Centralized-distributed Spectrum Access for Small

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Centralized-distributed Spectrum Access for Small Cell Networks: A Cloud-based Game Solution

arXiv:1502.06670v1 [cs.IT] 24 Feb 2015

Yuhua Xu, Member, IEEE, Jinlong Wang Senior Member, IEEE Yitao Xu, Liang Shen, Qihui Wu, Senior Member, IEEE and Alagan Anpalagan, Senior Member, IEEE

Abstract—By employing the advances in wireless cloud technologies, this letter establishes a centralized-distributed optimization architecture for spectrum access in small cell networks. In the centralized aspect, the spectrum access problem is solved in the cloud. In the distributed aspect, the problem is solved by a game-theoretic solution, aiming to address the challenges of heavy computation complexity caused by the densely deployed small cells. The properties of the game, including the existence of pure Nash equilibrium and the achievable lower bound, are rigorously proved and analyzed. Furthermore, a cloudassisted best response learning algorithm with low computational complexity is proposed, which converges to the global or local optima of the aggregate interference level of the small cell network. It is shown that the cloud-assisted algorithm converges rapidly and is scalable when increasing the number of small cells. Finally, simulation results are presented to validate the proposed cloud-based centralized-distributed solution. Index Terms—5G networks, small cell networks, wireless cloud, centralized-distributed spectrum access, potential game.

I. I NTRODUCTION MALL cell is an enabling technology for 5G networks, since it has been regarded as the most promising approach for providing a thousand-fold mobile traffic over the next decade [1], [2]. Technically, the use of very dense and lowpower small cells exploits two fundamental effects [3]: i) the distance between the small cell base station and the user is reduced, leading to higher transmission rates, and ii) the spectrum is more efficiently exploited due to the improved spectrum reuse ratio across multiple cells. However, as the network becomes denser, mutual interference among the cells becomes more serious [4], and hence it is timely important to develop efficient spectrum access approaches for the small cell networks to alleviate mutual interference. Recently, cloud radio access network (C-RAN) has drawn great attention as it can address a number of new challenges when trying to support growing end-user’s needs. The core idea behind C-RAN is to shift the baseband units (BBUs) from separate base stations to a centralized BBU pool for statistical multiplexing gain [5]. Some efforts, e.g., coding design [6]–[8] and performance analysis [9]–[11], have been made for such an innovative framework. Also, cloud technologies are now

S

This work was supported by the National Science Foundation of China under Grant No. 61401508 and No. 61172062. Yuhua Xu, Jinglong Wang, Yitao Xu, Liang Shen and Qihui Wu are with the Institute of Communications Engineering, PLA University of Science and Technology, Nanjing, 21007, China. (e-mail: [email protected]) A. Anpalagan is with the Department of Electrical and Computer Engineering, Ryerson University, Toronto, Canada (e-mail: [email protected]).

developed and strengthened for wireless virtualization [12] beyond the BBU pool, and some preliminary considerations can be found in the literature, e.g., decision making with delayed channel state information [13], MAC coordination for ambient living communications [14], radio resource management for heterogeneous networks [15] and cloud-based heterogeneous network selection [16]. In this work, we study cloud-based spectrum access for small cell networks. To the best of our knowledge, such an investigation has not been addressed in previous work. The key task of spectrum access is to determine the access channels for the small cells, which is a combinatorial problem. In the framework of could technologies, centralized optimization approaches can be applied. However, as the small cells are to be densely deployed, this optimization problem becomes extremely complicated. For example, consider a network with 20 small cells and five channels. In the simplest scenario in which each small cell choosing only one channel for transmission, the number of all possible channel selection profiles is 520 ≈ 9.53 × 1013, for which it is hard to achieve the optimal solutions using conventional optimization approaches. To cope with this challenge, we resort to game model [17], which is a powerful optimization tool involving multiple interactive decision-makers. To summarize, the contributions of this letter are: 1) We formulate the spectrum access problem for the small cells as a non-cooperative game in the cloud. It is proved that it is an exact potential game with the aggregate interference level as the potential function; furthermore, the Nash equilibrium (NE) of the game corresponds to the global or local optima of the spectrum access problem. Also, a lower bound of the achieved aggregate interference level is rigorously derived. 2) We propose a cloud-assisted best response algorithm to converge towards NE of the game. Compared with the conventional best response algorithm, the cloud-assisted algorithm converges rapidly and is scalable when the number of small cells becomes large. 3) By employing the great potentials of cloud, the centralized-distributed optimization architecture admits centralized and distributed decisions simultaneously. In the centralized aspect, the is solved in the cloud; in the distributed aspect, the methodology used in the could is based on game model, in which the cognitive agents behave as distributed decision-makers. Such an architecture provides new insights for future wireless

