Feedback Reduction for Multiuser OFDM Systems
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Feedback Reduction for Multiuser OFDM Systems Jeongho Jeon, Student Member, IEEE, Kyuho Son, Student Member, IEEE, Hyang-Won Lee, Member, IEEE, and Song Chong, Member, IEEE
Abstract
Feedback reduction in multiuser OFDM systems becomes an important issue due to the excessive amount of feedback required to run opportunistic scheduling, especially when the number of users and carriers in the system increases. In this paper we propose a novel feedback reduction scheme for efficient downlink scheduling. In the proposed scheme, each user determines the amount of feedback based on so called feedback efficiency factor in a distributed manner. The key idea is to give more feedback opportunity to users who are more often scheduled. Simulation results demonstrate that the proposed scheme can substantially decrease feedback load while achieving almost the same scheduling performance as in the case of full feedback. In addition, the proposed scheme offers unique advantages over existing ones. First, it is not tailored to a specific scheduling policy; thus, its performance is insensitive to underlying scheduling policy. Secondly, total feedback load can be maintained below a target level regardless of the number of users in the system.
Index Terms
OFDM system, opportunistic scheduling, CQI feedback reduction.
This research was supported by the Ministry of Knowledge Economy, Korea, under the ITRC(Information Technology Research Center) support program supervised by the IITA(Institute of Information Technology Advancement), (IITA-2008-C1090-0801-0037). Jeongho Jeon, Kyuho Son and Song Chong are with the School of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea (e-mail: {jjeon, skio}@netsys.kaist.ac.kr, [email protected]). Hyang-Won Lee is with the Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail:[email protected]).
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I. I NTRODUCTION
O
PPORTUNISTIC scheduling has been introduced as an effective means of increasing system throughput in modern wireless systems [1]. It exploits independently-fading wireless channels
of users in order to take advantage of multiuser diversity gain, but CQI (Channel Quality Indicator) information of all users must be known to the scheduler residing at base station (BS) every time slot. The amount of CQI feedback from users to the BS increases as the number of users increases, consuming non-negligible amount of radio resource. The CQI feedback load becomes a more serious issue in a multicarrier system such as up-to-date OFDM-based IEEE 802.11/16/20 systems since every time slot the CQI information must be fed back to the BS for all users and for all carrier frequencies (interchangeably, subchannels). Therefore, a proper CQI feedback reduction scheme must be incorporated into a system to prevent extravagant amount of channel feedback. There have been some works on CQI feedback reduction. In [2], best-M feedback scheme was proposed for multi-carrier systems. In this scheme, each user selects M (≤ N ) best subchannels where N is the number of subchannels in the system and feeds back the CQIs of these subchannels to the BS. Since this scheme always gives an equal number of feedback opportunity M to all users irrespective of whether they are in deep fading or not, it has potential to work reasonably well with scheduling policies concerning fairness such as PFS (Proportional Fairness Scheduler) [3] and MFS (Maximum Fairness Scheduler)1 [4]. However, it will not work well with MRS (Maximum Rate Scheduler) [4], [5], since some subchannels can be scheduled to a user in deep fading thereby yielding throughput loss unless M is closer to the number of subchannels N . In [6], selective multiuser diversity concept was proposed for a single-channel system, in which each user sends the CQI feedback based on the absolute SNR threshold scheme. More specifically, user k sends the feedback every time slot t only if γk (t) ≥ γth where γk (t) denotes the instantaneous received SNR of user k at time slot t and γth is the threshold. This scheme can be easily extended to multi-channel systems
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The MFS aims to allocate resources such that the minimum user’s data rate is maximized [4].
