, while the capacity scales as Θ (n)

Transcription

1 Scalability of Wireless Networks Capacity with ower Control Wang Hei Ho and Soung Chang Liew Department of Information Engineering he Chinese University of Hong Kong {whho4, Abstract eference [] showed that wireless networks with capacity scalable with the number of nodes, n, are achievable in theory. he transport capacity scales as Θ ( n), while the capacity scales as Θ (n). eference [2], on the other hand, showed that the capacity of IEEE 802. networks does not scale with n due to its carrier sensing mechanism. he previous work, however, has not considered the use of power control. he main contributions of this paper are two-fold: ) the demonstration that 802. networks are scalable with power control; 2) however, with power control, an enhanced MAC protocol called Selective Disregard of NAVs (SDN) can achieve substantially higher capacity. Specifically, capacity within 90% of the theoretical optimal capacity of infrastructure networks is achievable. Keywords Wireless Networks, WLAN, ower Control, 802., Network Capacity, Scalability, CSMA/CA, Ad-hoc Networks I. INODUCION hysical-interference and protocol carrier-sensing constraints against simultaneous transmissions by links [] [2] limit spatial re-use in wireless networks. he physical-interference constraints are due to a receiver s inability to decode its signal when the powers received from other interfering signals are large. he protocol carrier-sensing constraints are due to the carrier-sensing mechanism of the multi-access protocol used [2]. he 802. carrier-sensing constraints, in particular, cause the 802. network capacity to be non-scalable [2]. A variant of 802. for achieving scalable capacity, Selective Disregard of NAVs (SDN), was introduced in [2]. he main idea of SDN is to eliminate those 802. carrier-sensing constraints that are extraneous. Without power control, the original 802. networks are non-scalable, but SDN networks are [2]. he previous work, however, has not considered the use of power control. his paper considers the implication of power control for capacity scalability in 802. and SDN wireless networks. II. SIMULANEOUS-ANSMISSION CONSAINS WIH OWE CONOL A. Constraints of SDN SDN removes the extraneous 802. carrier-sensing constraints so that the capacity is largely limited only by the physical-interference constraints [2]. So, the constraints in SDN are basically physical-interference constraints. We model the interference constraints using a pair-wise interference model []. Consider the power relationship from node a to node b, ( a, b) = k a / r, where (a, b) is the power received by node b from the transmission by node a, a is the transmit power of node a, r is distance between the two nodes, >2 is the path-loss exponent, and k is a constant. Let i and i denote the transmitter and receiver of link i. For brevity, we will also use i and i to denote their positions in the following. Consider two links, and 2. With no power control (i.e., = 2 = = 2 ), link 2 can interfere with link if 2 < K () 2 < K (2) 2 < K (3) 2 < K (4) where K > (e.g., 0dB) is the power margin required for proper signal detection. Constraints () (4) correspond to DAA-DAA collision, DAA-ACK collision, ACK-DAA collision and ACK-ACK collision, from link 2 to link respectively. Similarly, link can also interfere with link 2 with four constraints of the same form. When power control is introduced, the transmit powers of nodes, 2, and 2 may be different. Assuming all nodes use the same receive power threshold, constraints () (4) become 2 < K 2 (5) 2 < K 2 (6) 2 < K 2 (7) 2 < K 2 (8) B. Constraints of 802. For 802. wireless networks, in addition to the constraints in Section A, protocol carrier-sensing constraints further limit the network capacity. Consider links and 2 again, and suppose that link is transmitting. hen, assuming all nodes use the same carrier-sensing power threshold, link 2 cannot transmit to link if any of the following constraints holds [2]: < VCSange( ) (9) 2 < VCSange( ) (0) 2 < CSange( ) () 2 < VCSange( ) (2) 2 < VCSange( ) (3) 2 < CSange( ) (4) 2 where VCSange( a ) is the virtual carrier sensing range due to the transmissions of S/CS by node a with transmit power /05/$ IEEE 70

2 a ; and CSange( a ) is the physical carrier sensing range due to the DAA transmission by node a. A similar set of inequalities can be obtained when we consider the case where link transmits first. he case of no power control is a special case in which VCSange( a )=VCSange and CSange( a ) =CSange, where VCSange and CSange are constants. III. LINK-INEFEENCE GAH FO HYSICAL CONSAINS Node Graph Carrier Sensing Link i-graph capacity per unit area will reach a limit eventually as the number of As increases. hat is, more As do not bring about higher capacity! Intuitively, if power control is introduced so that VCSange and CSange are also scaled by the size of grids, some of the inter-grid interferences can be prevented and we should observe network capacity that scales with node