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MessagePassingResourceAllocationfortheUplinkofMulti-CarrierMulti-FormatSystems

ARTICLEinIEEETRANSACTIONSONWIRELESSCOMMUNICATIONS·JANUARY2012

ImpactFactor:2.5·DOI:10.1109/TWC.2011.112311.101766·Source:DBLP

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MarcoBelleschi

Ericsson

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MarcoMoretti

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A Min-Sum Approach for Resource Allocation inCommunication Systems

Abstract—This paper considers distributed protocol designfor resource allocation (RA) problems. We propose a fullydecentralized RA scheme based on the max-sum message passing(MP) approach in which each message is the solution of smalldistributed allocation problems. Due to the presence of cyclesin the network graph, the MP routine may not converge to afixed point. To this end, we introduce ReMP, a reweighted MPalgorithm that perturbs the standard max-sum algorithm bysuitably reweighting messages. ReMP distributes the computa-tional effort of achieving an optimal RA among nodes. Such afeature makes ReMP particularly attractive in wireless networks.In particular, in single-cell scenarios we prove that ReMP allowsthe convergence to a fixed and provably optimum point withoutemploying any central controller.In this paper, we also propose a novel method that leveragesReMP to deal with RA in multi-cellular environment.

I. INTRODUCTION

In this paper, we propose a distributed and low-complexityresource allocation (RA) scheme, based on the max-summessage passing (MP) algorithm. RA problems often arisein radio resource management, combinatorial optimization,communication theory, organizational management, facilitylocation, machine learning, auction modeling. The formulationof RA as a MP problem entails exchanging messages, thatrepresent the solution of small distributed allocation problems,between nodes defined over a specific graphical model (GM)such as Bayesian networks, pairwise Markov random field orfactor graphs. Such feature makes MP particularly attractive inthose applications in which the presence of a central controlleris not desired. In particular, in this paper we investigatehow MP can find a natural application in radio multi-carriersystems based on the orthogonal frequency multiple access(OFDMA) scheme such as the 3GPP Long Term EvolutionAdvance (LTE-A) standard. In LTE-A, because of the lackof a central controller [1], it is desirable that the base stationsautonomously allocate resources such as subcarriers and trans-mission powers. In this scenario, the radio resources can beassigned on the basis of some optimality criterion and thustaking advantage of the so called multi-user diversity [2].Inspired by statistical physics, MP paradigms (e.g. sum-product, max-product, min-sum) have became a powerful toolthat efficiently adapts to manifold domains: data-mining [3],image processing and computer vision [4], [5], [6], bioinfor-matics [7], [8], signal processing [9], [10], error-correctingcoding (e.g. LDPC and turbo codes) [11], [12] etc. A commonissue to all of these applications is to make probabilisticinference of the unobserved variables given the observed ones,i.e. the so called marginalization problem that can be in theory

a NP-hard task (e.g. the computation of maximum a posteriori(MAP) assignment) [13]. It is well known that the convergenceof MP to the correct and optimal solution strictly dependson the structure of the underlying GM, e.g. the girth of thegraph 1. In other words, while the convergence is ensured insingle-cycle and cycle-free graphs (e.g. Markov chains andtrees) [14], MP is no longer guaranteed to output the correctsolution in loopy graphs. As such, considerable research effortsin recent years have tried to derive convergence criteria for MPalgorithms in arbitrary graphs [15], [16], [13], [17].In this paper, we present ReMP a reweighted version of thestandard max-sum MP algorithm. ReMP belongs to the familyof reweighted MP algorithms firstly applied to the sum-productalgorithm by Wainwright at. al. in [18] and successivelyextended to the max-product in [19]. A significant effort in thispaper is devoted to prove that in the single-cell (isolated-cell)case ReMP converges in linear time to fixed messages (andhence to fixed marginals) when a suitable weight is applied.As a consequence, the optimal assignment of the original RAformulation is given as an output of the ReMP procedure. Tothe best of the authors’ knowledge this is a novel result in thisarea. Moreover, since the algorithm converges not only in theassignment but also in the exchanged messages, no stoppingcriteria for the algorithm need to be devised.To this end, in order to better assess the practical feasibilityof the algorithm, the paper also provides some insights on themapping of the MP algorithm on the current LTE-A standard.Building on this insight, in a multi-cell scenario, we devisea distributed algorithm that is ReMP-based. Specifically, thealgorithm (X-ReMP) combines a cross-cell signaling and theordinary ReMP routine that still runs within each cell. Therationale is that the cross-cell signaling between cells aidsReMP to deal with the inter-cell multiple-access interference.Results show that X-ReMP outperforms similar algorithmsand achieve faster convergence. TO BE REFINED AFTEREDITING THE RESULT SECTION.

The paper is organized as follows. The next section for-mulates the RA task as a rate maximization problem in asingle cell multicarrier system. Next, Section III developsa message passing scheme that assigns subcarriers to userterminals in a distributed fashion. Section IV presents theproposed reweighted MP scheme and develops an algorithmthat solves the multicarrier RA problem. In Section V, weapply ReMP in the multicell environment by suitably takinginto account the impact of intercell interference in the objective

1The length of the shortest cycle contained in a graph.

function. In this section we also show how the messagesof ReMP can be naturally mapped into the existing uplinkscheduling messages of an LTE system. Section VI focuseson the convergence properties of ReMP, while Section VIIpresents and discusses numerical results. Finally, Section VIIIconcludes the paper.