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by applying the same threshold to all subchannels, but it can cause serious fairness problem when PFS or MFS is employed because users in deep fading have no chance of being scheduled at all due to their low received SNR at all carrier frequencies. In order to overcome this problem, [7] proposed a modified SNR thresholding scheme, referred to as the normalized SNR thresholding scheme. In this scheme, user k sends the CQI feedback when the normalized SNR value, γk (t)/γ k , exceeds a certain threshold A where γ k is the average of γk (t) from time slot 0 to t. In the average sense, however, the behavioral characteristic of the normalized SNR thresholding scheme is similar with that of the best-M scheme and consequently it may work well with PFS or MFS but not with MRS. A common drawback of all existing schemes mentioned above is that the total amount of feedback required at each time slot tends to increase in proportion to the number of users in the system, which significantly undermines their practicality for a large system. In order to overcome this problem, a random access based feedback protocol, named as opportunistic feedback, was proposed in [8]. Instead of giving feedback opportunity to all the users whose received SNR is greater than a threshold, it limits the opportunity to the users who become winners in the underlying random access process among them. Thereby it can explicitly control the total number of feedback per time slot regardless of number of users in the system. However, it inherently possesses the same drawbacks as the absolute SNR thresholding scheme, and feedback delay will be much longer due to the random access process. In contrast to the previous works, our proposed scheme uses a new metric called feedback efficiency factor as the feedback decision metric instead of received SNR. The key idea behind the proposed scheme is to give more feedback opportunity to users who are more often scheduled. The rest of the paper is organized as follows. In Section II, we describe the system model and define some notations. Section III proposes and analyzes the efficiency-based feedback algorithm in detail. In Section IV, we show through simulation that the proposed scheme reduces total feedback load substantially while maintaining almost the same scheduling performance as in the case of full feedback. In addition, we also demonstrate that the proposed scheme offers two unique advantages over the existing ones. First, its performance is insensitive to underlying scheduling policy and hence it works well with a broad class of scheduling policies ranging
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between two extreme policies: MRS aiming at extreme efficiency and MFS aiming at extreme fairness. Secondly, it explicitly keeps total feedback load below a desired level regardless of the number of users in the system. We conclude the paper in Section V.
II. S YSTEM M ODEL We consider the downlink of a single cell in an OFDM system with K users and N subchannels. Denote by K and N the set of all users and subchannels respectively, i.e., K = {1, 2, ..., K} and N = {1, 2, ..., N }. A time-slotted system is considered, where channels are assumed to hold their state for the duration of a time slot. Let ~π (t) = [~πk , ∀k ∈ K] be the channel state vector at time slot t. In a single-channel system (N = 1), each element of ~π (t) can be replaced with a parameter which reflecting thermal noise, path-loss, shadowing and multi-path fading in a single value. Let Γ~π (t) be the feasible region of the transmission rate vector ~r(t) = [rk (t), ∀k ∈ K] given channel state vector ~π (t) at time slot t, i.e., ~r(t) ∈ Γ~π (t). Note that Γ~π (t) can be differed according to the underlying physical layer model. The transmission rate of user k at time slot t is determined by a scheduler as rk (t) =
PN n=1
xnk (t)rkn (t), where rkn (t) is the achievable
rate of user k over subchannel n during time slot t, and xnk (t) is the scheduling indicator function such that xnk (t) = 1 if user k is chosen to be served over subchannel n during time slot t and xnk (t) = 0 otherwise. Because at most one user can be assigned to each subchannel at each time slot, the scheduling indicator function should satisfy
PK k=1
xnk (t) ≤ 1, ∀n, ∀t.
Let Rk (t) be the average throughput of user k up to time slot t, i.e., Rk (t) =
1 t
Pt
τ =1 rk (t),
and
~ let R(t) = [Rk (t), ∀k ∈ K]. Each user is associated with a utility function Uk (Rk (t)) of their average P ~ throughput, and let U (R(t)) = K k=1 Uk (Rk (t)). We assume that the utility function Uk (·), ∀k ∈ K is an increasing, strictly concave, and continuously differentiable function on R+ . Define the long-term average ~ ~ = limt→∞ R(t), and let Γ be the long-term feasible region of the long-term throughput vector as R ~ considering all the possible scheduling policies. It can be easily shown that average throughput vector R, the long-term feasible region Γ is convex, coordinate convex, and compact set. We finally denote by Rkmax the intercept of the long-term feasible region Γ along the k-th dimension. This is simply the long-term
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average throughput user k would get if user k was the only one scheduled for all subchannels and for all time slots.
