Dynamic Uplink/Downlink Resource Management in Flexible Duplex-Enabled Wireless Networks
DDynamic Uplink/Downlink Resource Managementin Flexible Duplex-Enabled Wireless Networks
Qi Liao
Nokia Bell Labs, Stuttgart, GermanyEmail: [email protected]
Abstract —Flexible duplex is proposed to adapt to the channeland traffic asymmetry for future wireless networks [1]. In thispaper, we propose two novel algorithms within the flexible duplexframework for joint uplink and downlink resource allocationin multi-cell scenario, named successive approximation of fixedpoint (SAFP) and resource muting for dominant interferer(RMDI), based on the awareness of interference coupling amongwireless links. Numerical results show significant performancegain over the baseline system with fixed uplink/downlink resourceconfiguration, and over the dynamic time division duplex (TDD)scheme that independently adapts the configuration to time-varying traffic volume in each cell. The proposed algorithmsachieve two-fold increase when compared with the baselinescheme, measured by the worst-case quality of service satisfactionlevel, under a low level of traffic asymmetry. The gain is moresignificant when the traffic is highly asymmetric, as it achievesthree-fold increase.
I. I
NTRODUCTION
Flexible duplex is one of the key technologies in fifthgeneration (5G) to optimize the resource utilization depend-ing on traffic demand [1]. The main objective is to adaptto asymmetric uplink (UL) and downlink (DL) traffic withflexible resource allocation in the joint time-frequency domain,such that the distinction between TDD and frequency divisionduplex (FDD) is blurred, or completely removed.Despite the advantage of adaptation to the dynamic trafficasymmetry, the drawback is the newly introduced inter-cellinterference (ICI) between duplexing mode DL and UL, here-inafter referred as inter-mode interference (IMI) . The DL-to-UL interference plays a more important role due to the largedifference between DL and UL transmission power. Manyworks focus on physical layer design to overcome IMI. In [2],special kinds of radio frames with different ratio of UL/DLare introduced to FDD, and heuristic approach is proposed tofind the most suitable one solely based on the traffic volume.A few studies target the problem of dynamic UL/DL resourceconfiguration. In [3], the authors formulate a utility maximiza-tion problem to minimize the per-user difference between ULand DL rates; while in [4] the problem is formulated as a two-sided stable matching game to optimize the average utility peruser. Both works consider a single cell system where IMI doesnot play a role. However, in a multi-cell system the optimalUL/DL configuration depends not only on the traffic volumebut also the interference coupling between all transmissionlinks. Although very few studies provide solutions within theflexible duplex framework, similar problem exists in dynamicTDD. A popular solution is the cell-cluster-specific UL/DL reconfiguration [5], but how to coordinate the clusters for inter-cluster IMI mitigation still remains a challenge.In this paper, we optimize UL/DL resource configurationin multi-cell scenario, by recasting max-min fairness probleminto a fixed point framework. Such framework is widely usedfor power control [6], [7] and load estimation [8], [9] for ULor DL systems independently. Our previous work [10] exploitsthe framework to tackle the joint UL/DL resource allocationand power control problem within flexible duplex, assumingthat ICI is simply proportional to the load. This assumption,however, is valid only when each resource unit has the samechance to be allocated to UL or DL, which may result in highprobability of generating IMI. We improved the model in thispaper. The main contribution is summarized in below. • A new interference model is defined, which allows toprioritize the positions of the resources for UL and DLtransmission, to reduce the probability of generating IMI. • We propose a novel algorithm SAFP to find algorithmicsolution to optimize UL/DL resource configuration. Un-like the models in previous works [6], [8], [9], the newinterference model is nonlinear and nonmonotonic . • Further we enhance SAFP to RMDI by detecting sequen-tially the dominant interferer in the system, and mutingthe partial resource in neighboring cells to reduce ICI. • We compare SAFP and RMDI numerically with twoconventional schemes: a) fixed UL/DL configuration, andb) dynamic TDD that adapts UL/DL configuration solelybased on traffic volume, and show a performance gainvarying from two to three fold depending on the trafficasymmetry.The rest of the paper is organized as follows. In Section II,the system model is described together with the correspondentnotation. The problem statement is given in Section III.The proposed algorithms SAFP and RMDI are introduced inSection IV and V, respectively. Finally, in Section VI, thenumerical results are presented.II. S