II. PROBLEM FORMULATION

We consider a cellular system that leverages an OFDMAscheme. Although the proposed method can be easily extendedto a downlink resource scheduling feature, we believe thatthe powerful of MP naturally adapts to the uplink. In fact,unlike the downlink, in the uplink the different UEs potentiallycompete to each other in order to be assigned to thoseresources that satisfy a given rate request. In this perspective,MP operates as a coordination tool that negotiates between theUEs in the network without the need for a central controller.As such, in the sequel we derive a mathematical frameworkthat is tailored for the uplink.We consider an LTE-A-like system featuring F availableOFDMA resources, i.e. physical resource blocks (PRB), be-longing to the set F . The system is populated by N UEsbelonging to the set N . UEs are distributed over K cellsbelonging to the set K. A generic cell k ∈ K contains N(k)UEs belonging to the set N (k). Each PRB spans 180KHzbandwidth and is made up of 12 contiguous subcarriersspanning 15KHz each to be put in the results section. UEsi ∈ N (k) are labeled with i = 1, . . . , N(k) and PRB j ∈ Fwith j = 1, . . . , F as in Figure 1. The rationale is that eachpair (i, j) in the GM represents a potential usage of PRB jby user i with a reward Ri,j .Our MP strategy originates from an optimization problemwhose objective function is a sum rate maximization. Hence,we consider the Shannon capacity as a measure of the achiev-able bit rate, and we represent the reward as the spectralefficiency

R(k)i,j = log2(1 +

P(k)i,j G

(k)i,j

I(k)i,j +N0

) (1)

where P(k)i,j is the transmitting power of UE i ∈ N (k), G(k)

i,j

is the channel gain between the UE i and the serving eNBin cell k when transmitting on PRB j, Ii,j is the interferenceperceived by user i on PRB j, and N0 is the thermal noise. Weassume a reuse-1 system in which PRBs can be fully reused byneighboring cells. As a consequence, it is possible to formulatethe interference perceived by any user i ∈ N (k) on any PRBj ∈ F as

I(k)i,j =

∑q∈K,q =k

∑v∈N (q)

P(q)v,jG

(k)v,j (2)

Thus, in line with the work [22], we disregard the effects oflink adaptation and power control and we focus on rate controlonly. The power level used by each user over each assignedPRBs are assumed to be pre-determined via slow powercontrol and can be regarded as fixed inputs to our schedulingalgorithms that eventually decides the PRBs allocation on a

much finer timescale [?].In this Section, we limit our analysis to the single-cell caseand derive the MP algorithm accordingly. We will discuss themulti-cellular case in Section V. Thus in the sequel, for theease of notation, we omit the cell index.Let rmax

i ∈ N and rmini ∈ N be nonnegative integer values

corresponding to the maximum and minimum number of PRBto be allocated to each node. For example, in the LTE-A UL,the PRBs that need to be allocated to a user equipment (UE)may be lower and upper bounded by some minimum rateconstraint and a maximum power budget respectively. We arenow in the position to formulate the problem of RA as follows

maximize∑

i,jRi,jxi,j (3)

subject to∑

jrmini ≤ xi,j ≤ rmax

i ∀i ∈ N , (i, j) ∈ E

(C1)∑ixi,j ≤ 1 ∀j ∈ F , (i, j) ∈ E

(C2)xi,j ≥ 0 ∀i ∈ N , j ∈ F , (i, j) ∈ E

(C3)

By exploiting the total unimodularity property of the matrixconstraints [21], the allocation variables xi,j in Problem (3) areinteger variables. Specifically, by imposing constraints (C2),xi,j can be either 1 or 0, indicating whether node i ∈ Nis assigned to node j ∈ F or not. As shown in [22] byappropriately introducing dummy PRBs and dummy sub-usersin the GM, we can always convert (C1) in equality constraints.Thus, in the sequel for the sake of simplicity, we considerconstraints (C1) as

∑j xi,j = ri with

∑i ri = F .

A. Resource allocation in multicarrier systems

In cellular wireless systems, Problem (3) can be interpretedas a cell-based optimization problem, where nodes i ∈ Nrepresent the users, nodes j ∈ F the available PRBs at thebase station (BS), and ri the number of requested PRBs.Let us focus on multi-carrier systems, e.g. OFDMA, in whichthe PRBs to be assigned are the available subcarriers. In thissetting, the weight Ri,j is the rate associated to each user-subcarrier pair in order to achieve a given target spectralefficiency. Hence, Problem (3) is a rate maximization problemin which constraints (C1) implicitly satisfy the request of eachuser in terms of data rate, and constraints (C2) represent theresource utilization constraints, i.e. a given subcarrier can beexploited by one user at most to avoid intra-cell interference.In the following section, we propose a MP approach that solvesto optimality Problem (3) by distributing the computationalload among users and BS, thus avoiding the usage of a centralcontroller.

III. A MESSAGE PASSING APPROACH

To solve Problem (3) via MP, we reformulate it as aminimum cost problem, where the unfulfillment of constraintsoutputs an infinite cost. Let us introduce the allocation vectorx = [x1,1, . . . , xN,F ] and define the cost functions Wi(x)

N 2 1

1 2 3 F-1 F

Fig. 1. Graph topology corresponding to a RA scenario.

and Cj(x) that account for the constraints (C1) and (C2) inProblem (3) respectively. For the sake of simplicity, withoutloss of generality, we assume a complete bipartite graph, i.e.each subcarrier j can be assigned to any of the N users andviceversa.In order to incorporate the rate constraints (C1), Wi(x)becomes

Wi(x) =

{ ∑j Ri,jxi,j if

∑j xi,j = ri

∞ otherwise(4)

Similarly, to deal with the resource utilization constraints (C2),Cj(x) is defined as follows

Cj(x) =

{0 if

∑i xi,j ≤ 1

∞ otherwise (5)

From Equation (4) and (5), the RA problem in (3) can berewritten as

maxx

(∑jCj(x) +

∑iWi(x)

