A radio resource management framework for multi-user multi-cell OFDMA networks based on game theory

Wireless Personal Communications manuscript No.(will be inserted by the editor)

A Radio Resource Management Framework for

multi-user multi-cell OFDMA Networks based on Game

Theory

Ioannis N. Stiakogiannakis · Dimitra I.

Kaklamani

Received: February 13, 2013/ Accepted: date

Abstract This work proposes a Radio Resource Management framework em-ploying game theoretic concepts for Orthogonal Frequency Division MultipleAccess, the most prevalent Multiple Access technique for the next generationwireless networks. The subcarrier allocation problem is encountered as a com-binatorial auction, where the base station auctions the subcarriers and theusers bid for and buy bundles of subcarriers, aiming at minimising their re-quired transmit power. Subsequently, each allocated subcarrier is loaded witha number of bits, decided by each user independently, and the power controlprocess is set up as a non-cooperative game. Each user responds to the inter-ference sensed in his environment and, through a best responses process, thegame converges to the unique, Pareto optimal, Nash equilibrium. In order toguarantee convergence, a limit is imposed to the maximum modulation levelfor each subcarrier. Simulation results show that the auction algorithm fol-lows closely the performance of the optimal algorithm, whereas it is of lowercomputational complexity and requires less feedback information. Similarly,the proposed distributed bit loading and power control scheme achieves lowertransmit power per offered bit rate unit. However, the distributed nature ofthe algorithm results in lower total offered bit rate, because of the partialknowledge and exploitation of channel state information.

Keywords OFDMA · Radio Resource Management · Margin Adaptive ·Subcarrier Allocation · Power Control · Game Theory · CombinatorialAuctions

Financial support by National Technical University of Athens under the program PEVE2009, grand no 65177200, is gratefully acknowledged.

I. N. Stiakogiannakis · D. I. KaklamaniNational Technical University of Athens,School of Electrical and Computer EngineeringE-mail: istiak@ieee.org, dkaklam@mail.ntua.gr

2 Ioannis N. Stiakogiannakis, Dimitra I. Kaklamani

1 Introduction

The development of Field-Programmable Gate Array (FPGA) technology andthe implementation of Fast Fourier Transform (FFT) on FPGAs made feasiblethe cost-effective employment of Orthogonal Frequency Division Multiplexing(OFDM). The adoption of OFDM and the corresponding Multiple Accesstechnique, Orthogonal Frequency Division Multiple Access (OFDMA), fromvarious standardisation bodies, including IEEE [1] and 3GPP [2], launched anenormous research effort on issues related to both PHY and MAC layers ofOFDM(A). This effort has provided with some seminal works, including [3–5],contributing to the establishment of Radio Resource Management (RRM) forOFDMA networks as a standalone research area and a number of fruitfulworks appeared during the next few years. In [6], a comprehensive survey onOFDMA RRM, it is noted that the RRM problem can be formulated as eitherMargin Adaptive (MA), where the objective is to minimise the total transmitpower, or Rate Adaptive (RA), where the objective is to maximise the sumcapacity of the network.

At the same time, a new analytical model is employed in telecommunica-tions research, the mathematically mature Game Theory. Game Theory aimsto model situations where a number of entities interact, by means of followingcertain strategies, in order to maximise their own profit. In telecommunica-tions, Game Theory has been employed to tackle problems such as powercontrol, routing, call admission control, etc [7, 8]. Game Theory encompassesa wide variety of games, among them cooperative and non-cooperative games,repeated games, auctions, and useful concepts such as Nash equilibrium, Stack-elberg equilibrium, truthfulness. Each of these game types and each of theseconcepts are appropriate for describing different types of interaction, in dif-ferent kinds of networks [9,10]. In this work, we focus on the MA formulationof the downlink OFDMA RRM problem, considering a multi-user multi-cellOFDMA network. Specifically, in the following, a subcarrier allocation algo-rithm based on combinatorial auctions is presented, along with a bit loadingand power control scheme based on non-cooperative games.

1.1 Related Work

In the past few years, several research works have proposed the employment ofauctions to tackle the subcarrier allocation problem. In [11], various schemesare presented, based on per-subcarrier auctions addressing both single-cell andmulti-cell resource allocation algorithms for the uplink OFDMA RRM prob-lem. In [12], the employment of Dutch auctions is proposed for the downlinkOFDMA RRM problem. The Base Station (BS) sets up simultaneous Dutchauctions, one per subcarrier. For each subcarrier, a starting price is announcedand, as time passes, this price is reduced according to a clock. Each user ob-serves his “best” subcarrier and, when he decides that its price is low enough,based on his valuation on the specific subcarrier, he bids for and buys it at its

An RRM Framework for multiuser multicell OFDMA based on Game Theory 3

current price. In [13–15], the RA formulation of OFDMA RRM is tackled anda subcarrier allocation algorithm is proposed based on per-subcarrier auctions.The users’ bidding and valuation functions are defined, so as to achieve a spe-cific goal, such as system throughput maximisation or fairness maximisation.The users bid for one subcarrier at a time and the user with the highest bidwins his requested subcarrier.

In [16], the subcarrier allocation problem is tackled in conjunction withthe scheduling problem. A multi-unit auction is proposed, which consists ofa multi-carrier proportional fairness algorithm as allocation method and theVickrey-Clarke-Groves payment scheme. To the best of authors’ knowledge, itis only in [17] that the employment of combinatorial auctions for the downlinkOFDMA subcarrier allocation problem is introduced. Therein, a combinatorialclock auction is proposed, according to which the BS announces the price foreach subcarrier and the users react to these prices by means of submittingthe bundle of subcarriers willing to buy. If there is excessive demand for anysubcarrier, the BS increases the price for this subcarrier and the users submittheir new bundle of subcarriers. The process is terminated when there is noexcessive demand for any subcarrier.

Since power control is a typical field where Game Theory has been applied,in the following, the presentation is limited to recent advancements. In [18], amulti-user network is examined, with parallel Gaussian channels, where eachuser tries to achieve a specific bit rate with the minimum transmit power.The analysis is based on the notion of generalised Nash equilibrium, the ex-istence and the uniqueness under certain circumstances of which are proved.Two distributed algorithms are presented, a sequential and a simultaneousiterative water filling algorithm, which converge to the generalised Nash equi-librium. [19] refers to the downlink of an OFDMA network with fractionalfrequency reuse, with a protected and a shared frequency band. The RA prob-lem is formulated as a non-cooperative game, which is proved to have a uniqueNash equilibrium under certain circumstances. A power control algorithm ispresented, which is proved to be incentive compatible.

In [20, 21], the power control problem is addressed from a different pointof view. In [20] at first, it is supposed that at least one user is able to collectinformation about the strategies and channel state information (CSI) of otherusers. Hence, a Stackelberg game is constructed, which is proved to lead to amore efficient equilibrium compared to the typical modelling of the game withonly myopic users. In [21], the notion of conjectural equilibrium is introducedand it is shown that the leader does not have to have a priori informationon the strategies and the CSI of the followers but this information can beacquired as the game evolves. It is proved that the game converges to an equi-librium comparable to the previous approach of Stackelberg equilibrium and,furthermore, that the Nash and Stackelberg equilibria of previous approachesare special cases of the conjectural equilibrium.