YSTEM M ODEL
In this paper, we use the following definitions. 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 x and y . Let diag( x ) denote a diagonal matrix with the elements of x on the maindiagonal. For a function f : R k → R k , f n denotes the n -foldcomposition so that f n = f ◦ f n − . The cardinality of set A is a r X i v : . [ ee ss . SP ] S e p ABLE I: NOTATION SUMMARY N set of BSs with | N | = N K set of UEs with | K | = K S set of services with | S | = S W set of MRUs with | W | = W S ( u ) ( S ( d ) ) set of UL (DL) services S n set of services served by the n th BS n s index of BS serving the s th service A UE-to-service association matrix B BS-to-service association matrix B ( u ) ( B ( d ) ) BS-to-UL (BS-to-DL) association matrix δ t ( δ f ) length of time duration (range of frequency) of an MRU W t ( W f ) number of smallest time (frequency) units in MRU set W w fraction of resource allocated to services ν cell load ν ( u ) ( ν ( d ) ) cell load in UL (DL) p transmit power allocated to services d traffic demand of services H channel gain matrix V link gain coupling matrix ρ s per service QoS satisfaction level ρ worst-case QoS satisfaction level denoted by | A | . The positive part of a real function is definedby [ f ( x )] + := max { , f ( x ) } . The notation that will be usedin this paper is summarized in Table I.We consider an orthogonal frequency division multiplexing(OFDM)-based wireless network system, consisting of a setof base stations (BSs) N := { n : n = 1 , , . . . , N } and aset of user equipments (UEs) K := { k : k = 1 , , . . . , K } .We assume that the network enables flexible duplex, wherethe resource in both frequency and time domains can bedynamically assigned to UL and DL. We define minimumresource unit (MRU) as the smallest time-frequency unit, thathas a length of δ t seconds in time domain and a range of δ f Hz in frequency domain. We consider a set of MRUs, denotedby W , consisting of W t smallest time units and W f smallestfrequency units, and we have W := | W | = W t · W f .We assume that K UEs generate a set of UL and DLservices S := S ( u ) ∪ S ( d ) within the time duration of W MRUs(i.e., W t δ t seconds). Let the UE-to-service association matrixbe denoted by A ∈ { , } K × S , where a k,s = 1 means thatthe s th service is generated by the k th UE, and otherwise.Let B ∈ { , } N × S denote the BS-to-service associationmatrix. To differentiate UL and DL services, we further defineBS-to-UL and BS-to-DL association matrices, denoted by B ( u ) ∈ { , } N × S and B ( d ) ∈ { , } N × S , respectively. Letthe set of services served by BS n be denoted by S n and letthe BS associated with service s be denoted by n s .Let w := [ w , . . . , w S ] T ∈ [0 , S be a vector collectingthe fraction of resource allocated to all services s ∈ S . The cell load , defined as the fraction of occupied resource withina cell, is denoted by ν = Bw ∈ [0 , N . The cell load inUL and DL are denoted by ν ( u ) = B ( u ) w and ν ( d ) = B ( d ) w respectively, and we have ν = ν ( u ) + ν ( d ) . We collect thetransmit power (in Watt) allocated to all services in a vector p := [ p , . . . , p S ] T . A. Link Gain Coupling Matrix
We assume that average channel gains over W MRUsfrom each transmitter (TX) to each receiver (RX) are known, Fig. 1: Example: Interference link gain.collected in H := ( h i,j ) ∈ R ( N + K ) × ( N + K )++ . Note that theTXs and RXs include both UEs and BSs. Let v l,s denote thechannel gain of the link between the TX of link l and the RXof link s . If l = s , v l,s is the channel gain of link s , otherwiseif l (cid:54) = s , v l,s is the channel gain of the interference link causedby service l to s . We define link gain coupling matrix ˜ V as ˜ V := (˜ v l,s ) ∈ R S × S + , with ˜ v l,s := v l,s /v s,s , (1)where ˜ v l,s is the ratio between the interference link gain fromservice