)(6)

whose optimal allocation vector x can be computed as

x = argmaxx

(∑jCj(x) +

∑iWi(x)

)(7)

The marginalization rule for the max-sum algorithm can bederived from [11] and involves the computation of eachcomponent of x as follows

xi,j = argmaxxi,j

[max∼xi,j

(∑jCj(x) +

∑iWi(x)

)](8)

where notation max∼xi,j

denotes the maximum over all the com-

ponents of x except xi,j . In Equation (8) the term

τi,j(xi,j) = max∼xi,j

(∑jCj(x) +

∑iWi(x)

)(9)

represents the marginal of Equation (6) with respect to thevariable xi,j . The marginalization (9) can be then distributivelycomputed by appealing to the MP dynamic in which each nodein G conveys the solution of a local problem to one otherby passing information messages according to the max-sumparadigm [23].In the sequel, we focus on the generic square node f ∈ F andthe circle node n ∈ N in Figure 1, i.e. the generic subcarrierand user. In line with the sum-product algorithm for bitwiseMAP channel decoding presented in [11], it is possible to

demonstrate that the message delivered by user n to subcarrierf embedding the cost function for the rate constraints (4) is

mn,f (xn,f ) =

Rn,fxn,f +maxxn,j

( ∑j,j =f

Rn,jxn,j + mj,n(xn,j)

)subject to

∑j xn,j = rn

(10)

where mj,n is the reverse direction message traversing theedge from subcarrier j to user n.Analogously, the message from subcarrier f to user n em-bedding the cost function for the resource utilization con-straints (5) is

mf,n(xn,f ) = maxxi,f

∑i,i=n

mi,f (xi,f )

subject to∑

i xi,f ≤ 1(11)

To elaborate, the message mn,f (1) piggybacks the informationrelative to the use of subcarrier f with transmitting rate Rn,f

by the user n, while mn,f (0) accounts for the lack of transmis-sion on subcarrier f by user n. Hence, each user actually de-livers a 2-sized real-valued vector mn,f = [mn,f (1),mn,f (0)]and each subcarrier the vector mf,n = [mf,n(1), mf,n(0)].From Equation (10) and (11), the marginal τn,f in Equation (9)can be evaluated as in [23] by invoking the generalizeddistributive law [24], i.e. by exploiting the formal equivalencebetween max

∑and

∑max

τn,f (xn,f ) = mn,f (xn,f ) + mf,n(xn,f ) (12)

Hence the computed allocation variable is

xn,f = argmaxxn,f

{τn,f (xn,f )} (13)

At each iteration of the MP algorithm, at most one messagepasses across any edge in any given direction and any suchmessage replaces the previous message sent on the sameedge in the same direction. When any user is aware of bothmessages (10) and (11), he can compute the marginal (12) andhence the allocation variable (13).It is worth noting that the nature of the proposed approachmakes it amenable to solve maximum weight b-matchingproblems (M-WBM) defined over arbitrary graph Φ, notnecessarily bipartite. M-WBM problems are well known incombinatorics [25] and their objective is to find a perfect b-matching of maximum weight, namely the subgraph Π in Φof maximum weight, in which the degree of each node i ∈ Πis bi. As such, M-WBM problem is a generalization of the RAproblem in (3), whose objective is to find the subgraph in Gof maximum weight in which each node i ∈ N has degree riand each node j ∈ F has degree 1.

IV. REMP: A REWEIGHTED MP SCHEME

In real-world communication systems, the MP schemeproposed in the preceding section might be limited by theburden of signalling required to transfer messages betweenBS and users. In fact, the exchange of messages actuallyinvolves effective transmissions on the channel. This can

be potentially exacerbated by the non convergence of thealgorithm in loopy graphs. In some circumstances, the nonconvergence of marginals does not necessarily correspond tothe non convergence of the allocation variables, hence a simpleheuristic that stops the algorithm after a predetermined amountof time may suffice. However in arbitrary graphs, the allocationvariables may keep on fluctuating indefinitely and no stoppingcriterion can be devised. In this section, we present ReMP areweighted MP algorithm that provably converges to the globaloptimum of Problem (3).We first observe that the amount of message signalling can besignificantly reduced. Nodes corresponding to the subcarriersare located at BS, hence the BS actually delivers only onemessage to each user embedding the information relative toeach of the F subcarriers. Moreover, the 2-sized messagevector mn,f , (mf,n) can be compressed into a real-valuedscalar.To elaborate, let us assume for each user rn = r for simplicity,and subtract the constant term

∑j,j =f mj,n(0) from both sides

of Problem (10)

mn,f (xn,f )−∑

j,j =f

mj,n(0) =

Rn,fxn,f +maxxn,j

( ∑j,j =f

Rn,jxn,j + mj,n(xn,j)− mj,n(0)

)subject to

∑j xn,j = r

(14)Let us now introduce the terms µj,n = mj,n(1) − mj,n(0)and observe that the terms within the summation in (14) are0 or Rn,j + µj,n, depending on whether the variable xn,j is0 or 1. In other words, since we request a total number of rassigned subcarriers for each user, only a few terms survive inthe summation of (14). In particular, if we consider the casexn,f = 1 the number of terms that survive is r−1, whereas inthe case xn,f = 0 the number of surviving (non zero) termsis r, since the summation runs through j = f .Given the above, the maximization in both cases (i.e., xn,f = 1and xn,f = 0) is straightforward. Specifically, let us define theset ψn = [Rn,1 + µ1,n, . . . , Rn,j + µj,n, . . . , Rn,F + µF,n]and denote with {υj,n}zth\f the zth sorted element of theset ψn without considering the term Rn,f + µf,n, so that{υj,n}(z−1)th\f ≥ {υj,n}zth\f ≥ {υj,n}(z+1)th\f . Hence, forxn,f = 1 the maximum is given by taking the summation overthe first r − 1 terms of {υj,n}zth\f , i.e.:

mn,f (1)−∑j,j =f

mj,n(0) =Rn,f +r−1∑z=1

{υj,n}zth\f (15)

while for xn,f = 0 the maximum is given by taking thesummation over the first r terms in {υj,n}zth\f , i.e.:

mn,f (0)−∑j,j =f

mj,n(0) =r∑

z=1

{υj,n}zth\f (16)