1.2 Paper Contributions and Outline

Contributing to the previous research efforts, the present work proposes a novelcombinatorial auctions algorithm to tackle the issue of subcarrier allocation.Contrary to the majority of previous approaches, the scheme of combinatorialauctions is employed, rather than per subcarrier auctions, in a completelydifferent setup compared to [17]. This algorithm is an appealing alternativeto the optimal algorithm for three reasons. Firstly, the proposed algorithmclosely follows the performance of the optimal one, as measured by all thefigures of merit examined. Secondly, it is of lower computational complexity,which results in faster decision making. Thirdly, due to its distributed nature,it causes less feedback overhead on the uplink.

Furthermore, a bit loading and power control algorithm is proposed. Theproposed scheme is based on non-cooperative games and it is proved to con-verge to the Pareto optimal Nash equilibrium if certain precautions are taken.Contrary to previous research efforts, the usage of infinity norm is proposed,which is accompanied with a proper channel probing scheme, achieving thatway to guarantee the convergence while maintaining the distributed nature ofthe algorithm. The proposed scheme is proved to be of lower computationalcomplexity, while at the same time it eliminates the feedback requirements.

One of the main advantages of this work is the fact that the efficiency andsuitability of the proposed schemes are evaluated in the frame of a multi-cellOFDMA network based on typical figures of merit for wireless networks. As amatter of fact, a series of simulations measures the performance of the networkunder the employment of the proposed algorithms based on offered bit rate,transmit power, and several other metrics.

The rest of this work is organised as follows. Section 2 presents a com-plete RRM framework, addressing the MA formulation for downlink multi-user multi-cell OFDMA. Thence, in section 3, a subcarrier allocation algo-rithm based on combinatorial auctions is presented. Section 4 presents a non-cooperative power control game. The existence under certain circumstances ofa unique Nash equilibrium is proved and a distributed bit loading and powercontrol algorithm is built upon this game. Section 5 includes the simulationresults and the corresponding analysis on them. Finally, the presented work issummarised and the conclusions are drawn in section 6.

2 Radio Resource Management Framework for multi-user

multi-cell OFDMA Networks

2.1 Problem Definition

The OFDMA system under consideration encompasses K BSs (k ∈ K ={1, . . . ,K}), with total available power Pmax

BS each, and U users (u ∈ U ={1, . . . , U}). Each user u requires bit rate Ru under Bit Error Rate (BER)Peu. The available bandwidth BW is divided into N OFDM subcarriers

(n ∈ N = {1, . . . , N}) and each subcarrier occupies bandwidth ∆f = BW/N .The MA formulation for the downlink RRM problem is based on the definitionof three allocation matrices:

– Subcarrier allocation matrixC = [Cu,n,k]. Cu,n,k = 1 (Cu,n,k = 0) indicatesthat subcarrier n of BS k is (not) allocated to user u.

– Bit loading matrixB = [bn,k]. bn,k ∈ {0, bmin, . . . , bmax} indicates the num-ber of bits that the BS k loads on subcarrier n where bmin (bmax) standsfor the minimum (maximum) modulation level that can be employed. Inthe scope of this work, it is assumed that the network employs exclusivelysquare M-QAM (Quadrature Amplitude Modulation) modulations, and, asa result, bn,k is an even integer.

– Power allocation matrix P = [Pn,k]. Pn,k ∈ R+ indicates the power that

the BS k transmits on subcarrier n.

As stated earlier, the MA formulation aims at minimising the total transmitpower of the network, while fulfilling the requirements of individual users. Thecorresponding optimisation problem is defined by (1)-(5).

minimise

Pn,k (1)

subject to

Ru ≤ Rou = ∆f

Cu,n,kbn,k, ∀u ∈ U (2)

Peu ≥ Peou = 4Cu,n,k Q

2bn,k − 1

Gu,n,kPn,k∑K

m=1m 6=k

Gu,n,mPn,m + σ2

∀u ∈ U , ∀n ∈ N , ∀k ∈ K (3)

PmaxBS ≥ P t

Pn,k, ∀k ∈ K (4)

Cu,n,k ≤ 1, ∀n ∈ N , ∀k ∈ K (5)

where Rou stands for the offered bit rate, Peou is the measured BER. Gu,n,k

is the channel gain for subcarrier n between the BS k and user u, includingtransmit and receive antenna gains, path losses, shadowing and fading effects,and σ2 is the received noise power in the bandwidth of subcarrier n. Sincethe network employs exclusively square M-QAM modulations, it has beenproven [22] that the BER can be closely approximated by (3), where Q (·)is the Q-function. Eq. (3) is the so-called link to system-level interface [23]employed herein. Finally, P t

k is the total transmit power of the BS k.It must be noted that each BS allocates each subcarrier to a unique user,

as constraint (5) implies, a practice that has been proved optimal for anyadaptive bit loading scheme [24].

2.2 RRM Framework

The problem (1)-(5) is a non-linear, non-convex, three-dimensional optimisa-tion problem that belongs to the NP-hard complexity class [25]. In order toachieve a solution within reasonable complexity, following the established ap-proach in the research field [4,5,25], the initial problem is tackled as a series ofsub-problems, which are solved sequentially, concluding thus to a sub-optimalsolution. Hence, the RRM framework has to define a solution for the BS se-lection problem, the subcarrier allocation problem and, finally, the bit loadingand power control problem [26].

Fig. 1 depicts the flowchart of the simulation process employed herein inorder to evaluate the performance of the proposed RRM framework.Up to V lim

new users ask for admittance for each BS k ∈ K. It must be noticed that, whena user is denied admittance, the network returns to the previous steady stateand the allocation matrices remain intact. On the contrary, when the user isadmitted, the allocation matrices are updated according to the decisions ofthe RRM framework. The components and the corresponding nomenclatureare explained in the following.

2.2.1 Base Station Selection

When a new user, denoted as u, requests access to the network resources, thevery first issue to tackle is the assignment of a BS from the set K. Towardsthis direction, diverse solutions have been proposed and evaluated. Amongthem, [25] proposes a selection criterion based on normalised Channel to In-terference plus Noise Ratio (nCINR), whereas [27,28] employ a criterion basedon average (over all subcarriers) channel gain. In the approach adopted herein,as it has been noticed that the Base Station Selection process plays a minorrole on system performance, the new user u is served by BS k if he lies in thegeographical area served by BS k. As a result, the terms BS and cell are usedinterchangeably. For notational convenience, the set of users served by cell k,including the new user u, is denoted as Uk with Uk elements.

2.2.2 Subcarrier Allocation

The second step on the RRM process is the subcarrier allocation. In multi-carrier networks, as OFDMA, the problem of subcarrier allocation is usuallydivided into two separate successive problems [4], where the first one dealswith the definition of the number of subcarriers to be allocated to each user,and the second one performs the subcarrier allocation, based on the outcomeof the former.

