Relay Station Placement for Cooperative Communications in WiMAX Networks

Transcription

1 Relay Station Placement for Cooperative Communications in WiMAX Networks Dejun Yang Xi Fang Guoliang Xue Jian Tang Abstract The recently emerging WiMAX (IEEE ) is a promising telecommunication technology to provide low-cost, high-speed and long-range wireless communications. To meet the growing demand for throughput, Relay Station is introduced by IEEE j to relay traffic for Subscriber Stations. By incorporating Cooperative Communications scheme in WiMAX, we can further improve the network capacity. In this paper, we study the Relay Station placement problem, which seeks to deploy a minimum number of Relay Stations to satisfy all data rate requests from Subscriber Stations via Cooperative Communications. We analyze the computational complexity of the problem and prove it to be NP-Complete. Then we present efficient algorithms with guaranteed approximation ratios. Extensive experiments show that the number of Relay Stations returned by our algorithms is close to those returned by optimal solution. I. Introduction WiMAX (based on the IEEE standard) has recently emerged as a telecommunications technology providing lowcost, high-speed and long-range wireless communications for a large variety of applications, such as civilian and military. Though promising broadband wireless access, the current existing standard IEEE e [1] still exhibits inherent problems in practice, such as low signal-to-noise-ratio (SNR) and coverage holes. To meet the growing demand for throughput and capacity, IEEE j [10] has been proposed to enhance IEEE e, which introduces a special kind of nodes, Relay Stations (RSs), to relay traffic for Subscriber Stations (s). Compared to the Base Station (BS), an RS has significantly simpler complexity and lower cost. Cooperative Communication (CC) [11] has been shown to have the potential to increase the capacity of wireless networks. The essence of CC is to exploit the broadcast nature of wireless communication and the relaying capability of other nodes to achieve higher throughput. There are two different cooperative communication modes: Amplify-and-Forward (AF) and Decode-and-Forward (DF). For AF mode, the relay node amplifies the received signal from the source node and forwards it to the destination node. For DF mode, the relay node decodes the received signal and re-encodes it before forwarding it to the destination node. For both modes, the location of the relay node is critical as it will directly affect the achievable data rate. Yang, Fang, Xue are all with the CSE Dept. at Arizona State University, Tempe, AZ Tang is with the EECS Dept. at Syracuse University, Syracuse, NY {dejun.yang, xi.fang, xue}@asu.edu. This research was supported in part by NSF grants and , ARO grant W911NF and Montana State MBRCT grant # The information reported here does not reflect the position or the policy of the federal government. Fig. 1. RS BS A WiMAX network with Cooperative Communication By incorporating CC technology in WiMAX networks, we can further improve the throughput. A WiMAX network with CC is illustrated in Fig. 1. In this paper, we study the Relay Station placement for CC problem, which is formally defined in Section III. The objective of this problem is to place a minimum number of RSs to satisfy the data rate requests from all Subscriber Stations via CC. The main contributions of this paper are the following: This paper is the first work to study the Relay Station placement for Cooperative Communications (RSPCC) problem in WiMAX networks. Specifically, we aim to satisfy the data rate requests from all s by deploying a minimum number of RSs and applying CC technology. We first develop an O(log n)-approximation algorithm to solve the RSPCC problem, wherenis the number of s. Then, for the case where the number of s connecting to an RS is bounded by a constant f, we present an fapproximation algorithm. We present a Linear Programming based approach to obtain a lower bound on the solution to the problem. Extensive experiments show that the number of Relay Stations returned by our algorithms is close to those returned by optimal solution. The rest of this paper is organized as follows. In Section II, we briefly review related work on relay node placement and cooperative communication. In Section III, we describe our system model and formally define the problem studied in this paper. In Section IV, we discuss the computational complexity of the problem and present our approximation algorithms. We then present a Linear Programming based approach to obtain a lower bound on the solution to the problem in Section V. In Section VI, we present numerical results to demonstrate the performance of our algorithms compared with the lower bounds. Finally, we conclude this paper in Section VII. RS

