Improving Resource Efficiency with Partial Resource Muting for Future Wireless Networks
IImproving Resource Efficiency with PartialResource Muting for Future Wireless Networks
Qi Liao ∗ and R. L. G. Cavalcante †∗ Nokia Bell Labs, Stuttgart, Germany † Fraunhofer Heinrich Hertz Institute and Technical University of Berlin, GermanyEmail: [email protected]; [email protected]
Abstract —We propose novel resource allocation algorithmsthat have the objective of finding a good tradeoff betweenresource reuse and interference avoidance in wireless networks.To this end, we first study properties of functions that relatethe resource budget available to network elements to the optimalutility and to the optimal resource efficiency obtained by solvingmax-min utility optimization problems. From the asymptoticbehavior of these functions, we obtain a transition point thatindicates whether a network is operating in an efficient noise-limited regime or in an inefficient interference-limited regimefor a given resource budget. For networks operating in theinefficient regime, we propose a novel partial resource mutingscheme to improve the efficiency of the resource utilization. Theframework is very general. It can be applied not only to thedownlink of 4G networks, but also to 5G networks equipped withflexible duplex mechanisms. Numerical results show significantperformance gains of the proposed scheme compared to thesolution to the max-min utility optimization problem with fullfrequency reuse.
I. I
NTRODUCTION
Future network architectures are envisioned to provideseamless connections anywhere and anytime with asymmetrictraffic. In particular, in the media access control (MAC)/linklayer, one of the key challenges to bring this vision to reality isto devise service-centric resource allocation mechanisms ableto find a good balance between interference avoidance andresource (spectrum) reuse. Promising approaches to addressthis challenge are the muting schemes, which include thealmost blank subframe (ABS) approach in time domain [1]and the mutually exclusive resource block allocation approachin frequency domain [2]. However, in these approaches twoquestions remain largely unanswered: • At which point full resource reuse becomes inefficient?In other words, when is it appropriate to activate resourcemuting? • How to develop efficient resource muting schemes thatdeal with complex interference patterns caused by dis-ruptive architectures such as flexible duplex in futurenetworks?In this paper, we address these questions by studying proper-ties of solutions to a large class of max-min utility fairnessproblems. In more detail, building upon the seminal workof Yates [3], which has introduced the concept of standardinterference functions (SIFs) in wireless networks (see Defi-nition 1), many researchers have devised efficient algorithmsable to solve a large class of max-min fairness problems with the signal-to-interference-plus-noise ratio (SINR) [4]–[7] asthe utility function. More recently, fairness problems withrate-related utilities characterized by nonlinear load couplingmodels have also been increasingly gaining attention [6], [8],[9]. Most studies devoted to these problems focus heavily onthe development of numerical solvers, and, by doing so, issuessuch as properties of the utility and the network efficiency asa function of the budget available to network elements haveremained largely unexplored. This gap has been addressedin the recent study in [10], which has used the concept ofasymptotic functions in nonlinear analysis [11] to derive tightbounds for the network performance obtained by solving max-min utility problems. That study has also derived a transitionpoint that, for a given resource budget, indicates whether anetwork operates in a resource efficient region.In this study we build upon the findings in [10] to developnovel efficient resource allocation algorithms for current andfuture network technologies. Our main contributions can besummarized as follows: • We develop partial resource muting schemes based on thecharacterization of resource-efficient regions of solutionsto a general class of resource allocation problems. • The above schemes, which are based on the solutionsto a series of subproblems using fixed point iterations,efficiently detect a set of bottleneck users that should beassigned to the muting region, and the schemes optimizethe service-centric resource allocation. • The framework proposed here can be applied both tothe downlink (DL) of 4G networks and to 5G networksequipped with flexible duplex mechanisms.This study is organized as follows. In Section II we introducenotation and mathematical background on i) max-min fairnessproblems and constrained eigenvalue problems (CEVPs) , andii) properties of the solutions to CEVPs. In Section III wepropose a resource muting scheme for DL resource allocation.This scheme is enhanced in Section IV to cater for complexinterference models in 5G networks. Conclusions are summa-rized in Section V.II. N
