,y K ) to be deterined. Our objective is to iniize the energy consuption in the syste. We denote by the set of obile terinals associated to base station BS i and by N i the nuber of obile terinals within that cell, i.e., the cardinality of the set.
2.1 Modification of the distribution function The probability density function and the throughput requireents of obile terinals both depend on the location. To siplify the proble resolution, we consider the following odification of the probability density function to have the location dependency in only one function. The probability density function of obile terinals considered in our work and denoted by f(x,y) is general. Instead of considering a particular probability density function, denoted by f(x, y), and an average throughput requireent, denoted by θ(x,y), in each location (x,y), we consider a constant throughput θ > 0 to be deterined and redefine the distribution of obile terinals f(x, y) as follows f(x,y) := f(x,y) θ(x,y) θ for all (x,y) D. Since f(x,y) ust be a probability density function, we need to ipose f(x,y)dxdy = 1, or equivalently, 1 θ D D f(x,y) θ(x,y)dxdy = 1. For this equation to hold, we have to ipose θ = f(x,y) θ(x,y)dxdy. D We have that the following equation holds: f(x,y)θ = f(x,y) θ(x,y). The previous equation siply states that, e.g., a obile terinal with double deand than another obile terinal would be considered as two different obile terinals both at the sae location with the sae deand as the other obile terinal. Since in a icrowave backhaul network, we need to consider the energy fro within base stations and fro base stations to obile terinals, we need to consider both the intra-cell costs and the inter-cell costs. This is the subject of the following two subsections. 2.2 Intra-cell costs The power transitted, denoted by P T, fro base station BS i to a obile terinal located at position (x,y) is denoted by Pi T(x,y) = P i(x,y). The received power, denoted by P R, at the obile terinal located at position (x,y) associated to base station BS i is given by Pi R(x,y) = P i(x,y)h i (x,y), where h i (x,y) is the channel gain between base station BS i and the obile terinal located at position (x,y), for every i {1,...,K}. We assue that neighboring base stations transit their signals in orthogonal frequency bands and that interference between base stations that are far fro each other is negligible. Consequently, instead of considering the SINR (Signal to Interference plus Noise Ratio), we consider as perforance easure the SNR (Signal to Noise Ratio). The SNR received at obile terinals at position (x,y) in cell is given by SNR i (x,y) = P i(x,y)h i (x,y) σ 2, where σ 2 is the expected noise power. We assue that the associated instantaneous obile throughput is given by the following expression, which is based on Shannon s capacity theore: θ i (x,y) = log(1+snr i (x,y)). The throughput requireent translates into θ i (x,y) θ. Thanks to our previous developent, we can consider a constant throughput requireent through the odification of the probability density function. Therefore, the throughput requireent becoes or equivalently, ( log 1+ P ) i(x,y)h i (x,y) σ 2 = θ, P i (x,y) = σ2 h i (x,y) (2θ 1). (1)
Therefore the intra-cell power required by base station i is given by Pi intra = P i (x,y)f(x,y)dxdy. (2) The previous equation provide us an energy cost function for the intra-cell requireents of the network. In the following subsection, our analysis will be focused on the inter-cell requireents. 2.3 Inter-cell costs Inordertotakeintoaccounttheroutingcost, weconsiderthepowertransittedp T frobasestationbs i to base station BS j denoted by Pij T = P ij. The received power P R at the receiving base station BS j fro the transitting base station BS i is given by Pij R = P ijh ij, where h ij is the channel gain between base station BS i and base station BS j. The SNR received at the receiving base station BS j fro the transitting base station BS i is given by SNR ij = P ijh ij σ 2, where σ 2 is the expected noise power. We assue that the associated instantaneous base station throughputatthereceivingbasestationbs j frothe transitting base station BS i is given by the following expression, which is based on Shannon s capacity theore: θ ij = log(1+snr ij ). Let us define by i the traffic requireent concentrated at base station BS i, i.e. i = θ f(x,y)dxdy. We assue that the traffic requireent i concentrated at base station BS i is sent at the other base stations proportionally to the traffic requireent at the other base stations. Therefore, the traffic between the receiving base station BS j and the transitting base