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networks such as 5G networks, and can be applied to other resource optimization problems. Related work: There are some centralized spectrum access approaches for small cell networks, e.g., the graph coloring approach [18]. Also, there are some distributed solutions found in the literature, e.g., sensing-based distributed approach [19], utility-based learning approach [20], reinforcement-learning based self-organizing scheme [21], coalitional game based scheme [22], evolutionary game based scheme [23] and hierarchical dynamic game approach [24]. In comparison, our work is differentiated in that a centralized-distributed optimization architecture is established, and a cloud-assisted game-theoretic distributed solution is proposed. The rest of this letter is organized as follows. In Section II, the system model and problem formulation are presented. In Section III, the centralized-distributed optimization architecture is established, the game model is formulated and analyzed, and the cloud-assisted best response algorithm is proposed. Finally, simulation results and discussions are presented in Section IV and conclusion is drawn in Section V. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION We consider a small cell network consisting of N small cell access points (SAPs). Each SAP serves several mobile users (MUs). There are M orthogonal channels available for the SAPs. For presentation, denote the SAP set as N , i.e., N = {1, . . . , N }, and the available channel set as M, i.e., M = {1, . . . , M }. It is shown that as the small cells become denser in 5G networks, the more spatial load fluctuation is observed by each SAP [3]. To capture such a fluctuation, it is assumed that each SAP chooses Kn channels for data transmission of the MUs. The number Kn can be regarded as the load of each SAP, which is jointly determined by the number of active MUs and their traffic demands1. Similar to previous work [20], [25], [26], we focus on the spectrum access problem and do not consider the problem of optimizing the required number of channels of each SAP. In practice, some simple but efficient approaches, e.g., the one proposed in [27], can be applied to estimate the cell load. Due to the spatial distribution and lower transmission power of SAPs, the transmission of a small cell only directly affects the neighboring small cells. To characterize the interference relationship among the small cells, the following interference graph is introduced. Specifically, if the distance dij between SAP i and j is lower than a threshold d0 , then they interfere with each other when transmitting on the same channel. Therefore, the potential interference relationship can be captured by an interference graph G = {V, E}, where V is the vertex set (the SAP set) and E is the edge set, i.e., V = {1, . . . , N } and E = {(i, j)|i ∈ N , j ∈ N , dij < d0 }. For presentation, denote the neighboring SAP set of SAP n as Jn , i.e., Jn = {j ∈ N : dnj < d0 }. If two or more neighboring small cells choose the same channel, mutual interference may occur. Thus, in order to 1 Furthermore, since the users in the small cells are always random and dynamic, it is not reasonable to allocate spectrum resources based on the instantaneous network state; instead, it is preferable to allocate spectrum resources according to their loads in a relatively longer decision period.

Small cell 4

Small cell 1

Small cell 3

Small cell 2

Fig. 1. An illustration for the considered interference model, in which different colors represent different channels. To reduce the interference in the network, i) for intra-cell spectrum access, it is mandatory to allocate different channels for the users in the same cell, ii) for inter-cell spectrum access, the number of overlapping channels should be minimized. According to (1), the interference levels of the cells are s1 = 1, s2 = 3, s3 = 2, s4 = 0.

mitigate interference among the small cells, it is desirable to allocate non-overlapping channels for them as soon as possible. Denote the choice of channels by SAP n as an = {c1 , c2 , . . . , cKn }, ci ∈ M, ∀1 ≤ i ≤ Kn . Note that an is a Kn -combination of M and the number of all possible chosen Kn n +1) . channel profiles of player n is CM = M(M−1)...(M−K Kn (Kn −1)...1 Motivated by the graph coloring for spectrum allocation problems [18], we define the experienced interference level as following: sn =

X X X

δ(e, f ),

(1)

j∈Jn e∈an f ∈aj

where δ(e, f ) is the following indicator function:  1, e = f δ(e, f ) = 0, e 6= f.

