A Scheduling Policy for Downlink OFDMA in IEEE 802.11ax with Throughput Constraints
aa r X i v : . [ c s . N I] S e p A Scheduling Policy for Downlink OFDMA inIEEE 802.11ax with Throughput Constraints
Konstantinos Dovelos and Boris Bellalta
Abstract —In order to meet the ever-increasing demand forhigh throughput in WiFi networks, the IEEE 802.11ax (11ax)standard introduces orthogonal frequency division multiple ac-cess (OFDMA). In this letter, we address the station-resourceunit scheduling problem in downlink OFDMA of 11ax subjectto minimum throughput requirements. To deal with the infea-sible instances of the constrained problem, we propose a novelscheduling policy based on weighted max-min fairness, whichmaximizes the minimum fraction between the achievable andminimum required throughputs. Thus, the proposed policy hasa well-defined behaviour even when the throughput constraintscannot be fulfilled. Numerical results showcase the merits of ourapproach over the popular proportional fairness and constrainedsum-rate maximization strategies.
Index Terms —IEEE 802.11ax, OFDMA, multi-user scheduling,max-min fairness, Lyapunov optimization.
I. I
NTRODUCTION
The ever-growing demand for fast and ubiquitous wirelessconnectivity poses the challenge of delivering high data rateswhile efficiently managing the scarce radio resources. Orthog-onal frequency division multiplexing (OFDM) has becomea mainstream transmission method for broadband wirelesssystems. Additionally, thanks to the independent fading ofusers’ channels, efficient spectrum utilization can be attainedby exploiting multiuser diversity. Specifically, OFDM sub-carriers can dynamically be allocated to multiple users ac-cording to their instantaneous channel conditions. Therefore,the multiuser version of OFDM, called orthogonal frequencydivision multiple access (OFDMA), has been recognized as akey technology for next-generation wireless systems.Towards this direction, the new IEEE 802.11ax (11ax)amendment for high efficiency wireless local area networks(WLANs) employes OFDMA in both downlink (DL) and up-link (UL) directions [1]. Specifically, DL OFDMA is expectedto boost DL throughput and alleviate the DL-UL asymme-try problem when the access point (AP) lacks transmissionopportunities compared to the stations [2]. The efficiency ofthe OFDMA transmissions, though, mainly hinges on how theAP selects the stations and allocates the available resources.Therefore, intelligent multi-user scheduling is crucial for at-taining the best possible system performance.The peculiarities of 11ax OFDMA implementation renderthe scheduling and resource allocation problem different fromthat in cellular networks. There are few recent works onthe OFDMA scheduling problem for 11ax networks. In [3],
The authors are with the Department of Information and CommunicationTechnologies, Pompeu Fabra University, Barcelona, Spain (e-mail: [email protected], [email protected]). This work has been partiallysupported by CISCO (CG the authors proposed a framework based on Lyapunov op-timization to dynamically adjust the OFDMA transmissionduration so that padding overhead is minimized. To do so,they assumed flat fading across the resource units (RUs) andconsidered fixed RU allocation in conjunction with round-robin user scheduling. The problem of joint user schedulingand RU allocation was firstly studied in [4], [5]. Specifically,D. Bankov et al . proposed a set of multiuser schedulers byformulating the unconstrained utility maximization problem asan assignment problem for the UL. However, they focused onmaximizing the utility of instantaneous station rates rather thantheir long-term throughput. In general, to provide fairness aswell as minimum quality-of-service among the stations, manyworks in the literature proposed to maximize a utility functionof the long-term time average rates under minimum averagerate requirements (see [6], [7], and references therein).In this work, we address the throughput-constrainedscheduling problem in OFDMA of 11ax. Specifically, wepropound a simple yet effective scheduling policy that appliesmax-min fairness in order to maximize the minimum fractionbetween the achievable and minimum requested throughputs.Thus, it minimizes the maximum throughput violation forinfeasible instances. Numerical simulations show that thepropounded policy increases substantially the throughput ofthe worst-case station, whilst scaling efficiently as the numberof stations increases in the network, with respect to existingmethods such as proportional fairness and constrained sum-rate maximization.
