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An Evolutionary Game Theoretic Framework for. Femtocell Radio Resource Management. Shangjing Lin, Student Member, IEEE, Wei Ni, Senior Member, IEEE,.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 11, NOVEMBER 2015

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An Evolutionary Game Theoretic Framework for Femtocell Radio Resource Management Shangjing Lin, Student Member, IEEE, Wei Ni, Senior Member, IEEE, Hui Tian, Member, IEEE, and Ren Ping Liu, Senior Member, IEEE

Abstract—Plug-and-play femtocells will be an integrating part of future cellular networks. Resource management and interference mitigation become challenging, suffering from severely delayed network control in large-scale deployments. We propose a new game theoretic framework, where fast interference suppression is decoupled from the relatively slow frequency allocation process to tolerate the delayed control. The key idea is to cast femtocell clustering as an outer-loop evolutionary game coupled with bankruptcy channel allocation, which drives the cells to spontaneously switch to less interfered clusters. Within each cluster, we design an inner-loop non-cooperative power control game, such that the requirement of prompt control is eliminated. The two loops interact recursively with analytically confirmed stability. Simulations show that our framework can improve the throughput by 13.2% in a network of 200 cells, compared to the prior art. The gain grows further with the network size. Index Terms—Femtocell, radio resource management, game theory, evolutionary game.

I. I NTRODUCTION

F

EMTOCELL is an emerging cellular technology, which is able to significantly improve coverage and throughput in indoor environments and hotspots. It complements and enhances existing macrocells by offloading mobile data traffic and saving radio/energy resources of macrocells [1]. Being an integrating part of future cellular networks, femtocells provide a new paradigm of network operation [2]. Particularly, plug-andplay femtocell base station (FBS) devices have been recently developed [3]. As a result, femtocells can be privately owned and randomly deployed (as opposed to the well-planned operators’ networks). Dedicated cabling may also be unavailable to connect the femtocells to the core network. Femtocell signals must go through public data networks to the core network,

Manuscript received November 3, 2014; revised February 28, 2015 and May 6, 2015; accepted June 29, 2015. Date of publication July 8, 2015; date of current version November 9, 2015. This work was supported by the National Natural Science Foundation of China under the Grant No.61471060, and the Foundation of China under the Grant No.61421061. The work was also partly supported by National Science and Technology Major Project of China under the Grant LTE-HI 2015ZX03001025-002. The associate editor coordinating the review of this paper and approving it for publication was Z. Han. S. Lin and H. Tian are with the State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China (e-mail: [email protected]; tianhui@ bupt.edu.cn). W. Ni and R. P. Liu are with Commonwealth Scientific and Industrial Research Organization (CSIRO), Sydney, NSW 2122, Australia (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TWC.2015.2453170

Fig. 1. An illustration on future femtocell deployment, where some femtocells are connected to the cellular S-GW via cellular infrastructure, as highlighted by the green path, while a larger number of femtocells would be connected to the S-GW through various ISP networks, as indicated by the red and the blue paths. The S-GWs disseminate control signalling to the femtocells via these colourcoded paths.

where critical cellular functionalities are coordinated, such as interference coordination and mobility control. Fig. 1 illustrates the architecture of future femtocell networks, where a number of femtocells are connected to the Servicing Gateways (S-GWs) within the core network through a variety of backhaul technologies, such as ADSL or Ethernet. The S-GWs are the anchors that provide interference coordination and mobility control to the femtocells. The backhaul connections can be provided by different Internet Service Providers (ISPs), resulting in substantial (signalling) delays to the cells. This makes it difficult for cellular operators to keep close and timely control on the cells. Nevertheless, close and timely control is necessary to suppress interference in existing cellular systems [4]. Specifically, power control operates at a fast pace of less than ten milliseconds to balance the transmit powers of interfering cells [4]. However, the backhaul delay of the femtocells can be as large as hundreds of milliseconds. As a result, excessive interference would be suffered in areas where femtocells are densely deployed while the frequencies licensed to the femtocells are limited. Radio resource management (RRM) becomes a challenge in femtocells, due to the severe backhaul delays. The network topology of femtocells is also likely to constantly change due to the plug-and-play devices, as well as the bursty nature of femtocell traffic [5]. Existing techniques that statically assign frequencies to cells (i.e., frequency-reuse

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techniques) or different parts of cells (i.e., fractional frequencyreuse techniques) become inadequate. Channel allocations adapting to varying traffic demands of femtocells are expected. This is also a challenge, given the heterogeneous femtocell architecture and the aforementioned severe delays over transport networks. Many existing works have tried to address these challenges of interference mitigation and channel allocation in a centralized manner. In [6], the femtocell RRM problem was formulated as linear programming, and solved at a central node with a lowcomplexity suboptimal method. In [7], network utility of admission control and resource allocation in a heterogeneous network was defined. A centralized linear programming problem was formulated to maximize the utility. In [8], a problem was formulated to jointly determine all co-channel users’ power spectral densities so as to maximize a system-wide utility function (e.g., weighted sum-rate of all users). The problem was rigorously proved to be (strongly) NP-hard, due to the non-convexity of the signal-to-interference-plus-noise (SINR) term [9]. Recently, the problem was extended to multiple channels in [9], where a user can switch between channels. Therefore, the NP-hard problem was further coupled with mixed integer non-linear programming. The solutions to the problem is centralised, and still requires the real-time global knowledge, including channel gains. In [10], users were proposed to associate with femtocells in a decentralised manner, where interference is mitigated by selecting users based on their quantized SINR reports. However, centralized optimization of SINR quantization levels is necessary to maximize the sum-rate of the system. More comprehensive RRM solutions were proposed in [5] and [11], where both the frequency and processing resources are allocated adapting to instantly changing traffic requirements of femtocells. Clique ideas were exploited, which require a centralized implementation. Unfortunately, all these works are unsuitable in general to large-scale, femtocell networks, where heterogeneous backhaul technologies will be involved and severe delays will be experienced for the signalling necessary to the centralised optimizations. Game theory has been extensively studied to enable distributed RRM. In [12], a Stackelberg game was formed between a number of macrocells and femtocells to suppress interference between the cells of the two types. The game requires the macrocells to calculate interference and send it back to each femtocell instantly. Significant overhead occurs. In [13], a twolevel Stackelberg game was formulated for distributed power control in cooperative communication networks. The source and the relays need to instantly and frequently exchange the price and the information about how much power to buy, whenever the power is adjusted. In [14], [15], evolutionary games were proposed to adjust the transmit powers of femtocells and mitigate interference. However, instant information exchange is required between the femtocells, which is hard to achieve in practice. In [16], a supermodular game was constructed for distributed power control. Unfortunately, the game does not consider the case of maximum power constraint, and therefore, it may converge to a point that is not feasible. In [17], [18], pricing was introduced to supermodular games of power control, to reduce the transmit powers at the Nash equilibria of the game.