density. CSange xange CSange xange Link Link 2 (a) Figure. Mapping of a network topology a) to b) i-graph. A Link-Interference Graph (i-graph) can be used to represent the physical constraints graphically. In an i-graph, an arrow-shape vertex represents a wireless link with the arrowhead pointing toward the receiver. here is a directional interference edge (i-edge) from vertex 2 to vertex if any of the constraints (5) (8) is satisfied. Figures a and b show an example of mapping a network topology to an i-graph. IV. SCALABILIY OF NEWOK CAACIY: ANALYICAL DISCUSSION Consider an infinitely large infrastructure-mode wireless network with multiple Access oints (A) laid out in a grid topology. Each grid is associated with an A, which serves as the base station for the clients located within the grid. Clients are randomly placed within the grid. What would happen to the total network capacity when we increase the number of As by reducing the grid size, while maintaining the same number of clients per A? he increase of node density as such is analogous to decreasing the scale on which a map is drawn (Figure 2). he location (x,y) is translated to (ax,ay), where a <. If the distance between two nodes before scaling is d, the distance after the transformation is ad. For SDN without power control, we only need to consider constraints () (4). hey are invariant to this scaling, as a on both sides of the inequality will cancel out. hus, the expected network capacity per unit area will scale with the number of As. his is because the expected capacity of a smaller grid is the same as the expected capacity in the original larger grid before transformation. d d3 d2 a* d (b) a* d3 Figure 2. Change in node density as network scales networks without power control are limited by constraints () (4) and (9) (4). Since the carrier sensing ranges are constants, they will cover more neighbor grids when the node density increases (Figure 3). hus, the network a* d2 Figure 3. Effect of CSange in 802. as network scales. We define two types of power control here. he first is Uniformly-Scaled ower Control (USC), in which all nodes use the same transmit power adjusted according to the node density (grid size). he second is Adaptive ower Control (AC), in which different nodes may use different transmit powers. he latter will be discussed in the next section. For SDN with USC, constraints (5) (8) need to be considered. However, with uniform transmit powers, constraints (5) (8) are the same as () (4). herefore, the network capacities of SDN with and without USC are the same. A non-uniform power control scheme (see Section V) is needed to boost the network capacity in SDN. For 802. with USC, in addition to constraints (5) (8), we have constraints (9) (4). Constraints (9) (4) will make the network capacity of 802. with USC less than that of SDN with or without USC. V. ADAIVE OWE CONOL (AC) FO SDN here are two shortcomings to USC: ) With respect to SDN, it does not bring about any capacity advantage. 2) It may be suboptimal in scenarios in which links are non-uniformly distributed. Specifically, in USC the transmit powers of all nodes are kept the same, and a link s transmit power is not adapted to what the link sees from other links in its neighborhood. he goal of AC is to overcome these limitations. his section addresses the AC for SDN, which aims to reduce the number of i-edges in the network. he algorithm consists of successive iterations. We assume that initially the transmit powers of nodes are high. Each iteration chooses a particular link and attempts to reduce the number of attacking i-edges emanating out of that link by reducing the transmit powers of its transmitter and receiver. here are two issues: in each iteration, (i) which link should be chosen; (ii) how to adjust the transmit powers. Addressing (i) and (ii) requires knowledge of the current transmit powers used by the nodes, { i }, and the power-gain matrix [G(i, j)], where G(i, j) = (i, j)/ i is the power-gain function from node i to node j. his section is divided into four parts. art A discusses the maximum power a link can be adjusted down by issue (ii) above. art B considers the order in which nodes are chosen for power adjustment issue (i) above. art C introduces an 7

3 algorithm for nodes to find out { i } and G(i, j) in their neighborhood for the computations in art A and art B. art D presents simulation results of the different strategies proposed in art B. A. er-iteration ower Adjustment We now consider the power adjustment of a chosen link. o reduce the i-edges from a link to other links, we may reduce the transmit powers of the transmitter and receiver of that link. he larger the power reduction, the more i-edges are likely to be eliminated. However, there is a bound on the power reduction, as explained below. In each per-iteration power adjustment, we assume the transmit powers of links other than the chosen link remain unchanged. When adjusting the transmit power of the chosen link, we must make sure that ) the connectivity