Indoor Wireless Planning using Smart Antennas

Transcription

1 Inoor Wireless Planning using Smart Antennas Ali Abbasi an Maji Ghaeri Department of Computer Science, University of Calgary s: {abbasi, Abstract This paper consiers the problem of inoor wireless planning using smart antennas. Smart antennas have gaine much attention in wireless networking because of their capability in proviing more spatial reuse an increase network capacity. Recent research has emonstrate their effectiveness in inoor environments where omni-irectional antennas have been traitionally the ominant technology. Much of the work, however, assumes that a network is alreay eploye an focuses on scheuling antenna patterns. In this work, we investigate fining a wireless plan for an inoor environment where the wireless plan specifies minimum number of antennas require to provie complete coverage of the environment as well as the location, transmission power an beam pattern for each antenna. This problem is more challenging than raio planning using omniirectional antennas because of the special shape of antenna beams. Both single-beam an multi-beam antenna patterns are consiere an Integer Linear Programming formulations are provie for computing the minimum cost wireless plan. Moreover, to solve large-scale instances of the problem an efficient polynomial-time heuristic is propose. I. INTRODUCTION Emerging popularity of WiFi-enable consumer evices in recent years, has resulte in growing eman for wireless Internet access in inoor environments such as university campuses an corporate builings. The large number of users connecte to such wireless LANs have to compete to gain access to the share wireless meium. The high-rate of contention reuces the overall throughput of these networks. Moreover, laptops an other powerful hanhel evices that are capable of running banwith-intensive applications constantly request higher ata rates from the network. To aress the limite banwith challenge, new technologies such as the smart antennas are being use to buil more efficient wireless LANs [1] [3]. Usually a single access point (AP) 1 is not capable of proviing satisfactory service for an inoor environment (e.g., university campus), thus multiple APs are eploye to cover the area. In a multi-ap wireless LAN, transmitte signals might be egrae or even corrupte ue to interference from transmissions of other APs. While accoring to Signal to Interference-plus-Noise Ratio (SINR) moel [10], noise can have the same negative effect, interference is the ominant limiting factor for the performance of multiple transmitter wireless networks [2]. Thus, to enhance the performance of a wireless network, either the average receive signal power shoul be increase or the interference power shoul be ecrease. Smart antennas can serve both goals well. 1 We use the terms access point an antenna interchangeably. Smart antenna refers to an antenna array consisting of multiple omni-irectional antenna elements, combine with smart signal processing algorithms [4]. Smart antennas have been extensively use in raar an aerospace systems, an most recently, in wireless communication systems. Perhaps the most important feature of a smart antenna is its beamforming capability. When beamforming to a user, the smart antenna creates a irectional beam towar the esire user an null the signal in the irections of unesire users by appropriately ajusting the magnitue an phase of the signal transmitte by each of its elements. In comparison to omni-irectional transmissions, beamforming reuces interference allowing more concurrent transmissions in the network. Moreover, by concentrating the transmission energy in a specific irection, beamforming creates a signal that is orers of magnitue stronger than that of the signals in other irections. One can further reuce interference in a network by combining beamforing feature with power control [5] an scheuling [1], [6]. While these techniques can significantly improve the performance of wireless networks, the amount of interference in a wireless network, an consequently its performance, funamentally epens on the network geometry [2]. The process of fining the optimal network geometry, which naturally happens uring the planning phase of a network, is what we call the wireless planning problem (WPP). In this work, we provie a mathematical optimization framework to systematically solve this problem. In our framework, the objective is to fin the optimal network geometry etermine by antennas configuration in terms of antenna location an beam pattern as well as transmission power so that a minimum pre-specifie SINR is observe across the network. There exist several work on wireless planning mainly in the context of omni-irectional antennas [7] [8]. Osais et al. [9] examine the connecte coverage problem in wireless sensor networks with the objective of proviing coverage over a set of control points with minimum number of sensors. A point is covere if it is within the range of at least one sensor. In contrast to [9], our goal is to provie a minimum require SINR throughout the network which is a more esirable property compare to just covering the network area. Moreover it consiers only a restricte form of sensors in which a sensors covers only a single irectional beam. It has been shown that such a restriction for antennas results in a sub-optimal network performance [6]. In this paper, we consier a general antenna moel an fin the optimal wireless plan that satisfies a prespecifie performance constraint for all the network users. The main contributions of this paper are as follows:

2 We efine WPP formally an establish its corresponence to NP-har set-cover problem. We then consier three variations of the problem representing ifferent antenna beam patterns, an provie Integer Linear Programs to fin the optimal wireless plan for each variation. We propose a polynomial-time heuristic to solve WPP an through numerical results we emonstrate its efficiency. The rest of the paper is organize as follows. Section II provies an overview of smart antennas an our system moel. In Section III, WPP is formally efine an its complexity is iscusse. Our optimization framework is formulate in Section IV. In Section V, a polynomial-time algorithm is presente to solve large-scale instances of the problem. Sample numerical results are provie in Section VI. Finally, Section VII conclues this paper. A. Antenna Moel II. SYSTEM MODEL AND ASSUMPTIONS A K-element smart antenna consists of K omni-irectional antenna elements with sophisticate signal processing algorithms capable of ientifying signal s irection of arrival an calculating beamforming weights w = [w 1, w 2..., w K ] base on that irection. The objective is to estimate the irection that makes highest Signal-to-Noise-Ratio (SNR) at the receiver an etermine amplitue an phase of each element so that the beam woul be create towars that irection. Each w i is a complex number of the form w i = w i e jφi, where w i an φ i enote, respectively, the amplitue an phase of the signal generate by antenna element i. There are two main types of smart antennas, namely aaptive an switche antennas. In the former, beamforming weights are compute base on receiver channel conitions, while in the latter, beam patterns are fixe an weights are precompute. Although switche smart antennas provie lower antenna gains for specific users, they o not nee instantaneous channel feeback from the receivers. Thus, in this work, we consier switche antennas, which achieve consierable performance at a lower complexity. A K-element switche antenna can prouce K istinct beams in ifferent irections. Let D = { 1, 2,..., K } enote the set of possible irections. For consumer evices, typically, omni-irectional antennas are use because of size limitations. Therefore, the system we consier in this paper consists of smart antenna APs an typical omni-irectional clients. Similar to [1], we consier a two-phase TDMA-base MAC protocol to synchronize ownlink an uplink traffic. The first phase is eicate to irectional transmissions from APs to clients, while the secon phase is reserve for clients omni-irectional transmissions. The majority of the traffic in enterprise WLANs tens to be ownlink, therefore, our goal is to fin a network coverage plan that meets ownlink traffic requirements. B. Network Moel The inoor area is iscretize into a set of n possible user locations L = {l 1, l 2,..., l n } that must be covere by the wireless plan. This is a natural approximation of the environment (as users cannot be in any arbitrary location) that gives the problem a iscrete structure rather than a continuous structure which is more ifficult to formulate. This approximation coul be mae as accurate as neee by consiering more possible user locations. There is also a set of m potential antenna locations enote by A = {a 1, a 2,..., a m }. Each of a i s is a possible location for placing an antenna an no more than one antenna can be locate at a point. C. Communication Moel Let P i P max enote the transmission power of antenna a i, i.e., the antenna place at location a i A. We assume that each antenna can choose a transmission power from a set of power levels P = {p 1, p 2,..., p l }, where p i P max for 1 i l. Let P ij enote the power receive by location l j L from antenna a i. Two propagation moels, namely the Protocol Moel an the Physical Moel [10], are wiely use in the literature for moeling the effect of interference in wireless networks. In this paper, we consier the more complicate Physical Moel as it provies a more accurate representation of the average behavior of receive signal power in a wireless network. In the Physical Moel, a user at location l j successfully receives information from its associate AP, a i if the SINR of the a i s transmission at l j is beyon a threshol β j, otherwise nothing can be receive. Let γ j enote the SINR at location l j an