(2)

Note that sn is the number of channels also chosen by the neighboring SAPs. For an individual SAP n, the interference level sn should be minimized. From a network-centric perspective, the aggregate interference level of all the SAPs, i.e., P s n∈N n should be minimized. The considered interference model is illustratively depicted in Fig. 1. Thus, we formulate the problem of load-aware spectrum allocation for cognitive small cell networks as follows: P1 :

min

X

sn .

(3)

n∈N

Remark 1: The definition of the experienced interference level is different from that of traditional PHY-layer interference. Here, the interference level is used to characterize the mutual influence among neighboring SAPs from a higherlevel view. Such an interference model has also been applied for single channel selection in opportunistic spectrum access networks [28]–[30]. In comparison, this work extends previous single channel selection to load-aware multiple channel selection. With the allocated channels, the small cell can perform power control to further reduce the mutual interference among different cells. However, this problem is beyond the scope of this letter.

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Virtual Distributed Decision Network in the Cloud

function of player n. Due to the limited interference range, the utility function can be expressed as un (an , aJn ), where an is the action of player n and aJn is the action profile of the neighboring players of n. Thus, the formulated spectrum access game belongs to graph game. As discussed before, each small cell prefers a lower interference level, which motivates us to define the utility function as follows: un (an , aJn ) = −sn ,

Disseminate decision results obtained by the game-theoretic distributed solution

Report individual information including location, load and preference

(4)

where sn is the experienced interference level of player n, as characterized by (1). The players in the game are selfish and rational to maximize their individual utilities, i.e., (F ) :

Physical Small Cell Network

Interference link among small cells Cognitive agent in the cloud

Fig. 2. The proposed centralized-distributed optimization architecture for small cell networks.

III. G AME M ODEL AND C LOUD - ASSISTED B EST R ESPONSE A LGORITHM Motivated by the advances in wireless cloud technologies for 5G networks [3], [13], [15], we establish a centralizeddistributed optimization architecture for small cell networks, which is shown in Fig. 2. In a system-level view, the physical small cell network is mapped into a virtual decision network in the cloud, in which the decision-makers are cognitive agents. Specifically, each cognitive agent corresponds to a small cell, which reports its individual information, e.g., location, load, and preference, to the cloud. The virtual decision network utilizes some optimization approaches to obtain the decision results. Finally, the cloud disseminates the decision results to the small cells. A. Non-cooperative Spectrum Access Game in the Cloud With the formed virtual decision network and information reported by the small cells, centralized optimization approaches can be applied. However, problem P1 is an integer optimization problem and NP-hard, which can not be efficiently solved by conventional centralized approaches when the number of small cells is large. To cope with this challenge, we resort to game model [17], which is a powerful tool for optimization problems involving multiple interactive players. In the following, we formulate P1 as a non-cooperative game in the cloud and propose a cloud-assisted distributed learning algorithm to solve it. Formally, the spectrum access game is denoted as F = [N , G, {An }n∈N , {un }n∈N ], where N = {1, . . . , N } is a set of players (small cells), G is the potential interference graph among the players, An = {1, . . . , M } is a set of the available actions (channels) for each player n, and un is the utility

max un (an , aJn ), ∀n ∈ N .

an ∈An

(5)