Number of Subcarriers Definition The Bandwidth Assignment Based on SNR(BABS) algorithm of [4], extended to the multi-cell scenario, is employedherein. In brief, the number of subcarriers su for each user u in cell k is

initialised with the minimum one sminu =

bmax∆f

. If there are not enough

Initialisations

Vk = V limk

,∀k ∈ K

New user u

Base Station Selection (§2.2.1)Output: k

Vk < V limk

Reject u

Vk ← Vk + 1

Channel Probing (§4.1)(Necessary for DST power control)

u∈Uksminu ≤ N

Number of Subcarriers Definition (§2.2.2)Output: su, ∀u ∈ Uk

Subcarrier Allocation

Hungarian (§2.2.2) or Auction (Sec. 3)

Output: Cu,∀u ∈ Uk

Bit Loading and Power Control

CNT (§2.2.3) or DST (Sec. 4)

Output: bn,k ,∀n ∈ N and

Pn,k, ∀n ∈ N , ∀k ∈ K

Rou ≥ Ru, ∀u ∈ Uk

P tk≤ Pmax

BS ,∀k ∈ K

Admit u

Fig. 1 Simulation process flowchart

subcarriers to serve the users of cell k, ie.∑

u∈Uksminu > N , the new user u is

rejected and the network returns to the previous steady state.

For the distribution of the remaining subcarriers, if any, another subcarrierwill be allocated to the user u that will gain the maximum power reduction,formally,

u = arg maxu∈Uk

Pu,k (su)− Pu,k (su + 1))

where,

Pu,k (su) = su2

Rusu∆f − 1

is a rough estimation on the total required power for user u. The averagenormalised Channel to Interference plus Noise Ratio (nCINR) Tu,k is derived

as Tu,k = 1N

n=1 Tu,n,k, whereas the nCINR Tu,n,k is defined as,

Tu,n,k =Gu,n,k

∑Km=1m 6=k

Gu,n,mPn,m + σ2

Γu is the SNR gap which defines the gap between a practical modulationscheme and the channel capacity, given by [22],

Γu =1

where Q−1 (·) is the inverse Q-function. Although (8) implies so, it is not nec-essary for the user to be able to distinguish the sources of interference in orderto calculate Tu,n,k but rather calculate the interference

∑Km=1m 6=k

Gu,n,mPn,m+σ2

as a whole.

The process is ended when the subcarriers are depleted or all users have

achieved the maximum number of subcarriers needed, smaxu =

bmin∆f

Subcarrier Allocation Algorithm Given su, the allocation problem can be seenas a matching problem. The “cost” for user u to be allocated subcarrier n isdefined as the average required power to be transmitted on this subcarrier,formally,

Pu,n,k =2

Rusu∆f − 1

Tu,n,k

Based on this definition, a cost matrix can be constructed of dimensions∑

u∈Uksu × N , where each user is represented by su rows whereas columns

represent the subcarriers. This transformation of the allocation problem to thematching problem allows to employ the optimal Hungarian algorithm [29,30],with computational complexity O

, which achieves to minimise the sum

average required power. If Cu ⊆ N is the set of subcarriers allocated to user u(n ∈ Cu ⇔ Cu,n,k = 1), the Hungarian algorithm solves optimally the problem,

minimise Pk =∑

u∈Uk

n∈Cu

Pu,n,k (11)

subject to

|Cu| = su (12)

The subcarrier allocation algorithm proposed by this work is presented insection 3.

2.2.3 Bit Loading and Power Control

The bit loading process is usually performed in conjunction with the powercontrol process, since the power transmitted on a subcarrier depends directlyon the bits per symbol to be loaded and vice-versa, under a BER constraint.

In [5], a greedy bit loading and power control process is proposed, based ona completely centralised scheme. This algorithm searches exhaustively amongthe subcarriers allocated to each user and increases the modulation level ofsubcarrier n that leads to the smallest power increment, until the requested bitrate is reached. Taking into consideration the fact that in a multi-cell networkthe transmit power from a BS on a subcarrier interferes to the users served byother BSs on the same subcarrier, specific precautions must be taken in orderto guarantee the BER of all users using the specific subcarrier network-wide.As a result, the increment on transmit power is calculated, considering all theco-channel BSs.

In the following, it is assumed that there are κ ≤ K co-channel users/BSson subcarrier n. For notational convenience, the co-channel users are enumer-ated as u1, . . . , uκ, and their serving BSs are denoted as k1, . . . , kκ, with anone-to-one correspondence (index i ∈ In = {1, . . . , κ}).

The subcarrier n is derived as,

n = arg minn∈Cu

bn,k + 2)

−κ∑

The derivation of the transmit power Pn,ki

is explained in the following.Since only square M-QAM modulations are considered, bn,k increases in stepof 2 bits per symbol.

In order for the network to guarantee the requested BER of the users, thereceived Signal to Interference plus Noise Ratio (SINR) on each subcarrier nmust be higher than a SINR threshold, imposed by the BER requirements.Formally,

Gui,n,kiPn,ki

∑κj=1j 6=i

Gui,n,kjPn,kj

+ σ2≥ γui,n,ki

, ∀n ∈ N , ∀i ∈ In (14)

where γui,n,kiis the SINR threshold, imposed by the requested BER and the

modulation scheme, derived as,

γui,n,ki= Γui

2bn,ki − 1)

As the aim is the minimisation of transmit power, it is adequate to ensurethat (14) is satisfied with equality for all co-channel users. The transmit powerof these users is mutually dependent and, by collecting the κ equations, thefollowing system of linear equations is built-up,

(I−DF)P = v (16)

where,

I = κ× κ identity matrix

D = diag {γu1,n,k1 , . . . , γui,n,ki, . . . , γuκ,n,kκ

F = [Fi,j ]1≤i,j≤κ, where Fi,j =

0 if i = j,Gui,n,kj

Gui,n,ki

if i 6= j

P = [Pn,k1 , . . . , Pn,ki, . . . , Pn,kκ

γu1,n,k1

Gu1,n,k1

, . . . ,γui,n,ki

Gui,n,ki

, . . . ,γuκ,n,kκ

Guκ,n,kκ

The aforementioned transmit power Pn,ki

is derived by solving (16).By Perron-Frobenius theorem [31], the system of linear equations (16) has aunique, non-negative solution, if and only if, ρ (DF) < 1, where ρ (DF) standsfor the spectral radius, ie. the largest eigenvalue of matrix DF.

The bit loading and power control algorithm proposed by this work ispresented in section 4.

3 Subcarrier Allocation Algorithm based on Combinatorial

Auctions

An auction requires the existence of three elements: owners willing to sell thegoods they own, goods for sale, and buyers willing to buy these goods. Incombinatorial auctions sets of goods are auctioned and the potential buyersbid for one or more subsets (combinations) of these goods. In the presentwork, the focus is on a subclass of combinatorial auctions, namely sealed-bid combinatorial auctions with single-minded bidders. Each bidder submits asingle bid (single-minded), which defines the subset of goods he is interested inand the price he is willing to pay for. The other participating users cannot beinformed about this bid (interception is not included in the model) and, thus,the bid is considered as sealed. The auctioneer collects all the bids, decides theallocation of goods to bidders and the corresponding prices to be paid, and

announces the result of the auction. The design of the proposed subcarrierallocation algorithm is based on the auction theoretic results of [32].