Since by definition µn,f = mn,f (1)−mn,f (0), by combiningEquation (15) and (16), we can derive

µn,f = Rn,f − {υj,n}rth\f = Rn,f − {Rn,j + µj,n}rth\f(17)

Note that in this case, messages µn,f sent from user n tosubcarrier f is a scalar quantity.Similarly, it is straightforward to demonstrate that the scalarmessages µf,n = mf,n(1)− mf,n(0) sent from subcarriers fto user n become

µf,n = −maxi,i =n

µi,f (18)

To summarize, the calculation of message µn,f by user namounts to:

1) Sort in descending order the whole set ψn.2) Pick the rth sorted element in ψn without considering

the term Rn,f + µf,n.3) Change the sign of the picked element and add the

transmitting rate Rn,f .Similarly, the computation of µf,n by subcarrier f amountsto:

1) Take the maximum of incoming messages µi,f withoutconsidering the term µn,f .

2) Change the sign of the picked message.Consequently, from Equation (17) and (18), at each step t wecan derive the MP routine as:

µ(t+1)n,f = Rn,f −

{υ(t)j,n

}rth\f

= Rn,f −{Rn,j + µ

(t)j,n

}rth\f

(19a)

µ(t+1)f,n = −max

i,i =nµ(t+1)i,f (19b)

Thus, the solution xn,f at (t+1) can then be easily retrievedby computing the node marginal as follows

τ(t+1)n,f = µ

(t+1)n,f + µ

(t+1)f,n (20)

and calculating

x(t+1)n,f =

{1 if τ (t+1)

n,f < 0

0 otherwise(21)

according to Equation (13).By introducing a suitable weight ρ, ReMP perturbs the stan-dard formulation (19). From a graphical perspective, reweight-ing amounts to approximate the original distribution, initiallydefined over a loopy GM, over an approximate combinationof trees [18], [19]. Accordingly, it is possible to formulate analternative reweighted max-sum MP algorithm for the ordinaryformulation (19) as

µ(t+1)n,f = Rn,f − ρ

(Rn,j + µ

(t)j,n

)rth\f

−

− (1− ρ)(Rn,f + µ

(t)f,n

) (22a)

µ(t+1)f,n = −ρmax

i,i =nµ(t+1)i,f − (1− ρ)µ

(t+1)n,f (22b)

2 1

1 2 3 4 6 5

1S

2S

Fig. 2. A RA scenario with 2 users (circle node) and 6 subcarriers (squarenodes). Let us focus on user 1 and assume that the subcarriers numberingreflects the subcarriers sorting at user 1. Hence, supposing that user 1 requiresr = 3 subcarriers, messages µ1,s with s ∈ S1 (red edges) depend on theelement R1,4 + µ4,1, i.e. the (r + 1)th bigger incoming element, whilemessages µ1,w with w ∈ S2 (blue edges) depend on the element R1,3+µ3,1,i.e. the rth bigger incoming element.

where for the sake of simplicity, we assume the same weightsρ ∈ (0, 1] for each edge. It is worthwhile to note thatEquation (22) reduces to the ordinary formulation (19) whenwe choose ρ = 1. In the next section, we will show thatwhen a ρ < 1 is adopted the reweighted ReMP (22) provablyconverges to the optimal solution of Problem (3) also in thepresence of cycles in the graph.By direct inspection of Equation (21), at each iteration eachuser n can distinguish between two different subsets of sub-carriers by sorting the marginals τn,f in increasing order. Wedefine the first subset as S1 ⊂ F given by the first r ≤ Fsubcarriers in the ordered list of marginals, and the secondsubset, denoted by S2 ⊂ F , given by the last F−r subcarriersof the list.Let us now focus on the subcarriers s ∈ S1 and w ∈ S2. Thefollowing proposition holds

Proposition IV.1. When considering the whole set of messagesthat user n receives from subcarriers, the MP scheme impliesthat at each iteration (t + 1) the message sent by user n tosubcarriers s ∈ S1 and w ∈ S2 are

µ(t+1)n,s = Rn,s − ρ

(Rn,j + µ

(t)j,n

)(r+1)th

−

− (1− ρ) (Rn,s + µs,n)(23a)

µ(t+1)n,w = Rn,w − ρ

(Rn,j + µ

(t)j,n

)rth

−

− (1− ρ) (Rn,w + µw,n)(23b)

Proof: The proof follows from the fact that messageµ(t+1)n,f depends on the rth bigger incoming element at iteration

t, i.e. (Rn,j + µ(t)j,n)rth with j = f . Hence, when considering

the whole set of incoming elements, i.e. (Rn,j+µ(t)j,n) ∀j ∈ F ,

message µ(t+1)n,w still depends on the rth bigger incoming

element at iteration t, while message µ(t+1)n,s actually depends

on the (r + 1)th bigger incoming element at iteration t (seeFigure 2 for an instrumental example).