2 II. Related Work The Relay node placement problem, which is similar to the RS placement problem, has been widely studied in the literature, especially in wireless sensor networks [4, 8, 9, 15 17, 19, 21]. In this problem, relay nodes are placed to meet certain requirements, such as coverage and connectivity. Usually, these problems can be categorized into two cases: singletiered (both relay nodes and sensor nodes can relay traffic) and two-tiered (only relay nodes can relay traffic). In [15], Lin and Xue proved the single-tiered problem with R = r and K = 1 is NP-hard, where R,r and K denote the transmission range of relay nodes, the transmission range of sensor nodes and the connectivity requirement respectively. They also presented a 5-approximation algorithm to solve this problem. Algorithms with better approximation ratios have been proposed for the cases where R r and/or K > 1 [4, 16, 21]. RS placement problem in j-based networks is also an active research area. Bhatnagar et al. [2] described an integer programming formulation and designed an efficient heuristic to solve a minimum cost deployment problem in WiMAX backhaul networks. Fu et al. [7] proposed a simple approach to place RSs in Manhattan-like environment. In [20], Yu et al. considered a clustering approach to solve a network planning problem. In [12], an optimal scheme was proposed to solve the problem of finding the location of a single RS and allocating resources for all the s. In [13], Lin et al. proposed a twophase heuristic algorithm to solve the dual-relay RS placement problem, where each is connected to the BS via exactly two RSs through a cooperative relaying scheme. In [14], they gave a heuristic algorithm to maximize the total capacity by placing a bounded number of RSs and using the same cooperative relaying scheme as in [12, 13]. III. System Model and Problem Formulation In this paper, we consider a static network consisting of three types of entities: the Base Station (BS), the Relay Stations (RSs) and the Subscriber Stations (s). Three types of RSs have been defined by the IEEE j standard, including Fixed Relay Station (FRS), Nomadic Relay Station (NRS) and Mobile Relay Station (MRS). An FRS is an RS permanently installed at a fixed location, an NRS is an RS that is intended to function at a fixed location for periods of time, and an MRS is an RS functioning while in motion. We focus on FRS in this paper. Let P r denote the transmission power of an RS. In the network, one BS B and a set S = {s 1,s 2,,s n } of n s are given and fixed. All the s and BS are located on a 2D Euclidean plane. For each s i S, let P i denote its maximum transmission power and c i 0 denote its data rate requirement. In order to mitigate interference, we assume Orthogonal Frequency Division Multiplexing (OFDM) is employed. The channel propagation model can be modeled as follows. Let N 0 denote the abient noise. When a node u transmits a signal to node v at power P u, the signal-to-noise ratio (SNR) at node v, denoted as SNR uv, is SNR uv = P u N 0 d(u,v) α, (1) where d(u,v) is the Euclidean distance between nodes u and v and α is the path loss exponent which is between 2 and 4 in general, depending on the characteristics of the communication medium. For transmission model, we first present a simple model without Cooperative Communication (CC). Direct Transmission When the source node directly transmits to the destination, the achievable data rate from s to d is C DT (s,d) = W log 2 (1+SNR sd ). (2) where W is the bandwidth of the channel. For CC strategy, there are two different modes, AF and DF. Here we only consider DF. DF mode can be best described with a well-known three-node example, where s is the source node, r is the relay node and d is the destination node, as shown in Fig. 2. s Fig. 2. r d A three-node example for CC Decode-and-Forward (DF) For decode-and-forward mode, the relay node decodes and estimates the signal transmitted by the source node in the first time slot and then transmits the estimated data to the destination in the second time slot. The achievable data rate from s to d is C DF (s,r,d) = W 2 min{ log 2 (1+SNR sr), log 2 (1+SNR sd +SNR rd )}.(3) We assume that C DT (s i,b) < c i for all s i N. In addition, we assume that it is possible for each s i N to achieve the required rate by placing a relay node between itself and BS. It is clear that there is no need to consider the s, which can achieve the required rates via directly transmitting to the BS, or cannot achieve it via CC no matter where we deploy the RSs. Our objective is to deploy a minimum number of such RSs to satisfy all the s data rate requirements via CC. We formally define the Relay Station Placement for Cooperative Communication (RSPCC) problem as follows. Definition 3.1: (Relay Station Placement for Cooperative Communication (RSPCC)): Given a BS B and a set S of s, each of which has a known location and a fixed rate requirement c i, a set of RSs R = {r 1,r 2,,r m } is said to be a feasible RS placement if for any s i S, there exists an RS r j R, such that C DF (s i,r j,b) c i. The Relay Station placement for cooperative communication problem seeks for a feasible RS placement with minimum size.