III. E FFICIENCY- BASED F EEDBACK R EDUCTION A. Proposed Algorithm As mentioned, our feedback scheme gives more feedback opportunity to users who are more often scheduled. To apply such a concept in determining each user’s feedback amount, we have introduced feedback efficiency defined as the ratio of the average number of allocated subchannels, sk (t), to the amount of feedback, fk (t), for an arbitrary user k at time slot t. More specifically, fk (t) denotes the number of subchannels of which the CQI information was sent back to the BS. A proper target efficiency factor will be predetermined from the system aspect, and all mobile users are expected to maintain the same efficiency by adjusting their feedback amount. The efficiency-based feedback reduction (EFR) algorithm is summarized in Algorithm 1. For the given target efficiency factor e, each user computes their feedback success probability pk (t) using (1) at time slot t. We denote by [·]+ 1 the orthogonal projection onto the interval [0, 1] since a probability is expressed on a linear scale from 0 (impossibility) to 1 (certainty). Intuitively, if one increases the target efficiency factor, then each user will decrease their feedback success probability in order to balance their own efficiency to the target value. With the success probability pk (t), each user performs totally N independent Bernoulli trials, and sends the CQIs of the best subchannels up to the number of successful trials at time slot t + 1. After that, according to the scheduling results sk (t), each user updates the average number of allocated subchannels. In this paper, we assume that each user can estimate their average number of allocated subchannels, which is a reasonable assumption; in the IEEE 802.16e system, the DL-MAP (downlink map) message contains such resource allocation information. We also assume the full queue condition that the outgoing queues residing at the BS for each user are full. It should be noted that each user determines their amount of feedback not based on the direction from the BS, but distributively based on the feedback efficiency factor.
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Algorithm 1 Efficiency-based Feedback Reduction Algorithm 1: At time slot t, – compute the feedback success probability: ·
1 sk (t) pk (t) = · N e
¸+ .
(1)
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– generate a finite Bernoulli sequence {Xki : i = 1, ..., N }, where each Xki is a binary random variable with the success probability pk (t), and the total number of successes is Xk = 2:
PN i=1
Xki .
At time slot t + 1, – select fk (t + 1) = Xk best subchannels and send back the CQIs of these subchannels to the BS. – update the average number of allocated subchannels: t+1
1X sk (t + 1) = sk (τ ) t τ =1
(2)
– set t = t + 1 and go to 1.
B. Interrelationship between Feedback Algorithms and Scheduling Policies To have a better understanding of the interrelationship between feedback algorithms and scheduling policies, it is worthwhile knowing how users share resources with scheduling policies of different fairness criterion. This is because it is appropriate to give more feedback opportunities to users who occupy more subchannels. However, existing schemes were designed without thoroughly considering the underlying scheduling policies. Many of the scheduling policies considered—including the PFS—can be viewed as a gradient-based ~ + 1)) − U (R(t)) ~ algorithm [9], [10]. It aims to attain sum utility maximization by maximizing U (R(t at each time slot t, which is equivalent to choosing the transmission rate vector having the maximum projection onto the gradient of the sum utility: T ~ arg max ∇U (R(t)) · ~r(t). ~ r(t)∈Γ~π(t)
(3)
Under the strict concavity of the utility function, it has been proved by several researchers that such a
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~ over the long-term feasible region Γ [9], [11]. gradient-based algorithm maximizes sum utility U (R) On the other hand, there was an attempt to develop utility functions to exploit a tradeoff between two different aims: total throughput maximization and fairness among users. For example, one class of utility functions applied for wireless packet schedulers was [12]2 : X (1 − α)−1 Rk (t)1−α , k∈K ~ U (R(t)) = X log(Rk (t)) ,
α > 0, α 6= 1, (4) α = 1,
k∈K
where α is a fairness exponent factor. With the utility functions in (4), (3) becomes: X
arg max
~ r(t)∈Γ~π(t)
(Rk (t))−α rk (t).
