Joint Range Adjustment and Channel Assignment ...

Joint Range Adjustment and Channel Assignment for Overlap Mitigation in Dense WLANs Shotaro Kamiya∗ , Keita Nagashima∗ , Koji Yamamoto∗ , Takayuki Nishio∗ , Masahiro Morikura∗ , and Tomoyuki Sugihara† ∗ Graduate School of Informatics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan † Allied Telesis Holdings K.K.

Abstract—Dense wireless networks require advanced interference management for the performance improvement. In carrier sense multiple access with collision avoidance (CSMA/CA) networks, the mutual interference can be modeled as the overlap area, and range adjustment, such as transmission power control (TPC) and dynamic channel assignment (DCA), is effective to reduce the overlap area. A game-theoretic framework is utilized to construct a joint TPC and DCA scheme which reduces the total overlap area. Although the evaluation of the overlap area requires topology of neighboring APs, i.e., received power levels among neighboring APs are required, the proposed scheme does not require such information because decomposed overlap areas between any two APs are utilized. Moreover, unilateral improvement dynamics of channel assignment are guaranteed to converge to a Nash equilibrium because the game is shown to be a potential game. We evaluate the proposed scheme through simulations and experiments. Simulation results reveal that the proposed joint TPC and DCA scheme effectively reduces the total overlap area compared to another game-theoretic interference management scheme particularly when the number of APs is large. It is because the proposed scheme can efficiently manage the coverage overlap even when there is a variation in transmission power level. Experimental results demonstrate the convergence of the proposed scheme in a real environment.

I. I NTRODUCTION The rapid growth in the popularity of wireless local area networks (WLANs) has caused very dense deployment of WLAN access points (APs), in which channel contention is serious issue. In such dense wireless environments, a simple increase of transmission power is no longer effective to improve the average data rate, and advanced interference management is required to achieve very high data rates [1]. The coverage overlap between APs operating on the same frequency channel results in channel contention on the basis of carrier sense multiple access with collision avoidance (CSMA/CA) [2]. In CSMA/CA networks, one AP has less transmission opportunities due to the number of the other transmitters whose signal at the AP exceeds the carrier sense threshold over the same channel. Therefore, to increase the transmission opportunities of each AP, reducing the coverage overlap is important. In terms of spatial reuse of a channel, controlling transmission power is important [3], and in terms of spectrum usage, it is necessary to assign an appropriate channel to each AP [4], [5]. The coverage overlap has been discussed in several ways. One of the most commonly used approaches is to consider total interference signal power, that is, the linear combination of received signal powers from neighboring transmitters [6]–[8].

We would like to point out that in these linear combination, the terms associated with nearby APs are often dominant because received signal power decreases exponentially with distance from transmitter. In another method, the overlap area is used to estimate the actual contention [9], [10]. The scheme proposed in [10] maximizes the area that each AP exclusively covers; however, it is difficult for each AP to estimate such area because the overlap area among more than three APs is complex. In this paper, we propose a joint scheme of transmission power control (TPC) and dynamic channel assignment (DCA) for mitigation of the total overlap area. These schemes are applied sequentially and independently. A novel feature is that both TPC scheme and DCA scheme are based on minimization of the overlap area. Both schemes are separately provided. TPC scheme is simple: each AP reduces its transmission power as long as the coverage hole does not occur. DCA scheme is formulated using game theory: each AP selects its channel that maximizes its payoff function. In the proposed scheme, assuming that each AP has a region of a circle centered at itself, each AP selects a channel which minimizes the sum of the overlap area between the AP and each other AP. It is a remarkable feature that each AP requires the overlap area between only two APs. As the overlap area between two circles can be readily determined, it is easy to calculate the sum of the overlap areas associated with neighboring APs. Moreover, the proposed scheme is formulated as a potential game [11], in which unilateral improvement dynamics are guaranteed to converge. Here we would like to emphasize that the purpose of the proposed scheme is a reduction in the total overlap area not system throughput enhancement. To show the effectiveness of the proposed scheme, we present both simulation and experimental results. In the simulation, we show that both TPC scheme and DCA scheme efficiently mitigate the total overlap area. In addition, the proposed scheme is observed to greatly reduce the total overlap area compared with the scheme based on the linear combination of receiver signal powers especially when the number of APs is large. In the experiment, we verify the convergence of the both TPC dynamics and DCA dynamics in a real environment. We implement a programmable AP platform based on IEEE 802.11g WLAN. The contributions of this paper are: 1) development of a joint TPC and DCA schemes mitigating the total overlap area with low complexity calculation; 2) elucidation of the effectiveness