between its transmitter and receiver can be maintained; 2) the VCSange is enough to cover interfering nodes; and 3) the power reduction does not create new i-edges from other links to the chosen link. Note that reducing power as such will not create new i-edges from the chosen link to other links.. Ensuring the reduced power satisfy the minimum decodable threshold: Suppose link is the chosen link. Assume that G(, ) and G(, ) are known (see the ower Exchange Algorithm in art C). hen, to guarantee connectivity from to, the minimum transmit power of is xth min ( ) = xth = (5) (, ) G(, ) where x th is the receive power threshold for signal detection. Similarly, the minimum transmit power of is xth min( ) = xth = (6) (, ) G(, ) 2. Ensuring the VCSange is enough to cover interfering nodes: his requirement is to ensure that virtual carrier sensing in SDN continues to work well. Let M (M ) denote the set of nodes whose transmissions can interfere with reception at ( ). So, before ( ) transmits, it has to be able to forewarn the nodes in M (M ) not to transmit via virtual carrier sensing. Otherwise, the ACK (DAA) from ( ) to ( ) might be corrupted by transmissions by the nodes in M (M ). his can be achieved in two ways. Either the S of or the CS of must reach the nodes in M or M. hus, we have adjusted ( ) x VCS th / G(, m) m M or M (7) O adjusted ( ) x VCS th / G(, m) m M or M (8) where x VCS th is the receiver sensitivity threshold for S/CS (i.e., carrier-sensing threshold) which is generally required to be smaller than x th so that VCSange is larger than xange. For example, if S/CS are transmitted at /r VCS the rate of DAA, as an approximation, we may set x VCS th = x th / r VCS. Note that in the above: (i) We have assumed the same transmit power is used to carry DAA/ACK and S/CS. (ii) (7) and (8) are an O relationship. When adjusting the transmit powers of and, as long as one of them is satisfied, the condition is fulfilled. 3. Ensuring the reduced power is stronger than neighbors pair-wise interferences: o ensure that no new i-edges are formed when and reduce their transmit powers, let N and N be respectively the set of transmitting and receiving nodes of those neighboring links without i-edges to link originally. We require adjusted ( ) K n G(n, ) / G(, ) n N (9) adjusted ( ) K n G(n, ) / G(, ) n N (20) Note that N and N do not need to cover all nodes in the network. In particular, they need to cover only nodes that can potentially interfere with and. Specifically, only node n that satisfies the following needs to be considered: (i) (n, ) > (, )/K x th /K if node n is to be included in N (ii) (n, ) > (, )/K x th /K if node n is to be included in N Steps, 2, and 3 are combined as follows. First, we set adjusted ( ) to the maximum allowed by (5) and (9). hen, we set adjusted ( ) to the maximum allowed by (6) and (20). hen, we see if either the resulting adjusted ( ) and adjusted ( ) satisfies (7) or (8). If yes, we are done. If not, we adjust either adjusted ( ) or adjusted ( ) upward until either (7) or (8) is fulfilled. In general, the computation time for each per-iteration power adjustment is O(n), where n is the number of nodes, thanks to steps 2 and 3. B. ower Control Scheduling Strategies We would like to study the importance of the order of links for power control, based on a link-by-link power adjustment nature. We consider the strategies for choosing a link for power adjustment in each iteration referred to as ower Control Scheduling Strategies. Specifically, two strategies are considered. heir performance results are presented in art D. In this paper, we assume there is a central node that knows the power transfer relationships among links and decides which link to control its power in each iteration. In real practice, we will implement AC in a distributed manner, in which no central node is needed. In distributed algorithms, every node only needs to monitor the local conditions surrounding them and multiple nodes may adjust their powers simultaneously. It turns out the distributed versions of the centralized algorithms discussed here can be easily devised. Due to space limitation, detailed discussions of the distributed algorithms are relegated to a separate paper. Generally speaking, good centralized algorithms also yield good distributed algorithm. In addition, it is also essential to understand centralized algorithms as benchmarks even though our ultimate goal is distributed versions of them. Let us denote the number of attacking i-edges of link l by i a (l) i.e., number of i-edges from l to other links; and the number of defending i-edges of link l by i d (l) i.e., number of i-edges from other links to l. Strategy : Choose the link with the largest i a : he intuition of this strategy is as follows. he link with the largest 72