3 IV. Algorithms for the RSPCC Problem In order to have a better understanding of the RSPCC problem, it is necessary to further investigate it. Plugging (1) into (3) and considering the rate requirement C DF c i, we have { W 2 log P 2(1+ N 0d(s i,r) ) c α i, W 2 log 2(1+SNR sib + P r N 0d(r,B) α ) c i, where SNR sib is known as both s i and B are given. After performing some simple algebraic manipulations, we have d(s i,r) α P i, N 0 (4ci/W 1) d(r,b) α P r. (5) N 0(4 c i /W SNR si B 1) Let R i = α P i N 0(4 c i /W 1) and R ib = α. In other words, we need to P r N 0(4 c i /W SNR si B 1) deploy a set R of RSs such that for each s i N, there exists at least one RS r j R located inside the intersection of the disk centered at s i with radius R i and the disk centered at B with radius R ib, as shown in Fig. 3. We call such an intersection the lens of s i, denoted by L i. Although a lens is determined by two disks in general, it can be represented by each s i in this paper as B is same for every. Let L = {L 1,L 2,,L n } denote the set of all lenses. In addition, we say an RS hits a lens L i if it is located in the area of L i. Fig. 3. R ib B s i Ri Region to deploy RSs A. Complexity of RSPCC Problem Before proposing any algorithms, we first analyze the computational complexity of the RSPCC problem. We prove the RSPCC problem to be NP-Complete by a reduction from Minimum Geometric Disk Cover Problem. It has been proved that the Minimum Geometric Disk Cover Problem is an NP- Complete problem [6]. Definition 4.1: (Minimum Geometric Disk Cover Problem): Given a set of points Q = {q 1,q 2,,q n } in the plane and a rational number R > 0, the minimum geometric disk cover problem is to find a set of centers O = {o 1,o 2,,o m } such that every point in Q will be covered by at least one disk with radius R and centered at one of the centers in O. Given any instance I M of the Minimum Geometric Disk Cover Problem, we construct an instance I R of the RSPCC problem as follows. For any point q i Q, we have a corresponding s i at the same location. The location of (4) the BS B is set to be the origin. We set W, N 0, c i and P i to appropriate values such that R i = R, for all s i S. The power P r of RS is chosen to be large enough such that the smallest disk centered at B with radius R ib can cover all the disks centered at s i, as shown in Fig. 4. Note that, for clarity, only the BS is shown in the figure. Fig. 4. B R NP-Complete proof Clearly, this construction can be done in polynomial time. Now we prove that I M has a solution if and only if I R has a solution. Suppose I M has a disk cover O such that every point q i Q is covered by at least one disk with radius R and centered at a center in O. Then I R has a solution by replacing the relay nodes at each center. By the way we set the values of W, N 0, c i and P i, we know that each s rate requirement is satisfied. On the other hand, supposei R has an RS deployment R such that C DF (s i,r j,b) c i is satisfied for some r j R, namely, Equation (5) holds for each s i N. Let O be the set of centers with same locations as RSs in R. Then O forms a disk cover of P. This proves the RSPCC problem is NPhard problem. Since it is clear that any solution to the RSPCC problem can be verified in polynomial time, we claim that the RSPCC problem is NP-Complete. B. Approximation Algorithms Since the RSPCC problem is NP-Complete, the best effort we can expect is to design approximation algorithms. Before proposing our approximation algorithms, we need to give some definitions and notations. A point is called a potential position (PP) for RSs if a RS could be placed at this position to hit at least one lens. Now we construct as follows a set P of PPs, from which it is sufficient to find an optimal solution to the RSPCC problem. Initially, P is empty. For any two intersecting lenses, we put their intersection points into P. For any isolated lens, namely the ones intersecting with no other lenses, we arbitrarily pick a point on the lens and put it into P. Note that the construction can be done in polynomial time, as there are at most n lenses and two lenses cross each other at most 4 times. We next show that it is sufficient to obtain an optimal solution from set P. Let R be an optimal RS placement. For each RS r j in R, if it is one of the PPs in P, we are done. Suppose an RS r j in P is not included in P. We can find a PP in P which hits at least those lenses hit by r j. This is demonstrated in Fig. 5. In the figure, an RS (square) not in P covers three lenses. Another RS (disk) from P hits all the lenses hit by the previous one.