of the proposed scheme in a large number of APs compared with another well-known scheme; and 3) demonstration of the convergence of the proposed scheme in a real environment through the implementation of APs. The rest of this paper is organized as follows. In Section II, we introduce the system model. In Section III, we propose TPC scheme and DCA scheme separately. In Section IV, the simulation setup and results are described. In Section V, the experimental setup and results are described. Section VI concludes this paper. II. S YSTEM M ODEL Let N be an index set of all managed APs. For each AP i ∈ N , let Pi denote the set of configurable transmission power levels, and let pi ∈ Pi denote the transmission power level. Similarly for each AP i ∈ N , let Ci denote the set of available channels and ci ∈ Ci denote the frequency channel. Let pi,max denote the maximum element of Pi . Let p denote (pi )i∈N and c denote (ci )i∈N . A cell Si (pi ) ⊂ R2 of AP i ∈ N is defined by the region where the received power is greater than a required power level T , i.e., Si (pi ) := { z ∈ R2 | Gzi pi ≥ T }, where Gzi is a link gain between AP i ∈ N and a receiver at z ∈ R2 . Note that a cell has no frequency selectivity; thus Si (pi ) does not depend on ci . Although here we assume a twodimensional Euclidean plane, the framework introduced in this paper can be easily extended to three-dimensional Euclidean space．Overlap region Oi (p, c) ⊆ Si (pi ) of AP i ∈ N is defined by ∪ Oi (p, c) := Si (pi ) ∩ Sj (pj ). (1) j∈N \{i}:cj =ci

We call |Oi (p, c)| the overlap area1 , where |·| represents the Lebesgue measure on R2 , i.e., area. III. G AME A NALYSIS AND S CHEMES Our objective is to set appropriate vectors p and c in a distributed manner. A distributed resource allocation among multiple decision makers can be formulated and analyzed using game theory. In this section, first, we introduce the gametheoretic framework to analyze (??). Subsequently, we provide TPC scheme and DCA scheme separately. A. Game-Theoretic Framework We introduce some definitions related to game theory [12]. A strategic form game is defined as a triplet G = (N , (Ai )i∈N , (ui )i∈N ), where N is the finite set of players, Ai is ∏ the set of strategies associated with player i, and ui : i∈N Ai → R is the∏ payoff function that player i wishes to maximize. We denote i∈N Ai by A, and let A−i denote ∏ A . A is called the strategy space. j∈N \{i} j A fundamental solution concept for strategic form games is a Nash equilibrium. A Nash equilibrium of game G is defined 1 A time slot selection game discussed in [10] is essentially the same as a channel selection game (N , (Ci )i∈N , (|Si (pi ) \ Oi (p, c)|)i∈N ), and has been proved to be a potential game. However, TPC is not considered in [10].

by a strategy profile a∗ = (a∗i , a∗−i ) ∈ A which satisfies ui (a∗i , a∗−i ) ≥ ui (ai , a∗−i ) for every i ∈ N and ai ∈ Ai . Hereafter, we state a theorem related to a potential game [11] which plays an important role in DCA scheme. Definition 1. A strategic form game G is a potential game if there exists a potential function V : A → R such that ui (a′i , a−i ) − ui (ai , a−i ) = V (a′i , a−i ) − V (ai , a−i ), (2) for every i ∈ N , ai , a′i ∈ Ai , and a−i ∈ A−i . It is known that if a strategic form game G is a potential game with a finite strategy space, then unilateral improvement dynamics are guaranteed to converge to a Nash equilibrium in a finite number of steps [11]. Definition 2. A strategic form game G is called a bilateral symmetric interaction (BSI) game [13] if there exist functions wij : Ai × Aj → R and si : Ai → R such that ∑ ui (a) = wij (ai , aj ) − si (ai ), (3) j∈N \{i}