V. X-REMP APPLIED TO THE MULTI-CELLULARENVIRONMENT

In multi-cellular scenarios a RA scheme must deal withthe effects of the multiple access interference (MAI) andachieving a stable allocation that maximizes Problem (3)strongly depends on the geographical position of the UEs inthe different cells. As pointed out in [26], it is possible toexperimentally show that MAI induces oscillation in resourceallocation and any given RA algorithm may eventually notconverge to a steady state. In practice, assuming perfect CSIavailable at the eNB, a “naive” RA algorithm assigns a “good”resource to a given user. If the same resource is considered“good” also in the neighboring cell, at the next iteration thatresource may become “bad” and both cells can free it up andselect an other resource. Afterwards, the “bad” resource maybecome again “good” because only far-apart UEs select it.This behavior may trigger a loop in which a given resource“bounces” between different sets of UEs and no terminationcan be ensured.Different strategies have been devised in the literature todeal with such problem. In general, coordinated/centralizedstrategies (see e.g. [27]) outperform uncoordinated/distributed(see e.g. [28]) ones at the expense of less scalability and highersignaling between cells. Other solutions investigate non-reuse1 systems(see e.g. [29], [30]) in which users are categorizedin ”interior” users and ”edge” users. Accordingly, specificconstraints are devised in order to reduce the interactionbetween ”edge” users in neighboring cells. The main drawbackof such approach is a less flexibility in the resource allocationthat might lead to a lower throughput and a reduced resourceutilization. Moreover, such methods should account for thecost associated with managing the reuse pattern by introducingfrequency planning and measurements to establish the boarderbetween interior and exterior users.In this section, we describe X-ReMP, namely an extensionof ReMP to a multi-cellular system. Our solution combinesReMP with low-complexity cross signaling between cells. Inwords, ReMP follows the procedure presented in Section IVbut the rewards are modified in order to take into account theeffects of MAI. As such, rather than the actual transmittingrate, we replace the objective function in Problem (3) withthe rewards En,f associated to each pair user-resource (n, f).The idea is to compute at any given iteration (t), the ratevariation induced by the usage of a given resource in cellq ∈ K. Accordingly, from Equation (1) we obtain

∂R(q,t)n,f

∂I(q,t)n,f

= −P

(q)n,fG

(q,t)n,f

log(2)(I(q,t)n,f +N0)2(P

(q)n,fG

(q,t)n,f /(I

(q,t)n,f +N0) + 1)

(24)Equation (24) is calculated by the receiver, i.e. the eNB,and broadcasted to the neighboring cells. In turn, each useri ∈ N (k) belonging to a different cell k can autonomouslycompute the interference variation ∆I

(q,t)i,j that the usage of a

given resource j induces to the cell q as

∆I(q,t)i,j = G

(q,t)i,j P

(k)i,j (25)

and calculate the new reward E(k,t+1)i,j at iteration (t+ 1) as

E(k,t+1)i,j = R

(k,t)i,j +

∑q∈K,q =k

∂R(q,t)n,f

∂I(q,t)n,f

∆I(q,t)i,j ∀n ∈ N (k)

(26)Hence, any user has to retrieve the term in Equation (24) foreach cell q ∈ K, compute the interference caused to eachcell q ∈ K (Equation (25)), and eventually update the rewardfor each resource (Equation (26)). The new reward is thenembedded in the MP procedure. As described in Section II,we note that for each cell k ∈ K there exists at most one usern ∈ N (k) allocated to the resource f .In Section VII-A, we show that X-ReMP outperforms existingalternative algorithms....

A. Message passing in evolving cellular systems

Message passing lends itself naturally to evolving 4G cellu-lar communication systems such as the 3GPP Long Term Evo-lution Advanced (LTE-A) system [31] both in single (isolated)cell and multicell deployment scenarios. In LTE-A systems,the subcarriers of a macro cell may be reused by user deployednodes such as a home Node B (HNB) [32], a femto base station[33] or device-to-device (D2D) communication nodes [34]. Inthese evolving architectures, there is a need for interferencecoordination even within a single LTE-A cell, since multiplenodes may reuse an overlapping set of OFDM subcarriers.Because of the potentially large number of nodes within acell, the growing complexity of schedulers in existing nodesand the user requirement on plug-and-play deployment, thereis a need for distributed algorithms for subcarrier allocation[35].

In this type of heterogeneous environment, according tothe message passing architecture of Figure 1, each subcarriercan be managed by a separate virtual base station (nodesj = 1 . . . F of Figure 1) that represents the node that isresponsible for the allocation of that subcarrier. The user nodes(i = 1 . . . N ) may correspond to user equipments (UE) orsensors and actuators in a D2D environment. Note that withina single femto/pico base station or HNB, a single subcarriercan still be allocated to at most one user and therefore theconstraint C2 still holds for every j ∈ F .

The m and m messages in uplink and downlink respectivelycan then be naturally be mapped onto the standard LTE-A scheduling control messages, as illustrated by Figure 3.In LTE-A, the scheduling and resource allocation of uplinktransmissions are controlled along a slotted time axis with thegranularity of the so called subframe which is one ms. UEsmay periodically sound the uplink channel by transmittingknown sequences as sounding reference signals (SRS). TheSRS allow the base station (BS) to perform channel dependentscheduling on the millisecond timescale, when a UE indicatesthat data is available for uplink transmission (schedulingrequest, SR). The BS, in turn, uses the scheduling grant (SG)message to allocate the appropriate subcarrier to a UE. Them and m messages of the message passing framework, asproposed in this paper, fit naturally in this framework.

SG

SRS: Sounding Reference Signal

SR: Scheduling Request

SG: Scheduling Grant

BS

UE

BS: Base Station

UE: User Equipment

SRSRS SRS

Fig. 3. The m and m messages in uplink and downlink respectively can thenbe naturally mapped onto the existing uplink and downlink scheduling requestand grant (SR and SG respectively) messages of evolving LTE-A systems.