4 number of attacking i-edges is the link that seriously interferes with neighboring links. By reducing its power first, we can increase the chance that more i-edges can be reduced in the iteration. Note that the power adjustment steps in art A is a defensive one in that it ensures that no new i-edges to the chosen link is created, rather than i-edges from it to others are eliminated. hus, even with the best intention of Strategy here, there is no guarantee that i-edges can be identified for elimination in each iteration. In case of a tie in which multiple links have the same i a, we will pick the one with the smallest i d. If there are multiple links with the same i a and i d,, one of the links will be chosen in random. he reader is referred to the description of Strategy 2 for the motivation for considering a link with the smallest i d. seudocode of Strategy : //LinkSet is the set for link waiting for power control LinkSet = all links; While (LinkSet!= NULL){ L = arg max_l in LinkSet (i a(l)); If L >, L = arg min_l in L (i d(l)) else m = link in L; If L >, m = a random link in L; erform per-iteration power adjustment on m; emove m from LinkSet } Figure 4. seudocode of Strategy. he overall AC algorithm of Strategy is shown in Figure 4. Once a link is picked for power adjustment, it will not be picked again. A round consists of the considerations of all links. We choose a link only once in each round because otherwise it is possible for the chosen link to be chosen again in the next iteration because it still has the largest i a. his will result in an infinite loop. We may run the algorithms for several rounds to continue to reduce the i-edges. In our simulation experiment, however, we have found that typically after one round, only a few additional i-edges can be eliminated in future rounds. Since Strategy loops for n iterations in each round, and in each iteration O(n) computations are needed, the computation time for one round of Strategy is O(n 2 ). Strategy 2: Choose the link with the smallest i d : he intuition of this strategy is as follows. With respect to step 2 of the per-iteration power adjustment algorithm in art A, having fewer defending i-edges to consider may allow us to lower the power by a larger amount. Hence, this may increase the likelihood of an attacking i-edge being eliminated. hat is, whereas Strategy maximizes the number of candidate i-edges for elimination, Strategy 2 maximizes that the chance that a candidate i-edge can be eliminated. seudocode of Strategy 2: //LinkSet is the set for link waiting for power control LinkSet = all links; While (LinkSet!= NULL){ L = arg min_l in LinkSet (i d(l)); If L >, L = arg max_l in L (i a(l)) else m = link in L; If L >, m = a random link in L; erform per-iteration power adjustment on m; emove m from LinkSet } Figure 5. seudocode of Strategy 2. In case of a tie in which multiple links have the same i d we will pick the one with the largest i a, If there are multiple links with the same i a and i d,, one of the links will be chosen in random. he overall AC algorithm for Strategy 2 is shown in Figure 5. Similar to Strategy, once a link is picked for power control, it will not be picked again in each round to avoid infinite looping of the algorithm. As with Strategy, the computation time of Strategy 2 is also O(n 2 ). C. ower Exchange Algorithm In [2], a ower Exchange Algorithm (E) has been proposed for establishing the i-graph of a network. Our per-iteration power adjustment procedure in art A requires not only the knowledge of the current i-edges, but also the power-transfer relationship between nearby nodes so that we can ensure no new i-edges are created after power adjustment. If we assume the presence of a central node, such information can be gathered for the algorithms in arts A and B. We extend the E in [2] for our purpose here. he E algorithm here is a local algorithm in that each node finds out the i-edges and potential i-edges in its neighborhood. he nodes then can send this information to the central node for execution of the algorithms in arts A and B. estart Mode in the receiver design is assumed [2]. In this mode, a receiver will switch to receive a stronger signal even if it is in the midst of sensing a weaker signal not targeted for it. he E packets are special packets periodically broadcasted by nodes to exchange power information with neighbors. We assume the transmit powers of these packets are the same as the transmit powers of regular packets like DAA/ACK/S/CS. Consider an arbitrary node a. he E packets sent by node a contain three types of information: () Active links: (a, b) or (b, a), where b is any other node which forms an active link with a; (2) ransmit power a of node a; (If node a is an A, we assume it uses different a for different client stations and establishes multiple links with clients) (3) ower set, as described below. he identity of the sender of a E packet is implicit in the MAC address of the E packet.. Each node a monitors the power it receives from other nodes and keep this information in a power set (a)={(b, a), (c, a), }. For this purpose, the powers of the E packets from nodes b, c, can be measured by node a. 2. Each node a periodically broadcasts a E packet at a rate lower than the data rate. 3. Node a gathers information from the E packets received from its neighbors. he proof of the correct operation of AC with E will be deferred to another paper due to the space limit here. D. Comparison of Scheduling Strategies We now present the performance results of the two strategies. We used MALAB to simulate an infrastructure topology with 25 As placed uniformly in a square-grid manner over a x km 2 domain. 25 client stations are placed randomly in the domain so that each A on average has five associated clients. In addition to the two strategies, for benchmarking purposes, we also considered the random 73