T = {t 1, t 2,..., t l } enote the set of active transmissions other than a i s transmission. Then γ j is given by P i / α ij γ j = t k T P k/ α kj + η j where, ij is the istance between a i an l j, α 2 is the pathloss exponent, an η j enotes the noise power at location l j. For notational simplicity, efine g ij as g ij = 1/ α ij. The bit-rate achievable by user locate at l j is a nonecreasing function of γ j. For instance, using Shannon s formula, the maximum error-free rate λ j achievable by user j is given by λ j = log(1 + γ j ). Thus by using ifferent threshols γ j across users, non-uniform traffic eman in the network coul be aresse. III. WIRELESS PLANNING PROBLEM In this section, we formalize our efinition of a wireless plan an efine the wireless planning problem that is consiere in this paper. Definition 1 (Wireless Plan). Let U = A B be the universe of possible beam pattern placements, where A enotes the set of potential antenna locations an B enotes the set of possible beam patterns that is efine as B = {(b 1,..., b K ) b i P; 1 i K b i P max }. A nonempty set W U s.t (1)

3 W = {(a i, b li ) a i = a j b li = b lj } is calle a wireless plan. Accoring to efinition of wireless plan, two antenna can not be place at the same locations. All the wireless plans o not satisfy the coverage requirements of the network. The following efinition specifies a plan by which the coverage conition is satisfie. Definition 2 (Desirable Wireless Plan). Given a prespecifie SINR threshol β j for each location l j L, A wireless plan W + is calle esirable if l j L, γ j > β j. Due to the cost associate with eployment an maintenance of a wireless network, the objective of network planning process is to minimize the number of antennas require to cover the entire network. Definition 3 (Wireless Planning Problem). The objective of the wireless planning problem is to fin a esirable wireless plan W that minimizes the number of require antennas. More formally, the wireless planning problem is efine as. W = arg min W + A. Computational Complexity (a i,.) W + 1 Wireless LAN planning problem has be shown to be NPhar [8] [7]. Thus there is no hope of fining any polynomial time algorithm for it. Actually this problem is an extension of NP-har set-cover problem [11]. WPP formulations that will be presente in the paper are all base on integer formulation of set-cover problem, thus in this section we present this problem an show that how a simpler form of WPP can be cast to this problem. Set-cover is efine as follows, Definition 4 (Set-cover Problem). Given a universe U, a collection of subsets of U enote by S = {S 1,..., S T }, an a cost function c : S R +, fin a minimum cost subcollection of S enote by C that covers all elements of U. Consier a simpler variation of WPP in which interference is ignore in computing SINR for all locations.we refer to this variation as WPP-S. We illustrate constructively the mapping of WPP-S to set-cover problem. Here, the universe U is the set of locations to be covere U = L an collection of subsets S is efine as S = {S im S im is the set of all locations l j L that are covere by (a i, b m ) A B}. Let the cost of a set S im be the price of installing an antenna at a i. Consiering equal cost for all the antennas, the objective of WPP-S is to fin the minimum number of a i s which covers the universe L. We say a point l j is covere by S im when it is within the area of one of the beams of b m such that the transmitte power p on that beam, meets the requirement of SINR moel. Since in WPP-S, we on t consier interference from other concurrent transmissions, equation (1) is reuce to (2) p α ijη j β j (3) Fig. 1. a i 2 v i Beam pattern of a irectional antenna. We moel the antenna beam as a circular sector in which the relationship between the beam an the set of points it covers is etermine by the Target In Sector (TIS) test [12]. This test states that a point l j is covere by beam of an antenna locate at a i if both of the following conitions are satisfie: 1) ij r, 2) v i ij ij cos( θ 2 ), where ij is the vector from location a i to l j, v i is a unit vector that enotes the beam irection, r is the transmission range (obtaine from 3) an θ is the central angel of the beam (see Fig. 1). The first conition in TIS checks if the point l j is within the transmission range of the antenna place at a i. The secon conition checks if the vector ij is within the central angle of the antenna beam by computing the inner prouct of v i an ij. The equality hols when point l j is along one of the two eges of the transmission sector of the antenna beam. A brute force solution is to consier all subsets of S in increasing orer of carinality, check the coverage conition 1, an output the first subset that satisfies it. However, there is an exponential number of subsets which results in worstcase exponential running time. In the following sections we formulate the problem as a integer linear program an present a polynomial-time heuristic to solve it. IV. WIRELESS PLANNING FORMULATION In this section, we erive integer linear programs for several instances of WPP. In erivation of these formulations we basically exten integer formation of set-cover problem 4 by aing constraints of WPP to it. min w i x Si s. t. S i S S i : u S i x Si 1 u U x Si {0, 1} S i S. Where w i is the weight assigne to set S i. The constraint inicates coverage of every member u of the universal set U by at least one covering subset. The objective function inicates the goal to minimize the cost of the cover. The SINR constraint escribe in section II is a nonlinear function, hence to simplify the formulation, we pre-compute (4)