where wij (ai , aj ) = wji (aj , ai ) for every (ai , aj ) ∈ Ai × Aj . Theorem 1 ( [13]). A BSI game with payoff function (2) is a potential game with the potential function ∑ 1∑ ∑ wij (ai , aj ) − si (ai ). (4) V (a) = 2 i∈N j∈N \{i}

i∈N

B. Suppressing Transmission Power Transmission power control is performed before channel allocation. The following scheme decreases g(p, c) by suppressing transmission power level of each AP as long as the coverage hole does not occur. We assume that a set of configurable transmission power levels Pi consists of finite #{P } elements; Pi1 , Pi2 , . . . , Pi i , which satisfies Pi1 < Pi2 < #{Pi } . . . < Pi = pi,max . #{·} represents the cardinality. We assume that more than two APs do not simultaneously change their transmission power levels and one AP i ∈ N reduces the transmission power from Pik to Pik−1 (k ≥ 2) if ∪ W ∩ Si (Pik ) \ Si (Pik−1 ) ⊂ Sj (pj ). (5) j∈N \{i}

Recall that W denotes the region that we want to cover. Lefthand side of (4) represents a subset of W not covered by AP i when pi = Pik−1 but covered when pi = Pik , while right-hand side of (4) represents a region covered by all APs except AP i. This inclusion relation means the other APs N \ {i} are able to make up the deficit of transmission power mitigation of AP i, so that the proposed scheme keeps the condition of H(p) = ∅. As each AP i ∈ N monotonously decreases the transmission power level pi and Pi is finite, the dynamics of TPC will converge. 2 2 This TPC scheme can be formulated as a strategic form game (N , (Pi )i∈N , (u′i )i∈N ) with u′i (p) := K 1(H(p) = ∅) − pi , where K > maxi∈N {pi,max }, and this game is a potential game under the condition of p = (pi,max )i∈N in the initial status. Note ∑ that the potential function of this game is V ′ (p) = 1(H(p) = ∅) − i∈N pi . A similar approach is found in [14].

TABLE I: Parameters in simulations.

C. Radio Model for Channel Assignment

j∈N \{i}

Number of APs #{N } Maximum cell radius ri,max Path loss exponent α Transmission power levels pi Number of channels #{Ci }

2, 4, 8, 12, 20, 32, 48, 64 50 m 3 0–10 dBm (Integer values) 3

10000 (m2)

We model the DCA problem as a strategic form game (1) G(1) := (N , (Ci )i∈N , (ui )i∈N ). The elements are as follows. The set of players N is the set of APs. The strategy set for each AP i ∈ N , Ci , is the set of available channels. We let bi (p, c) be the payoff function to be maximized, i.e., −O ∑ (1) ui (c) = − |Si (pi ) ∩ Sj (pj )| 1ci =cj . (6)

8000

Regarding game G(1) , we obtain the following theorem. Proof. Let wij (ci , cj ) = −|Si (pi )∩Sj (pj )| 1ci =cj . As we find wij (ci , cj ) = wj (cj , ci ), G(1) is a BSI game and thus G(1) is a potential game by Theorem 1.

Total overlap area:

6000

Theorem 2. The strategic form game G(1) is a potential game.