l to service s and the serving link gain of s .An example is shown in Fig. 1, where we consider a systemenabling downlink and uplink decoupling in 5G [11]. Theinterference caused by UL service (link l ) to DL service (link l ) has a link gain of v , = h , , i.e., the link gainbetween TX (transmitter of l ) and RX (receiver of l ).Given that the channel gain of l is h , , the interferencecoupling ratio is given by ˜ v , = h , /h , . Remark 1 (Incorporating different interference conditions) . Without loss of generality, we can modify ˜ V to take intoaccount different interference conditions. For example, toallow self-interference cancellation we can define ˜ v s,s := 0 for every s ∈ S , while to allow zero intra-cell interference wehave ˜ v l,s := 0 if l and s are associated with the same BS.B. Quality of Service Metric In [10] we assume that the probability that l causes ICI to s associated with a different BS is approximated by the fractionof its allocated resource w l , which leads to Pr { l interferes s | n l (cid:54) = n s } ≈ w l for l, s ∈ S . (2)The average signal-to-interference-plus-noise ratio (SINR) of s ∈ S is approximated by SINR s ≈ p s (cid:80) l ∈ S ˜ v l,s p l w l + σ s v s,s = p s (cid:104) ˜ V T diag( w ) p + ˜ σ (cid:105) s , (3)where ˜ σ := (cid:2) σ /v , , σ /v , , . . . , σ S /v S,S (cid:3) T , σ s denotes thenoise power in the receiver of s . Note that in (3) w l serves asa probability. The interference condition is taken into accountin ˜ v l,s as illustrated in Remark 1. Note that ˜ v l,s is computed with average channel gain over W MRUs.Thus, (3) is the ratio between average received signal strength and averagereceived interference, rather than the actual average SINR. Since we do notassume to know the distribution of the channel gain, here we use (3) toapproximate the average SINR. owever, the approximations (2) and (3) are only validunder the assumption that each MRU is considered to be“equal” for all the services to be allocated, namely, the positionof resource is not specified for UL or DL. Unfortunately,such assumption results in a high probability of IMI. In thefollowing we introduce an improved SINR model based on asimple UL/DL resource positioning strategy to reduce IMI.Recall that conventional TDD or FDD specifies a set ofresource for UL and DL respectively to prevent IMI. Withflexible duplex, the challenge is to allow different resourcepartitioning between UL and DL in each cell, while limitingthe probability of generating IMI. Let us take an example,cell m with UL load ν ( u ) m and cell n with DL load ν ( d ) n share same set of available resource. It is obvious that theminimum overlapping area between UL resource in cell m and DL resource in cell n is (cid:104) ν ( u ) m + ν ( d ) n − (cid:105) + , which canbe easily achieved by allocating the set of resource to ULtraffic in cell m in some priority order while allocating thesame set of resource to DL traffic in cell n in reverse order.Given the aforementioned strategy, to derive the interferencecoupling matrix that incorporates the probability that a linkcauses ICI to another, we introduce a reuse factor couplingmatrix C ( w ) depending on w . Let x s ∈ { u , d } denote the ULor DL traffic type of service s ∈ S , and recall that n s denotesthe serving BS of s , C ( w ) is defined as C ( w ) := C := ( c l,s ) ∈ R S × S + , (4) c l,s := (cid:104)(cid:16) ν ( x l ) n l + ν ( x s ) n s − (cid:17) /ν ( x s ) n s (cid:105) + if x l (cid:54) = x s min (cid:110) , ν ( x l ) n l /ν ( x s ) n s (cid:111) if x l = x s , where the load of cell n s occupied by traffic type x s iscomputed by ν ( x s ) n s := (cid:104) B ( x s ) w (cid:105) n s . In general, c l,s is definedas the ratio of the overlapping area on the resource planebetween the load of cell n l serving traffic type x l and theload of cell n s serving traffic type x s to the load of cell n s serving traffic type x l .With C ( w ) in hand, given the power vector p , we canmodify (3) and derive the SINR of service s ∈ S as SINR s ( w ) ≈ p s (cid:20)(cid:16) C ( w ) ◦ ˜ V (cid:17) T diag( p ) w + ˜ σ (cid:21) s , (5)where with a slight abuse of notation, X ◦ Y denotes theHadamard (entrywise) product of matrices X and Y . Notethat the first term in the denominator is the interference powerreceived by service s divided by the channel gain of s , and it isequivalent to (cid:80) l c l,s w l v l,s p l /v s,s , where c l,s · w l approximatesthe probability that service l causes interference to service s .The maximum achievable number of bits for service s ∈ S within the time span of resource set W is η s ( w ) = δ t δ f W w s log (1 + SINR s ( w )) , (6)where the unit of δ t δ f is Hz · s/MRU, while W w s is the numberof MRUs allocated to s .Assuming that the nonzero traffic demands d :=( d , . . . , d S ) T ∈ R S ++ is known, where d s is defined as number of required bits of s during the time span of W ,we introduce per service quality of service (QoS) satisfactionlevel , written as ρ s ( w ) = η s ( w ) /d s , s ∈ S . (7)III. P ROBLEM F ORMULATION