In [17], to retain the supermodularity requires the feasibility region of the transmit power to reduce. As a consequence, the game is not an optimum power solution [17]. In [18], there can be multiple Nash equilibria, and the optimality cannot be verified. In [19], a non-cooperative power control game was formulated among uplink cellular users, where fairness, throughput, and their trade-off can be pursued by defining different utility functions. However, the game requires every user to know its uplink channel gain, the total received power plus noise at the base station, and the total number of active users. A large amount of overhead is required in a real-time manner. In this paper, we propose a new game theoretic framework, where fast interference suppression is decoupled from the relatively slow frequency allocation process to tolerate the delayed control. The key idea is to cast femtocell clustering as an outer-loop evolutionary game coupled with bankruptcy channel allocation, which drives the cells to spontaneously switch to less interfered clusters. Within each cluster, we design an inner-loop non-cooperative power control game, such that the requirement of prompt control is eliminated. The two loops interact recursively with analytically confirmed stability. Simulations show that our framework can improve the throughput by 13.2% in a network of 200 femtocells, compared to the prior art. The gain will grow further with the network size. Our proposed algorithm is a new, non-straightforward design of the three components, i.e., power control, channel selection, and clustering, to address the critical issue of delayed control in future pervasive wireless cellular networks. Our major technical contributions include the following. 1) We decouple the network control of femtocells into fast power control and slow topology/clustering control, accommodate the fast power control in the slow control framework, and specify the iterative interactions between the two parts. 2) We design an evolutionary game theoretic framework for slow topology control (clustering), where base stations spontaneously select or switch between clusters until interference within every cluster is balanced and the system is stabilised with proportionally fairly allocated bandwidths to the cells. 3) We design a distributed fast power control game, which enables each base station to adjust its transmit power adapting to interference and enables the bandwidths granted to the cells to converge in a proportionally fair manner. As the result of our new design, the system will be able to tolerate substantially delayed control of practical deployments, stabilise and enhance fairness in a decentralised manner. This is of practical value, and is crucial to cost-efficiently extend network scale and coverage. The rest of the paper is organized as follows. In Section II, the system model is presented. In Section III, the proposed evolutionary framework is elaborated on, followed by stability analysis in Section IV. In Section V, simulations are carried out and the effectiveness of the proposed game theoretic framework is demonstrated, followed by concluding remarks in Section VI.

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II. S YSTEM M ODEL In this section, the system model of future femtocell networks is presented, followed by the channel model. The network structure we consider is as shown in Fig. 1, where there are N active femtocells in an area of A m2 and each cell only serves one user for illustration convenience. N = req {1, · · · , N} collects the cell indices; Ri denote the bandwidth requirement of cell i. The N cells are managed by the S-GWs for mobility control and interference coordination. We assume the signalling exchange between the S-GWs is delay-free, since the S-GWs are interconnected within the cellular operator’s core network. Some of the N cells are connected to the S-GWs via cellular infrastructure (as highlighted by green link), whereas a larger number of cells are connected to the S-GWs through transport networks (e.g., edge networks) and ISP networks. In this sense, real-time centralized coordination is intractable, due to the delayed control that the cells undergo and void real-time operations. Let K denote the number of available channels in the network. K = {1, · · · , K} collects the channel indices. The physical bandwidth of each channel is denoted by ωc . Practical fading effects are considered, including path loss and Rayleigh fading. A cell user equipment (FUE) ˆi receives its desired signals from its serving femtocell base station (FBS) i and the interference from other FBSs j (j ∈ N , j = i). At a given channel, i.e., channel k, the channel gain from FBS j to FUE ˆi can be given by k Gkj,ˆi = Kj,ˆi d−α ˆ gj,ˆi , j,i

where Kj,ˆi is the path loss coefficient from FBS j to the FUE ˆi; d ˆ is the distance between FBS j and FUE ˆi; α is the path loss j,i

exponent; gk ˆ is the Rayleigh fading component from FBS j to j,i

FUE ˆi on channel k. As a result, the downlink SINR of FUE ˆi on channel k can be given by: γik = 

P i Gk ˆ i,i

k 2 j∈N \i Pj G ˆ + σ

(1)

j,i

where Pi is the transmit power (to be specific, power spectral density) of cell i, and σ 2 is the variance of the additive white Gaussian noise. III. P ROPOSED E VOLUTIONARY F RAMEWORK In this section, we propose a new game theoretic framework which overcomes the critical issue of delayed control over transport networks. Specifically, we decouple the RRM into a fast inner-loop power control and a progressive outerloop topology/clustering control. For the inner loop, the fast power control is decentralised by formulating a distributed noncooperative game within each cluster. The signalling required for power control is therefore eliminated; the fast power control is unaffected by the excessive delay over the transport networks. For the outer loop, channels are assigned to the clusters, and an evolutionary game is designed to incentivise reclustering