MAYBE ADD SOMETHING ABOUT MULTI-CELLULAR SIGNALING, I.E. HOW EQ. (24) CAN BEBROADCASTED....

VI. ON THE CONVERGENCE AND OPTIMALITY OF REMP

• MOVE THIS SEC. TO APPENDIX??• EQUATION SIGNS MUST BE MODIFIED AC-

CORDING TO THE MAX-SUM!!!In this section, we demonstrate that ReMP converges to fixedpoint messages and hence to fixed marginals. As such, ReMPconverges to x = x⋆ that provably corresponds to the optimalassignment of Problem (3).For the sake of clarity, let us rewrite the updates (22) as follows

µ(t+1)n,f = Rn,f − ρ

(R

n,u(t)n

+ µ(t)

u(t)n ,n

)−

− (1− ρ)(Rn,f + µ

(t)f,n

) (27a)

µ(t+1)f,n = −ρmax

i,i =nµ(t+1)i,f − (1− ρ)µ

(t+1)n,f (27b)

where u(t)n indicates the subcarrier such that R

n,u(t)n

+ µ(t)

u(t)n ,n

occupies the rth sorted position in ψn neglecting the elementRn,f + µf,n at iteration t.

A. Convergence

Let us stack all µ(t)n,f messages into the NF -dimensional

vector µ(t) =[µ(t)1,1, . . . , µ

(t)1,F , . . . , µ

(t)N,F

]. The ReMP rule

(27) translates into the mapping µ(t+1) = T(µ(t)) =[T1,1

(µ(t)

), . . . , TN,F

(µ(t)

)]where the generic mapping

µ(t+1)n,f = Tn,f

(µ(t)

)is computed as

Tn,f

(µ(t)

)=

Rn,f − ρ

[R

n,u(t)n

−(ρmax

j =nµ(t)

i,u(t)n

+ (1− ρ)µ(t)

n,u(t)n

)]−

− (1− ρ)

[Rn,f −

(ρmax

i=nµ(t)i,f + (1− ρ)µ

(t)n,f

)](28)

Theorem VI.1. The mapping T : RNF → RNF is a blockcontraction [36, Appendix A.6] under the block-maximumnorm:

∥T(µ)∥∞ = max1≤n≤N,1≤f≤F

|Tn,f (µ)| . (29)

Proof: The mapping T is a block contraction if

∥T (y)−T (x)∥∞ ≤ α ∥(y)− (x)∥∞ ∀x,y (30)

and the modulus of the mapping is α < 1.Let ν and γ be the user and the subcarrier such as

|Tν,γ(y)− Tν,γ(x)| = ∥T(y)−T(x)∥∞. By applying theMP rule (27), one obtains

Tν,γ(y)− Tν,γ(x) = −(1− ρ) (yγ,ν − xγ,ν)−

− ρ[(

Rν,u

(y)ν

+ yu(y)ν ,ν

)−(R

n,u(x)ν

+ xu(x)ν ,ν

)](31)

where u(y)ν and u

(x)ν are the subcarriers such that R

ν,u(y)ν

+

yu(y)ν ,ν

and Rν,u

(x)ν

+ xu(y)ν ,ν

occupy the rth ordered positiongiven the vectors x and y, respectively. Let us now focus onthe right hand side (RHS) of (31) which can be simplified byusing the following Lemma, where for the ease of notation wehave set α = u

(y)ν and β = u

(x)ν .

Lemma VI.2. It is always possible to find two subcarriers ξ1and ξ2 such that

(yξ1,ν − xξ1,ν) ≥ (Rν,α + yα,ν)− (Rν,β + xβ,ν)

≥ (yξ2,ν − xξ2,ν)(32)

Proof: If it is α = β, then it is ξ1 = ξ2 = α = β andthe lemma is proved. Otherwise, if it is α = β there are fourdifferent cases to consider:

C1 : Rν,α + yα,ν < Rν,β + yβ,νC2 : Rν,α + yα,ν > Rν,β + yβ,νC3 : Rν,α + xα,ν > Rν,β + xβ,ν

C4 : Rν,α + xα,ν < Rν,β + xβ,ν

(33)

Since the elements Rn,f + yn,f are sorted in descending order,when the number of requested subcarriers is r = 1 onlycases C1 and C3 are valid. For r ≥ 2, cases C1 and C2 andcases C3 and C4 are mutually exclusive, and all their possiblecombinations must be considered.

• In case C1 and C3 hold, it is

yβ,ν − xβ,ν =Rν,β + yβ,ν − (Rν,β + xβ,ν) > (34a)Rν,α + yα,ν − (Rν,β + xβ,ν) >

Rν,α + yα,ν − (Rν,α + xα,ν) =

yα,ν − xα,ν (34b)

and it is ξ1 = β and ξ2 = α.• In case C1 and C4 hold, it is Rν,β + yβ,ν > Rν,α + yα,ν

and Rν,β + xβ,ν > Rν,α + xα,ν , but the rth orderedposition is occupied by subcarrier α for the messages yand by subcarrier β for the messages x.As shown in Figure 4, this implies that there must be atleast one subcarrier δ such that{

Rν,δ + yδ,ν < Rν,α + yα,νRν,β + xβ,ν < Rν,δ + xδ,ν

(35)

( )β

( )α

( )β( )δ

( )δ( )α

, ,n f f nP x+ ɶ

, ,n f f nP y+ ɶ

Sorted order

Sorted order

rth position

(a)

(b)

Fig. 4. Sorted values of Rn,f + yf,n and Rn,f + xf,n when cases C1 andC4 hold.