The objective is to partition the resource set W in each cell n ∈ N into three subsets: resource for UL, resource for DL,and blanked resource , respectively, to maximize the worst-case QoS satisfaction level , defined as ρ ( w ) := min s ∈ S ρ s ( w ) . (8)All demands of the services are feasible, when ρ ( w ) ≥ .We formulate the problem in Problem 1, where (9a) and(9b) imply the objective of maximizing the worst-case QoSsatisfaction level ρ ∗ , and (9c) is the per-cell load constraint. Problem 1 max . w ∈ R S + ,ρ ∈ R + ρ (9a)s.t. w ≥ ρ f ( w ) , (9b) g ( w ) := (cid:107) Bw (cid:107) ∞ ≤ , (9c)where the vector-valued function f is defined by f : R S + → R S ++ : w (cid:55)→ [ f ( w ) , . . . , f S ( w )] T , (10a)where f s ( w ) := d s δ t δ f W log (1 + SINR s ( w )) . (10b)In [10], we show that with conventional model of SINR(3), Problem 1 is equivalent to solve a nonlinear system ofequations such that w = ρ f ( w ) , g ( w ) = 1 and that ρ ismaximized. It is worth mentioning that, with the modifiedmodels of interference coupling (4) and SINR (5), Problem1 is a multi-variate nonconvex optimization problem. More-over, the constraint (9b) is neither convex nor continuouslydifferentiable, and Problem 1 is not necessarily equivalent tothe nonlinear system of equations.In Section IV we provide algorithmic solution to Problem 1,denoted by w ∗ . The per-cell fraction of resource to allocatedto UL and DL are then obtained as ν ( u ) , ∗ = B ( u ) w ∗ and ν ( d ) , ∗ = B ( d ) w ∗ , respectively. If ρ ∗ := ρ ( w ∗ ) ≥ , alldemands are feasible. However, if ρ ∗ < , the solution toProblem 1 is not a good operating point, since the demandsof all services are infeasible. In other words, all users areunsatisfied. Therefore, a further question arises: how can wetransform the desired demands in Problem 1 from infeasibleto feasible? One of the factors causing infeasible demand isthe bottleneck services. In Section V we modify Problem1 by dedicating partial resources for bottleneck services,while muting them for others, and develop an algorithm withheuristic strategies. Under certain conditions, enhanced interference mitigation can be achievedby muting partial resources in some cells. However, it is also possible thatthe optimal solution returns an empty set of the blanked resource. emark 2 (New challenge due to complex interferencecoupling) . Problem 1 is formulated along similar lines toour previous work [10, Problem 2a]. However, in [10], thereceived interference in SINR (3) is an affine function of w ,which further leads to some nice properties of f (as shown inLemma 1). In this paper, because we introduce more complexinterference coupling (4) and the resulting modified SINRmodel (5) , the desired properties of f do not exist, which bringsnew challenge with developing efficient algorithmic solution. IV. S
UCCESSIVE A PPROXIMATION OF F IXED P OINT
In this section, we first provide background informationabout the mathematical tool to solve the problem. Then, wepropose a novel efficient algorithm SAFP to find a feasiblepoint of w with good, if not optimal, objective value of ρ ∗ . A. Background Information and Previous Results
With the conventional SINR model in (3), f defined in (10)has the following property. Lemma 1 ([10, Lemma 1]) . With SINR defined in (3) , f : R S + → R S ++ is a standard interference function (SIF) (seeAppendix A for definition). Knowing that f is SIF, and that g : R S ++ → R ++ in (9c)is a monotonic norm, we encounter the same type of problemas [10, Problem 2a]. The following proposition is providedbased on the previous result [10, Theorem 1], which givesrise to an algorithmic solution to Problem 1 with conventionalSINR model based on the fixed point iteration scheme. Proposition 1.