4 Fig. 5. An RS (square) can be replaced by another one (disk) For each PPp P, letl(p) denote the set of lenses hit byp. Moreover, for a subsetp P of points,l(p ) = p P L(p). Here a simple solution is that we can think of the RSPCC problem as a Set Cover problem, with L being the universal set and {L(p) p P} being a family of subsets of L. Then we apply the approximation algorithm in [3] to solve it. As the algorithm in [3] is more general and to make our paper self-contained, we list the algorithm in Algorithm 1. Algorithm 1 O(ln n) Approximation Algorithm for the RSPCC problem 1: R. 2: Compute the set L of all lenses, the set P of PPs and L(p) for each p P. 3: while L = do 4: Find a PP p P with maximum L(p). 5: R R {p}. 6: L L L(p). 7: L(p ) L(p ) L(p) for all p P. 8: end while 9: return R. Theorem 4.1: Algorithm 1 guarantees to compute a solution R to the RSPCC problem in polynomial time. Furthermore, we have R O(lnn) R, where n is the number of s and R is an optimal RS placement. Proof. Since the proof of this theorem is similar to the one in [3], we omit it in this paper. The approximation ratio in Theorem 4.1 is very conservative. In practice, the number of s that can be connected to an RS is usually bounded by a constant f, as the density of s is bounded. In this case, we can apply the algorithm in [18] to achieve a significantly better approximation ratio. We illustrate the pseudo code in Algorithm 2. Theorem 4.2: Assume that the number of wireless nodes that can be connected to a relay node is bounded by f. Algorithm 2 guarantees to compute a solution R to the RSPCC problem in polynomial time. Furthermore, we have R f R, where R is an optimal RS placement. Proof. Since the proof of this theorem is similar to the one in [18], we omit it in this paper. Algorithm 2 f-approximation Algorithm for the RSPCC problem 1: R. 2: Compute the set L of all lenses, the set P of PPs and L(p) for each p P. 3: while L = do 4: Let L i be one of lenses in L. 5: R R {p} for all p P such that L i L(p). 6: L(p ) L(p ) L(p) for all p P. 7: L L L(R). 8: end while 9: return R. V. A Lower Bound on the RSPCC Problem As the RSPCC problem is proved to be NP-Complete in Section IV, it is unlikely to obtain optimal solutions efficiently. To have a suitable comparison with our algorithms, we present a Linear Programming (LP) based approach to obtain a lower bound of the RSPCC problem in polynomial time. First, we formulate this problem using an Integer Linear Programming (ILP) as follows. Minimize Subject to L i L(p j) p j P z j z j 1 (L i L) (6) z j {0,1} (p j P) (7) For each PP p j P, we introduce a corresponding binary variable z j, 1 j P. Specifically, { 1 if p j is selected to deploy an RS, z j = 0 otherwise. For each lens L i, we need to guarantee that at least one PP hitting it is selected. We characterize this with Constraints (6). The objective is to minimize the total number of selected PPs. Though this ILP can solve the problem optimally, it is proved that solving ILP itself is also NP-hard [18]. Instead, we solve the corresponding LP, where Constraints (7) are relaxed to 0 z j 1, for any p j P. Obviously, by relaxing the constraints, we can only obtain a lower bound of the optimal solution. VI. Numerical Results No previous algorithms have been proposed for the RSPCC problem. Thus to evaluate the performances of our approximation algorithms, we compared the results with the lower bounds obtained from the LP presented in Section V. In our simulations, we set W = 10 MHz for the channel. The transmission power P s is 200 mw for all s and the transmission power P r of RS is set to 1 W. We used 2 as the path loss exponent and assume that the ambient noise is We considered two cases in this experiment: constant deployment region and constant density. In both cases, the BS was located at the center of the square and the s were