Notation : A is a set; a is a vector; ( · ) + = max( · , ; max( A ) denotes the maximum element of set A ; and ( A ) i,j is the ( i, j ) th entry of matrix A .II. S YSTEM M ODEL
A. DL OFDMA Model
Consider the DL of a 11ax network consisting of an AP anda set K , { , . . . , K } of stations. We assume a throughput- constrained traffic model where each station k has a minimumDL throughput requirement denoted by r min k . To this end, theAP periodically commences a DL OFDMA transmission ofduration T DL , as shown in Fig. 1. Specifically, the time axis isdivided into scheduling periods of equal duration, with period t corresponding to the normalized time interval [ t, t + 1) . Ineach scheduling period, an OFDMA transmission takes placewithout collisions; if there is an ongoing transmission, thescheduled OFDMA transmission is deferred until the channelis sensed idle. Let N , { , . . . , N } be the set of RUs, witheach RU comprising of multiple consecutive subcarriers. Thevector of channel gains of station k over RU n in period t isdenoted by g k,n [ t ] . The channel state in scheduling period t isthen defined as G [ t ] , { g k,n [ t ] } k ∈K ,n ∈N , and is assumed toevolve according to a block-fading process. Thus, G [ t ] remainsconstant during period t but is independent and identicallydistributed across different scheduling periods. B. Scheduling Policy
Let p k,n [ t ] denote the transmit power assigned to the k thstation over RU n in scheduling period t . For simplicity, weassume equal power allocation across the N RUs. Hence, p k,n [ t ] = P total /N , where P total is the power budget of theAP. The scheduling decisions are then specified by the binaryvariables s k,n [ t ] ∈ { , } , with s k,n [ t ] = 1 if RU n is assignedto the k th station, and s k,n [ t ] = 0 otherwise. A schedulingpolicy controls the decisions of the AP at each period t , whichare given by the matrix S [ t ] defined as ( S [ t ]) k,n , s k,n [ t ] .More particularly, at the beginning of each period t , the APobserves the random channel state G [ t ] , and selects S [ t ] whoseelements satisfy the 11ax RU allocation constraints K X k =1 s k,n [ t ] ≤ , ∀ n ∈ N , (1) N X n =1 s k,n [ t ] ≤ , ∀ k ∈ K . (2)Constraint (1) ensures that stations cannot share the same RU,while constraint (2) guarantees that every station is assignedto one RU at most. Finally, the set of all feasible schedulingdecisions is defined as S , (cid:8) S ∈ { , } K × N | (1) − (2) (cid:9) . (3) C. Rate Allocation and Throughtput
For a given channel realization g k,n [ t ] and transmit power p k,n [ t ] , the AP can transmit f ( p k,n [ t ] , g k,n [ t ]) bits per OFDMsymbol to station k over RU n . The function f ( · , · ) modelsthe rate selection scheme, and has to conform with the 11axrestriction that a single modulation and coding scheme (MCS)is employed over all the subcarriers of a RU [13]. Next, assumethere are L MCSs, and let ρ l denote the bit rate of MCS l . This is suitable for elastic applications, such as file transfers and Webbrowsing sessions [8], [9]. If RU n consists of S n data subcarriers and all subcarriersare used for transmission, then the set of achievable bit rateson RU n ∈ N is given by R n = { S n ρ , . . . , S n ρ L } . Thenumber of bits transmitted to station k over RU n during thescheduling period t is denoted by r k,n [ t ] , and is calculated as r k,n [ t ] = f ( p k,n [ t ] , g k,n [ t ]) T DL T OFDM , (4)where T OFDM is the duration of an OFDM symbol, and f ( p k,n [ t ] , g k,n [ t ]) ∈ R n . Then, the instantaneous transmissionrate r k [ t ] associated with the k th station is given by r k [ t ] = N X n =1 s k,n [ t ] r k,n [ t ] ( bits/period ) . (5)Finally, the throughput of station k is defined as the long-termtime average ¯ r k , lim T →∞ sup 1 T T − X t =0 E { r k [ t ] } , (6)where the expectation is taken with respect to the channelrealizations as well as the scheduling decisions.III. T HROUGHPUT -C ONSTRAINED S CHEDULING
A. Problem Description
It is customary to assess the efficacy of a scheduling policyin terms of fairness. This is modeled by a utility function U ( · ) ,which is a concave, continuous and entrywise non-decreasingfunction of the stations’ throughput. Let ¯ r = (¯ r , . . . , ¯ r K ) . Thethroughput-constrained scheduling problem is then formulatedas the utility maximization problem [9]maximize { S [ t ] ∈S} U (¯ r ) s.t. ¯ r k ≥ r min k , ∀ k ∈ K . (7) Infeasible Instances and Admission Control : Even thoughthe aforementioned problem can be solved using tools instochastic network utility optimization, there is no guaranteethat (7) is feasible, i.e., the minimum throughput requirementswill be fulfilled. This situation can occur when some stationshave too week channel conditions, and/or too high throughputrequirements. Traditional approaches assume that (7) has afeasible solution, but it might not yield a good performance forinfeasible instances. One can deal with the infeasible instancesby employing admission control. However, it is not trivial toidentify which station to remove from the system in order toturn (7) into a feasible problem.