The subcarrier allocation problem can be encountered as a combinato-rial auction problem, where the role of auctioneer is played by the BS, thesubcarriers are the goods for sale and the users are the bidders, anticipatingfor the subcarriers. The BS charges for the transmit power, employing a flatrate, denoted herein as α monetary units per power unit [12]. This chargingscheme may be conjuncted with the pricing and billing system that the serviceprovider employs. However, the analysis of such a perspective is out of scopefor this work. Each user has an infinite amount of these monetary units but,as a rational entity, he tries to satisfy his needs with the lowest possible cost.

3.1 Definition and Algorithm Presentation

From the game theoretic perspective, this auction constitutes a game

k ∪ Uk,V ∪ {Bru} , gok (·) ∪ {gou (·)}

The players participating in this game are the BS k and the users u ∈ Uk,thus, the set of players is k ∪ Uk. On the one hand, the BS defines the priceper power unit α ∈ R

+, thus, the strategy space for the participating BSis V = R

+. On the other hand, each user u ∈ Uk expresses his preferencesby reporting a subset of the available subcarriers Cr

u ⊆ N and his valuationvru ∈ R

+ on this subcarrier set. The bid, designated as the pair bru = 〈Cru, v

ru〉,

is the strategy for player u. Taking into consideration the fact that Cru can

be any subset of N with su elements (|Cru| = su, cf. §2.2.2), it follows that

Cru is member of the su-subset of N that contains all the subsets of N with

cardinality su, formally Cru ∈ Psu(N ). As a result, the strategy space for user

u is Bru = Psu(N ) × R

+ and bru ∈ Bru, and the strategy space for the game is

designated as V ∪ {Bru}. Finally, if pu (br), br =

br1, . . . , bru, . . . , b

, is the

payment paid by user u ∈ Uk, the payoff function gou (br) = [vru]

+ − pu (br)

describes the outcome for each user participating in the auction, where,

[vru]+=

vru if bru is granted,

0 otherwise.(17)

Accordingly, the payoff function for the BS is the total income from users’payments, ie. go

k(br) =

u∈Ukpu (b

r). The payoff for both the BS and theusers depends on the bids of all users br and the rules of the auction, theallocation and payment schemes, as described in Alg. 1.

At the initialisation phase, the set of allocated subcarriers Cu is empty (ln.1) and the payment is zero (ln. 2) for all users. At the user side, each usermakes a set Cr

u, which contains the su best available subcarriers, in terms ofnCINR Tu,n,k (ln. 5-6). This is the set of subcarriers to apply for. Thence,the average power needed to be transmitted on these subcarriers in order to

Algorithm 1: Subcarrier Allocation Algorithm based on CombinatorialAuctionsInput: k, Uk, su,∀u ∈ Uk , Ru,∀u ∈ Uk , Tu,n,k,∀u ∈ UkOutput: Cu,∀u ∈ Uk

1 Cu ← ∅, ∀u ∈ Uk2 pu ← 0, ∀u ∈ Uk3 repeat

4 foreach u ∈ Uk do

5 N ← arg sortin∈N Tu,n,k

6 Cru ←{

N (i) : 1 ≤ i ≤ su

7 P ru ←

n∈Cru

su∆f −1Tu,n,k

8 vru ← αP ru

9 Uk ← arg sortivru√su

10 W ← ∅11 for i = 1 to |Uk| do12 u← Uk(i)13 if Cru ⊆ N then

14 Cu ← Cru15 N ← N\Cru16 Uk ← Uk\{u}17 W ←W ∪ {u}18 else

19 for j = 1 to |W| do20 w ←W(j)21 if Cru ∩ Cw 6= ∅ then22 pw ←

vru√su

23 W ←W\{w}24 break

25 until Uk = ∅

meet the bit rate constraint Ru is derived as P ru =

n∈Cru

su∆f −1Tu,n,k

(ln. 7).

Since the distribution of bits to subcarriers has not been defined yet, this isthe best approximation on the required power. Consequently, the valuationfor the bundle to request is defined as vru = αP r

u (ln. 8). Thence, each usersubmits his bid bru = 〈Cr

u, vru〉.

At the BS side, the BS collects the submitted bids and makes an ordered

list Uk, which contains the users in descending order of ratiovru√su

(ln. 9). This

ratio is a measure expressing a kind of average value per subcarrier [32]. TheBS maintains also a winner list W (ln. 10) containing the users whose requesthas been accepted at the auction, whereas their payment has not been definedyet. Then, the list Uk is scanned sequentially (ln. 11-12). If all the subcarriersthat the user under consideration requires are in the set of available subcarriers(ln. 13), then they are allocated to the user (ln. 14) and they are removed fromthe set of available subcarriers (ln. 15). Since the user has covered his needs

in bandwidth, he is removed from the set of anticipating users (ln. 16), andhe is added in the winner list (ln. 17).

If the user’s request is not granted (ln. 18), this is due to the fact that one ormore of his requested subcarriers have already been allocated to a user/winnerwith higher priority. The winner list is scanned sequentially (ln. 19-20) in orderto find the first winner w that caused the denial of bid of the current user u(ln. 21). The payment for this winner is then defined as pw =

vru√su

(ln. 22).

Since the payment for this winner is decided, he is removed from the winnerlist (ln. 23) and the scan of the winner list is terminated (ln. 24). The paymentscheme employed by the BS could be summarised as follows: the winner willpay per subcarrier the average value of the first bid that is denied because ofhim, reminding thus the second-price auctions [33].

Reaching the end of the ordered list of participating users Uk, the algorithmis repeated with the updated set of available subcarriers (ln. 3) and only theusers that have not been allocated yet the defined number of subcarriers (ln. 4)take part in the auction. The allocation process ends when each user u ∈ Uk

has been allocated su subcarriers, which means that the set of anticipatingusers is empty (ln. 25).

3.2 Remarks

It must be highlighted that the subcarrier allocation process described pre-viously is distributed to both the users and the BS. The first part (ln. 4-8)is executed by each user independently. The subcarriers that form the set Cr

are selected based on the simple but rational concept of selfishness; each userseeks the best available subcarriers in order to cover his needs in bit rate at thelowest possible cost. The definition of the metrics P r

u and vru is imposed by thenature of the problem under consideration; the objective of each user is notthe maximisation of his bit rate but the fulfilment of his bit rate requirementswith the minimum required power, and consequently at the minimum cost.

The second part of the algorithm (ln. 9-24) is executed at the BS. The BScollects the bids from all the participating users and starts serving users in

descending order ofvru√su. From the auction perspective, this is equivalent to

prioritising the bidder willing to pay the most per subcarrier whereas, fromthe network perspective, this results in giving priority to the user who requiresmore power. This choice is based on two reasons. The first one is relatedto the cell-edge users. Usually, these users encounter problems due to lowchannel quality and, consequently, require high power. Therefore, this choiceaims at prioritising users with worse channel conditions. The second reasonis related to the structure of the algorithm. The model that Alg. 1 followsconsists of successive combinatorial auctions. If a user u is not satisfied at aniteration, this is due to the fact that one or more subcarriers from the set Cr

have already been allocated to a user with higher priority. As a result, user uwill come back at the next iteration with an updated request Cr

u, where theunavailable subcarriers will be replaced with worse ones. Consequently, the

required average power P ru will be higher. In other words, the algorithm gives

priority to users with high required power, because if they are not served, theywill come back requiring even higher power.