Suppose SINR is modeled with (3) , and • f : R S + → R S ++ is SIF, • g : R S ++ → R ++ is monotonic, and homogeneous withdegree (i.e., g ( α x ) = αg ( x ) for all α > )There exists a unique solution to Problem 1, denoted by { w ∗ , ρ ∗ } , where w ∗ can be obtained by performing thefollowing fixed point iteration: w ( t +1) = f (cid:0) w ( t ) (cid:1) g ◦ f (cid:0) w ( t ) (cid:1) , t ∈ N , (11) where with a slight abuse of notation, g ◦ f denotes the com-position of functions g and f . The iteration in (11) convergesto w ∗ , and we have ρ ∗ = 1 /g ◦ f ( w ∗ ) and g ( w ∗ ) = 1 .Proof. The proof is omitted here since it uses our previousresult [10, Theorem 1] and is along the same lines as [10,Proposition 1].
B. Successive Approximation of Fixed Point
Proposition 1 provides an algorithmic solution to Problem1 with SINR (3), by utilizing the properties of SIF. Unfortu-nately, with the modified SINR in (5), f is not SIF because thecoupling matrix C ( w ) depends on w in a non-monotonic andnon-differentiable manner. However, it is easy to show that byreplacing C ( w ) in (5) with some approximation C (cid:48) := C ( w (cid:48) ) computed with fixed w (cid:48) , the SINR in (5) falls into the same class as (3), and the approximated problem can be solved byProposition 1 with f ( w ) replaced by f C (cid:48) ( w ) := f ( w , C ( w (cid:48) )) .Therefore, our essential, natural idea is to efficiently com-pute a suboptimal solution of Problem 1 by solving a sequenceof (simpler) max-min fairness subproblems whereby the non-contractive mapping f is replaced by suitable contractionapproximation f C (cid:48) . These subproblems can be solved withProposition 1.More specifically, the proposed SAFP algorithm consists insolving a sequence of approximations of Problem 1 in the form max . w ∈ R S + ,ρ ∈ R + ρ ; s.t. w ≥ ρ f C (cid:48) ( w ) ; g ( w ) ≤ , (12)where f C (cid:48) ( w ) represents approximation of f ( w ) at the currentiterate w (cid:48) . The unique solution to (12) can be obtained by thefixed point iteration (11), with C ( w ) replaced by C ( w (cid:48) ) .Unfortunately, due to the complexity of C ( w ) , the con-vergence of SAFP to a limit point cannot be guaranteed,since multiple fixed points can exist in the system wherethe inequality sign in (9b) is replaced by the equality sign.Different initial values of ˆ w may lead to different fixed points. Moreover, the solution to the system of nonlinear equationsmay not be the optimal solution to the original problem ofmaximizing the minimum, due to the nonmonotonicity of themapping f when including C into the interference model. Thus, we design the searching algorithm to guarantee theutility increase with initial values of { ρ ∗ , w ∗ } , maximumnumber of random initiation N max , and algorithm stoppingcriterion depending on the maximum number of iterations N iter and the distance threshold (cid:15) , illustrated as below. • The algorithm runs for N max times, each with a differentrandom initialization of ˆ w and the corresponding C ( ˆ w ) . • For each initialization ˆ w n , n = 1 , , . . . , N max , weiteratively perform the fixed point iteration in (11) with f ( w ) replaced by f ˆ C n ( w ) where ˆ C n := C ( ˆ w n ) . Theiteration stops if the number of iterations exceeds N iter orthe distance yields (cid:107) w (cid:48) − w (cid:107) ≤ (cid:15) and returns the solution { w (cid:48) , ρ (cid:48) } with respect to the n th random initialization. Thesolution is updated with w ∗ ← w (cid:48) , ρ ∗ ← ρ (cid:48) if ρ (cid:48) > ρ ∗ .The proposed SAFP algorithm is summarized in Algorithm 1.Although the convergence of SAFP to a global optimumcannot be guaranteed and heuristics are introduced, numericalresults in Section VI (e.g., Fig. 2b) show that each randominitialization converges to a fixed point, and with limitednumber of initializations, the algorithm finds a suboptimal,if not optimal, solution among multiple fixed points.V. R ESOURCE M UTING FOR D OMINANT I NTERFERER