(5)
k∈K
For simplicity of the exposition, consider a single-channel system (N = 1) in which the resource allocation problem in (5) turns into a situation where a single user selection is needed who has the maximum weighted rate where the weights are the marginal utilities3 : arg max (Rk (t))−α rk (t). k∈K
(6)
As α → 0, the total throughput is maximized (MRS), and when α = 1, the proportionally fair throughput allocation is achieved (PFS). As α increases, fairness is improved at the cost of reduced total throughput, and especially as α → ∞, the throughput allocation becomes max-min fair (MFS). With the scheduler in (6), each user shares the single-channel as in Observation 1 below. Observation 1: For the two-user case, each user receives time slots in a long-term sense as in the following ratio: θ10 =
µ 1+
1 R1max R2max
¶(α−1)/α ,
θ20 = 1 − θ10 ,
(7)
where Rkmax is the long-term maximum achievable rate of user k. Proof: See Appendix A for the proof.
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The utility functions in (4) were originally introduced in the context of wireline window-based congestion control [13].
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The scheduler form in (6) is often referred to as Generic Proportional Fair Scheduler in the IEEE 802.16 Task Group m (TGm) [16].
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In the single-channel system (N = 1), the long-term average number of allocated subchannels is equal to each user’s time sharing ratio on the channel, θk0 . Thus, the feedback success probability is given by h 0 i+ θk in the long-term sense, and the CQI information of the channel will be sent back to the BS if the e 1
outcome of a Bernoulli trial is successful. In this context, one can easily notice that the expected feedback opportunity of each user is proportional to their resource sharing ratio under our proposed scheme. This result can be readily extended to a multi-channel system without loss of generality. Fig. 1 plots (7) when the long-term maximum achievable rate of user 1 is twice than that of user 2, i.e., R1max /R2max = 2. Under our proposed scheme, each user’s feedback amount will be determined according to each sample instance of the resource sharing ratio at the given value of α from 0 to ∞. However, existing schemes have their single ideal operating point. For example, when α = 0 (MRS), best-M and normalized SNR thresholding schemes will reveal redundant feedback, because the strong user monopolizes all the resources, and the weak user does not need to send feedback at all. However, the absolute SNR thresholding scheme will be well-matched with MRS because the scheme lets the strong user send more feedback. On the other hand, when α = 1 (PFS), each user shares an equal fraction of time slots4 . Hence, the absolute SNR thresholding scheme can cause a serious fairness problem because the weak user has no chance of being scheduled at all due to the lack of the CQI information. Yet the bestM and normalized SNR thresholding schemes will be harmonious with PFS because these schemes give an equal or approximately equal number of feedback opportunities irrespective of their relative channel strengths between users. The case of α → ∞ (MFS) also can be explained analogously.
C. Feedback Load Estimation A common drawback of existing schemes is that the total feedback load increases in proportion to the number of users, which significantly undermines their practicality for a large system. However, under our proposed scheme, the total feedback load can be maintained below a target level regardless of the number
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By simply extending the result in (7), if there are totally K users in the system, then each user will equally receive fraction 1/K of the
time slots referred to as the equal time sharing property of the PFS [15].
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of users in a cell. Intuitively, if there are a large number of users in a cell, then each user will receive a relatively small number of subchannels. Consequently, each user will reduce their feedback amount in order to keep the target efficiency factor. Under our proposed scheme, the feedback success probability is given by pk =
£ s ¤+ k
Ne 1
, k ∈ K, where
we dropped the time index t for brevity. With the success probability pk , each user repeatedly performs totally N independent but identical Bernoulli trials, and the resulting number of successful trials, Xk , follows the binomial distribution with parameters N and pk , i.e., Xk ∼ B(N, pk ). Thus, the k-th user’s expected number of subchannels, of which the CQI information will be sent back to the BS, is given by: E[Xk ] = N pk ≤
sk , e
and the total feedback load is obtained by summing the feedback amount of all the individual users: X k∈K
E[Xk ] ≤
1X N sk = , e k∈K e
(8)
where the equality in (8) is because the available number of subchannels is always N regardless of the number of users in a cell. Finally, the total feedback load per channel of our efficiency-based feedback scheme is bounded by: 1 FEFR ≤ . e
(9)
Note that the estimated total feedback load per channel of our proposed scheme in (9) is not a function of the number of users in a cell, but a function of the target efficiency factor. We now look into the statistics of the feedback load of existing schemes by assuming that users experience independent identically distributed (i.i.d) fading. For a Rayleigh fading channel, the instantaneous received SNR γk , k ∈ K is distributed exponentially with a probability density function [17]: µ ¶ 1 γk g(γk ) = exp − , γk γk
γk ≥ 0,
where γ k is the average SNR. Under the absolute SNR thresholding scheme, each user sends the CQI of a channel if the instantaneous received SNR exceeds a certain threshold γth . Thus, the total feedback load
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per channel of the absolute SNR thresholding scheme is: ¶ X µ ¶ Z γth K µ K X γth FABS = 1− g(γk )dγk = exp − . γk 0 k=1 k=1
(10)
On the other hand, under the normalized SNR thresholding scheme, each user sends the CQI of a channel when the normalized SNR value, γk /γ k , exceeds a certain threshold A. By simply replacing γth in (10) by γ k A for each user, the total feedback load per channel of the normalized SNR thresholding scheme is obtained by: FNOR =
K X
exp (−A) = K exp (−A) .