station BS i is given by i ( j /). We ake the siplifying assuption log(1+snr ij ) SNR ij. Then the throughput requireent translates into or equivalently P ij h ij σ 2 = i j, P ij = σ2 h ij i j. (3) The power cost to transit the traffic i is thus given by σ 2 i j h ij, (4) where = = θ j = θ D C j f(x,y)dxdy f(x,y)dxdy. Siilar to eq. (2) of the previous subsection, equation (4) provides us a power cost function for the inter-cell requireents of the network. In the next section, we consider both inter-cell and intra-cell cost functions to deterine the total power cost and obtain the asyptotic location of base stations to iniize this total power cost. 3 Results Fro the previous section, the total power of the network is equal to the su of intra-cell power (the su of the power used within each cell in the network) and the inter-cell power (the su of the power used over the pairs of counicating base stations in the network), i.e. P total = P intra i + j i P inter ij,
where Pi intra = P i (x,y)f(x,y)dxdy, is the intra-cell power consuption in cell and fro eq. (1), we obtain Pi intra σ = Ci 2 h i (x,y) (2θ 1)f(x,y)dxdy, and fro eq. (3) the inter-cell power consuption is P inter ij = σ2 h ij i j. In the following subsection, thanks to the freespace path loss approxiation, we are able to find an expression for the channel gain and the total power cost. 3.1 Free-space path loss approxiation Let d i (x,y) denote the Euclidean distance between obile terinal at position (x,y) and base station BS i located at (x i,y i ), i.e. d i (x,y) = (x i x) 2 +(y i y) 2. Siilarly, let d ij denote the Euclidean distance between base station BS i located at (x i,y i ) and base station BS j located at (x j,y j ), i.e. d ij = (x i x j ) 2 +(y i y j ) 2. The free-space path loss approxiation gives us that the channel gain between base station BS i and the obile terinal located at position (x, y) is given by h i (x,y) = d i (x,y) 2, and analogously the free-space path loss approxiation give us that the channel gain between base station BS i and base station BS j is given by h ij = d 2 ij. The total power cost is therefore given by Ci (2 θ 1)σ 2 d i (x,y) 2 f(x,y)dxdy+ σ2 i j d 2 ij. j i (5) When the nuber of base stations K is very large, in our setting tends to infinity, instead of looking at the locations of base stations (x i,y i ), we will look into the liit density ν of the locations (x i,y i ). To do that, we identify each set of K points with the easure ν K = 1 K δ (xi,y i), where δ (xi,y i) is the delta Dirac function at location (x i,y i ). The asyptotic analysis of these functions has been perfored (see e.g. [11, 12]) within the context of optial transport theory with the extensive use of Γ-convergence. The inter-cell power cost in ters of the easure ν K is given by σ 2 (x i,y i ) (x j,y j ) 2 dν K dν K, D D which is equivalent to σ 2 (x i,y i ) (x j,y j ) 2 d(ν K ν K ). D D We notice that the discrete su of eq. (5) becoes an integral by considering the liit of easures {ν K } K N. The total cost taking into account both cost functions gives the proble Min + σ2 D D (2 θ 1)σ 2 (x,y) (x i,y i ) 2 f(x,y)dxdy (x i,y i ) (x j,y j ) 2 d(ν K ν K ). We denote the function V(x) = x 2.
The necessary conditions of optiality (see [13]) are (2 θ 1)σ 2 φ+ 2σ2 V ν = c ν a.e. (6) where φ is the Kantorovich potential for the transport fro f to ν and c is the Lagrange ultiplier of the ass constraint on ν. A connection between the Kantorovich potential φ and the transport ap T fro f to ν is given by the Monge-Apère equation f = ν(t)det( T). Fro equation (6), we obtain Since (2 θ 1)σ 2 φ+ 2σ2 V ν = 0. T(x) = x φ(x). Therefore we have the syste { (2 θ 1)σ 2 (x T(x))+ 2σ2 V ν = 0 f = ν(t)det( T). (7) We can proceed by an iterative schee, fixing an initial ν 0 and obtaining T 0 fro the first equation of the syste (7) and obtaining ν 1 fro the second equation and proceed iterating the schee above. The previous syste of equations allows us to find the optial asyptotic base stations placeent ν as a function of the distribution of obile terinals f when the solution exists. 