The proposed SAFP finds a feasible point of w ∗ withsuboptimal, if not optimal, objective value of ρ ∗ . If ρ ∗ ≥ ,the obtained w ∗ provides fairness on the services, and thedemands of all services are feasible. However, if ρ ∗ < , w ∗ is not a good operating point since the traffic demands ofall services are infeasible. Therefore, in this section we focusthe following question: how can we transform the desireddemands in Problem 1 from infeasible to feasible? lgorithm 1: SAFP algorithm for resource partitioning input : i ← , N max > , N iter > , (cid:15) > , ρ ∗ ← , w ∗ ← output: { w ∗ , ρ ∗ } while i ≤ N max do random initialization of w (cid:48) ; C (cid:48) ← C ( w (cid:48) ) ; j ← , w ← ; ∆ ( j ) ← (cid:107) w (cid:48) − w (cid:107) ∞ ; w ( j ) ← w (cid:48) ; while j ≤ N iter or ∆ ( j ) ≥ (cid:15) do % solving approximated subproblem with C (cid:48) ; while (cid:107) w (cid:48) − w (cid:107) ∞ ≥ (cid:15) do w ← w (cid:48) ; w (cid:48) ← f C (cid:48) ( w ) /g ◦ f C (cid:48) ( w ) ; % Update C with optimized w (cid:48) ; w ( j +1) ← w (cid:48) ; C ( j +1) = C (cid:48) ← C ( w (cid:48) ) ; ∆ ( j +1) ← (cid:107) w ( j +1) − w ( j ) (cid:107) ∞ ; j ← j + 1 ; ρ (cid:48) = ρ (cid:48) ( w (cid:48) ) ← min s ∈ S w (cid:48) s /f C (cid:48) ,s ( w (cid:48) ) ; % update the solution if ρ (cid:48) exceeds the stored value ; if ρ (cid:48) > ρ ∗ then ρ ∗ ← ρ (cid:48) ; w ∗ ← w (cid:48) ; i ← i + 1 ;In [12], the authors propose a removal selection criterionfor an infeasible DL power control problem, that removessequentially the bottleneck services until the demands forall the remaining services are feasible. However, is there amethod of further increasing ρ ∗ without removal of services?Motivated by coordinated muting using almost blank subframe(ABS) for time domain intercell interference coordinationintroduced in [13], we are interested in exploring the tradeoffbetween resource utilization and interference reduction byintroducing the resource muting in flexible duplex. A. Modified Load Constraints Incorporating Resource Muting
The key concept is to sequentially reserve some resource ina cell for the dominant interferer, while muting them in thecells strongly impacted by the interferer. To this end, we rankthe services based on the interference level that they generateto others, given by I s ( w ) := (cid:16) c (cid:48) s ˜v (cid:48) sT (cid:17) p s w s , for s ∈ S , (13)where c (cid:48) s := row s C ( w ) denotes the s th row of C ( w ) , and ˜v (cid:48) s := row s ˜ V denotes the s th row of ˜ V .Moreover, to prevent the waste of resource, we select thestrongly affected cells to mute their resource. The set of cellsto mute the resource reserved for s is selected by M s := { m ∈ N \ { n s } : J s,m ( w ) ≥ α } , (14)where α is a threshold and J s,m ( w ) is the interferencegenerated from service s to a cell m (cid:54) = n s , defined as J s,m ( w ) := (cid:104) B (cid:0) c (cid:48) s ◦ ˜v (cid:48) s (cid:1) T (cid:105) m p s w s . (15) If a set of dominant interferers S is chosen, and for each s ∈ S a subset of the cells M s is selected to mute resource w s , then, in each cell we have the load constraint g (cid:48) m ( w ) := (cid:88) s ∈ S { m ∈ M s } w s + (cid:88) l ∈ S m w l ≤ , for m ∈ N , (16)where {·} is the indication function, the first term is the totalamount of resource to be muted in cell m , and the secondterm is the amount of available resource for services in m .Since g (cid:48) m ( w ) ≤ needs to be held for every m ∈ N , theload constraint can be rewritten as g (cid:48) ( w ) := max m ∈ N g (cid:48) m ( w ) ≤ . (17)Note that without the muting scheme, i.e., if ¯ S = ∅ , the firstterm in (16) is zero and (17) is equivalent to the per-cell loadconstraints in (9c). B. Design of Heuristic Algorithm