Fig. 2. Illustration on the proposed evolutionary game theoretic framework for distributed RRM, where a single evolution cycle of the framework is demonstrated, including distributed power control, channel allocation, and cluster evolution (from top to bottom).

towards balanced efficient use of the channels. We keep the topology/clustering control (along with channel allocation) under centralised coordination. This is due to the slow-varying nature of these aspects. The outer loop tolerates the excessive delay over transport networks. Fig. 2 illustrates the operations of the proposed framework, where the inner-loop power control and the outer-loop channel/ clustering control operate sequentially and repeat iteratively until the stabilisation of the network. Specifically, we sequentially carry out 1) distributed (iterative) power control, given the previous results on clustering and channel allocation, until the powers are stabilised; 2) the bankruptcy channel allocation at the S-GWs to the cells with the stabilised transmit powers; and 3) the evolutionary clustering where the S-GWs calculate the probabilities to steer the cells to spontaneously switch clusters. These operations repeat until the entire network is stabilised, i.e., no cells switch clusters. The operations can be initialized by using the a priori knowledge of the cells, such as the geographical information acquired when the base stations are purchased and registered to the operator. Graph theoretic methods, e.g., the one proposed in [20], can be used to get an initial off-line clustering result, {D1 , · · · , DM }, where Dm is the m-th cluster and M is the number of clusters. After that, Dm evolves spontaneously in the proposed framework, as shown in the figure. The cycle of the operations illustrated in Fig. 2 can be adjusted, adapting to the delays over transport networks. The cycle can be specified to cover the largest round-trip delay of all the cells, as shown in Fig. 3, i.e., t3 ≥ maxN i=1 (vi ). As a result, the problem of unbalanced control delay is addressed. The signalling required for our framework is low, only occurring in the outer-loop clustering control, as shown in Fig. 3. Also, the signalling is delay tolerant due to the slow-varying nature of the outer-loop operations (as mentioned earlier). Details will be provided later in Section III-D. Note that game theory has been widely applied to the channel allocation (as discussed earlier), where the game players (whom

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Fig. 3. An example of the time schedule of the proposed evolutionary game theoretic framework, where t1 and t3 are the beginning and the end of a cycle of an evolutionary stage, respectively. ui indicates the time when BS i receives the clustering control message from the S-GWs. t2 indicates the pre-agreed time when all the cells report their stabilised powers. vi indicates the time when the S-GWs receive the power report from cell i.

the channels are to be assigned to) are usually required to be rational (or in other words, they are able to adopt the optimal responses, given the information available to them). Unfortunately, such rationality is unachievable in a network with a large number of cells (e.g., hundreds). One reason is because the uncertainty of all other cells’ decisions/future behaviours is so high and there is no single optimal response that a cell shall adopt at any instant. Another reason is that the rationality results from precise real-time knowledge on all the other cells, while the knowledge may be unavailable in practical deployment due to large delays. Our proposed evolutionary framework does not require rationality at individual cells. The cells (i.e., player) can evolve separately, and their evolution processes can be adjusted with limited information in a decentralized manner. Therefore, our framework is of practical value.

Given a cluster Dm and its assigned channels φm , we formulate the distributed power control of the cells as a noncooperative game, where each cell i ∈ Dm recursively adjusts and adaptively decides its own transmit power Pi ∈ (0, Pmax ]. Pmax is the maximum transmit power of each FBS.   Let P−i = P1 , · · · , Pi−1 , Pi+1 , · · · , P|Dm | collect the transmit powers of the rest of the cells in Dm , and Ii,ˆi (P−i ) =  k j∈Dm \{i} Pj Gj,ˆi (i ∈ Dm , k ∈ φm ), where | · | denotes cardinality. The SINR of the user in cell i can be rewritten as Gk ˆ P i i,i

Ii,ˆi (P−i ) + σ 2

,

(2)

  and we let γ −i = γ1 , · · · , γi−1 , γi+1 , · · · , γ|Dm | collect the SINRs of the rest of the cells in Dm . We define a utility function Ui (Pi , P−i ) for cell i ∈ Dm as  req  (3) Ui (Pi , P−i ) = arctan αm γi /γi req

Ci (Pi , P−i ) = Pi ,

where γi is the SINR of the FUE in cell i; γi is the target SINR that is required to achieve the bandwidth requirement of cell i (given the current allocated channels to Dm ); αm is the steep coefficient that can be used to adjust the convergence speed of the proposed non-cooperative power control in Dm . The rationale of designing the utility function (3) is as follows. First, our use of the arctan(·) function to design the utility

(4)

which is used to prevent the proposed non-cooperative power control game converging to a fixed point (i.e., a Nash equilibrium) that requires unnecessarily high transmit power of every FBS and therefore is not efficient [17]. As a result of designing the utility and price functions of (3) and (4), the non-cooperative game of distributed power control has a unique Nash equilibrium, as will be discussed in Section IV. Now, the surplus function of the formulated non-cooperative power control game in Dm can be given by Si (θm , Pi , P−i ) = Ui (Pi , P−i ) − θm Ci (Pi , P−i ),

A. Non-Cooperative Power Control Game

γi (Pi , P−i ) =

is due to the monotonic increasing feature of the function, which incentivizes each base station to drive up the SINR as requested. Second, our use of arctan(·) is also due to the asymptotic convergence of the function to a constant (i.e., π2 ). This brings down the incentive for a base station to thrive for excessively high SINR, and prevents the base station from producing excessive interference to others. Third, the use of arctan(·) provides a concave structure to the power control, which enables the decentralisation of power control and ensures the convergence of the proposed power control game. Last but not least, our use of arctan(·) along with the linear penalty function ensures that the actual achievable SINRs are equally proportional to the requested SINR across the cells within a cluster. In dense deployment scenarios, this leads to an equal ratio of the achievable to the required spectral efficiency, which is critical to decide the proper bandwidth for the cells and to improve fairness among the cells. We also design a linear price function Ci (Pi , P−i ):