It is worth noticing that the auction algorithm is able to reduce the feed-back overhead, in comparison to schemes that require full channel knowledge atthe BS, thanks to multi-user diversity and the consequent differentiated pref-erences of users on subcarriers. In the proposed solution, a processed, reducedform of CSI is communicated only when there is need for it. The structure ofthe algorithm implies that the users must inform the BS for their bids by send-ing Cr

u and vru. Furthermore, at a next iteration, the users that still take partin the auction must update their bids, by replacing the unavailable subcarrierswith available ones and updating their valuation. It is clear that the successof the proposed algorithm is based mainly on the multi-user diversity. If thephysical environment does not contribute towards this direction, for examplein rural areas, multi-user diversity can be achieved using dumb antennas, asdescribed in [34].

Finally, the allocation algorithm guarantees truth revelation, ie. the antic-ipating users have no interest to lie about their preferences and valuations.In order to establish this claim, it can be shown that the allocation algorithmsatisfies the four sufficient conditions for truth revelation for single-minded bid-ders, described in [32, Theorem 9.6], namely Exactness, Monotonicity, Criticaland Participation. Concluding, a thorough analysis on the proposed algorithmshows that its computational complexity can be described as O

underthe assumption that logN ≤ Uk ≤ N .

3.3 Two users - Two subcarriers - One BS

In order to gain deeper comprehension on the different approaches to thesubcarrier allocation process, this section examines the simple scenario of asingle BS that serves two users (u = 1 and u = 2). The BS has two subcarriers(n = 1 and n = 2) and each user requires one subcarrier (s1 = s2 = 1). For

convenience, it is assumed that R1 = R2 = ∆f bpsHz

, so that 2Ru

su∆f −1 = 1, u =1, 2.

In this case, the subcarrier allocation algorithm must choose between thetwo possible allocations:

– User 1 gets subcarrier 1 whereas user 2 gets subcarrier 2 (PT = 1T1,1

+ 1T2,2

– User 1 gets subcarrier 2 whereas user 2 gets subcarrier 1 (PT = 1T1,2

+ 1T2,1

where PT = P1+ P2 stands for the total average transmit power, and Pu, u =1, 2 denotes the average power transmitted for user u. The index of BS hasbeen omitted for clarity.

Apart from the Hungarian and the proposed auction algorithm, two morealgorithms are considered in this section, namely the adaptive and the randomone. The adaptive algorithm was presented in [35]. In this approach, each user

takes the token in a round-robin fashion, based on a sequence defined bytheir introduction to the system. When the user has the token, he is allocatedthe best, in terms of nCINR, subcarrier from the subcarriers that have notbeen allocated yet, and then passes the token to the next user. The processis ended up when each user u has been allocated the predefined number ofsubcarriers su. The algorithmic complexity of this approach is O

[28].Both the Hungarian and the adaptive algorithms require full knowledge ofChannel State Information, ie. it is necessary for the BS k to know the nCINRfor each user u ∈ Uk on each subcarrier n ∈ N . In order to avoid this overheadin uplink due to feedback, a widely used solution is the random distributionof subcarriers among users, in order to achieve averaging of channel quality.In this approach, each user is allocated su subcarriers randomly selected fromthe set of available subcarriers, an algorithm with complexity O (N) [28].

Table 1 describes how the subcarrier allocation is performed and the totalaverage power is derived for each of the algorithms under consideration. Asfar as notation is concerned, nu, u = 1, 2 is the subcarrier allocated to user u,while p is a binary random variable, equal to 0 or 1.

Table 1 Subcarrier allocation algorithms for the simple scenario of two users - two subcar-riers - one BS

Hungarian PT ← min(

+ 1T2,2

, 1T1,2

+ 1T2,1

Adaptive n1 ← argmaxn T1,n

n2 ← 3− n1

PT ← 1T1,n1

+ 1T2,n2

Auction n1 ← argmaxn T1,n, n2 ← argmaxn T2,n

if 1T1,n1

≥ 1T2,n2

if n2 = n1 then n2 ← 3− n1

if n1 = n2 then n1 ← 3− n2

PT ← 1T1,n1

+ 1T2,n2

Random PT ← p(

+ 1T2,2

+ (1− p)(

+ 1T2,1

In order to evaluate the performance of the algorithms, 105 Monte Carloiterations of the subcarrier allocation process have been conducted, accordingto the parameters given in Table 2, unless it has already been defined differ-ently. The two subcarriers, for which the two users anticipate, are randomlychosen from the set of subcarriers.

The remark drawn from fig. 2 is the ranking of the algorithms on thetotal average transmit power. The optimal Hungarian algorithm requires theleast power, the auction algorithm follows, very close to the optimal curve,the adaptive algorithm comes third, and finally, the random algorithm is theone that requires the most power. Furthermore, from the numerical results,it is shown that the adaptive algorithm performs the same allocation as theoptimal one in 75% of the cases, this percentage raises to 91% for the auction

−20 0 20 400

PT (dBm)

HungarianAdaptiveAuctionRandom

Fig. 2 Cumulative Distribution Function of total average transmit power for subcarrierallocation algorithms for the simple scenario of two users - two subcarriers - one BS

algorithm, whereas it falls to 50% for the random algorithm, as it is statisticallyexpected. For further analysis and comparative results, cf. [36].

4 Bit Loading and Power Control based on Non-Cooperative

Considering the power control from another perspective, if the initiative isleft to the users, it is expected that each user would aim at minimising hisown transmit power, while meeting his requested BER. This behaviour canbe imposed by a power pricing scheme, similar to the one described in theprevious section, where the BSs charge per transmit power unit. Formally,this could be written as a minimisation problem for each subcarrier n,

minimise Pn,ki∀i ∈ In (18)

subject to

Gui,n,kiPn,ki

∑κj=1j 6=i

Gui,n,kjPn,kj

+ σ2≥ γui,n,ki

∀i ∈ In (19)

For convenience and clarity, a simplified notation is used in the following,

Pi = Pn,ki, Gi,j = Gui,n,kj

Γi = Γui, bi = bn,ki

, γi = γui,n,ki= Γi

2bi − 1)

Furthermore, in order to rewrite the per subcarrier minimisation problem in amore compact form, we define the interfering power vector P−i and interferinggain vector Gi,−i as,

P−i = [Pj ]κj=1j 6=i

= [P1, . . . , Pi−1, Pi+1, . . . , Pκ]T

Gi,−i = [Gi,j ]κj=1j 6=i

= [Gi,1, . . . , Gi,i−1, Gi,i+1, . . . , Gi,κ]

Utilising this notation, constraint (19) can be written as

Pi ≥ Pmini =

γiGi,i

Gi,−iP−i + σ2)

, ∀i ∈ In (20)

where Pmini = Pmin

i (P−i) stands for the minimum transmit power to meetBER constraint. The user on the one hand has to meet constraint (19), whileon the other hand tries to minimise his transmit power. This can be inter-preted as an attempt to reduce the distance between Pi and Pmin

i . In or-der to follow the typical game theoretic notation, where each player triesto maximise his utility function, the utility function for each user is definedas vi(P) = −

∣Pi − Pmini (P−i)

∣. Furthermore, the action of each user is thetransmit power he asks for from his serving BS, Pi ∈ Pi = R

+. Summarising,the power control problem on each subcarrier can be formulated as a non-cooperative game G = [{ui}κi=1, {Pi} , {vi(·)}], where the parameters stand forthe players, the strategy space and the utility functions respectively.