OTATIONS , D
EFINITIONS , AND P RELIMINARIES
The following definitions are used in this paper. The non-negative and positive orthant in K dimensions are denoted by R K + and R K ++ , respectively. Let x ≤ y denote the component-wise inequality between two vectors. A norm (cid:107) · (cid:107) on R K + is a r X i v : . [ ee ss . SP ] J a n aid to be monotone if ( ∀ x ∈ R K + )( ∀ y ∈ R K + ) ≤ x ≤ y ⇒(cid:107) x (cid:107) ≤ (cid:107) y (cid:107) . By diag( x ) we denote a diagonal matrix withthe elements of x on the main diagonal. The cardinality of set A is denoted by | A | . The positive part of a real function isdefined by [ f ( x )] + := max { , f ( x ) } . Standard interferencefunctions (SIFs) are defined as follows: Definition 1. [3] A function f : R K → R ++ ∪ {∞} is an SIF if the following axioms hold: 1) (Monotonicity) (cid:0) ∀ x ∈ R K + (cid:1) (cid:0) ∀ y ∈ R K + (cid:1) x ≤ y ⇒ f ( x ) ≤ f ( y ) ; 2)(Scalability) (cid:0) ∀ x ∈ R K + (cid:1) ( ∀ α > αf ( x ) > f ( α x ) ; and 3)(Nonnegative effective domain) dom f := { x ∈ R K | f ( x ) < ∞} = R K + . A vector function f : R K + → R K ++ : x (cid:55)→ [ f ( x ) , . . . , f N ( x )] is called an SIF if each of the componentfunctions is an SIF. We now turn our attention to the general description of theproblems we address in this study. In more detail, a large arrayof utility maximization problems in wireless networks canbe seen as particular instances of the following optimizationproblem [4]–[6], [8]: maximize x ∈ R K + min k ∈ K u k ( x ) (1a) subject to (cid:107) x (cid:107) ≤ θ (1b)where K := { , . . . , K } is the set of network elements, u k : R K + → R + is the utility function of network element k ∈ K , and (cid:107) · (cid:107) is a monotone norm used to constrainthe resource utilization x = [ x , . . . , x K ] to a given budget θ > . In the above problem formulation, the k th component x k of the optimization variable x is the resource utilizationof network element k ∈ K . Now, consider the followingconditional eigenvalue problem (CEVP):( The conditional eigenvalue problem :) Given a monotonenorm (cid:107) · (cid:107) , a budget θ ∈ R ++ , and a mapping T : R K + → R K ++ : x (cid:55)→ (cid:2) T (1) ( x ) , . . . , T ( K ) ( x ) (cid:3) , where T ( k ) : R K + → R ++ is an SIF for each k ∈ K , the CEVP is stated as follows:Find ( x , c ) ∈ R K + × R ++ such that T ( x ) = 1 c x and (cid:107) x (cid:107) = θ. (2)As an implication of the results in [7], if the utility functionsin (1) and the SIF T in (2) are related by ( ∀ k ∈ K )( ∀ x ∈ R K ) u k ( x ) = x k /T ( k ) ( x ) , then x (cid:63) solves (1) if ( x (cid:63) , c (cid:63) ) solves (2). Furthermore, we have ( ∀ k ∈ K ) u k ( x (cid:63) ) = c (cid:63) .Therefore, to solve (1), we only need to devise efficientalgorithmic solutions to (2). To this end, [7] has proved that(2) has a unique solution ( x (cid:63) , c (cid:63) ) ∈ R K ++ × R ++ and that x (cid:63) ∈ R K ++ is the limit of the sequence (cid:0) x ( n ) (cid:1) n ∈ N generatedby x ( n +1) = θ (cid:13)(cid:13) T (cid:0) x ( n ) (cid:1)(cid:13)(cid:13) T (cid:16) x ( n ) (cid:17) , with x (0) ∈ R K + . (3)With knowledge of x (cid:63) , we recover c (cid:63) from the equality c (cid:63) = θ/ (cid:107) T ( x (cid:63) ) (cid:107) .Note that later studies have also established the connectionbetween Problem (1) and Problem (2) by using arguments withdifferent levels of generality [6]. Recently, [10] has studied the influence of the budget θ on the solution to these problems,and we summarize some of the results of that study becausethey are crucial to the contributions that follow.One of the key tools used in the analysis in [10] is thenotion of asymptotic functions associated with SIFs: Proposition 1. [10, Prop. 1] In R K + , the asymptotic function T ∞ : R K → R ∪ {∞} associated with an SIF T : R K → R ++ ∪ {∞} is given by (cid:0) ∀ x ∈ R K + (cid:1) T ∞ ( x ) = lim h →∞ T ( h x ) /h ∈ R + . (4)For convenience, before we show relations between proper-ties of asymptotic functions and properties of the solution toProblem (2), let us the recall the following definitions: Definition 2 (Utility and resource efficiency) . [10, Def. 4]Let ( x θ , c θ ) ∈ R K ++ × R ++ denote the solution to Problem (2) for a given budget θ ∈ R ++ . The utility and the resourceefficiency function are defined by, respectively, U : R ++ → R ++ : θ (cid:55)→ c θ and E : R ++ → R ++ : θ (cid:55)→ U ( θ ) / (cid:107) x θ (cid:107) . We can now state selected properties of the solution toProblem (2) (and hence to Problem (1)):