To analyze the properties of the formulated spectrum access game, we first present the following definitions. Definition 1 (Nash equilibrium [31]). An action profile a∗ = (a∗1 , . . . , a∗N ) is a pure strategy Nash equilibrium (NE) if and only if no player can improve its utility by deviating unilaterally, i.e., un (a∗n , a∗Jn ) ≥ un (an , a∗Jn ), ∀n ∈ N , ∀an ∈ An , an 6= a∗n (6) Definition 2 (Exact potential game [31]). A game is an exact potential game (EPG) if there exists an ordinal potential function φ : A1 × · · · × AN → R such that for all n ∈ N , all an ∈ An , and a′n ∈ An , the following holds: un (an , aJ −n ) − un (a′n , aJn ) = φ(an , aJn ) − φ(a′n , aJn ) (7) In other words, the change in the utility function caused by the unilateral action change of an arbitrary player is exactly the same with that in the potential function. It is known that EPG admits the following two promising features: (i) every EPG has at least one pure strategy NE, and (ii) an action profile that maximizes the potential function is also a NE. The properties of the proposed spectrum access game are characterized by the following theorems. Theorem 1. The formulated spectrum access game F is an EPG, which has at least one pure strategy Nash equilibrium. In addition, the global optima of problem P 1 are pure strategy Nash equilibria of F . Proof: To prove this theorem, we first construct the following potential function: 1 X sn (a1 , . . . , aN ), (8) Φ(an , a−n ) = − 2 n∈N

where sn is characterized by (1). Recalling that the chosen channels of player n is denoted as an = {c1 , c2 , . . . , cKn }, define In (ci , aJn ) as the set of neighboring players choosing a channel ci , 1 ≤ i ≤ Kn , i.e., In (ci , aJn ) = {j ∈ Jn : ci ∈ aj },

(9)

where Jn is the neighbor set of player n. Then, we denote sn (ci , aJn ) = |In (ci , aJn )|

(10)

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as the experienced interference level on channel ci , where |A| is the cardinality of set A, i.e., the number of elements in |A|. Accordingly, the aggregate experienced interference level of player n is also given by: X (11) sn (e, aJn ) sn (an , aJn ) =

which satisfies the definition of EPG, as characterized by (7). Thus, the formulated spectrum access game F is an EPG, which has at least one pure strategy Nash equilibrium. Furthermore, according to the relationship between the potential function and the network-centric optimization objective, Theorem 1 is proved.

Now, suppose that an arbitrary player n unilaterally changes its channel selection from an = {c1 , c2 , . . . , cKn } to a∗n = {c∗1 , c∗2 , . . . , c∗Kn }. For presentation, we classify the channels into the following three sets: ∗ • C0 = an ∩ an . That is, the channels in set C0 are chosen by player n both before and after its unilateral action change. Note that C0 may be a null set. ∗ • C1 = an \{an ∩an }, where A\B means that B is excluded from A. That is, the channels in C1 are only chosen by player n before its unilateral action change. ∗ ∗ • C2 = an \{an ∩ an }. That is, the channels in C2 are only chosen by player n after its unilateral action change. From the above classification, the change in utility function of player n caused by its action unilateral action change is given by: X X un (a∗n , aJn )−un (an , aJn ) = sn (e, aJn )− sn (e, aJn )

Theorem 2. For any network topology, the aggregate interference level P of all the P players at any NE point is bounded by

e∈an

e∈C1

e∈C2

(12) Also, the change in the potential function caused by the unilateral change of player n is as follows: Φ(a∗nn, a−n ) − Φ(an , a−n ) =

1 2

un (a∗n , aJn ) − un (an , aJn ) P  uk (ak , a∗Jk ) − uk (ak , aJk ) + k∈D P1  uk (ak , a∗Jk ) − uk (ak , aJk ) + k∈D2  o P + uk (ak , a∗Jk ) − uk (ak , aJk ) , ∪ In (e, aJn ), D2 =

e∈C1

(13)

(14) (15)

un (ak , a∗Jk ) − un (ak , aJk ) = 0, ∀k ∈ D3 , k 6= n

(16)

Based on (14) and (15), we have X  X uk (ak , a∗Jk ) − uk (ak , aJk ) = |D1 | = sn (e, aJn ) k∈D2

.