Each user maximises his utility function by playing his minimum transmitpower, ie. when (20) is satisfied with equality. As a result,

P ∗i = br (P−i) =

γiGi,i

Gi,−iP−i + σ2)

, ∀i ∈ In (21)

where br (P−i) stands for the best response of user ui to the actions P−i ofhis opponents. In a Nash equilibrium point, all users’ actions are their bestresponses to the actions of their opponents [37,38]. As a result, in equilibrium,

P ∗i = br

P∗−i

=γiGi,i

Gi,−iP∗−i + σ2

, ∀i ∈ In (22)

Writing (22) in matrix form, it derives that,

P∗ = DFP∗ + v (23)

From (23), it becomes clear that if the sequence{

P(l)}∞l=0

defined by,

P(l) = DFP(l−1) + v (24)

converges, it converges to the Nash equilibrium of the power control game.As derived from [31, Theorem 7.19], the aforementioned sequence converges

to the unique Nash equilibrium if and only if ρ (DF) < 1. However, if the valueof ρ (DF) is to be used as the convergence criterion, a centralised control entityis necessary for collecting from all the co-channel users the information neededto built matrices D and F, calculating the eigenvalues of DF and decidingwhether ρ (DF) < 1 holds or not. On the contrary, taking into considerationthe fact that ρ (DF) ≤ ‖DF‖ for any natural norm ‖·‖ [31, Theorem 7.15], theconvergence condition ρ (DF) < 1 can be replaced by the stricter condition‖DF‖ < 1, which is sufficient but not necessary. In order to guarantee thedistributed nature of power control, it is necessary to employ a natural norm

that considers only the lines of the matrix DF (own CSI information). Anatural norm that meets this constraint is the infinity norm ‖·‖∞, defined as,‖A‖∞ = max1≤i≤κ

j=1 |Ai,j |. Consequently,

‖DF‖∞ = max1≤i≤κ

γiGi,i

j=1j 6=i

Gi,j (25)

In order to achieve convergence based on the infinity norm, it is sufficient todemand,

γiGi,i

j=1j 6=i

Gi,j < 1, ∀i ∈ In (26)

From (26),

γi <Gi,i

∑κj=1j 6=i

(15)⇔

bi < blimi = log2

1 +Gi,i

∑κj=1j 6=i

Concluding, if each user ui respects the limit blimi = blimui,n,kiposed by (27)

during the bit loading phase, then the power control game converges to theunique Nash equilibrium.

Furthermore, from (23) and (24), using the infinity norm, it can be shownthat,

∥P∗ −P(l)

∞≤ ‖DF‖l∞

∥P∗ −P(0)

∞(28)

proving, thus, that the convergence to the Nash equilibrium is exponentiallyfast. The error between the power calculated at current iteration of (24) andthe equilibrium one is bounded as,

∥P∗ −P(l)

∞≤ ‖DF‖∞

1− ‖DF‖∞

∥P(l) −P(l−1)

∞(29)

which is also valid element-wise, in case of infinity norm,

∣P ∗i − P

∣≤ 2bi − 1

2blimi − 2bi

(l)i − P

(l−1)i

∣(30)

The iterative process (24) and the power control game will terminate when allusers stop to update their transmit power, based on a predefined upper boundfor the estimation error, as designated by (30).

4.1 Channel Probing

When a BS k decides to use a subcarrier n that was not in use, the set ofco-channel BSs on subcarrier n is altered and, as a consequence, the currentestimation of blimu,n,k is invalidated for all the co-channel users. In order to main-tain the distributed nature of the algorithm, all the co-channel users must beinformed that BS k is intended to use subcarrier n and respond appropriatelyto this intention. The employed scheme for tackling this problem is as follows:

– The BS k transmits on all the subcarriers that were not in use up to now.For this procedure a low spectral efficiency modulation should be used,for example Binary Phase Shift Keying (BPSK), so as to minimise powerwaste.

– The co-channel users sense that the number of co-channel BSs has changedand re-calculate the limit blimu,n,k, based on the updated set of co-channelBSs.

– If (27) is violated for a user on subcarrier n, this user asks his serving BSto block the usage of this subcarrier n from BS k.

– The BS that receives such a message from one of its serving users, forwardsthe message to BS k through the backbone network.

– The BS k accepts the request and does not allocate subcarrier n.

4.2 Algorithm Presenentation

The proposed distributed bit loading and power control algorithm is given inAlg. 2. The algorithm follows the greedy approach of the centralised algorithmof [5]. The main difference is that the proposed algorithm is performed by eachuser independently, by exploiting the CSI knowledge on his own subcarriers.The Input of the algorithm indicates all the necessary quantities that must bedefined prior to the execution of the distributed bit loading and power control.It must be clarified that, thanks to the distributed nature of the algorithm,there is no need for communicating all this information among BSs and users.On the contrary, much of the CSI information is exploited only locally, atusers’ end.

Initially, all the subcarriers of the BS k are unloaded (ln. 1-2). The al-gorithm is executed independently by each user of BS k (ln. 3). While therequested bit rate has not been reached yet for the user u under consideration(ln. 4), he tries to increase the modulation level for each of his allocated sub-carriers from bn,k to bn,k+2. The set Cf

u (ln. 5) is the subset of the subcarriersallocated to user u that it is feasible to increase their modulation level by 2bits per symbol, ie. the new modulation level does not exceed the maximummodulation level (bn,k+2 ≤ bmax) or the convergence limit (bn,k+2 < blim

u,n,k).

If Cfu is empty (ln. 6), the requested bit rate cannot be provided, as there

is no subcarrier able to increase its modulation level. As a result, the BS krejects the newly added user u (ln. 7), since the latter cannot be served with-out causing problems to already accepted users. For the subcarriers in Cf

u , the

increment in transmit power is calculated in order to support the increasedmodulation level (ln. 9). The increment in transmit power is calculated as,

∆Pn = Pn,k

bn,k + 2)

− Pn,k

=2bn,k+2 − 1

Tu,n,k

− 2bn,k − 1

Tu,n,k

= 32bn,k

Tu,n,k

The necessary increment in transmit power is calculated considering the CSIinformation available at this phase, before the execution of power control. Thesubcarrier requiring the least transmit power increment is selected (ln. 10) andits modulation level is increased by 2 bits per symbol (ln. 11).