B. Proposed Policy
We present a low-complexity solution to the throughput-constrained scheduling problem (7), which does not dependon admission control and has a well-defined behaviour forinfeasible instances. First, let ¯ r π denote the throughput vectorattained by scheduling policy π . Then, (7) is feasible whenevera policy π exists with ¯ r πk /r min k ≥ , ∀ k ∈ K . Therefore, wepropose to directly maximize the minimum ¯ r k /r min k , leadingto the optimization problemmaximize { S [ t ] ∈S} min k ∈K (cid:26) ¯ r k r min k (cid:27) . (8) Note that the weighted max-min problem (8) admits a well-defined solution. More particularly, if ¯ r k /r min k ≥ , k ∈ K ,then the policy that solves (8) maximizes the minimum surplus ¯ r k − r min k ≥ . Likewise, if there is at least one station k ′ ∈ K with ¯ r k ′ /r min k ′ < , the policy that solves (8) minimizes themaximum constraint violation r min k ′ − ¯ r k ′ ≥ .IV. S OLUTION VIA L YAPUNOV O PTIMIZATION
In the sequel, we resort to Lyapunov optimization to derivea near-optimal solution.
Definition 1.
Let U ⋆ denote the maximum utility value of (8) ,and ¯˜ r , (¯ r /r min1 , . . . , ¯ r K /r min K ) . A scheduling policy π issaid to produce an O ( ǫ ) -optimal solution to (8) if U (cid:0) ¯˜ r π (cid:1) ≥ U ⋆ − O ( ǫ ) , and all the associated constraints are satisfied.A. The Transformed Problem Following the approach in [11, Ch. 5] for solving stochasticnetwork optimization problems, we transform the problem (8)into a form involving only time averages rather than functionsof time averages. To this end, let γ [ t ] = ( γ [ t ] , . . . , γ K [ t ]) bea vector of auxiliary variables chosen within a set Γ . The set Γ must bound both the auxiliary and rate variables, and henceis selected as Γ = (cid:8) γ ∈ R K | ≤ γ k ≤ R max , ∀ k ∈ K (cid:9) , (9)where R max = max( S Nn =1 R n ) is the maximum transmissionrate over a RU. Now consider the transformed problem:maximize { S [ t ] ∈S} , { γ [ t ] ∈ Γ } min k ∈K { γ k [ t ] } s.t. ¯ γ k ≤ ¯ r k /r min k , ∀ k ∈ K . (10)The connection between (8) and (10) is established as fol-lows. Suppose an arbitrary scheduling policy π solves theproblem (10). The maximum utility value, denoted by U ( γ π ) ,is then attained, whilst satisfying all the associated constraints.Because U ( · ) is concave, it also holds ¯˜ r π ≥ ¯ γ π ⇒ U (cid:0) ¯˜ r π (cid:1) ≥ U (¯ γ π ) ≥ U ( γ π ) , (11)where the last inequality is Jensen’s inequality for concavefunctions. According to (11), if U ( γ π ) ≥ U ⋆ − O ( ǫ ) , then U (cid:0) ¯˜ r π (cid:1) ≥ U ⋆ − O ( ǫ ) as well, hence yielding the desired result.Next, we detail the algorithm the produces such a solution. B. The Drift-Plus-Penalty Algorithm
In Lyapunov optimization, each time average constraint isassociated with a virtual queue, and constraint satisfaction isexpressed as a queue stability problem. More particularly, forthe constraint ¯ γ k ≤ ¯ r k /r min k , we consider a virtual queue thatobeys the recursion Q k [ t + 1] = (cid:0) Q k [ t ] − r k [ t ] /r min k + γ k [ t ] (cid:1) + , (12)where Q [ t ] , r k [ t ] /r min k , and γ k [ t ] correspond to the virtualqueue size, service rate, and arrival rate for the schedulingperiod t , respectively. The drift-plus-penalty (DPP) algorithmis described in Algorithm 1; the parameter V affects theconvergence speed and accuracy of the algorithm [11]. Algorithm 1
Weighted Max-Min SchedulingFor each scheduling period t ∈ { , , , . . . } do: Observe { Q k [ t ] } k ∈K and the channel state G [ t ] . Choose γ ( t ) ∈ Γ such thatmaximize V min k ∈K { γ k [ t ] } − K X k =1 Q k [ t ] γ k [ t ] . Choose S [ t ] ∈ S such thatmaximize P Kk =1 Q k [ t ] r k [ t ] r min k . Update the virtual queues using (12).