It is obvious that the modified g (cid:48) is also monotonic andhomogeneous with degree , which enables leverage of Propo-sition 1 to solve the modified Problem 1, with g ( w ) replacedby g (cid:48) ( w ) to incorporate the resource reservation and mutingstrategy.Compared to the solution to the original Problem 1, resourcemuting may not necessarily improve the desired utility ρ ,because muting of w s in cell m ∈ M s may lead to waste ofresource. Therefore, we develop a heuristic algorithm RMDIto guarantee a utility that is no less than the ρ derivedin Algorithm 1. The Algorithm is described briefly in thefollowing steps.1. Derive w (0) = w ∗ to Problem 1 with Algorithm 1 andcompute the corresponding ρ (0) = ρ ∗ .2. Compute I s ( w ∗ ) and rank the services based on I s . Let q s denote the rank of s , e.g., the maximum interferer ˆ s := arg max s I s has a rank of q ˆ s = 1 . Set k = 1 .3. Add the service with highest rank into ¯ S ( k ) , e. g., ¯ S ( k ) = { s : q s ≤ k } .4. Solve modified Problem 1 with ¯ S ( k ) using Algorithm 1(with g replaced by g (cid:48) ), derive w ( k ) and ρ ( k ) .5. If ρ ( k ) ≥ ρ ( k − , increment k and go back to Step 3;otherwise stop the algorithm.6. Obtain solution w (cid:63) = w ( k − .VI. N UMERICAL R ESULTS
In this section, we analyze the performance of the proposedalgorithms SAFP and RMDI, by considering the asymmetry ofUL and DL traffic in two-cell scenario. The distance betweenthe two BSs is km. The transmit power of BS and UEare and dBm respectively and all the other simulationparameters mainly related to channel gain can be found in[14, Tab. A2.1.1-2]. We define the minimum time unit δ t as . ms and the minimum frequency unit δ f as kHz. Furtherwe have W t = 20 and W f = 300 , i. e., a resource plane thatspans a time duration of . seconds and frequency of MHz(including the guard band).e defined a fixed total traffic demand
Λ = (cid:80) s d s = 50 kbits within W t δ t = 0 . seconds, which implies a to-tal serving data rate of Mbit/s. The total traffic can beasymmetrically distributed between the two cells with dif-ferent ratios among T inter := { / , / , / , . . . , / , / } .Within each cell, the traffic can be asymmetrically distributedbetween UL and DL traffic with ratios among T intra := { / , / , / , . . . , / } . UEs with either UL or DL trafficare generated with uniform distribution within the intersectionof two balls with radius km, and with BS 1 and 2 as theircenters respectively, to analyze the scenario of high inter-cellinterference. Without loss of generality, we can place oneUL and one DL service in each cell with the traffic demandcomputed by the traffic ratio mentioned above.
1) Algorithm convergence of SAFP.
Let us first examine theconvergence of Algorithm 1, and compare it with Algorithm“FP” that is summarized in Proposition 1 with conventionalSINR model (3). The parameters are set as N max = 30 , N iter = 1000 , (cid:15) = 10 − . In Fig. 2a we show the con-vergence of the SAFP with one particular initialization of w (cid:48) and C ( w (cid:48) ) and compare it with FP. The magenta circleindicates the starting point with an updated C (cid:0) w ( j ) (cid:1) , andthe green dashed line shows that with each fixed C (cid:0) w ( j ) (cid:1) ,by performing fixed point iteration, ρ monotonically increasesand converges to the fixed point with respect to C (cid:0) w ( j ) (cid:1) .Note that the green dashed line is not the “actual” utility ρ ,since it is computed with updated w ( i ) and the approximation C (cid:0) w ( j − (cid:1) . Therefore, we plot the red line to show theconvergence of the actual utility at each step of updating C ,computed with w ( j ) and C (cid:0) w ( j ) (cid:1) . By comparing the redcurve and the blue curve (convergence of FP algorithm), weobserve a significant increase of utility ρ by using SAFP.This is because, comparing with FP that randomly placesthe UL and DL resource, SAFP is based on an improvedinterference model, where ICI only appears in the intersectionof the sets of allocated MRUs between different cells. Fig.2b illustrates that with each random initialization of w (cid:48) , theproposed SAFP converges to a fixed point. The example showsthat initializations converge to two different fixed pointswith utilities . and . respectively. w ∗ corresponding tohigher utility is chosen as the final solution.
2) Performance comparison.