4 Siulations We follow the developent done in [13]. We siulate the exaple for the one-diensional case. If we suppose that the barycenter of ν is in the origin, we obtain: V ν = x 2 + y 2 dν(y), so that Aφ (x)+4bx = 0, which gives φ 4 (x) = (2 θ x and T(x) = 1) ( 1+ ) 4 (2 θ. 1) Putting previous expressions in the one-diensional Monge-Apère equation and indicating by v the density of ν, we obtain f(x) = ν (( 1+ and changing variables Probability distribution 0.6 0.5 0.4 0.3 0.2 0.1 v(y) = MT pdf BS pdf 4 (2 θ 1) 1 1+ 4 f (2 θ 1) ) )( x 1+ ( y 1+ 4 (2 θ 1) ) 4 (2 θ, 1) ). (8) 0-3 -2-1 0 1 2 3 Area Figure 1: Optial probability distribution function of the base stations given the probability density function of the obile terinals. We consider that obile terinals are distributed over the line as a noral distribution function with zero ean and standard deviation equal to one, i.e. N(0, 1). We consider that the throughput requireent θ is constant and equal to 24 Kbps. We notice that asexplainedinsubsection2.1wecouldhaveconsidered a non-constant throughput requireent and redefine the obile terinal probability density function for the throughput requireent to be constant. Fro eq. (8), we can copute the optial base station distribution given by Fig. 1. We notice that
Probability distribution 0.6 0.5 0.4 0.3 0.2 0.1 MT pdf BS pdf 0-3 -2-1 0 1 2 3 Area Figure 2: Optial probability distribution function of the base stations given the probability density function of the obile terinals. the optial base station distribution corresponds to a soother noral probability distribution. Motivated by the previous siulation, we consider a second scenario where obile terinals are distributed as a truncated noral distribution function between [ 1,1]. Fro eq. (8), we can copute the optial base station distribution given by Fig. 2. Fro Fig. 2, we notice that surprisingly the optial base stations distribution has a support that does not coincide with the obile terinals distribution. If we only consider the intra-cell cost we would have obtained the exact sae probability distribution of obile terinals (it is easy to see since in that case the cost would have been zero). We have thus verified that the optial base station probability density function would have been odified by the intra-cell cost as given by Fig. 2. Siilar to the first scenario, the optial base station distribution corresponds to a soother probability distribution. 5 Conclusions In this work, we investigated the asyptotic optial placeent of base stations with icrowave backhaul links. We considered the proble of iniizing the total power of the network while aintaining a required throughput. Using optial transport theory, we provided the optial asyptotic base station placeent. Moreover, the case where routing cost is taken into account is also analyzed. Acknowledgent The work of A. Silva has been partially carried out at LINCS (http://www.lincs.fr). References [1] H. Hotelling, Stability in copetition, The Econoic Journal, vol. 39, no. 153, pp. 41 57, 1929. [2] F. Plastria, Static copetitive facility location: An overview of optiisation approaches, European Journal of Operational Research, vol. 129, no. 3, pp. 461 470, 2001. [3] J. J. Gabszewicz and J.-F. Thisse, Location, in Handbook of Gae Theory with Econoic Applications, vol. 1, ch. 9, pp. 281 304, Elsevier, 1992. [4] E. Altan, A. Kuar, C. K. Singh, and R. Sundaresan, Spatial SINR gaes cobining base station placeent and obile association, in IEEE INFOCOM, pp. 1629 1637, 2009. [5] E. Altan, A. Kuar, C. K. Singh, and R. Sundaresan, Spatial SINR gaes of base station placeent and obile association, IEEE/ACM Transactions on Networking, pp. 1856 1869, 2012. [6] A. Silva, H. Tebine, E. Altan, and M. Debbah, Optiu and equilibriu in assignent probles with congestion: Mobile terinals association to base stations, IEEE Transactions on Autoatic Control, vol. 58, pp. 2018 2031, Aug 2013. [7] A. Silva, H. Tebine, E. Altan, and M. Debbah, Spatial gaes and global optiization for the obile association proble: The downlink
case, in Proceedings of the 49th IEEE Conference on Decision and Control, CDC 2010, Deceber 15-17, 2010, Atlanta, Georgia, USA, pp. 966 972, 2010. [8] A. Silva, H. Tebine, E. Altan, and M. Debbah, Uplink spatial gaes on cellular networks, in Proceedings of the 48th Annual Allerton Conference on Counication, Control, and Coputing (Allerton), pp. 800 804, Septeber 2010. [9] F. Baccelli and B. Blaszczyszyn, Stochastic geoetry and wireless networks, volue 1: Theory, Foundations and Trends in Networking, vol. 3, no. 3-4, pp. 249 449, 2009. [10] F. Baccelli and B. Blaszczyszyn, Stochastic geoetry and wireless networks, volue 2: Applications, Foundations and Trends in Networking, vol. 4, no. 1-2, pp. 1 312, 2009. [11] G. Bouchitté, C. Jienez, and M. Rajesh, Asyptotique d un problèe de positionneent optial, Coptes Rendus Matheatique, vol. 335, no. 10, pp. 853 858, 2002. [12] G. Buttazzo and F. Santabrogio, A odel for the optial planning of an urban area, SIAM J. Math. Anal., vol. 37, no. 2, pp. 514 530,2005. [13] G. Buttazzo, S. G. L. Bianco, and F. Oliviero, Optial location probles with routing cost, arxiv preprint arxiv:1306.6070, 2013.