C. Maximization Subproblems
According to Algorithm 1, we can address the weightedmax-min problem by solving a set of deterministic subprob-lems at every scheduling period t . More specifically, thefirst subproblem regards the auxiliary viarables. The optimalauxiliary variables are obtained by observing that max γ [ t ] ∈ Γ V min k ∈K { γ k [ t ] } − K X k =1 Q k [ t ] γ k [ t ] ≤ max γ [ t ] ∈ Γ γ min [ t ] V − K X k =1 Q k [ t ] ! , (13)where γ min [ t ] = min k ∈K { γ k [ t ] } . Based on (13), it is straight-forward to show that γ ⋆k [ t ] = R max , if V > P Kk =1 Q k [ t ]0 , otherwise . Next, the maximization subproblem for the scheduling deci-sion S [ t ] is recast asmaximize S [ t ] ∈S K X k =1 N X n =1 s k,n [ t ] φ k,n [ t ] , (14)where φ k,n [ t ] , Q k [ t ] r k,n [ t ] /r min k . The transmission rate r k,n [ t ] is calculated usinig (4). The above problem determinesthe optimal RU assignment for the scheduling period t , andis a classical assignment problem. Hence, it can be solved in O (max( K, N ) ) using the Hungarian method [10].V. P ERFORMANCE E VALUATION
In this section, we assess the performance of the proposedscheduling policy. For this purpose, we consider the followingbenchmark strategies: • Proportional Fairness (PF): In each period t , the sched-uler selects the stations that maximize the instantaneousweighted sum-rate, where the weight associated withstation k is equal to the inverse of the exponential movingaverage of its throughput [5]. • Ergodic Sum-Rate Maximization (ESRM): We solve theproblem in (7) for U (¯ r ) = P Kk =1 ¯ r k . Since U (¯ r ) is a linear function of the rates in ESRM, we have U (¯ r ) = U ( r ) . Therefore, we can readily employ the DPP
16 17 18 19 20 21 22 23
Minimum Throughput (kb/sch. period) CD F ESRMPFWMMConstraint (a)
16 17 18 19 20 21 22 23
Minimum Throughput (kb/sch. period) CD F ESRMPFWMMConstraint (b)
Fig. 2: Empirical CDF of the minimum throughput for K = 12 stations: (a) single RU pattern ( N , S ) ; (b) multiple RU patterns. algorithm without using auxiliary variables [11]. To thisend, we have a virtual queue for the minimum throughputconstraint r min k ≥ ¯ r k , which obeys the recursion Z k [ t + 1] = (cid:0) Z k [ t ] − r k [ t ] + r min k [ t ] (cid:1) + , ∀ k ∈ K . (15)Then, in each period t , the scheduler selects the stationsby solving (14) for φ k,n [ t ] = V ESR r k,n [ t ] + Z k [ t ]( r k,n [ t ] − r min k ) , (16)where V ESR denotes the control parameter of the DPPalgorithm for the ESRM scheme.
TABLE I: Main simulation parameters.
Notation Description Value f c Carrier frequency GHz d max Radius of the WLAN area m P total Maximum transmit power dBm T OFDM
Duration of OFDM symbol µ s T DL Duration of DL OFDMA transmission . msTABLE II: MCS for the 20 MHz channel [13]. Index MCS Minimum SNR (dBm) / − / − / − / − / − / − / − / − / −
10 256-QAM, / − A. Simulation Setup
The area of the WLAN is modeled by a circle of radius d max meters. The AP is located at the center of the circle, andstations are uniformly distributed inside the circle with mini-mum distance from the AP of meter. The path attenuationis calculated using the 11ax path-loss model for a residentialscenario [14]PL k = 40 .
05 + 20 log ( f c / .