(5)

where θm is the price coefficient that is used to adjust the weight of the price for Dm . Note that (5) is formulated to leverage the throughput gained by each individual cell and the interference that the cell is to generate towards other cells. Particularly, the transmit power of a cell acts as the cost while the cell thrives for high throughput. Raising the transmit power, the cell can increase its throughput; meanwhile the cost rises and discourages the cell from raising the power. As a result, the transmit power will be stabilised at a balanced point, leveraging throughput and interference. This prevents a cell from transmitting excessively high power and releasing excessive interference. Also note that θm can be jointly designed with αm , so that the formulated non-cooperative game of distributed power control has a unique Nash equilibrium. Details will be provided in Section IV. The non-cooperative game of distributed power control essentially now becomes the following optimisation problem: max Si (θm , Pi , P−i ) s.t. 0 < Pi ≤ Pmax , ∀ i = 1, · · · , N. (6)  Note that Ii,ˆi (P−i ) = j∈Dm \{i} Pj Gj,ˆi can be replaced by the measurement results of the femtocell users, denoted by I˜i,ˆi. In other words, FBS i does not need to know explicitly P−i ; no inter-femtocell signalling is required at this power control

LIN et al.: AN EVOLUTIONARY GAME THEORETIC FRAMEWORK FOR FEMTOCELL RADIO RESOURCE MANAGEMENT

stage. P−i can be dropped from (6). As a result, γi =

Gk ˆ Pi i,i

. I˜i,ˆi +σ 2

Si (θm , Pi ) can be rewritten as a function of γi through variable substitution. To solve (6) turns to determine the optimum γi that maximizes Si under the power constraint. To solve this problem, the first-order partial derivative of Si with respect to γi needs to be equal to zero, i.e., ∂Si αm = − θi = 0.  req 2 ∂γi 1 + αm γi /γi

(7)

We note that the second-order partial derivative of Si with respect to γi is negative, i.e., ∂ 2 Si ∂γi2

=−

2αm γi Gk ˆ i,i

req I˜i,ˆi γi

  < 0.  req 2 2 1 + αm γi /γi

(8)

As a result, we can achieve the optimum of γi , i ∈ Dm , as given by req γi 1 − θm ∗ . (9) γi = αm θm Accordingly, we can obtain ⎛ ⎞ req ˜ γ I 1 − θ ˆ m i i,i P∗i = min ⎝ , Pmax ⎠ . θm αm Gk ˆ

(10)

i,i

For mathematical tractability, we assume here that P∗i  Pmax which is reasonable in densely deployed femtocell scenarios.  req γ I˜ ˆ m Therefore, P∗i = i ki,i 1−θ θm , and (9) is achievable. αm G

i,ˆi

In fact, Si in (5) and (6) can be generalized as a function of (γi , γ −i ), since γi (i ∈ Dm ) can be uniquely presented in terms of Pk (k ∈ Dm ) by solving a set of equations specified by (2). It is therefore reasonable for the proposed power control game to first optimize γi∗ followed by calculating P∗i , as described above. Our later analysis also confirms the convergence and the stability of the power control game. Using (10), each FBS i independently adjusts P∗i based on FUE ˆi’s measurement result of SINR, I˜i,ˆi , until P∗i stabilises. Then, the FBS reports its stabilized transmit power P∗i and its SINR γi∗ to the S-GWs. Note that, once the distributed power control is stabilised, req the ratios of the achievable SINR γi∗ to the required SINR γi  m converge to α1m 1−θ θm within each cluster Dm (m = 1, · · · , M), as shown in (9). This is key to our channel allocation design, as will be discussed in Section III-B, and is also crucial to the stability of the entire proposed framework, as will be discussed in Section IV. B. Bankruptcy Channel Allocation Game Our proposed channel allocation starts by specifying the number of channels required by a cluster, Dm , as given by   θm req rm ≈ φm αm (11) 1 − θm

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where · denotes the ceiling function. The continuous physical req bandwidth required by the cluster is rm ωc ; ωc is the physical bandwidth of a channel. The channels allocated to a cluster will be efficiently utilized by all cells of the cluster. This bandwidth requirement is reasonable in densely deployed femtocell scenarios, and can be obtained by evaluating req req γi and γi∗ . Particularly, γi (i ∈ Dm ) is not a fixed threshold. It is updated every cycle (as illustrated in Fig. 2) based on the current allocated bandwidth φm , and is equal to the SINR required to guarantee the basic quality of service given the bandwidth φm . In dense deployment scenarios, the number of cells per cluster, i.e., |Dm |, is large. The interference within a cluster Dm is high. To meet the basic quality requirement of the cells i ∈ Dm , the bandwidth to be allocated to the cells needs req to be enlarged, which in turn reduces the required SINR γi . req req req γi becomes small and log(1 + γi ) ≈ γi . For the same reason, we can approximate log(1 + γi∗ ) ≈ γi∗ . As a result,  req req log(1+γi ) γ θm ≈ γi ∗ = αm 1−θ for i ∈ Dm , as shown in (9). log(1+γ ∗ ) m i

i

φ ω log(1+γ

req

)

c i ≈ The bandwidth requirement of cell i ∈ Dm is m log(1+γ ∗ i )  θm φm ωc αm 1−θ , which is equal across the cluster Dm . The m number of channels required for the cells is as given by (11). req Through such iterative adjustments of φm and γi (i ∈ Dm ), our proposed evolutionary framework leverages the achievable SINRs and the available bandwidth, and converges to a balanced point. The densely deployed femtocells are the focus of this paper, because they are the case where strong inter-cell interference needs to be suppressed. req In the case of sparsely deployed femtocells, log(1 + γi ) ≈ req req ∗ ∗ ∗ log γi < γi and log(1 + γi ) ≈ log γi < γi . We can still use (11) to update the bandwidth requirement  of an entire req