and as a consequence, it is

Rν,α + yα,ν − (Rν,β + xβ,ν) >

Rν,δ + yδ,ν − (Rν,δ + xδ,ν) =

yδ,ν − xδ,ν

(36)

Thus, due to (34a) it is ξ1 = β and, due to (36), ξ2 = δ.• If C2 and C3 hold, then there must be at least one

subcarrier ϵ such that{Rν,ϵ + yϵ,ν > Rν,α + yα,ν > Rν,β + yβ,νRν,α + xα,ν > Rν,β + xβ,ν > Rν,ϵ + xϵ,ν

(37)

so that it is

yϵ,ν − xϵ,ν =Rν,ϵ + yϵ,ν − (Rν,ϵ + xϵ,ν) >

Rν,α + yα,ν − (Rν,β + xβ,ν)(38)

and it is ξ1 = ϵ and, due to (34b), ξ2 = β.• Eventually, if C2 and C4 hold, it is

yα,ν − xα,ν =Rα,ν + yα,ν − (Rα,ν + xα,ν) > (39a)Rν,α + yα,ν − (Rν,β + xβ,ν) >

Rβ,ν + yβ,ν − (Rβ,ν + xβ,ν) =

yβ,ν − xβ,ν (39b)

and it is ξ1 = α and ξ2 = β.

Let subcarrier η1 = argmaxγ,ξ1 (yγ,ν − xγ,ν , yξ1,ν − xξ1,ν)and η2 = argmaxγ,ξ2 (yγ,ν − xγ,ν , yξ2,ν − xξ2,ν), byapplying Lemma 1 to (31), we have

− (yη1,ν − xη1,ν) < (40a)− (1− ρ) (yγ,ν − xγ,ν)− ρ (yξ1,ν − xξ1,ν) ≤ (40b)Tν,γ(y)− Tν,γ(x) ≤− (1− ρ) (yγ,ν − xγ,ν)− ρ (yξ2,ν − xξ2,ν) < (40c)− (yη2,ν − xη2,ν) (40d)

where the strict relationship in (40a) and (40c) depends onthe fact that: a) the events in which the equivalences yγ,ν −xγ,ν = yξ1,ν − xξ1,ν and yγ,ν − xγ,ν = yξ2,ν − xξ2,ν holdcan be neglected in practical propagation environments; b) itis 0 < ρ < 1.

To proceed further we need now the following Lemma

Lemma VI.3. Given the difference − (yη,ν − xη,ν), it isalways possible to find two users θ1 and θ2 such that

yθ1,η − xθ1,η < − (yη,ν − xη,ν) < yθ2,η − xθ2,η (41)

Proof: Let us define users θy = argmaxκ =ν yκ,ηand θx = argmaxκ=ν xκ,η, then applying rule (27) to− (yη,ν − xη,ν) yields

− (yη,ν − xη,ν) =(ρyθy,η + (1− ρ) yν,η

)−

− (ρxθx,η + (1− ρ)xν,η)

= ρ(yθy,η − xθx,η

)+ (1− ρ) (yν,η − xν,η)

(42)

Now, since it is by definition xθx,η ≤ xθy,η and yθy,η ≤ yθy,η ,the following set of inequalities holds true

yθy,η − xθy,η ≤ yθy,η − xθx,η ≤ yθx,η − xθx,η (43)

In (43), the equality relation holds only if θx = θy .By defining θ1 = argmaxθy,ν

(yθy,η − xθy,η, yν,η − xν,η

)and θ2 = argmaxθx,ν (yθx,η − xθx,η, yν,η − xν,η) and substi-tuting (43) in (42), it is

yθ1,η − xθ1,η < − (yη,ν − xη,ν) < yθ2,η − xθ2,η (44)

Now, let us consider inequality (40) and use Lemma 41to define users ζ1 and ζ2 such that yζ1,η1 − xζ1,η1 <− (yη1,ν − xη1,ν) and − (yη2,ν − xη2,ν) < yζ2,η2−xζ2,η2 , thenwe can write

yζ1,η1 − xζ1,η1 < Tν,γ(y)− Tν,γ(x) < yζ2,η2 − xζ2,η2 (45)

Thus, since it is |Tν,γ(y)− Tν,γ(x)| = ∥T(y)−T(x)∥∞,taking the absolute value of all the term of the inequality (45)yields

∥T(y)−T(x)∥∞ <

max(ζ1,η1),(ζ2,η2)

(|yζ1,η1 − xζ1,η1 | , |yζ2,η2 − xζ2,η2 |)(46)

and since it is

max(ζ1,η1),(ζ2,η2)

(|yζ1,η1 − xζ1,η1 | , |yζ2,η2 − xζ2,η2 |) ≤

max1≤ν≤N,1≤γ≤F

(|yν,γ − xν,γ |) = ∥y − x∥∞(47)

combining (46) and (47), yields

∥T(y)−T(x)∥∞ < ∥y − x∥∞ . (48)

Since (48) is a strict inequality, it is always possible to find areal number ϵ ∈ (0, 1) such that

∥T(y)−T(x)∥∞ < ϵ ∥y − x∥∞ . (49)

Q.O.D.

B. Optimality

In order to demonstrate the optimality of the fixed pointsolution, we need to derive the Lagrangian relaxation of theoriginal Problem (3) and its slackness conditions. Let us define

Lemma VI.4. The slackness conditions of Problem (3) are

Ri,j − λ∗j = max

1≤k≤F

{Ri,k − λ∗

k

}(50)

where λ = (λ1 . . . , λF ) is the vector of Lagrange multipliers.