Proposition 2. [10, Prop. 3] Assume that the followingCEVP: Find ( x ∞ , λ ∞ ) ∈ R K + × R + such that T ∞ ( x ∞ ) = λ ∞ x ∞ , (cid:107) x ∞ (cid:107) = 1 (5) has a unique positive solution ( x ∞ , λ ∞ ) ∈ R K ++ × R ++ , whereeach coordinate T ( k ) ∞ of T ∞ : R K + → R K + is an asymptoticfunction associated with the SIF T ( k ) in (2) . Then the solutionto (2) has the following properties: (i) Asymptotic utility and resource efficiency: sup θ> U ( θ ) = lim θ →∞ U ( θ ) = 1 /λ ∞ , (6) sup θ> E ( θ ) = lim θ → + E ( θ ) = 1 / (cid:107) T ( ) (cid:107) . (7)(ii) Upper bound for the utility: ( ∀ θ ∈ R ++ ) , U ( θ ) ≤ (cid:26) θ/ (cid:107) T ( ) (cid:107) , if θ ≤ θ (trans) /λ ∞ , otherwise , (8) where θ (trans) is the transition point defined by θ (trans) := (cid:107) T ( ) (cid:107) /λ ∞ . (9)(iii) Upper bound for the resource efficiency: ( ∀ θ ∈ R ++ ) , E ( θ ) ≤ min { / (cid:107) T ( ) (cid:107) , / ( λ ∞ θ ) } . (10)The transition point θ (trans) in (9) serves as a coarse indicatorof whether we can obtain substantial gains in utility byincreasing the budget θ . More precisely, if the given budget θ is greater than θ (trans) , then the network is likely operatingin a regime where the performance is limited by interference,so increasing θ even by orders of magnitude typically bringsonly marginal gains in utility. In contrast, if θ < θ (trans) , thennoticeable gains in utility can be obtained by increasing thebudget θ . Fig. 1 illustrates this observation, which is heavilyexploited by the algorithms proposed in the next sections. -4 -3 -2 -1 log( θ ) -2 -1 l og ( U ( θ )) log(U( θ ))Bound log( θ /||T(0)||)Bound log(1/ λ ∞ )Transition point ||T(0)||/ λ ∞ Noise-limited regime Interference-limited regimeTransition Point Bound log(1/ λ ∞ )Bound log( θ /||T(0)||) Fig. 1: Example: utility as a function of the power budget θ .III. P ROPOSED ALGORITHMS FOR DL RESOURCEALLOCATION
In this section, we build upon the results in [7] andProposition 2 to devise novel algorithms for DL resourceallocation. Then, in Section IV, we study problems withmore complex interference models incorporating disruptivearchitectures envisioned for 5G networks.
A. System Model and Problem Formulation
We consider an orthogonal frequency division multiplex-ing (OFDM)-based wireless network system, consisting of aset of base stations (BSs) N := { , . . . , N } and a set ofcommunication links K := { , . . . , K } in DL. Hereafter, weuse the terms “communication links”, “services”, and “users”interchangeably (without loss of generality, we can assume thateach user sets up one communication link for one service at anunit of time). Let W denote the total bandwidth in Hz, and let w ∈ [0 , K be a vector collecting the fraction of bandwidthallocated to the services. We assume that the transmit powerspectral density (in Watts/Hz) of all services are given andcollected in a vector p ∈ R K ++ . Let the data rate demandof all services (in bit/s) be collected in ¯ r ∈ R K ++ . The matrix A ∈ { , } N × K denotes the BS-to-service assignment matrix,where a n,k = 1 if k is served by BS n , and otherwise.By v k,l we denote the channel gain between the transmitterof service l and the receiver of service k . For k (cid:54) = l , v k,l is the interference channel gain, which is positive if service l causes interference to service k , otherwise v k,l = 0 . The scalar v k,k > is the channel gain of link k . By σ k we denote thenoise spectral density (in Watt/Hz) in the receiver of service k .By interpreting w as the probability of generating interferencefrom the transmitter of a link to the receiver of another link (onany resource block) [8], [12], the SINR that link k experiencescan be approximated by SINR k ( w ) ≈ p k v k,k (cid:80) l (cid:54) = k v k,l p l w l + σ k = p k (cid:104) ˜ V diag( p ) w + ˜ σ (cid:105) k , (11)where ˜ V ∈ R K × K + denotes the interference coupling ma-trix with the ( k, l ) -th entry defined as v k,l /v k,k , and ˜ σ :=[ σ /v , , . . . , σ K /v k,k ] ∈ R K ++ . The achievable rate (in bit/s) of service k is computed by r k ( w ) = w k W log (1 + SINR k ( w )) . (12)The objective of the first algorithm proposed here is tomaximize the worst-case service-specific quality of service(QoS) satisfaction level , defined as the ratio of the achievablerate r k ( w ) to the rate demand ¯ r k , subject to a per-BS loadconstraint. Formally, the problem is stated as follows: maximize w ∈ R K + min k ∈ K r k ( w ) / ¯ r k (13a) subject to (cid:107) Aw (cid:107) ∞ ≤ θ, (13b)where (cid:107) · (cid:107) ∞ denotes the L ∞ -norm. Note that (13b) impliesa resource reuse factor of , and with full load constraint wehave θ = 1 ; i.e., ( ∀ n ∈ N ) (cid:80) k ∈ K a n,k w k ≤ .In this work, we are interested not only in the solution to theabove problem, but also in answering the following question: Is the load limit θ = 1 with resource reuse factor an efficientoperation point?B. Optimal Resource Allocation and Performance Limits To study the resource efficiency of the solution to Problem(13) as a function of the budget θ , we start with next technicalresult. The proof is omitted because it is similar to that in [13,Ex. 2]. Lemma 1.