cluster, because req γi

γi∗

log(1+γi ) log(1+γi∗ )



req

log γi log γi∗



req

γi γi∗

= αm

θm 1−θm

given

≥ with carefully designed αm and θm . In other words, (11) specifies the maximum requirement for the cluster. However, the actual bandwidth requirements of individual cells within the cluster can be unequal, as opposed to the case of densely deployed femtocells. The actual bandwidth requirements depend on the current SINRs of the individual cells,  log(1+γ

req

)

log γ

req

log(αm

θm 1−θ

)

m based on (9). As since log(1+γi ∗ ) ≈ log γi ∗ = 1 + log γi∗ i i a consequence, the satisfaction levels of the cells with respect to the bandwidth allocated to the cluster become unequal, if the allocated bandwidth is less than the one specified in (11) due to the limited available spectrum. On the other hand, the inter-cell interference is weak and can be independently suppressed at individual cells in the case of sparsely deployed femtocells. Graph theoretic approaches have been extensively studied to optimally cluster femtocells with negligible inter-cell interference in a static manner [21]. For this reason, we do not consider the case in this paper. We can formulate the channel allocation process as a cooperative bankruptcy game in the case where the available channels are insufficient to satisfy all the cells. The game is carried out at the S-GWs, since the S-GWs of the same operator are interconnected and can operate collaboratively. It provides a practical means to distribute insufficient channels among the clusters in a fair and decentralized fashion.

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In the bankruptcy game, we define that each cluster of cells, i.e., Dm (m = 1, · · · , M), is a player. We also define a coalition, denoted by S, which is a set of clusters. Players of the same coalition are collaborative, competing for channels with other players (or coalitions). Let = {D1 , · · · , DM }, and W = {S : ∀S ⊆ } denote the set of all possible coalitions of

. |W| = 2M . We also define the characteristic function, denoted by v(S), associated with each coalition S, which is the number of the channels unclaimed by the players outside S. v(S) can be mathematically formulated, as given by ⎛ ⎞  req ⎠ rm v(S) = max ⎝0, K − . (12)

cept is suitable for the fair allocation of discrete channels. The bankruptcy game can also be solved by using other classical methods, e.g., the Nucleolus [23], game quadratic programming [24], cost gap allocation [25], and minimum cost-remaining saving [26]. C. Evolutionary Clustering Adjustment The S-GWs proceed to review the current clustering based on P∗i (i ∈ Dm , m = 1, · · · , M), and steer the future evolution of the clustering in the next evolution cycle. We start by specifying the ratio of the cells that need to switch clusters, as given by

m∈ \S

This bankruptcy game can be solved using the Shapley value method [22] which can distribute the available channels among the clusters with fair compensations to the clusters. The Shapley value, φm (m = 1, · · · , M), which is the number of channels to be allocated to Dm , can be written as [22]  M − 1 1 φm = [v (S ∪ {Dm }) − v(S)] (13) |S| M S⊆ \{Dm }

where coalitions are formed in  respect of Dm . Given  each formed coalition, Dm demands v(S ∪ {Dm }) − v(S) number of channels as a compensation. The Shapley value takes the average of the demands over all possible coalitions which can be formed, as shown in (13). It is worth pointing out that the application of the bankruptcy concept to channel allocation is new, though the concept itself is not. The new application is enabled by our new design of distributed power control in Section III-A. Our new design of distributed power control is important to implement the bankruptcy concept computationally efficiently. Particularly, it provides fairness within each cluster by getting the ratios of the achievable and required SINRs of the cells in the cluster to converge. Channels allocated to a cluster can be efficiently utilized by every cell in the cluster, without affecting the (proportional) fairness amongst the cells. This, in turn, allows the bankruptcy channel allocation to take place on a cluster basis (rather than to individual cells), i.e., fairly distributing the channels among the clusters to achieve the fairness of the entire network. Our design of distributed power control is also necessary to implement the bankruptcy concept. Without our design, the cells would have different bandwidth requirements (as opposed to the equal requirements in our design). The channels would be allocated on the basis of individual cells, i.e., different bandwidths allocated to different cells (even within a cluster). This would result in a chaotic interference situation, where each cell suffers an inconsistent interference level across the bandwidth allocated to the cell. The stabilisation design of such system would become tedious and intractable. It is also worth mentioning that our proposed framework is not limited to the bankruptcy concept. It can use other allocation criteria as well, such as proportional fair. The bankruptcy con-

xm (t + 1) = xm (t) =

x2m (t)

π(t) ¯ πm (t)   k =m



j∈Dm

∗ j∈Dk Pj

 (14)

P∗j

where xm (t) is the ratio of the cluster size |Dm | to the total cell number N at the clustering review time t, πm (t) is the payoff of ¯ is the average payoff of all the the cluster Dm at time t, and π(t) clusters at time t. Here, t and (t + 1) correspond to t1 and t3 in Fig. 3, respectively. Note that here we design the payoff of Dm to be the average transmit power of the cluster, i.e., πm (t) = 

∗ j∈Dm Pj xm (t)N .