an ∈ An , a ¯n 6= a∗n , an , a∗Jn ), ∀¯ un (a∗n , a∗Jn ) ≥ un (¯

a ¯ n ∈An

Kn where CM is the number of Kn -combinations of the channel set A n (Note that |An | = M ). It is seen that P u (¯ a , a∗Jn ) represents the aggregate experienced n n a ¯n ∈An interference level of player n as if it would access all possible channel profiles simultaneously while the neighboring users still only transmit on their chosen channels. As a result, it can be calculated as follows: X X Kn −1 un (¯ an , a∗Jn ) = −CM−1 Kj , (22) j∈Jn

where |Jn | is the number of neighboring users of user n. Thus, equation (21) can be re-written as: Kn −1 X CM−1 Kn CM

Kj = −

j∈Jn

e∈C2

(18) Now, combining (12), (16), (17) and (18) yields the following equation: Φ(a∗n , a−n ) − Φ(an , a−n ) = un (a∗n , a−n ) − un (an , a−n ), (19)

1 X Kn Kj , M j∈Jn (23)

Finally, it follows that: U (aN E ) =

X

n∈N

un (a∗n , a∗Jn ) ≥ −

P P

n∈N j∈Jn

M

Kn Kj (24)

which proves Theorem 2. Theorem 2 characterizes the achievable interference bound of the formulated spectrum access game. Some further discussions are given below: •

e∈C1

(17) X uk (ak , a∗Jk )−uk (ak , aJk ) = −|D2 | = − sn (e, aJn )

(20)

which is obtained according to the definition given in (6). Based on (20), it follows that: X Kn (21) un (¯ an , a∗Jn ), CM × un (a∗n , a∗Jn ) ≥

e∈C2

− un (ak , aJk ) = −1, ∀k ∈ D2

X 

Kn Kj

Proof: For any pure strategy NE aNE = (a∗1 , . . . , a∗N ), the following inequality holds for each player n, ∀n ∈ N :

∪ In (e, aJn ), D3 =

un (ak , a∗Jk ) − un (ak , aJk ) = 1, ∀k ∈ D1

k∈D1

j∈Jn

M

un (a∗n , a∗Jn ) ≥ −

N \{D1 ∪D2 }, and uk (ak , a∗Jk ) is the utility function of player k after n’s unilateral action change. Note that player n belongs to the neighboring player set of player k, i.e., n ∈ Jk . Since the action of player n only affects its neighboring players, the following equations hold: un (ak , a∗Jk )

n∈N

a ¯n ∈An

k∈D3 ,k6=n

where D1 =

U (aNE ) ≥ −

•

If all the players choose only one channel for transmission, i.e., Kn = 1, ∀n ∈ N , we have U (aN E ) ≥ P |Jn | n∈N − . M When the number of available channels increases, the bounded aggregate interference level decreases. In particular, if the number of channels becomes sufficiently large, i.e., M → ∞, we have U (aN E ) → 0. In this case, the spectrum resources are abundant and mutual interference among the players are completely eliminated. Also, when the network becomes sparse, i.e., decreasing |Jn |, the bounded aggregate interference level also decreases.

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B. Cloud-assisted best response learning algorithm

Algorithm 1: Cloud-assisted best response algorithm

Initialization: Set the iteration index i = 0, let each player n, ∀n ∈ N , randomly select an action ak (0) ∈ Ak . 1). Find the updating user set: Set B = ∅ Loop for i = 1, . . . , With the assistance of the cloud, find the best action selection for each player n : a(b) n (i − 1) = arg max un (an , aJn (i − 1)) an ∈An

(25)

(b)

if an (i − 1) 6= an (i − 1), include n in B, i.e., B = B ∪ n. End loop If B = ∅ The learning procedure terminates. End if 2). Action update: Randomly select a user k form B. Let user k update its action according to the best response rule, i.e., (b)

ak (i) = ak (i − 1),

(26)

and all other users keep their action unchanged, i.e., an (i) = an (i − 1), ∀n 6= k. 3). Go to step 1).