Suppose that the SINR is modeled as in (11) , then the mapping T := (cid:2) T (1) ( w ) , . . . , T ( K ) ( w ) (cid:3) ,where ( ∀ k ∈ K ) T ( k ) ( w ) : R K + → R ++ : w (cid:55)→ ¯ r k / (cid:0) W log (1 + SINR k ( w )) (cid:1) , is an SIF. By using Lemma 1 and [14, Prop. 1] we verify that theoptimal solution w ∗ to (13) satisfies ( ∀ k ∈ K ) u k ( w ∗ ) = c ∗ [14, Prop. 1] for some c (cid:63) ∈ R ++ . This result can be alterna-tively obtained as follows. Assuming that each BS serves atleast one user and that every user is served by a BS, whichguarantees that each BS serves a nonempty and unique set ofusers, we have that all rows of the assignment matrix A arelinearly independent. Therefore, A is a nonnegative full (row)rank matrix, so the function g : R K → R + : w (cid:55)→ (cid:107) A | w |(cid:107) ∞ (which in particular implies g ( x ) = (cid:107) Aw (cid:107) for w ∈ R K + ),where | · | denotes the coordinate-wise absolute value of avector, is a monotone norm. Thus, the problem in (13) is aninstance of that in (1), and the solution to (13) can be easilyobtained with the iterations in (3) as explained in Section II.Furthermore, by using Proposition 1, we can compute theasymptotic mapping T ∞ associated with T . By doing so, asshown below, we are able to compute ( w ∞ , λ ∞ ) defined inProposition 3, and the performance limits in Proposition 2become readily available. Proposition 3.
Let T : R K + → R K ++ : w (cid:55)→ (cid:2) T (1) ( w ) , . . . , T ( K ) ( w ) (cid:3) be as defined in Lemma 1. Supposethat p ∈ R K ++ and ˜ V is irreducible, implying that each serviceis interfered by at least another of the services. The asymptoticmapping T ∞ : R K + → R K + : w (cid:55)→ [ T (1) ∞ ( w ) , . . . , T ( K ) ∞ ( w )] associated with T is given by: T ∞ ( w ) = diag( p ) − Φ ˜ V diag( p ) w (14) where Φ := ln(2) W diag (¯ r , . . . , ¯ r K ) . urthermore, there exists a unique positive solution ( w ∞ , λ ∞ ) ∈ R K ++ × R ++ to the CEVP T ∞ ( w ∞ ) = λ ∞ w ∞ , (cid:107) Aw ∞ (cid:107) ∞ = 1 , (15) which is given by λ ∞ = λ ∗ , w ∞ = w ∗ / (cid:107) Aw ∗ (cid:107) ∞ , (16) where λ ∗ and w ∗ are, respectively, the unique largest realeigenvalue and a corresponding real eigenvector of the matrix G := diag( p ) − Φ ˜ V diag( p ) .Proof. Applying Proposition 1, we have ( ∀ k ∈ K )( ∀ w ∈ R K + ) T ( k ) ∞ = lim h →∞ T ( k ) ( h w ) /h = lim h →∞ ¯ r k / (cid:16) hW log (cid:16) p k / (cid:104) h ˜ V diag( p ) w + ˜ σ (cid:105) k (cid:17)(cid:17) . Defining g ( h ) := log (cid:16) p k / (cid:104) h ˜ V diag( p ) w + ˜ σ (cid:105) k (cid:17) and f ( h ) := ¯ r k / ( W h ) , we have that lim h →∞ g ( h ) = 0 , lim h →∞ f ( h ) = 0 , g (cid:48) ( h ) (cid:54) = 0 for h ∈ R + , and that lim h →∞ f (cid:48) ( h ) /g (cid:48) ( h ) exists. By using L’Hˆopital’s rule , weverify that T ( k ) ∞ = ln(2)¯ r k (cid:104) ˜ V diag( p ) w (cid:105) k / ( W p k ) . We ob-tain (14) by writing T ∞ := (cid:104) T (1) ∞ , . . . , T ( K ) ∞ (cid:105) in matrixform. We can also verify that G is nonnegative and ir-reducible. As a result, by Perron-Frobenius theory [15, p.673], G has a simple positive eigenvalue λ ∗ ∈ R ++ as-sociated with a positive right eigenvector w ∗ . Furthermore,any other real eigenvalue λ satisfies | λ | ≤ λ ∗ , and if T ∞ ( w ) = Gw = λ w for some w ∈ R K + \{ } , then we have λ = λ ∗ and w = c w ∗ for some c ∈ R ++ . In particular,with c = 1 / (cid:107) Aw ∗ (cid:107) ∞ , w ∞ = c w ∗ , and λ ∞ = λ ∗ , wededuce T ∞ ( w ∞ ) = Gw ∞ = λ ∞ w ∞ and (cid:107) Aw ∞ (cid:107) ∞ = (cid:107) A ( w ∗ / (cid:107) Aw ∗ (cid:107) ∞ ) (cid:107) ∞ = (cid:107) Aw ∗ (cid:107) ∞ / (cid:107) Aw ∗ (cid:107) ∞ = 1 . Theabove implies that (16) is the unique solution to the CEVPin (15), and the proof is complete. Remark 1.