j∈Dm

|Dm |

P∗j

=

This is because overcrowded clusters require excessive transmit power to offset interference and achieve the required SINRs, as shown in (10). Our evolutionary clustering design of (14) incentivizes cells to progressively switch out of overcrowded clusters to less crowded clusters until the payoff converges across the clusters. To achieve this, the S-GWs broadcast (xj (t + 1) − xj (t)) for j = 1, · · · , M to all the cells at the beginning of the next evolution cycle (t + 1). The S-GWs also send the power threshold for the clusters with the average power larger than ¯ to indicate the most interfered π(t), say D  m (πm (t) > π(t)), ¯ π(t) ¯ πm (t) − 1 xm (t) cells of the cluster. In response, the most in  terfered ππ¯m(t) (t) − 1 xm (t) cells of Dm switch out of the cluster at the evolution cycle (t + 1). All the cells that switch out of their current clusters will spontaneously and randomly choose one of the clusters with the average transmit power lower than ¯ will be chosen at the π(t). ¯ Any cluster Dk with πk (t) < π(t) probability, as specified by xk (t + 1) − xk (t)  . xj (t + 1) − xj (t) ∀j:πj (t)

j∈Di





P∗j +maxj∈D  {P∗j } (a)

|Di |  ∗ j∈Di Pj |Di |

i



, where the

indicate before and after the most interfered  cell, arg maxj∈Di {P∗j }, switches out of the cluster, respectively. (a) is due to the fact that the largest of a set of scalars is larger than the average of the rest of the set; (b) is due to the reduced transmit powers after the switching. Meanwhile, the transmit power of the other cells increases due to the increased cluster sizes. In this sense, the average transmit power of a given cluster is monotonic with respect to the size of the cluster. We proceed to evaluate the stability of our evolutionary clustering game by employing the concept of Lyapunov stability [30]. Lyapunov stability is an important measure to evaluate the stability of solutions of differential equations near to a point of equilibrium. If all solutions that start out near an equilibrium point stay near the point forever, then the equilibrium point is Lyapunov stable [30]. For illustration convenience, we let M = 2, as often considered, e.g., [31] and [32]. Now, (14) can be rewritten with continuous relaxation, as given by f1 (x1 ) + f2 (x2 ) d x1 (t) = x21 − x1 (16) dt f1 (x1 )  where we define fi (xi ) = j∈Di P∗j for i = 1, 2. fi (xi ) is monotonic with respect to xi , as discussed earlier; x1 + x2 = 1. We are also able to calculate the equilibrium xe by solving f1 (x1 ) f2 (1 − x1 ) = . x1 1 − x1 1) 2) Given the monotonicity of f1 (x and f2 (x x1 x2 , xe is unique, as illustrated by Fig. 4, where we assume an infinite number of cells, such that the curves are continuous. We choose a Lyapunov function as given by

V(t) =

1 (x1 − xe )2 ≥ 0. 2

Fig. 4. An illustration on the equilibrium point xe .

where the equality holds if and only if x1 = xe , or in other f1 (x1 ) words, x1 = f1 (x1 )+f . 2 (1−x1 ) The first-order derivative of the Lyapunov function with respect to t can be given by   dV f1 (x1 ) + f2 (x2 ) = (x1 − xe ) x21 − x1 dt f1 (x1 )   x2 (1 − x1 ) f2 (1 − x1 ) f1 (x1 ) (x1 − xe ) . (17) − = 1 f1 (x1 ) 1 − x1 x1   f1 (x1 ) 1) Obviously, ddVt ≤ 0, since (x1 − xe ) and f2 (1−x − 1−x1 x1 always take opposite signs, as can be visualised in Fig. 4. The equality yields if and only if x1 = xe . This conclusion can be generalized for M ≥ 2, where Fig. 4 will become multidimensional. For example, in the case of M = 3, the curves will be replaced by planes with an intersecting point of xe , and dV d t ≤ 0 will still hold. It is worth pointing out that fi (xi ) depends on the geometry of Di , as well as the geometry of other clusters. This is because our proposed framework allows the cells within a cluster to reuse the same spectrum, exploiting SDMA techniques. As a consequence, the general expression for fi (xi ) is unavailable, varying with the initial clustering and geometry of the cells. Our stability analysis (17) confirms that the evolutionary clustering is stable under any initial clustering and geometry, due to the monotonicity of the average transmit power. The equilibrium xe can be different for different initial clustering and cluster geometries though. It is also worth mentioning that a ping-pong effect may be observed within a small neighbourhood of xe , when the number of cells is finite and xe is not an integer. This is common to evolutionary games [31], [33], due to the discrete nature of cluster size. Our analysis on Lyapunov stability indicates that the ping-pong effect will stay within a small local area and will never deviate from the area. In such case of finite cells, the centralised coordination can intervene, terminate the evolution, and eliminate the ping-pong effect. V. S IMULATION R ESULTS In this section, extensive MATLAB simulations are conducted to evaluate the proposed evolutionary game theoretic approach. The case where the evaluation is carried out is a femtocell network with a square area of A = 500 × 500 m2 ,