4 g ij for every pair (a i, l j ) A L. This computation takes O( A L ) time, where A an L are the carinalities of the sets A an L respectively. We also nee to know the set of points that is covere by each antenna beam-pattern pair. As we have alreay mentione we use TIS test to fin this set. Although the power raiate from an antenna in the form of an electromagnetic signal ecays as the signal travels in the environment but oes not isappear completely over finite istances of an inoor environment. Thus, we assume that within the central angle of the antenna beam the antenna coverage range is unboune. Obviously, the bounaries of the inoor environment(set L) put a limit on the carinality of the covere set by each beam. Matrix C[c j,im ] L, A B represents coverage relationship between all (l j ; a i, b m ) pairs such that 1, If location l j is uner the coverage of antenna a i c ji = when it raiates base on beam pattern m, Computing matrix C takes O( A L B ) time. We consier three ifferent beam pattern types, namely Constant Power Single-Beam, Dynamic Power Single-Beam, Dynamic Power Multi-Beam. These antenna patterns iffer in the number of beams that an antenna can raiate an the set of available power levels for each beam. A. Constant Power Single-Beam System In this system, antennas raiate a single-beam with maximum power P max. There are D beam patterns totally. Thus, we only nee to fin the optimal antenna locations an orientations but not the power levels. To formulate WPP in this case, we introuce a set X = {x i a i A, D} of binary ecision variables to represent the ecision space of the problem where x i is efine as follows 1, If an antenna is place at location a i an x i = irecte towars irection, Another set of ecision variables H = {h ji l j L, a i A, D} is efine to capture the associations of l j L to x i X as follows { h 1 If location l j is associate to antenna beam x i ji = 0 otherwise. Using the above efinitions, WPP can be represente as the following Integer Linear Program: min s. t. i, x i (5a) x i 1, a i A (5b) i, h ji x i P ij i, (1 h ji )c ji x ip ij + η j γ j, l j L (5c) h ji 1, l j L (5) i, h ji x i, h ji c ji, (l j, a i, ), l j L, a i A, D (5e) (l j, a i, ), l j L, a i A, D (5f) The objective function (5a) captures our esire to minimize the number of antennas use in the plan. (5b) enforces existence of at most one antenna at each AP location; by constraint (5c), coverage of each location l j accoring to SINR moel is guarantee; the numerator in the constraint takes the receive power from beam of antenna a i if it is installe an l j is associate to it. The enominator takes the sum of receive powers from all existing antennas which cover l j an l j is not associate to them plus η j. (5) inicates that each location l j shoul be associate to at most one antenna beam. Constraints (5e) an (5f) enote that when l j is associate to beam of the antenna a i, such beam must exist an cover l j. Due to inclusion of h ji x i term, the constraint (5c) is quaratic. However, because of constraints (5e) an (5f) an the fact that h ji, x i, an c jk all take {0, 1} values, we have h ji x i = h ji an h ji c ji = h ji. Thus, this constraint can be represente as follows: i, h ji P ij i, c ji x i P ij i, h ji P γ j (6) ij + η j or, equally as the following linear constraint: h jip ij γ j ( c jix i P ij ) + i, i, γ j ( i, B. Dynamic Power Single-Beam System h jip ij ) γ j η j 0 (7) With fixe power assignment, the only available options to cover all the points in the area are installing new antennas or changing the locations an irections of current available antennas. Nevertheless, base on the SINR capture moel, it is possible to change transmission powers of eploye antennas an cover new points. Since the transmission power of an antenna can take any value p l P, there exist P D beam patterns in this case. To inclue variable power assignment in our optimization framework, ecision variables x i an h ji are reefine as 1, If an antenna is place at a i, irecte towars x i [l] = irection an transmit at power p l, an h ji[l] = { 1, If location l j is associate to antenna x i [l], These variables are substitute in 5a,5b,5c,5, 5e,5f, an P ij is also ajuste to obtain the formulation of ynamic power single-beam system.