The bit loading is followed by the power control process. For each loadedsubcarrier (ln. 12), a power control game is initiated. Each co-channel user(ln. 16) updates his transmit power, based on the interference he undergoes(ln. 17). As previously noticed, it is not necessary for the user to be able todistinguish the sources of interference but rather calculate the interference asa whole. Thence, the co-channel BS checks whether it is possible to transmitthe required power by checking the maximum available power limitation (ln.18). If the required power overcomes this limit, the BS k is asked to rejectuser u (ln. 19). The process is terminated when the users has approximatedthe equilibrium by a factor ǫ, by applying the convergence criterion (30) (ln.20). Specifically, herein, it is chosen to evaluate the convergence based on

the relevant error

n,ki−P

(l)n,ki

P(l)n,ki

rather than the absolute error of (30). The

iteration counter l (ln. 13,15) is not necessary in practice, but it is used hereinfor presentation reasons. In fact, each user responds spontaneously, throughthe process of best responses, to the increment of undergoing interference upto achieving the desired convergence precision.

Concluding, it must noticed that the computational complexity of the pro-posed, distributed algorithm can be described as O

, whereas the com-

plexity for the centralised algorithm of [5] is O(

5 Simulation Results

5.1 Parameters

The basic parameters of the simulated OFDMA network are summarised inTable 2, mainly derived from [39]. At this point, the approximation of equi-librium in power control game must be addressed. Since lp bits are used fortransmit power report, 2lp different states can be encoded. Due to large cellradius, large variations are expected in transmit power, thus a direct associa-tion between transmit power value and a codeword would lead to low accuracyin power reporting. Instead, differential reporting is adopted [1]. Initially, the

Algorithm 2: Distributed Bit Loading and Power Control Algorithm

Input: k, Uk, Cu,∀u ∈ Uk, Ru, ∀u ∈ Uk, blimu,n,k,∀u ∈ Uk, ∀n ∈ Cu,

Tu,n,k, ∀u ∈ Uk, ∀n ∈ Cu, γu,n,k ,∀u ∈ U ,∀n ∈ N , ∀k ∈ K,Gu,n,k, ∀u ∈ U ,∀n ∈ N ,∀k ∈ K, Pn,k,∀n ∈ N ,∀k ∈ K, bn,k ,∀n ∈ N ,∀k ∈ K,blimu,n,k

,∀u ∈ U ,∀n ∈ N , ∀k ∈ KOutput: bn,k, ∀u ∈ Uk, Pn,k, ∀n ∈ N ,∀k ∈ K

1 bn,k ← 0, ∀n ∈ N2 Pn,k ← 0, ∀n ∈ N3 foreach u ∈ Uk do

4 while Rou < Ru do

5 Cfu ←{

n ∈ Cu :(

bn,k + 2 ≤ bmax

bn,k + 2 < blimu,n,k

6 if Cfu = ∅ then7 Reject u

8 else

9 ∆Pn ← 3 2bn,k

Tu,n,k,∀n ∈ Cfu

10 n← argminn∈Cf

u∆Pn

11 bn,k ← bn,k + 2

12 foreach n ∈ Cu do

13 l← 014 repeat

15 l← l+ 116 foreach i ∈ In do

17 P(l)n,ki← γui,n,ki

Gui,n,ki

∑κj=1j 6=i

Gui,n,kjP

(l−1)n,kj

18 if∑N

n=1 P(l)n,ki

> PmaxBS

19 Reject u

20 until

2bn,ki −1

2blimui,n,ki −2

P(l)n,ki

−P(l−1)n,ki

P(l)n,ki

< ǫ,∀i ∈ In)

user sends explicitly his required power (in dBm), based on a scale of length lp.In the following iterations, he sends the required increment in transmit power(in dB), relevant to the previous value. By choosing to encode changes up to1 dB, the power can be changed by a step of 2−lp dB. Considering this limi-tation, it becomes clear that the Nash equilibrium cannot be approximated asclose as desired, but the approximation is subject to this limitation of quan-

tisation. From (30), the relative error

n,ki−P

(l)n,ki

P(l)n,ki

< ǫ can be limited up to

the point the desirable accuracy ǫ satisfies ǫ ≥ 1 − 10−2−lp

10 . In the following,the equality is chosen in order to achieve the maximum accuracy, thus, withlp = 6 bits, ǫ = 3.6 · 10−3.

Table 2 OFDMA Network Simulation Parameters

Number of Cells K = 7

Cell Radius R = 1 km

Central Frequency fc = 2.5 GHz

Available Bandwidth BW = 10 MHz

Number of OFDM Subcarriers N = 128

Subcarrier Spacing ∆f = 78.125 kHz

Modulation Types

QPSK (bmin = 2 bitssymbol

16-QAM (b = 4 bitssymbol

64-QAM (bmax = 6 bitssymbol

Requested Bit Rate Ru = 2048 kbps

Bit Error Rate Peu = 10−4

BS maximum transmit power PmaxBS = 43 dBm

BS Antenna Gain GTxBS = 16 dBi; Omnidirectional

MS Antenna Gain GRxMS

= 0 dBi; Omnidirectional

Propagation Model COST-Hata-Model [40]

BS Antenna Height hBS = 32 m

MS Antenna Height hMS = 1.5 m

Shadowing σshadowing = 8.9 dB

Channel Model ITU Pedestrian B [41]

Noise Spectral Power Density N0 = −174 dBmHz

MS Noise Figure FRxMS = 7 dB

Report length lp = 6 bits [1, Sec. 8.4.12.3]

5.2 Results

The following results have been obtained through Monte Carlo simulations,with 5000 iterations and the presented quantities are average values of the cor-responding metrics. Although the operation of the multi-cell OFDMA networkhas been simulated, in the following, the metrics presented correspond to thecentral cell of the network (k = 1), in order to study the performance of a cellthat operates under realistic interference conditions. As far as notation is con-cerned, ’Hungarian-CNT’ and ’Auction-CNT’ stands for the Hungarian andthe proposed auction algorithm with centralised bit loading and power control,whereas ’Hungarian-DST’ and ’Auction-DST’ represent respectively the twoalgorithms with the proposed distributed bit loading and power control.

Offered bit rate Fig. 3 depicts the offered bit rate of the central cell k Ro =∆f

n∈N bn,k versus the requested bit rate R =∑Vk

u=1 Ru for the differentsimulated RRM schemes. R increases as the number of users Vk asking foradmittance in cell k increases from 1 to V lim

k= 20. Comparing the subcarrier

allocation algorithms under the same power control scheme, it can be notedthat the auction algorithm closely follows the optimal Hungarian algorithm.Specifically, under centralised power control, the auction algorithm reaches atleast the 94.24% of the offered bit rate of the optimal algorithm, whereas thispercentage is 91.59% for the distributed power control. Comparing the dif-ferent power control schemes under the same subcarrier allocation algorithm,

2 6 10 14 18 22 26 30 34 38 422

R (Mbps)

Hungarian−CNTAuction−CNTHungarian−DSTAuction−DST

Fig. 3 Offered bit rate vs. requested bit rate

it is noticed that the distributed scheme is inferior to the centralised one atsaturation by 3.31Mbps (79.20%) for the Hungarian algorithm and 3.45Mbps(76.98%) for the auction algorithm. As previously noticed, the distributedscheme tends to be more conservative in power consumption. Hence, the apriori exclusion of specific subcarriers, because they violate the convergencecriterion, reduces the available bandwidth of the cell, which leads eventuallyto reduction in offered bit rate. The strictness of the convergence criterion,imposed by the partial knowledge of CSI, causes the inferior performance ofthe distributed power control compared to the centralised scheme.