Proposition 3 shows an efficient method to com-pute ( w ∞ , λ ∞ ) as the solution to the CEVP (15) . Note that, in this specific case, the asymptotic mapping associated witha nonlinear mapping (defined in Lemma 1) becomes linear ,as shown in (15) . Moreover, to compute the eigenvalues of G = diag( p ) − Φ ˜ V diag( p ) , we can equivalently computethe eigenvalues of the matrix Φ ˜ V , which is independent of p .C. Partial Resource Muting Having ( w ∞ , λ ∞ ) in hand, we are now able to compute θ (trans) with (9) and answer the following question related tothe resource efficiency raised in Section III-A: If the transition point yields θ (trans) < , full resource reuse(i.e., θ = 1 ) may not be an efficient operation point becausewe are likely operating in an interference limited region wherethe resource availability θ can be decreased without noticeablechanges in utility – see Fig 1. Suppose that D ∈ R K × K is an invertible matrix and X ∈ R K × K , theeigenvalues of the matrices X and DXD − are the same [16, Rem. 1]. BS 1BS 2BS 3 DetectedbottleneckusersResource allocated to non bottleneck users served by BS 3 Resource allocated to bottleneck users served by BS 3 and its neighboring BSs 1 and 2
Fig. 2: Resource muting region in downlink.The new challenge arises:
How to improve the resource efficiency if the network isoperating in an interference-limited inefficient region?
Since the bottleneck users usually consume most of theresources and impair the performance, we consider mutingpartial resources in the neighboring cells to mitigate theinterference received in (and generated by) the bottleneckusers. Based on ( λ ∞ , w ∞ ) obtained in (16) and the derivedtransition point θ (trans) , we propose a resource muting schemeconsisting of the following steps.
1) Triggering the Resource Muting Scheme: θ (trans) < implies inefficient usage of resources in the region [ θ (trans) , due to heavy interference. Instead of allocating all resourcesin at least one BS to achieve only a slight increase in utility,the network may better benefit from muting partial resourcesto reduce the interference of bottleneck users. Therefore, theresource muting scheme can be triggered if θ (trans) < .
2) Modifying the Interference Pattern and Load Con-straints:
Suppose that a set of bottleneck users, denoted by K ( b ) , is selected (details about the selection of bottleneck usersare given in the next subsection). The motivation is to allocate K ( b ) to a muting region, such that each user k ∈ K ( b ) neitherreceives interference from the neighboring cells nor generatesinterference to those cells, as shown in Fig. 2. Since thereceived/generated interference of the bottleneck users from/tothe neighboring cells is canceled, the interference couplingmatrix is updated as follows ( ∀ k ∈ K ( b ) )( ∀ m ∈ N k )( ∀ l ∈ K m ) v k,l = v l,k = 0 , (17)where N k denotes the set of neighboring cells of cell k .On the other hand, to incorporate the muting region in theconstraint, we modify the load constraint in (13b) as shownbelow: g ( w ) ≤ , (18a) g : R K → R + : w (cid:55)→ max n ∈ N (cid:88) k ∈ K n | w k | + (cid:88) m ∈ N n (cid:88) l ∈ K ( b ) m | w l | , (18b)where K ( b ) m denotes the set of bottleneck users in cell m .Constraint (18) implies that, for each cell n , the sum ofresources allocated to its serving users K n (including boththe bottleneck and non bottleneck users) and the resourcesallocated to all the bottleneck users served by its neighboringcells m ∈ N n is limited to the total amount of resources. Notehat the modified function g is a monotone norm. Thus, theproblem with the modified constraint (18) is still an instanceof that in (1).
3) Detecting Bottleneck Users:
The asymptotic behavior ( λ ∞ , w ∞ ) provides a good guide to detect efficiently thebottleneck services, because it indicates the existing limits ofthe utility and the fraction of allocated resources as θ → ∞ .More precisely, let the k th entry of w ∞ be denoted by w ( k ) ∞ .The larger the value of w ( k ) ∞ , the higher the chance that thecorresponding user impairs the system performance owing tothe large amount of occupied resources, and, consequently, thehigher the possibility that this user causes heavy interference.To show how w ∞ can be useful in the development ofefficient heuristics for selecting the set of bottleneck users, wecompare the following two approaches: exhaustive search and successive selection . As we discuss below, the latter approachis suboptimal, but its computational complexity is subtantiallysmaller than that of the former approach. • Exhaustive search . Given a set of candidate users, denotedby K ( c ) , we find an optimal subset K ( b ) ⊆ K ( c ) thatprovides the maximum utility. This problem poses aserious computational challenge: the number of subsetsis exponential in the size of the domain. It might requirean exhaustive search over (cid:80) i =1 ,..., | K ( c ) | (cid:0) | K ( c ) | i (cid:1) possiblesubsets. • Successive selection : By sequentially selecting users withthe highest values w ( k ) ∞ , the resulting optimized utility hasa general trend of first increasing and then decreasing,as shown in Fig. 3. Therefore, instead of exhaustivelysearching for the best set of bottleneck users, we proposean efficient heuristic to approximate the solution. Wesuccessively add the users with the next highest valueof w ( k ) ∞ until the utility does not increase.Fig. 3 illustrates the motivation for successive selection . Itshows the optimized utility depending on the number of bot-tleneck users for four simulation instances, each of them with adifferent distribution for the user locations and traffic demands.Suppose that the selection of bottleneck users is based on therank order the components of w ∞ . Let us sort the entriesin w ∞ in descending order, with the order of w ( k ) ∞ denotedby o k . For example, if the number of bottleneck users is ,then, the highest ranked users, i.e., { k ∈ K : o k ∈ { , , }} are selected. The curves generated by different instances showa common trend of utility increase when the users with thehighest values of w ( k ) ∞ are allocated in the muting region. Then,if too many bottleneck users are selected, the utility decreases.Such pattern reflects the tradeoff between resource utilizationand interference mitigation. Therefore, although the curves arenot always piecewise monotonic, numerical results show thatadding users successively until the utility starts to decreaseprovides a simple means of identifying bottleneck users.