5 C. Dynamic Power Multi-Beam System Smart antennas can raiate multi-beam beam patterns by combining multiple beams in ifferent irections an ajusting amplitue an phase of them. In this case, the supplie power is istribute among active beams either uniformly or nonuniformly, but the sum of allocate powers is boune above to P max. If the ifference of each two consecutive power levels is the same p i+1 p i = P max /l, 1 i P 1, the number of these patterns is equal to ( ) P + D D. Following the metho of previous subsections, new binary variable are efine. We efine x b i as the binary variable which takes value 1 when there is an antenna at location a i raiating beam pattern b. Also binary variable h b ji captures association of location l j to x b i. By substituting these variables in 5a,5b,5c,5, 5e,5f, we obtain the formulation for a ynamic power multi-beam system. D. Discussion The above formulations all can be solve using integer programming algorithms like branch an cut implemente in various optimization softwares. Branch an cut employs a linear programming algorithm such as Simplex to fin the optimal solution to integrability-relaxe version of the problem. Since the solution may not be integral, through techniques like cutting plane an branching suitable inequality constraints are foun an augmente to the problem to forbi the same fractional results happen again. These steps are one successively until an optimal integer solution is achieve. This approach may nee exponential number of iterations to explore the entire integer omain of the problem. In aition, even meium-size instances of WPP results a large number variables an constraints. Thus, fining the optimum solution for large instances of WPP using integer programming solvers may become impractical. However, even a suboptimal solution might be much better than a hoc planning an be close to optimal one. In the following section we present our polynomial-time heuristic to fin such a solution. V. GREEDY ALGORITHM Branch an cut consiers all possible pairs of (antenna, beam patterns) an prunes the space successively to fin the optimal solution. Base on set-cover heuristic [11], we present a heuristic in which WPP is solve for one antenna at a time, choosing appropriate place an beam pattern for it. This leas to a greey approach in which we make a locally optimum ecision at the moment with hope that sequence of locally optimum ecisions achieves globally optimal solution. The algorithm is calle GreeySelect an shown in algorithm 1. We show the list of selecte (antenna,beam patterns) as S an covere locations by C. At first both sets are empty. In each iteration, the pair of antenna location an beam pattern that covers maximum uncovere user locations is selecte an ae to S. This process continues until all user locations are covere or aition of new antennas isn t possible or oesn t improve coverage. When an antenna enters S never leaves it, however introuction of new antennas may remove some locations from C. Since, there are at most A antennas available, the number of iterations is O( A ). Fining an upating information at each iteration takes O( A L ), so the time complexity of GreeySelect is O( A 2 L ). Algorithm 1: GreeySelect Algorithm Input: A, L Output: S, C begin S ; C ; while S < A an C < L o Cnum C ; foreach (a i, b k ) / S o S S (a i, b k ); foreach l j L o Compute γ j; Covere j 1 {γ j >β j }; if Covere > Cnum then (besta, bestb) (a i, b k ); Cnum Covere; S S (a i, b k ); if Cnum > j 1 {γ j >β j } then S S (besta, bestb); C {j γ j > β j}; VI. NUMERICAL RESULTS In this section, we provie numerical results to compare ifferent antenna placement scenarios. A. Network Setup User locations are istribute accoring to 2D Poisson istribution with parameter λ = 1 in a square area. A user location is selecte with probability p = 0.7 as a possible antenna location. K is set to 4, thus there are four possible beam irections each with the central angle of π 2. The set of power levels P inclues four values of 1, 2, 3, 4. The path loss exponent α is assume to be 2. For each location l j, SINR threshol β j, an noise η j are set to 1. In the results, Optimal refers to the results of integer formulations an Greey refers to the results of GreeySelect algorithm. We use Fixe, Variable, an Multi as abbreviations of constant power single-beam system, ynamic power single-beam system, an ynamic power multi-beam system respectively. B. Results we increase area of the network from 9 to 36. λ remains fixe, so the average number of user locations will be increase from 9 to 36. For each network size, we average the results of Greey an Optimal over 10 runs. The optimal number of antennas require to cover all locations is emonstrate in figure 2. These results compare ifferent beam pattern types. As observe, ynamic power multi-beam pattern constantly outperforms other types of beam patterns. We also measure the results of greey algorithm for