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where there are K = 30 available frequency channels and each channel has a bandwidth of ωc = 180 KHz. We assume that the N femtocells yield a random uniform distribution in the square area A. In other words, the actual coordinates of the i-th cell, (xi , yi ), are given by √ √ xi = ui A; yi = vi A, i = 1, · · · , N where ui , vi ∼ U(0, 1). U(0, 1) denotes the uniform distribution between 0 and 1. The assumption of uniformly distributed femtocells does not affect our design of the evolutionary game theoretic framework. The distribution is only employed for simulation evaluations. Our proposed framework is general, and applicable to any spatial geometry/distribution of cells (e.g., a spatial Poisson point (SPPP) model [34, Sec. 2.2.1]. Consider that the privately owned/installed plug-and-play FBSs are most likely to be indoors and operate at a low power level. We assume that the maximum transmit power of each base station Pmax = 200 mW. We also assume a path loss exponent of 3.7. The receive noise at the FUEs is set to be −174 dbm/Hz. The traffic demand is set to be 5 Mb/s per cell, uniform across all the cells, unless otherwise specified. For comparison purpose, we simulate the off-line centralised approach proposed in [20], where cells are statically and sequentially clustered and the closest cells are put into different clusters. For fair comparison, we also enhance the off-line centralised approach by performing our proposed distributed power control game over the statically clustered cells. Such enhanced approach is tolerant to severely delayed control over transport networks, and therefore it is feasible in practice. As far as we know, there is no other comparable approach which can adaptively switch cells between clusters and automate the transmit powers. Our comparison study to the off-line centralised counterpart is therefore fair and of interest. (Other optimal centralized algorithms [6]–[9] are computationally intractable in a practical network with hundreds of cells, as discussed in Section I, and therefore not considered here.) We also consider a practical situation, where the coordinates of the cells registered at the operator are inaccurate. They are within the range r of (xi , yi ). This is due to the fact that in most cases only coarse locations of the cells, e.g., the addresses of the properties equipped with the cells, are available. Fig. 5 shows the overall throughput of the network with the increasing number of cells, where the number of clusters is M = 4, and r = 0, 20, 30, and 40 m. The figure demonstrates that, in general, the proposed evolutionary game theoretic framework provides higher throughput than the off-line centralised approach. The gain of the proposed approach increases with the growth of r and N. Particularly, the gain grows from 5.8% to 13.2%, when N = 200, and r increases from 0 to 40 m. The reason for the increasing gain is because our evolutionary approach is robust to the initial clustering. The throughput of the approach degrades marginally due to inaccurate information of the network geometry. In contrast, the existing off-line approach relies on accurate network geometry information, and its throughput decreases increasingly with the growth of r. Fig. 6 plots the average transmit power required to achieve the throughput performance in the previous figure, with respect

Fig. 5. Overall system throughput versus the number of cells, where the number of clusters M = 4, and r = 0, 20, 30, and 40 m.

Fig. 6. Average transmit power per base station versus the number of cells, where the number of clusters M = 4, and r = 0, 20, 30, and 40 m.

to the increasing number of cells. The transmit power is the average of all the FBSs after they are stabilised from the distributed power control game. We can see that the stabilised transmit power increases with the number of femtocells, due to the increased interference. We also see that our proposed evolutionary framework requires substantially lower transmit powers than the off-line centralised approach. Particularly, the average transmit power of our approach is 36% lower in the case of N = 100 and r = 0. This is because the evolution of clusters helps alleviate interference and in turn reduces the requirement of transmit power. Fig. 7 plots the overall throughput under different numbers of clusters, where M = 2, 3, and 4; and r = 20 m since such cell location ambiguity is often practically experienced. We can see that the proposed evolutionary approach consistently outperforms the existing off-line centralized solution with a gain increasing along with N. This is because when M is small (in other words, the frequency reuse factor is small), every

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Fig. 7. Overall system throughput versus the number of cells, where the number of clusters M grows from 2 to 4, and r = 20 m.

femtocell suffers strong interference no matter which cluster the femtocell switches to. For example, when M = 2, every cell is in the same cluster as one of its two closest neighbouring cells. The gain of evolutionary clustering is marginal. When M increases, evolutionary clustering allows each cell to switch to the least interfering clusters, and therefore the gain of our evolutionary approach enlarges. A close look at each of the three solid curves shows that for every given number of clusters, the throughput first grows fast with N, due to the increased traffic demand. Then, the throughput growth slows down. The reason is that the network is so dense, the clusters become overcrowded, and excessive interference is suffered. Switching between clusters is not as effective to suppress the interference as it was for less dense femtocells. In this sense, the number of cells where the throughput growth starts to slow down corresponds to the case where the network density and interference are balanced for the given number of clusters. A joint look across the three solid curves reveals that the number of clusters which balances the network density and interference grows with N, continuously driving up the throughput. It is noted that clustering alleviates interference by reducing cells in overcrowded clusters, meanwhile decreasing the bandwidth that every cell can use. The continuous growth of throughput indicates that the reduced interference of evolutionary clustering can compensate for the reduced bandwidth. Fig. 8 shows the phase plane of the proposed evolutionary game of clustering, where N = 150 and M = 3 according to the result of Fig. 5. The population proportions of two of the three clusters, i.e., x1 and x2 , are plotted, since x3 can be straightforwardly obtained by x3 = 1 − x1 − x2 . The phase plane of x1 and x2 lies in the lower triangular part of the plane, due to the fact that 0 ≤ x1 + x2 ≤ 1. In Fig. 8, we can see that the cluster sizes evolve towards the evolutionary equilibria, and no clusters deviate from the equilibria. This confirms experimentally that our proposed evolutionary game theoretic framework is stable, as discussed in Section IV. In the figure,

Fig. 8. Phase plane of the proposed evolutionary clustering to reach the evolutionary equilibria, where a randomly generated network topology with N = 150 is considered. We consider M = 3, r = 0, and x1 and x2 are the proportions of clusters 1 and 2 with regards to the entire population of the cells, respectively.

we also see that the number of the evolutionary equilibria is not unique. It depends on the initial clustering of the cells. This is a common phenomenon in evolutionary games, as shown in [31, Fig. 4] and [33, Fig. 3]. It does not invalidate the evolutionary stability of the proposed scheme. We proceed to evaluate our proposed evolutionary game theoretic framework in the case of non-uniform bandwidth requirements of the cells. Two classes of bandwidth requirement are considered: a high bandwidth requirement of 6.0 Mb/s per cell and a low bandwidth requirement of 4.0 Mb/s per cell. XH denotes the proportion of the cells with the high bandwidth requirements with respect to the entire population of the cells. We vary XH from 0 to 100%, to evaluate the impact of unbalanced traffic demand on our framework. It is worth mentioning that, as far as we know, our proposed approach is the only approach which is able to incorporate unbalanced traffic demands into the clustering process through our new design of power control. Other algorithms, such as the graph theoretic approaches proposed in [20], cannot do so, and clustering is carried out only on a geometric basis. Fig. 9 shows the average satisfaction level of the cells which are the average ratio of the actually achieved data rate to the required data rate of every cell. We can see that the average satisfaction level decreases with the increasing number of highrequirement cells. This is because the traffic demand which increases along with the high-requirement cells results in the rise of interference. The interference gets even severer when the density of cells grows. Fig. 10 plots the Jain index of the satisfaction levels of the individual cells, where the proposed evolutionary approach is considered, and r = 0, 20, and 30. The Jain index is an important measure to indicate fairness among the cells [35]. The index is over 0.87 for all r = 0, 20, and 30 in the figure, confirming that our evolutionary approach can achieve good fairness among the cells.