4) Fixed Point Iteration:
Once the set of bottleneck usersis updated, the load constraint is modified as shown in (18) toincorporate the muting scheme, and the fixed point iteration(3) (with θ = 1 and (cid:107) · (cid:107) = g ( · ) as defined in (18)) can Number of selected bottleneck users O p t i m i z ed u t ili t y Simulation instance 1Simulation instance 2Simulation instance 3Simulation instance 4
Fig. 3: Example of successive selection of bottleneck users.
Algorithm 1:
Partial Resource Muting input : n ← , K ( b ) (0) = ∅ output: K ( b ) ∗ , w ∗ Compute w ∗ (0) , c ∗ (0) using (3) with θ = 1 ; Compute w ∞ , λ ∞ and transition point θ (trans) usingProposition 2 and 3; if θ (trans) < then Sort w ( k ) ∞ in descending order; while n = 0 or c ∗ ( n ) − c ∗ ( n − ≥ do K ( b ) ∗ ← K ( b ) ∗ ( n ) , w ∗ ← w ∗ ( n ) ; n ← n + 1 ; K ( b ) ( n ) ← K ( b ) ( n − ∪ { k | o k = n } ; Update interference pattern and load constraintswith (17) and (18); Compute c ∗ ( n ) , w ∗ ( n ) corresponding to K ( b ) ( n ) with fixed point iteration (3); else if θ (trans) ≥ then K ( b ) ∗ ← K ( b ) ∗ (0) , w ∗ ← w ∗ (0) be applied to solve Problem (1). Algorithm 1 summarizes theproposed resource muting mechanism. D. Numerical Results
We consider a real-world scenario with 15 three-sectormacro BSs and 10 pico BSs in the city center of Berlin,Germany. The locations of the macro BSs are given by thereal data set [17], while the pico BSs are placed uniformly atrandom in the playground. The macro BSs are equipped withdirectional antennae with transmit power of dBm, whilethe pico BSs have omnidirectional antennae with transmitpower of dBm. The total bandwidth is 10 MHz. Themacrocell pathloss is obtained from the real data set [17],and the picocell pathloss uses the 3GPP LTE model in [18].Uncorrelated fast fading characterized by Rayleigh distributionis implemented on top of the pathloss. A fixed number of usersare randomly and uniformly distributed on the playground. Thetraffic demand per user is uniformly distributed between [0 , MBit/s with an average value of MBit/s.
1) Resource Efficiency with or without Resource Muting:
Fig. 4 illustrates how the transition point of the utility and theresource efficiency change when the resource muting schemes activated. Fig. 4a shows that the resource efficient regionincreases (or, equivalently, the interference-limited region de-creases) when applying resource muting. The achieved utility U ( θ ) and resource efficiency E ( θ ) with the muting scheme aresignificantly higher at θ = 1 (with the load constraint (18)).
2) Efficient Selection of Bottleneck Users:
To show that w ∞ is useful to identify bottleneck users, we use Monte Carlosimulations to generate random locations and demands ofusers, and we compare the number of bottleneck users (cid:12)(cid:12) K ( b ) (cid:12)(cid:12) selected by exhaustive search and by successive selection asdescribed in Section III-C3. In the upper subfigure of Fig.5, points of (cid:16)(cid:12)(cid:12)(cid:12) K ( b )( exh ) (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) K ( b )( suc ) (cid:12)(cid:12)(cid:12)(cid:17) are plotted (note thatthere are overlapping points), with each corresponding to adistribution of user locations and traffic demands, where K ( b )( exh ) and K ( b )( suc ) denote the set of bottleneck users selected by exhaustive search and by successive selection , respectively.The lower subfigure shows the empirical probability densityof the number of bottleneck users. Although Fig. 5 shows that successive selection generally selects less members than ex-haustive search , it is shown in Fig. 6 that successive selection achieves similar performance to exhaustive search .