5 randomly distributed. Let ALP denote the LP based algorithm, ALN denote the O(ln n) approximation algorithm and ACT denote the f-approximation algorithm. Number of RSs used ALP ALN ACT Number of s Fig. 6. Results with increasing density: 7000m 7000m deploy region; n = 20, 40, 60, 80, 100 and 120. For the case of constant deployment region, we randomly generated a number of s, from 20 to 120, in a fixed region of size 7000m 7000m. Fig. 6 shows the average number of RSs required by ALP, ALN and ACT. We note that the number of RSs obtained by ALN is close to the lower bound, and thus to the optimal value. Though performing not as well as ALN, ACT still requires at most three times of RSs as required by the optimal solutions. The gap between the results returned by ALN and ACT is the consequence of locally selecting PPs hitting a single lens. Number of RSs used ALP ALN ACT Size of Region Fig. 7. Results with constant density: from to For the case of constant density, we kept the density of s at 10 per km 2 while increasing the region size from 2000m 2000m to 4000m 4000m. Fig. 7 shows the average number of RSs required by ALP, ALN and ACT. Similar with the results of previous case, ALN also performs closely to ALP. VII. Conclusions and Future Work We studied the problem of placing minimum number of RSs to satisfy all the data rate requests from s. We analyzed the complexity of this problem and proved it to be NP-Complete. To solve the problem, we have presented an O(ln n)-approximation algorithm. For a more practical scenario, where the number of s connecting to an RS is bounded, we presented a constant approximation algorithm. We also presented linear programming based formulations to obtain lower bounds for comparison purpose. Numerical results show that the solutions returned by our algorithms are at most three times of the optimal solutions. For future work, we aim to design approximation algorithms with better approximation ratios. REFERENCES [1] IEEE e Working Group, Part 16: Air interface for fixed and mobile broadband wireless access systems amendment 2: Physical and Medium Access Control layers for combined fixed and mobile operation in licensed bands, IEEE Standard, [2] S. Bhatnagar, S. Ganguly, and R. Izmailov; Design of IEEE based multi-hop wireless backhaul networks; AccessNet 06. [3] V. Chvatal; A greedy heuristic for the set covering problem; Math. Operations Research, vol.4(1979), pp [4] A. Efrat, Sándor P. Fekete, P. Gaddehosur, J. Mitchell, V. Polishchuk and J. Suomela; Improved Approximation Algorithms for Relay Placement; ESA 08; pp [5] A. Efrat, M.J. Katz, F. Nielsen and M. Sharir; Dynamic data structures for fat objects and their applications; Comput. Geom. Theory Appl., vol. 15(2000), pp [6] R.J. Fowler, R.M. Paterson and S.T. Tanimoto; Optimal packing and covering in the plane are NP-complete; Information Processing Letters, vol.12(1981), pp [7] I-K. Fu, W-H. Sheen, and F-C. Ren; Deployment and radio resource reuse in IEEE j multi-hop relay network in Manhattan-like environment; ICICS 07, pp [8] X. Han, X. Cao, E.L. Lloyd and C.-C. Shen; Fault-tolerant relay node placement in heterogeneous wireless sensor networks; Infocom 07, pp [9] B. Hao, J. Tang and G. Xue; Fault-tolerant relay node placement in wireless sensor networks: formulation and approximation; HPSR 04;pp [10] IEEE j-06/014r1; Harmonized definitions and terminology for j Mobile Multihop Relay, October [11] J.N. Laneman, D.N.C. Tse and G.W. Wornell; Cooperative diversity in wireless networks: efficient protocols and outage behavior; IEEE Trans. Inform. Theory, vol.50(2004), pp [12] B. Lin, P-H. Ho, L-L. Xie, and X. Shen; Optimal relay station placement in IEEE j net-works; IWCMC 07, pp [13] B. Lin, P-H. Ho, L-L. Xie, and X. Shen; Relay station placement in IEEE j dual-relay MMR networks; ICC 08, pp [14] B. Lin, M. Mehrjoo, P.H. Ho, L.L. Xie and X. Shen; Capacity enhancement with relay station placement in wireless cooperative networks; WCNC 09, pp [15] G. Lin and G. Xue; Steiner tree problem with minimum number of Steiner points and bounded edge-length; Information Processing Letters, vol.69(1999), pp [16] E. Lloyd and G. Xue; Relay node placement in wireless sensor networks; IEEE Transactions on Computers; Vol. 56(2007), pp [17] S. Misra, S. Hong, G. Xue and J. Tang; Constrained relay node placement in wireless sensor networks to meet connectivity and survivability requirements; Infocom 08. [18] V. Vazirani; Chapter 2, Approximaition Algorithms, Springer-Verlag, [19] D. Yang, S. Misra, X. Fang, G. Xue and J. Zhang; Two-tiered constrained relay node placement in wireless sensor networks: efficient approximations; IEEE SECON 10, accepted. [20] Y. Yu, S. Murphy and L. Murphy; A clustering approach to planning base station and relay station locations in IEEE j multi-hop relay networks Communications; ICC 08, pp [21] W. Zhang, G. Xue, and S. Misra; Fault-tolerant relay node placement in wireless sensor networks: problems and algorithms; Infocom 07; pp