4000

2000

0

The potential function of game G(1) is given by 1∑ ∑ V (1) (c) = − |Si (pi ) ∩ Sj (pj )| 1ci =cj . 2

0

(7)

We find that V (1) (c) is equal to minus the objective function gb(p, c) in the optimization problem (??). This means that, if we increase V (1) (c), then gb(p, c) accordingly decreases. Finally, we describe how each AP i ∈ N selects channel ci . We define the best response BRi : C−i → 2Ci of player i ∈ N to strategy profile c−i ∈ C−i by ci

(8)

(1)

In G , best response dynamics, which refer that only one player i ∈ N acts at each time and player i ∈ N chooses an element of BRi (c−i ) as the strategy, are guaranteed to converge to a Nash equilibrium by the property of potential games. IV. N UMERICAL E VALUATION In this section, we describe how APs calculate their payoff functions and reveal the following two points through simulations: how g(p, c) behaves when V (1) (c) increases; the performance of the proposed joint scheme of TPC and DCA compared to the scheme introduced in Section III-B. A. Estimation of the Overlap Area In channel assignment dynamics, each AP calculates its (1) payoff function ui (c). Here we present how to estimate |Si (pi )∩Sj (pj )|, the area of overlap between two cells of APs i and j. We use the path loss model with path loss exponent α. We assume that the path loss does not depend on frequency. For each AP i ∈ N , let ri (pi ) (or simply ri ) denote a radius of cell Si (pi ) and let ri,max denote a radius of cell Si (pi,max ). By considering the relation between path loss and received signal power, we obtain the following equation: pi T /Gij pi,max = α = , α ri,max ri dα ij

40 60 Time step

80

100

0 5 10 15 20 25 30 35 40 45 Time step

(a) TPC dynamics.

i∈N j∈N \{i}

BRi (c−i ) := { ci ∈ Ci | arg max ui (ci , c−i ) }.

20

(9)

(b) DCA dynamics.

Fig. 1: Transition of total overlap area for 20 pattens of random distribution (#{N } = 12). where dij is the distance between APs i and j. With (8), managed AP i can estimate ri and dij assuming the knowledge of ri,max , pj , and Gij pj . Subsequently, |Si (pi ) ∩ Sj (pj )| is estimated, for example under the condition of |ri − rj | < dij < ri + rj , by |Si (pi ) ∩ Sj (pj )| = ri2 θij + rj2 θji − dij ri sin θij ,

(10)

where θij and θji are interior angles of a triangle formed by ri , rj , and dij , i.e., θij is the angle between ri and dij , while θij between rj and dij . B. Comparison Scheme As a comparison with the proposed method, we use DCA scheme for interference management proposed in [15]. We utilize the same TPC scheme as described in Section II. The compared method is based on strategic form game and employs the following payoff function: ∑ (2) ui (c) = −pi Gij pj 1ci =cj . (11) j∈N \{i} (2) Game G(2) := (N , (Ci )i∈N , (ui )i∈N ) has been shown to be a potential game with potential function as follows: 1∑ ∑ V (2) (c) = − pi Gij pj 1ci =cj . (12) 2 i∈N j∈N \{i}

Further discussion of this type of game can be found in [8], [16]–[20]. C. Simulation Model and Results The performance of the proposed scheme is evaluated through simulations. We assume that W is the square region of dimension 100 m × 100 m and that each AP i ∈ N is uniformly and randomly deployed in W. We consider 20

(m2)

AP9

AP1

AP4

30000 25000

AP8 AP5

11.25 m

Average total overlap area:

20000 15000

5000

AP6

Compared scheme Proposed scheme

0 0

10

20

30

40

50

60

18.75 m

Compared scheme Proposed scheme

TABLE II: Parameters in experiments. Configurable transmission power level Pi , ∀i Available channels Ci , ∀i Initial channel Transmission power of STAs Path loss exponent α Required RSSI T

1500 Average total overlap area:

：STA

Fig. 3: Geographic distribution of APs and STAs.

1000

500

0, 3, 6, 9 dBm 1, 6, 11 1 10 dBm 3 −75 dBm

0 0

10

20

30

40

50

60

Number of APs:

(b) w/ TPC.