3) Performance Improvement:
In Fig. 6 we compare theperformance of four protocols for K = 200 users and K =400 users. The performance is characterized by the cumulativedistribution function (CDF) of the achievable utility derivedfrom Monte Carlo simulations with random distributionof user locations and traffic demands. The four protocols aredescribed below:(I)
Non-muting : We solve Problem (13) with θ = 1 byapplying the fixed point iteration in (3).(II) Muting based on w ∞ , successive selection : We solveProblem (13) with Algorithm 1. The interference couplingand the load constraints are modified according to (17)and (18), respectively.(III) Muting based on w ∞ , exhaustive search : Similar toAlgorithm 1, but the bottleneck users are selected byexhaustive search.(IV) Muting based on I , successive search : Similar toAlgorithm 1, but the bottleneck users are selectedsuccessively based on a different indicator I k := (cid:80) l (cid:54) = k ( p l v k,l w ∗ l + p k v l,k w ∗ k ) , which reflects the sum ofthe received and generated interference by user k .Comparing the performance of Protocol I with the othermuting schemes in Fig. 6, we observe that the muting schemessignificantly improve the desired utility. Comparing ProtocolII and III, we verify that successive selection performs onlyslightly worse than exhaustive search, but the computationaleffort is significantly reduced. Furthermore, comparing Pro-tocol III and IV, we verify that w ∞ is a more appropriateindicator for bottleneck selection than the indicator based oninterference measurements.IV. E XTENSION OF THE F RAMEWORK TO
5G N
ETWORKS
The framework can be extended to optimize 5G networksequipped with flexible duplex mechanisms. Flexible duplex -4 -2 log( θ ) -2 -1 l og ( U ( θ )) Non-muting: Bound log( θ /||T(0)||)Non-muting: Bound log(1/ λ ∞ )Non-muting: Transition point θ (trans) Non-muting: Empirical log(U( θ ))Muting: Bound log( θ /||T(0)||)Muting: Bound log(1/ λ ∞ )Muting: Transition point θ (trans) Muting: Empirical log(U( θ )) (a) Log-log plot of utility depending on θ . -4 -2 log( θ ) -3 -2 -1 l og ( E ( θ )) Non-Muting: Bound log(1/||T(0)||)Non-Muting: Bound log(1/( λ ∞ θ ))Non-Muting: Empirical log(E( θ ))Muting: Bound log(1/||T(0)||)Muting: Bound log(1/( λ ∞ θ ))Muting: Empirical log(E( θ )) (b) Log-log plot of resource efficiency depending on θ . Fig. 4: Comparison between the performance limits of non-muting scheme and muting scheme, K = 200 . |K (b)(exh) | | K ( b ) ( s u c ) | |K (b) | E m p i r i c a l p r obab ili t y Exhaustive searchSuccessive selection
Fig. 5: Number of bottleneck users selected by exhaustivesearching and by successive selection.is one of the key technologies in 5G to adapt to asymmetricuplink (UL) and DL traffic with flexible resource allocationin joint time-frequency domain. The main challenge is thenewly introduced inter-mode interference (IMI) between ULand DL as shown in Fig.7, which makes Problem (13) farmore complex. In previous work [14] we have shown howto incorporate IMI into the interference model, and we havedeveloped a novel algorithm called successive approximationof fixed point (SAFP) to approximate the solution to (13). Inthis study we briefly describe the interference model and theSAFP algorithm, and we put more focus on a new resource
Utility E m p i r i c a l CD F o f u t ili t y K=200: without mutingK=200: muting based on I , successive searchK=200: muting based on w ∞ , successive selectionk=200: muting based on w ∞ , exhaustive searchK=400: without mutingK=400: muting based on I , successive searchK=400: muting based on w ∞ , successive selectionk=400: muting based on w ∞ , exhaustive search Fig. 6: Performance comparison between max-min utilityfairness with and without muting for K = 200 and .muting scheme based on the spectral properties of asymptoticmappings. A. Joint UL/DL System Model
We now generalize the DL system model defined in SectionIII-A by considering a set of services K , including both ULand DL services. As in previous sections, let the BS-to-serviceassignment be denoted by A ∈ { , } N × K . Without resourcemuting, the utility max-min fairness problem can also bewritten in the general form (13). However, the SINR modelneeds to be modified to take IMI into account. As shownin Fig. 7, inter-cell interference appears in the overlappingresource region. We introduce the overlapping factors ( c k,l ) ,collected in C ( w ) ∈ R K × K ++ , to incorporate the probabilitythat intra- and inter-mode interference appear for a given re-source allocation w . Let ν ( x ) := (cid:104) ν ( x )1 , . . . , ν ( x ) N (cid:105) , x ∈ { u , d } denote the UL or DL load (i.e., the fraction of occupiedresources) of all cells that can be computed with w . Forexample, ν ( u ) n := (cid:80) k ∈ K ( u ) n w k denotes the load of BS n inUL, where K ( u ) n ⊆ K n is the set of UL services in BS n . TheSINR of service k incorporating IMI is approximated by SINR k ( w ) ≈ p k (cid:104)(cid:16) C ( w ) ◦ ˜ V (cid:17) diag( p ) w + ˜ σ (cid:105) k (19)with C ( w ) := ( c k,l ) ∈ R K × K + , (20) c k,l := (cid:104)(cid:16) ν ( x l ) n l + ν ( x k ) n k − (cid:17) /ν ( x k ) n k (cid:105) + if x l (cid:54) = x k min (cid:110) , ν ( x l ) n l /ν ( x k ) n k (cid:111) if x l = x k , where ◦ denotes the Hadamard product, x k ∈ { u , d } denotesthe duplex mode of service k , and n k denotes the serving BSof k . The approximation is based on two probabilities: • The overlapping factor c k,l is the ratio of the overlappingarea between load ν ( x l ) n l and ν ( x k ) n k to the load ν ( x k ) n k , whichroughly indicates the probability that an arbitrary resourceunit allocated to mode x l in BS n l causes interference toBS n k in mode x k . Note that the first case x l (cid:54) = x k corresponds to the IMI between UL and DL, while x l = x k corresponds to the intra-mode interference. • w l collected in w serves as the probability that anarbitrary resource unit in n l is allocated to service l .The multiplication of c k,l by w l loosely approximates theprobability that a resource unit allocated to l in duplex mode x l in BS n l causes interference to service k in duplex mode x k in BS n k . B. Successive Approximation of Fixed Point
Introducing C into (20) removes the properties of T givenin Lemma 1, which further leads to possibly more than onefixed point of the resulting CEVP (2). In [14] we developed anovel algorithm SAFP to approximate the near-optimal fixedpoint of the CEVP. The novel proposed algorithm assistedwith random initialization and successive approximation issummarized below: • The algorithm runs for Z ( max ) times, where at the i -thtime different random initializations of w (0) := ˆ w i andthe corresponding ˆ C := C ( ˆ w i ) are used. • For each initialization, we iteratively perform the follow-ing two steps: 1) we use the fixed point iteration (3) withrespect to the approximated ˆ C and derive w ( t ) , and 2)update ˆ C = C ( w ( t ) ) and increment t . The iteration stopsif (cid:107) w ( t ) − w ( t − (cid:107) ≤ (cid:15) , where (cid:15) is a distance threshold. • Each random initialization converges to a fixed point (notnecessarily different from those derived from other ini-tializations). We choose the solution with the maximumutility.As shown in [14], the algorithm converges for each randominitialization. With a limited number of random initializations,the algorithm is able to find a solution with low computationalcomplexity.