Concluding the analysis on the offered bit rate, from the numerical results,it is shown that when the introduction of new users is terminated at V lim

k =20, ∀k ∈ K and R = 40.96Mbps, the rate of change of the offered bit rate∆Ro

∆Rfor the simulated schemes varies from 3.38% to 4.70%, indicating, thus,

that the network could hardly serve more incoming traffic, in other words thenetwork is saturated.

Transmit Power The transmit power of the central cell P t =∑

n∈N Pn,k

versus the offered bit rate Ro is depicted in fig. 4. Comparing the Hungarianand auction algorithms under the same power control scheme, it can be noticedthat the auction algorithm is slightly worse than the Hungarian algorithm.Specifically, the auction algorithm requires at worst case 0.46dB more transmitpower than the Hungarian algorithm under centralised power control, whereasthis difference is 0.15dB under distributed power control. Comparing the twopower control schemes, it is clear that the distributed power control leads tolower power consumption than the centralised scheme. As the offered bit rateincreases, the distributed algorithm, more conservative on power consumption,increases the required transmit power per offered bit rate unit less rapidly thanthe centralised scheme, leading, however, at the same time to lower offered bitrate. Specifically, the Hungarian algorithm under distributed power control

2 4 6 8 10 12 14 1626

Ro (Mbps)

Pt (dB

Fig. 4 Transmit power vs. offered bit rate

requires up to 3.34dB less transmit power than the centralised scheme. Thisdifference is 2.70dB for the auction algorithm.

A final remark concerns the gap between the total transmit power at satu-ration and the available transmit power of the cell Pmax

BS = 43dBm. From thenumerical results, it is shown that there is a power surplus of approximately3.5dB for the centralised scheme and 9.5dB for the distributed one. This powersurplus could lead to the employment of lower power radio frequency (RF) am-plifiers and thus lower cost, and, if the available spectrum can also be usedsecondarily by cognitive networks, this could lead to lower cost for bandwidthoccupation for the network operator [42].

2 4 6 8 10 12 14 160

Ro (Mbps)

Fig. 5 Uplink overhead vs. offered bit rate

Uplink overhead Fig. 5 depicts the total overhead caused by the feedback infor-mation for each of the examined schemes. This overhead includes the requiredinformation that the users send back to their serving BS, both at subcarrierallocation and bit loading and power control processes. A first remark refersto the fact that the Hungarian algorithm requires greater amount of feedbackinformation in comparison to the auction algorithm, whereas the gap betweenthem broadens as the offered bit rate increases. This gap is due to the fullCSI knowledge that the Hungarian algorithm requires, contrary to the partialknowledge required by the auction algorithm. Comparing the schemes underthe same subcarrier allocation algorithm, first for the Hungarian algorithm,it is observed that the centralised power control requires greater amount offeedback, compared to the distributed scheme, a gap which broadens as theoffered bit rate increases. The smaller requirements of the distributed powercontrol are attributed to two facts. The first one concerns the subcarrier allo-cation process. Since the set of available subcarriers is smaller than the one inthe case of the centralised scheme, the algorithm feedbacks a smaller amountof CSI. The second fact is related to the bit loading and power control pro-cess. The centralised scheme requires the users to feedback the channel gainon their subcarriers, as perceived from all the BSs. At the distributed scheme,the user simply informs the serving BS about the modulation level for eachallocated subcarrier and the required transmit power per allocated subcarrierfor each iteration of the power control game. As far as the auction algorithmis concerned, it is noticed that the curve of the distributed scheme is slightlyhigher, compared to the one of the centralised scheme. Considering that atthe subcarrier allocation process the algorithm requires reporting CSI only forthe required subcarriers, the reduction of available subcarriers does not seemto affect the uplink overhead. On the contrary, the power control process con-tributes mainly to the increment of uplink overhead. Hence, from fig. 5, it isshown that the overhead caused by the feedback of the modulation level foreach allocated subcarrier and, mainly, the iterative feedback of the updatedtransmit power during the power control game results in a greater amount ofoverhead overall.

Number of auctions Fig. 6 depicts the required number of auctions for thecompletion of the subcarrier allocation process. A first remark is that thenumber of auctions increases linearly with the offered bit rate, with a slopeapproximately 0.38 auctions per Mbps. Comparing the two curves, a slightincrement in number of auctions occurs for the distributed power control. Thisincrement is attributed to the smaller set of available subcarriers. A smaller setleads to more overlaps among the required subcarrier sets, increasing, thus,the number of the necessary auctions. This slight increment in the numberof auctions contributes, in minor, to the increment of feedback, as alreadyexplained.

Power control iterations The number of iterations required for the convergenceof the power control game is depicted in Fig. 7. First of all, it can be noticed

2 4 6 8 10 12 14 161

Ro (Mbps)

Fig. 6 Number of auctions vs. offered bit rate

2 4 6 8 10 12 14 160

Ro (Mbps)

Fig. 7 Power control iterations vs. offered bit rate

that the number of iterations varies form 2.4 to 14.4. It should be noted thatthe minimum number of iterations is 2, since in the first iteration the user ofcell k defines his power and at the next iteration the co-channel users fromthe other cells respond by updating their transmit power. In the followingiterations, all the users respond simultaneously to the updated interferenceenvironment. The upper limit that exceeds 14 iterations is mainly definedby the desirable accuracy in the approximation of the equilibrium. In thesimulations conducted, the maximum accuracy was demanded, leading, thus,to a relatively large number of iterations. Finally, the relatively large numberof iterations is the major factor that burdens the uplink overhead, due to thetransmission of the updated power at each iteration.

6 Conclusions

In this work, an RRM framework, tackling the MA problem for the downlinkof multi-user multi-cell OFDMA networks, has been presented. Specifically,the present work proposes a subcarrier allocation algorithm based on com-binatorial auctions and a bit loading and power control algorithm based onnon-cooperative games. Firstly, the auction-based subcarrier allocation algo-rithm is shown to be an appealing alternative to the optimal Hungarian sub-carrier allocation algorithm. It shows comparable performance with slightlylower offered bit rate and slightly higher required transmit power, whereas itis of lower computational complexity and alleviates the overhead on reverselink, since it requires less feedback information. Secondly, the main advantageof the proposed bit loading and power control scheme is the fact that it canbe executed in a distributed way, transferring, thus, part of the complexity tothe users end and resulting in a reduced feedback overhead. The distributednature of the proposed solution contributes also to the scalability of the net-work, since there is no need for a single entity to control and coordinate ina centralised way all the network links. The proposed scheme achieves lowertransmit power per offered bit rate unit but the distributed nature of the algo-rithm results in a lower total offered bit rate, because of the partial knowledgeand exploitation of the channel state information.

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A radio resource management framework for multi-user multi-cell OFDMA networks based on game theory

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