5 strategy in which a random link is chosen in each iteration. Figure 6 shows the number of remaining i-edges versus the number of iterations. We iterated only one round (see art B) so that the total number of iterations is equal to the number of links (i.e., 25). It can be seen that the number of remaining i-edges of the random case is always more than that of the two strategies, with Strategy 2 having the best performance. It is interesting to note that the gaps between the remaining i-edges for different strategies widen and then narrow as the number of iterations increases. Specifically, the remaining i-edges for the different strategies converge to values that differ by only 20 to 30 i-edges at the end. his means that at convergence, the different strategies do not yield significantly different network capacities. increasing at around 5 Mbps. he other three schemes capacities increase almost linearly. For 802. with USC, the total network capacity is around 32% of the optimal capacity. It is clear that SDN performs much better than % of the optimal capacity can be achieved with SDN without power control. With AC, SDN can achieve about 93% of the optimal capacity. Figure 6. Number of remaining i-edges versus number of iterations. VI. SCALABILIY OF NEWOK CAACIY: NUMEICAL ESULS We generated grid topologies in a x km 2 domain with randomly placed clients in MALAB. Initially, there are four As with transmit power 28.8mW. ogether the four As cover the whole domain. For 802., we set VCSange = CSange = 2.78 x xange. We vary the number of As in the domain while fixing the client-to-a ratio to 5:. We connect each client to its closest A to form a link and identify the contentions between the links based on constraints (5) (8) for SDN and (5) (8) and (9) (4) for In generating the network capacity, we go through the As in the domain randomly; and for each A, we randomly choose a link to transmit, provided that its ability to transmit successfully is permitted by the underlying constraints. We define network capacity as the total number of links that can be selected as such. For USC, we scale the xange of nodes so that As just cover a grid area. For AC, we adopt Strategy 2 as the power control scheduling strategy. he performance results here pertain to the power assignments after one round of the execution of the Strategy-2 AC algorithm. Figure 7 shows the simulation results. he dashed line is the ideal maximum achievable capacity, which is the total number of As in our case. We see that 802. without power control saturates very quickly. Specifically, its capacity stops Figure 7. Simulation results of total network capacity against number of As. VII. CONCLUSION able I. Capacity scalability of 802. and SDN with and w/o power control. IEEE 802. SDN w/o power control Non-scalable 78% scalable with power control 32% scalable 93% scalable his paper has investigated the capacities of 802. and SDN wireless networks with power control. able I summarizes our findings. In essence, although power control can solve scalability problem, different scalable networks may have different degrees of scalability. Although two schemes may both have network capacities that scale linearly with the number of As, the slope of the linear curve may be different. he ratio of the slope of the curve to the slope of the optimal curve gives the degree of optimality. hus, we may say that 802. with power control is 32% scalable; while SDN with power control is 93% scalable. Due to space limitation, our investigations on other power control strategies, including distributed implementations of the power control algorithms will be reported elsewhere. EFEENCES []. Gupta,.. Kumar, he Capacity of Wireless Network, IEEE rans. On Information heory, Vol. 46, No. 2, pp , Mar [2]. C. Ng, S. C. Liew, L. Jiang, Achieving Scalable erformance in Large-Scale IEEE 802. Wireless Networks, IEEE Wireless and Communications and Networking Conference (WCNC 05), Mar [3] L. Jiang and S. C. Liew, emoving Hidden Nodes in IEEE 802. Wireless Networks, IEEE Vehicular echnology Conference, Sep [4] Vikas Kawadia and.. Kumar, rinciples and rotocols for ower Control in Ad Hoc Networks, IEEE Journal on Selected Areas in Communications, pp , vol.23, no. 5, January [5] J.. Monks, V. Bharghavan, and W. Hwu, A power controlled multiple access protocol for wireless packet networks, IEEE INFOCOM. 200, vol.20, pp., April