C. Partial Resource Muting
We proposed partial resource muting scheme for 4G DLwireless networks in Section III-C to improve the resourceefficiency. Along similar lines, the resource muting schemecan be tailored for IMI mitigation in 5G networks enablingflexible duplex. To deal with the complex interference causedby IMI, we have introduced SAFP as the tool to find efficientlythe approximated solution to the CEVP with the existenceof possibly multiple fixed points. Hence, by replacing thefixed point iteration with SAFP (line in Algorithm 1),the same steps proposed in Algorithm 1 can be applied tooptimize the joint UL/DL resource allocation, particularly theset of bottleneck users allocated to the muting region and theresource fraction allocated to them.It is worth mentioning that the steps used for triggering themuting scheme and for detecting the bottleneck users requirethe computation of the transition point θ (trans) and of the tuple ( λ ∞ , w ∞ ) . However, with the modified SINR model (19), T ( w ) is not an SIF, and Proposition 3 cannot be appliedin a straightforward manner to compute the desired values.Therefore, we propose to approximate T ( w ) by replacing C ( w ) with the converged approximation of ˆ C achieved bySAFP. Let the approximation of T ( w ) be denoted by T ˆ C ( w ) .Since ˆ C is a matrix with known entries, T ˆ C ( w ) is an SIF. S 1BS 2BS 3 Inter-mode interference Intra-mode interference
DLUL
Fig. 7: Regions where inter-cell interference appears.Hence, we can use Proposition 1 to compute T ˆ C , ∞ ( w ) ,and further derive ( λ ˆ C , ∞ , w ˆ C , ∞ ) with respect to T ˆ C , ∞ ( w ) along similar lines to Proposition 3. Using the same techniquedescribed in Section III-C, we obtain the triggering parameter θ (trans) and the set of bottleneck users from (cid:16) λ ˆ C , ∞ , w ˆ C , ∞ (cid:17) . D. Numerical Results
We consider the same network introduced in Section III-D.In addition, flexible duplex is enabled such that resources canbe dynamically allocated to UL or DL services. Owing to thespace limitation, we only present one exemplary numericalresult in Fig. 8. We compare the performance of the followingfour protocols: (I)“FIX” for fixed ratio between UL and DLresources; (II)“SAFP” for dynamic UL/DL resource configura-tion with SAFP without muting; (III)“Resource muting basedon I ” for the resource muting scheme based on interferenceindicator I as introduced in Section III-D, and (IV)“Resourcemuting based on w ∞ ” for the resource muting scheme basedon the asymptotic behavior w ∞ . The performance is obtainedby averaging Monte Carlo simulations with randomdistribution of user locations and traffic demands conditionedon the given traffic asymmetry for K = 200 . We define ameasure called inter-cell traffic distance to reflect both theinter-cell traffic asymmetry and the intra-cell UL/DL trafficasymmetry between a pair of cells m, n , computed as D m,n := (cid:107) d n − d m (cid:107) , where (cid:107)·(cid:107) denotes the L -norm, and the per-cellUL/DL traffic demands d n := (cid:104) d ( u ) n , d ( d ) n (cid:105) are normalized. Fig.8 shows that dynamic UL/DL configuration achieves a twofoldincrease in the average utility compared to fixed UL/DL ratio.Resource muting brings further improvement, varying from % to %, by adapting to the traffic asymmetry. Note thatthe performance gains increase with the traffic asymmetry.V. C ONCLUSION
We have characterized an efficient resource utilization re-gion by studying the asymptotic behavior of solutions to max-min utility optimization problems with interference modelsbased on the load imposed by services. Building upon thisresult, we have developed a partial resource muting schemethat is suitable for both conventional 4G DL networks and flex-ible duplex enabled 5G networks. Simulations show significantperformance gains in both scenarios, measured by the worst-case QoS satisfaction level, compared to the optimal solutionto the max-min optimization problem without resource mutingmechanisms. We have also shown that the gain obtainedwith the proposed muting scheme increases when the trafficbecomes highly asymmetric.
Average inter-cell traffic distance A v e r age u t ili t y FIXSAFP
Resource muting based on I Resource muting based on w ∞ Fig. 8: Average utility with different inter-cell traffic distances.R
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