Fig. 2: Average total overlap area vs. number of APs.

patterns of random distribution. We assume that each AP has knowledge on the region W. We assume that all APs have the same configurable transmission power levels, the same available channels, and the same initial channel. The simulation parameters are summarized in Table I. In addition, to evaluate the total overlap area in W, we introduce the following evaluation function: ∑ ∪ gW (p, c) = Oi (p, c) . (13) W ∩ f ∈F i∈N :ci =f This function derives from the objective function in (??) and represents the total overlap area limited in W. Fig. 1 shows the mitigation of the total overlap area gW (p, c) for 20 deployment patterns when #{N } is 12. Both TPC and DCA mitigate the total overlap area gW (p, c). In particular, note that DCA scheme with TPC indirectly decreases the total overlap area gW (p, c) although this scheme actually decreases gb(p, c). Fig. 2 shows the impact of the number of APs on the decrease of the average total overlap area gW (p, c) in the comparison of two methods. Without TPC, i.e., all APs use the maximum transmission power level, 10 dBm, there is no difference between the proposed scheme and the compared scheme (Fig. 2a). By contrast, with TPC, the proposed scheme more effectively decreases the average total overlap area gW (p, c) than the compared scheme, especially when the number of APs #{N } is large (Fig. 2b).

V. E XPERIMENTAL S ETUPS AND R ESULTS To verify the convergence of the proposed scheme in a real environment, we built a testbed. The testbed consists of 10 APs and 14 STAs distributed in a rectangular region of dimension 11.25 m × 18.75 m as shown in Fig. 3. In this testbed, we use a set of points where STAs exist in place of W. Managed APs and STAs are implemented by Linux machines (Raspberry Pi 1 model B/B+) with a WLAN adapter (Buffalo WLI-UCGNM2). Each AP i ∈ N is based on the IEEE 802.11g standard and has knowledge of every other neighboring AP N \ {i}; channels, transmission power levels, and received power level at the point where AP i ∈ N exists. The key parameters are shown in Table II. 9 Transmission power level (dBm)

(m2)

(a) w/o TPC.

AP10

AP2

Number of APs:

2000

AP3

AP7

10000

6 AP 1 AP 2 AP 3 AP 4 AP 5 AP 6 AP 7 AP 8 AP 9 AP 10

3

0 0

5

10

15

20

25

Time index

Fig. 4: Transition of transmission power levels for every AP i ∈ N. Fig. 4 shows the transition of transmission power level pi for every AP i ∈ N and Fig. 5 shows the transition of channel ci for every AP i ∈ N . These figures show the convergences of the proposed dynamics of both TPC and DCA, and we confirm that the proposed scheme performs successfully even when in a real environment.

Channel

11

6

1 0

5 AP 1 AP 2 AP 3

10

15 AP 4 AP 5 AP 6

20 25 Time index

30

AP 7 AP 8 AP 9

35

40

45

AP 10

Fig. 5: Channel updates for every AP i ∈ N . VI. C ONCLUSION In this paper, we have proposed the joint method of TPC and DCA reducing the overlap area for dense WLANs. First, each AP suppresses the transmission power in TPC phase, then each AP selects a channel in which the AP has less overlap area with neighboring APs in DCA phase. These dynamics always converge, in particular, unilateral improve dynamics in the proposed DCA scheme are guaranteed to converge because it is shown to be a potential game. Simulation results have revealed three points: (1) in the proposed scheme, both TPC and DCA schemes mitigate the total overlap area although they do not directly reduce the total overlap area; (2) when combined with the proposed TPC scheme, the proposed DCA scheme has less interference, in terms of the overlap area, than the compared scheme based on the linear combination of received signal power; and (3) when the number of APs is large, the proposed scheme has much less interference than the compared scheme. Experiment has demonstrated the convergence of the proposed scheme in a real environment. ACKNOWLEDGMENT This work was supported in part by JSPS KAKENHI Grant Number 24360149. R EFERENCES [1] M. Di Renzo, A. Guidotti, and G. E. Corazza, “Average rate of downlink heterogeneous cellular networks over generalized fading channels: a stochastic geometry approach,” IEEE Trans. Commun., vol. 61, no. 7, pp. 3050–3071, May 2013.

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