We compare the performanceof SAFP and RMDI to the performance of the other threeprotocols, described in below. • FIX:
Fixed ratio and same position of the UL and DLresource in different cell. IMI does not exist due to theorthogonal frequency band for UL and DL. The amountsof the UL and DL resource are fixed to be the same. • dTDD: Adaptive UL and DL resource proportional to thetraffic volume in each cell independently. • FP:
Proposed algorithm in [10] (summarized in Proposi-tion 1) that solves Problem 1 with old SINR model (3).To compare the performance of protocols FIX, dTDD, FP,SAFP, and RMDI under different traffic asymmetry, we definea measure inter-cell traffic distance , given by D m,n := (cid:107) ϑ n − ϑ m (cid:107) , where ϑ n := (cid:104) ϑ ( u ) n , ϑ ( d ) n (cid:105) T characterizes the UL andDL traffic distribution in cell n , and ϑ ( x ) n := (cid:104) B ( x ) d (cid:105) n / Λ , n = 1 , , x ∈ { u , d } denotes the fraction of the total traffic Λ that traffic of type x in cell n accounts for, such that (cid:80) n ∈ N (cid:80) x ∈{ u , d } ϑ ( x ) n = 1 . For example, if ϑ = ϑ =[0 . , . T , we have D , = 0 .Fig. 3a and 3b show the cumulative distribution function(CDF) of utility ρ derived by applying the five protocols underlow and high inter-cell traffic distance, respectively. The CDFis derived from simulation run times, each with differentuser locations and channel propagation, for every combinationof the inter-cell traffic distribution ratio in set T inter and intra-cell traffic distribution ratio in set T intra . All cases with D , ≤ . are considered as low inter-cell traffic distance , while with D , > . as high inter-cell traffic distance .Both Fig. 3a and 3b show that CDF F ( dTDD ) d (1) > . for dTDD, implying that service outage probability , i.e., theprobability that at least one service cannot be served withsatisfied QoS requirement, is above %. The performanceis worse than protocol FIX with F ( FIX ) d (1) > . . Thisis because although UL/DL resource splitting is adapted tothe traffic volume, the full occupation of the resource maycause severe IMI to some services. Such observation encour-ages the application of our proposed algorithms, which areable to reduce the interference coupling among services. Bycomparing FP, SAFP and RMDI, we show that FP furtherdecreases the outage probability to below %, and SAFP andRMDI significantly outperform FP, with the outage probabilityfor low traffic distance below %. Among the three, RMDIprovides the best performance of the utility distribution. Bycomparing Fig. 3a and 3b, we observe that SAFP and RMDIprovides even higher performance gain under high trafficasymmetry.
3) Performance gain depending on traffic asymmetry.
To an-alyze the performance gain depending on the traffic asymme-try, we average the utility obtained from simulation runtimes for D , falling into the intervals [0 , . , [0 . , . , [0 . , . , [0 . , . , [0 . , . , [0 . , , respectively.Let us consider FIX as the baseline. Fig. 3c shows that theperformance of FIX decreases with the traffic asymmetry, andthe average utility is below (infeasible QoS target) whentraffic distance D , > . . Although dTDD adaptively splitsthe UL/DL resource, the full occupation of the resource causessevere IMI, leading to the worst performance. On the otherhand, FP reduces interference coupling among services, andprovides % gain when traffic asymmetry is low, and almost -fold gain when the asymmetry is ultra high. The proposedSAFP incorporates interference coupling with UL/DL resourcelocalization, which improves the gain to -fold when the trafficasymmetry is low while . -fold when asymmetry is high. Theenhanced version RMDI further improves the gain by mutingpartial resource for interference cancellation. The gain is moresignificant when the traffic is highly asymmetric, achieving . -fold increase when D , ≥ . . Number of iterations U t ili t y ρ FP: updated ρ at each FP iterationSAFP: updated ρ at each FP iterationSAFP: optimized ρ for each SASAFP: actual ρ at each iteration of SA (a) Comparison between FIX and SAFP. Number of iterations U t ili t y ρ (b) Examination of the random initialization. An example: With randomly initialized ˆ w , SAFP converges to two local optima with ρ ∗ (1) = 4 . and ρ ∗ (2) = 1 . . Fig. 2: Examination of SAFP.A
PPENDIX A Definition 1.
A vector function f : R k + → R k ++ is a standardinterference function (SIF) if the following axioms hold:1. (Monotonicity) x ≤ y implies f ( x ) ≤ f ( y )
2. (Scalability) for each α > , α f ( x ) > f ( α x ) In Definition 1 we drop positivity from its original definition[6] because it is a consequence of the other two properties [15].R
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