4) + 20 log (min( d k , { d k > } ·
35 log ( d k / , ( dB ) where f c is th carrier frequency in GHz, d k is the distancebetween the AP and the k th station, and {·} denotes theindicator function. The channel bandwidth is MHz, and isdivided into N RUs. Without loss of generality, we assume thateach RU consists of S subcarriers. We also consider Rayleighfading across the RUs. Let g k,n denote the channel gain ofstation k over RU n . The power p k,n of station k is uniformlydistributed among the subcarriers of RU n , and therefore thereceived signal-to-noise ratio (SNR) at each subcarrier isSNR k,n = 10 log (cid:16) p k,n S (cid:17) − PL k + 10 log ( g k,n ) ( dBm ) . Based on the received SNR, the maximal MCS is selected,which is denoted by l ∗ . The bit rate and SNR threshold ofeach MCS are given in Table II of the Appendix. Finally, It isworth noting that 256-QAM with / code rate is the highestavailable MCS for 26-tone RUs; 1024-QAM is an optionalfeature for RUs with at least 242 tones each [13]. Next, therate of station k over RU n is calculated as r k,n = Sρ l ∗ T DL T OFDM ( bits / sch. period ) . The values of the main simulation parameters are given inTable I. For the auxiliary variables, the option set is
Γ = { Sρ , . . . , Sρ L } ; ρ is the bit rate of BPSK with code rate / , and ρ L is the bit rate of -QAM with code rate / . B. Numerical Results1) Single versus Multiple RU Patterns:
We first investigatethe benefits of employing multiple RU patterns. We use theminimum throughput achieved by a station as a figure of merit.To this end, we evaluate the empirical cumulative distributionfunction (CDF) of the minimum throughput for 100 networkrealizations. Specifically, for each network realization, theminimum throughput is calculated by averaging over 1000small-scale fading realizations. Regarding the 11ax multipleRU patterns, we consider the following three options: • N = 9 RUs with S = 24 data subcarriers each. • N = 4 RUs with S = 48 data subcarriers each. • N = 2 RUs with S = 102 data subcarriers each. Number of Stations M i n i m u m A v e r age T h r oughpu t ( k b / sc h . pe r i od ) ESRMPFWMMConstraint (a)
Number of Stations M i n i m u m A v e r age T h r oughpu t ( k b / sc h . pe r i od ) ESRMPFWMMConstraint (b)
Fig. 3: Minimum average throughput as a function of the number of stations: (a) single RU pattern ( N , S ) ; (b) multiple RU patterns. We then solve (14) for each pattern under equal powerallocation across RUs, and choose the one that yields themaximum objective value. We plot the results in Fig. 2 fora minimum throughput requirement r min k = 20 kb/sch. period, ∀ k ∈ K . The control parameters of the DPP algorithm forthe WMM and ESMR are V WMM = 900 and V ESRM = 10 ,respectively. As we observe, the WMM outperfoms both thePF and ESRM policies. Note that, although the use of multipleRU patterns might increase the computational complexityof the scheduler, it yields substantial performance gains forthe WMM policy. In particular, the minimum throughputrequirement r min corresponds to the th percentile whena single RU pattern is employed, whilst it drops to the thpercentile for multiple RU patterns.
2) Number of Stations:
We now examine how the perfor-mance of the schedulers under consideration scales as thenumber of stations increases. To this end, we calculate theaverage of the minimum throughput for the same numberof network and small-scale fading realizations as previously.From Fig. 3, we see that the WMM delivers the highestminimum throughput even when the throughput constraintof kb/sch. period cannot be fulfilled. Consequently, theperformance of the proposed scheduler scales more efficientlycompared to the PF and ESRM schedulers. Finally, we stressthat the WMM policy benefits the most from the utilization ofmultiple RU patterns.VI. C ONCLUSIONS
In this letter, we provided a solution to the DL OFDMAscheduling problem in 11ax with minimum throughput re-quirements. First, we introduced a scheduling and rate al-location model which conforms with the 11ax implemen-tation constraints. Then, we formulated the throughput-constrained scheduling problem as an unconstrained problemusing weighted max-min fairness. Relying on Lyapunov op-timization, we derived a dynamic scheduling policy whichminimizes the maximum throughput constraint violation forinfeasible instances, and maximizes the minimum throughputsurplus otherwise. We finally provided numerical results show-casing that our approach outperforms the popular proportional fairness and constrained sum-rate maximization strategies interms of the minimum throughput.R
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