Theorem 3. The proposed cloud-assisted best response learning algorithm converges to a pure strategy NE point of the formulated spectrum access game F in finite steps. Therefore, the aggregate interference level of the small cell networks is globally or locally minimized. Proof: From the learning procedure, it is seen that the updating user in each iteration makes its utility function increasing. Accordingly, the potential function of the game, as specified by (8), is increasing. Since the potential function

0.9 Cumulative distribution function (CDF)

As the distributed spectrum access problem now formulated as an exact potential game, the best response (BR) algorithm [31] can be applied to achieve Nash equilibria of the game. However, there is a limitation of conventional BR algorithm: one player is randomly selected to update its action in each iteration. Such a random scheduling mechanism is desirable for distributed systems; however, the convergence speed is very slow when the network becomes dense. To overcome this problem, we exploit the benefits of centralized-scheduling in the cloud, and propose a cloud-assisted best response learning algorithm. In the conventional BR, the randomly selected user may not update its action; in comparison, the randomly selected user in the cloud-assisted BR updates its action in each iteration. Thus, it can be expected that the convergence would speed up. Formally, the could-assisted best response learning algorithm is described in Algorithm 1.

0.8 N=20, Conventional BR 0.7 N=30, Conventional BR

0.6 0.5

N=20, Could−assisted BR 0.4 N=30, Could−assisted BR

0.3 0.2 0.1 0

0

50

100 150 200 250 300 Iterations needed for converging to NE

350

400

Fig. 3. The convergence speed comparison between the conventional BR and the cloud-assisted BR.

is up bounded (the maximum value is zero), the learning algorithm will finally converge to a global or local maximum point of the potential function in finite steps. Thus, Theorem 3 is proved. IV. S IMULATION R ESULTS AND D ISCUSSION We consider a small cell network deployed in a square region. When there are 20 small cells, the square region is 200m × 200m. When the number of small cells increases, the square region increases proportionally to keep the same density. The coverage distance of each small cell is 20m, and the interference distance is 60m. For presentation, the load of each cell is randomly chosen from a load set L = {1, 2, 3}. There are five channels available in the network. To begin with, we compare the convergence speed of the cloud-assisted BR and the conventional BR. The companion results of the cumulative distribution function (CDF) of the iterations needed for converging are shown in Fig. 3. The results are obtained by simulating five different network topologies and 1000 independent trials for each network topology. It is noted from the figure that for the same size network, e.g., N = 20 or N = 30, the iterations needed for converging of the cloud-assisted learning algorithm is significantly decreased. Furthermore, when the network scales up from N = 20 to N = 30, the convergence speed of the could-assisted BR is slightly decreased while that of the conventional BR is largely decreased. This implies that the proposed cloud-assisted algorithm is especially suitable for large-scale networks. Secondly, the aggregate interference level when varying the number of small cells is shown in Fig. 4. It is noted from the figure that as the network scale increases, the aggregate level increases, as can be expected. More importantly, it is noted that the performance of the proposed cloud-based best response algorithm is close to the optimum solution. Also, the game-based solution significantly outperforms the random selection strategy. V. C ONCLUSION In this letter, we established a centralized-distributed spectrum access approach for small cell networks. In the central-

6 −20

Aggregate interference level

−40 −60 −80 −100 −120 −140 −160 −180 20

Fig. 4. cells.

Best NE (Optimum) Proposed Could−assisted BR Worst NE Random selection 25

30 35 40 Number of small cells (N)

45

50

The aggregate interference level when varying the number of small

ized aspect, the spectrum access problem is solved in the cloud. In the distributed aspect, the problem is solved by a game-theoretic solution, aiming to address the challenges caused by the heavy computational complexity when the small cells are densely deployed. The properties of the game, including the existence of pure Nash equilibrium and the achievable lower bound, were rigorously proved and analyzed. In addition, a cloud-assisted best response learning algorithm with low computational complexity was proposed, which converges to the global or local optima of the aggregate interference level of the small cell network. Furthermore, it is shown that the cloud-assisted algorithm converges rapidly and is scalable with the increase in the number of small cells in 5G networks. Note that there are still some new challenges and open issues to be studied in the future. First, cheat-proof mechanisms are needed to prevent the users from reporting their loads and demands greedily and maliciously. Secondly, new game models and learning algorithms with imperfect information constraints, e.g., delayed information and uncertain information caused by the limited reporting link, should be developed to achieve robust spectrum access in the cloud.

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