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also affects the fairness. When the traffic demand is high, the interference becomes excessive. Not only do the satisfaction levels of the cells decrease (as shown in Fig. 9), but also the fairness between the cells degrades (as shown in Fig. 10). The reason for the degraded fairness is due to the fact that req req the approximation log(1 + γi ) ≈ γi for (11) becomes less accurate when the traffic demands are too high and do not yield req 0 < γi  1. VI. C ONCLUSION

Fig. 9. The average satisfaction level of the cells with respect to their achievable data rate, where N = 150, M = 3, and the population of the cells with high bandwidth requirement of 6.0 Mb/s, XH , grows from 0% to 100%, and the remaining (1 − XH ) cells have the low requirement of 4 Mb/s.

In this paper, we proposed a new game theoretic framework which enables femtocells to evolutionarily form clusters and allocate channels, and independently control transmit powers. Our framework tolerates severely delayed network control, and the stability of the framework is theoretically verified. Simulation results show that our framework is able to improve the network throughput by 13.2% in a network of 200 cells, and save the transmit power by 36% in a network of 100 cells. The gain of our framework will increase further with the growth of the network size. R EFERENCES

Fig. 10. Jain index of the satisfaction levels of individual cells with the increasing proportion of the cells with high bandwidth requirement of 6 Mb/s—XH , where the remaining (1 − XH ) cells have the low requirement of 4 Mb/s, N = 150, and M = 3.

In the figure, we can see that the impact of unbalanced bandwidth requirements on the fairness is obvious. Particularly, the Jain index is convex with respect to XH . The lowest (turning) points appear at XH = 60% for all r = 0, 20, and 30. Before the turning points, the Jain index decreases due to the increasing traffic demand and the subsequently increasing interference. After the turning points the index grows, because of the increasing number of cells which have the same bandwidth requirements and have similar satisfaction levels. The turning points are therefore where the effects of the increasing interference and the unbalanced traffic demands are most strongly coupled. We also see that the Jain index is lower at XH = 100% than it is at XH = 0, even though both the XH values correspond to uniform and balanced bandwidth requirements among the cells. This reveals that the volume of the balanced traffic demands

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Shangjing Lin (S’11) received the B.E. degree in electronics information engineering from Communication Command College, Wuhan, China, in 2008, and the M.E. degree in communication engineering from the Wuhan Institution of Posts and Telecommunications, Wuhan, China, in 2011. She is currently pursuing the Ph.D. degree at Beijing University of Posts and Telecommunications, Beijing, China. She was a visiting Ph.D. student at the CSIRO, Australia, from 2014 to 2015. She is a recipient of Rohde & Schwarz Scholarship. Her research interests include radio resource management, mobile social network, and game theory.

Wei Ni (M’09–SM’15) received the B.E. and Ph.D. degrees in electronic engineering from Fudan University, Shanghai, China, in 2000 and 2005, respectively. Currently, he is a Senior Scientist with CSIRO, Australia. He also holds adjunct positions with Macquarie University (MQ), Sydney, and the University of Technology, Sydney (UTS). Prior to this, he was a Research Scientist and Deputy Project Manager at the Bell Labs R&I Center, Alcatel/AlcatelLucent (2005–2008), and a Senior Researcher at Devices R&D, Nokia (2008–2009). His research interests include radio resource management, software-defined networking (SDN), network security, and multiuser MIMO. Dr. Ni serves as Editor for Hindawi Journal of Engineering since 2012, Secretary of IEEE NSW VTS Chapter since 2015, PHY Track Chair for IEEE VTC-Spring 2016, and Publication Chair for BodyNet 2015. He also served as Student Travel Grant Chair for WPMC 2014, a Program Committee Member of CHINACOM 2014, a TPC member of IEEE ICC’14 workshop on body area networks, ICCC’15, EICE’14, and WCNC’10.

Hui Tian received the M.S. degree in microelectronics and Ph.D. degree in circuits and systems both from Beijing University of Posts and Telecommunications, Beijing, China, in 1992 and 2003, respectively. Currently, she is a Professor at BUPT, the Network Information Processing Research Center Director of State Key Laboratory of Networking and Switching Technology, and the MAT Director of Wireless Technology Innovation Institute (WTI). Her current research interests mainly include radio resource management, cross layer optimization, M2M, cooperative communication, and mobile social network.

Ren Ping Liu (M’09–SM’14) received the B.E. and M.E. degrees from Beijing University of Posts and Telecommunications, Beijing, China, in 1985 and 1988, respectively, and the Ph.D. degree from the University of Newcastle, Callaghan, Australia, in 1996. He is a Principal Scientist of networking technology in CSIRO. He is also an Adjunct Professor at Macquarie University and the University of Technology, Sydney. His research interests include MAC protocol design, Markov analysis, QoS scheduling, TCP/IP Internetworking, and network security. He has over 100 research publications in leading international journals and conferences. Prof. Liu is the founding chair of the IEEE NSW VTS Chapter. He served as TPC chair, as OC co-chair, and on the technical program committees of a number of IEEE Conferences. He has also been heavily involved in and led commercial projects delivering networking solutions to government and industry customers.