6 Number of antennas Antenna ratio GreeyFixe GreeyVariable GreeyMulti 2 OptimalFixe OptimalVariable OptimalMulti Network size Network size Fig. 2. Optimal number of antennas to cover the network Fig. 4. Antenna ratio of GreeySelect algorithm Coverage ratio GreeyFixe GreeyVariable GreeyMulti Network size Fig. 3. Coverage ratio of GreeySelect algorithm ifferent beam patterns to compare them with the associate optimal ones. Greey algorithm may fail to fin a total coverage plan when it is available. We efine coverage ratio as the ratio of the number of locations covere by the greey algorithm to the number of locations. The coverage ratio achieve by greey algorithm is shown in figure 3. For small network sizes where maximum istance between two locations in network is comparable to antenna transmission raius, we observe total coverage. However for larger network sizes when there isn t such a relationship, GreeySelect almost achieves 75% coverage. As apparent from the plot, we expect this ratio continues to hol. This plot suggests that for a ranom istribution of noes GreeySelect fins a plan with constant coverage ratio(near 1). When GreeySelect algorithm covers all locations, it typically nees more antennas than the optimal one. We efine antenna ratio as the ratio of the number of antennas employe in GreeySelect to the optimal case. It is another criterion which together with coverage ratio show the behavior of GreeySelect. The antenna ratio is epicte in figure 4. As apparent for larger network sizes, antenna ratio is almost 0.8. All in all, antenna ratio of 0.8 an coverage ratio of 0.75 confirm the efficiency of the greey algorithm. VII. CONCLUSION The problem of inoor wireless planning using smart antennas is investigate in this stuy. We formulate single-beam an multi-beam cases of WPP as integer linear programs. Due to complexity of fining the optimal solution, we also present a greey heuristic. Although numerical results confirm efficiency of greey algorithm, optimal solutions are always esirable. Thus in the future, we inten to apply integer programming ecomposition methos to make fining optimal solution more tractable. Consiering fairness issues in the planing problem is another interesting area for future research. REFERENCES [1] X. Liu, A. Sheth, M. Kaminsky, K. Papagiannaki, S. Seshan, an P. Steenkiste, Dirc: Increasing inoor wireless capacity using irectional antennas, in Proc. ACM SIGCOMM, Barcelona, Spain, Aug [2] M. Haenggi, J. Anrews, F. Baccelli, O. Dousse,, an M. Franceschetti, Stochastic geometry an ranom graphs for the analysis an esign of wireless networks, IEEE J. Sel. Areas Commun., vol. 7, no. 7, Sep [3] M. Blanco, R. Kokku, K. Ramachanran, S. Rangarajan, an K. Sunaresan, On the effectiveness of switche beam antennas in inoor environments, in Proc. PAM, Clevelan, USA, Apr [4] M. Chryssomallis, Smart antennas, IEEE Antennas Propag. Mag., vol. 42, no. 3, Jun [5] Z. Huang, Z. Zhang,, an B. Ryu, Power control for irectional antenna-base mobile a hoc networks, in Proc. IWCMC, Vancouver, CAN, Jul [6] K. Sunaresan, K. Ramachanran, an S. Rangarajan, Optimal beam scheuling for multicasting in wireless networks, in Proc. ACM Mobi- Com, Beijing, China, Sep [7] E. Amali, A. Capone, M. Cesana, an F. Malucelli, Optimizing wlan raio coverage, in Proc. IEEE ICC, New york, USA, Jun [8] M. C. S. Bosio, A. Capone, Raio planning of wireless local area networks, IEEE/ACM Trans. Netw., vol. 15, no. 6, Dec [9] Y. Osais, M. St-Hilaire, an F. R. Yu, On sensor placement for irectional wireless sensor networks, in Proc. IEEE ICC, Dresen, Germany, Jun [10] P. Gupta an P. R. Kumar, The capacity of wireless networks, IEEE Trans. Inf. Theory, vol. 46, no. 2, Mar [11] V. V. Vazirani, Approximation algorithms. Springer Verlag, [12] J. Ai an A. A. Abouzei, Coverage by irectional sensors in ranomly eploye wireless sensor networks, Journal of Combinatorial Optimization, vol. 11, no. 1, Feb