Proof: Let us define the set

X =

{x|∑

jxi,j = ri, xi,j ≥ 0 ∀(i, j)

}and derive the Lagrangian of Problem (3) as

L(x,λ) =∑

i,jRi,jxi,j +

∑jλj(1−

∑ixi,j) (51)

Since X is bounded and hence compact the corresponding dualfunction q(λ) = infx∈X L(x,λ) can be attained for any λ byWeierstrass’ theorem [36, Appendix A.2, Proposition A.8].The dual function can be derived asq(λ) = inf

x∈XL(x,λ) =∑

jλj + inf

x∈X

∑i

(∑j(Ri,j − λj)xi,j

)=∑

jλj +

∑i

(∑jinfx∈X

(Ri,j − λj)xi,j

)=∑

jλj +

∑imax

j(Ri,j − λj)ri

(52)

where the last equalities follows from the fact that if (Ri,j −λj) > 0 the infinum above is attained for xi,j = 0, while if(Ri,j − λj) < 0 the infinum is attained for xi,j = ri. Thecorresponding dual problem is

maximize q(λ)subject to λ ∈ ℜF (53)

Since constraints of Problem (3) are affine, Slater’s conditionholds implying strong duality of the dual problem [37, Section5.2.3]. Hence if Problem (3) has an optimal solution x⋆, itsdual has an optimal solution λ⋆ and

q(λ⋆) =∑

i,jRi,jx

⋆i,j

Thus,∑jλ⋆j +

∑imax

j(Ri,j − λ⋆

j )ri =∑

i,jRi,jx

⋆i,j (54)

Since x⋆ is the optimal solution, we have∑jλ⋆j =

∑jλ⋆j

∑ix⋆i,j

hence from Equation (54), we obtain∑imax

j(Ri,j − λ⋆

j )ri =∑

i,j(Ri,j − λ⋆

j )x⋆i,j (55)

Since∑

j x⋆j = ri, Equation (55) becomes∑

i,j

(Ri,j − λ⋆

j −maxj

(Ri,j − λj)

)x⋆i,j = 0 (56)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

200

400

600

800

1000

1200

1400

1600

1800

2000Ite

ratio

ns

ρ

Fig. 5. Effects of ρ ∈ (0, 1] on the convergence of ReMP to the fixedpoint µ⋆ with N = 8 users and F = 32 subcarriers.

Thus, if x⋆i,j > 0 we have

Ri,j − λ⋆j = max

1≤k≤F(Ri,k − λ⋆

k) (57)

Let us now focus on subcarriers s ∈ S1 and w ∈ S2.

Lemma VI.5. At convergence, the MP scheme implies that

Rn,s + µs,n < Rn,w + µw,n ∀s ∈ S1, w ∈ S2 (58)

Proof: The proof follows combining Equation (20) withEquation (22a) and appealing to Proposition IV.1, i.e.:

τn,s = Rn,s − ρ (Rn,w + µw,n)(r+1)th −− (1− ρ) (Rn,s + µs,n) + µs,n

= ρ[(Rn,s + µs,n)− (Rn,w + µw,n)(r+1)th

]< 0

(59)

according to the fact that s ∈ S1 and w ∈ S2.From Lemma VI.4 and VI.5, it is possible to derive the

following Theorem

Theorem VI.6. If ReMP converges to a fixed point µ⋆, thefixed point respects the slackness conditions (50), i.e. it is anoptimal solution for the original Problem (3).

Proof: We note that the condition

Rn,s − λ⋆s < Rn,w − λ⋆

w ∀s ∈ S1, ∀w ∈ S2 (60)

implies the slackness condition (50). As such, Equation (60)holds by appealing to Lemma VI.5 and imposing λ⋆

s = −µ⋆s,n

and λ⋆w = −µ⋆

w,n.

VII. PERFORMANCES OF REMP

In this section, we explore the relations between ReMP anda similar assignment algorithm presented in the literature.

A. Simulations

To study the behavior of ReMP as a RA method, wehave carried out an extensive set of simulations in randomlygenerated single-cell OFDMA systems assuming a frequencyselective Rayleigh channel model.In Figure 5, we report the number of iterations required byReMP to converge to the fixed point µ⋆ as a function of ρ.Note that the number of iterations is minimum when ρ ∼ 0.8,while tends to infinity as ρ → 1. In Figure ??, we comparethe performances of ReMP and CM as the number of availablesubcarriers varies adopting ρ = 0.8 for the ReMP algorithm.Although both methods converge to the optimal assignmentx⋆, CM experiences an almost linear increase in the numberof required iterations. To the contrary, ReMP converges also inthe exchanged messages and achieves an optimal assignmentmuch faster than CM as the number of subcarriers grows. Inparticular, the required iterations only slightly increase with F ,i.e. the iterations gap between the first scenario with F = 8and the last scenario with F = 120 subcarriers is about 3.5iterations.

VIII. CONCLUSIONS

In this paper we presented ReMP a reweighted max-summessage passing algorithm that adapts to RA problems. ReMPrelies on a MP procedure that entails passing informationmessages, i.e. the solutions of small distributed allocationproblems, between users and resources until an allocationdecision is taken. While ordinary MP algorithms no longerguarantee to converge when RA is defined over loopy graphs,ReMP allows the exchanged messages to converge to a fixedpoint in a finite number of iterations. Once the ReMP updatesconverge to some fixed point message, then the fixed pointis used to compute the marginals and derive the assignmentthat in turn corresponds to the optimal solution of the originaloptimization problem. Simulation results show that ReMPoutperforms a similar MP-based algorithm for RA in OFDMAscenarios. Moreover, compared with conventional assignmentalgorithms, ReMP converges to some fixed message µ⋆ (andhence fixed marginal τ ⋆) and thus each user locally recognizeswhen the optimal assignment has been attained. Hence, ReMPappears particularly attractive in those applications in whichthe presence of a central coordinator is not desired.

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