(11)
k=1
Admittedly, the total feedback load per channel of the best-M feedback scheme is deterministically given by: FBest-M =
KM . N
(12)
Both best-M and normalized SNR thresholding schemes let the users send equal or approximately equal amounts of feedback irrespective of their relative channel strengths to other users. By simply setting A = − ln M , each user’s feedback amount and the total feedback load of the normalized SNR thresholding N scheme becomes identical with those of the best-M feedback in the average sense. To verify the analysis, we simulated our proposed and existing schemes for the 24-subchannel system. In the case of the absolute SNR thresholding or normalized SNR thresholding scheme, the same threshold was applied to the multiple subchannels respectively, and resulting sum feedback load was averaged over 24 subchannels to obtain the per channel feedback load. For the efficiency-based feedback scheme, PFS was performed. Fig. 2 plots the total feedback load per channel of our proposed and existing schemes. It can be seen that the analytic results agree well with the simulation ones. Note that the slopes of existing schemes and the upper bound of our efficiency-based feedback scheme can be adjusted by tuning the feedback control parameters. To sum up, the total feedback loads of existing schemes are very sensitive to variation in the number of users in a cell, and thus there is a possibility that existing schemes can cause not only the underutilization of downlink resources due to insufficient feedback when the number of users is small, but also the extravagant use of uplink resources due to excessive feedback when the number
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of users is large. However, the total feedback load of our proposed scheme is maintained below a target level regardless of the number of users in a cell by adjusting each user’s feedback load adaptively. Also, it should be noted that our proposed scheme does not require any underlying contention mechanisms, like random access based opportunistic feedback [8], to control the total feedback load.
D. Selection of the Feedback Efficiency Factor With imperfect knowledge of the channel state due to the partial CQI feedback, the scheduler selects ˆ ~π(t) at each time slot t as follows: the transmission rate vector ~r(t) over the reduced feasible region Γ T ~ˆ arg max ∇U (R(t)) · ~r(t), ˆ ~π(t) ~ r(t)∈Γ
(13)
~ˆ ~∗ where R(t) is the average throughput vector up to time slot t with imperfect channel knowledge. Let R ~ˆ ∗ be the long-term throughput vector obtained by the scheduler with perfect and imperfect channel and R knowledge respectively. Because the gradient-based scheduler aims to maximize sum utility, we define the performance ratio of the scheduling with imperfect channel knowledge to perfect channel knowledge as follows:
PK ρ = Pk=1 K
ˆ∗ ) Uk (R k
∗ k=1 Uk (Rk )
.
(14)
However, finding the closed-form of (14) under a certain feedback scheme is a daunting task. It requires to formulate the throughput achieved with a specific scheduling policy under a certain feedback scheme, but even formulating the throughput of the multi-channel systems with a specific scheduling policy under perfect channel knowledge itself is not an easy task. Fig. 3 plots the empirically obtained performance ratio of our efficiency-based feedback scheme for various number of users. PF scheduling was performed and all the other simulation environments are identical with that of Section IV. It can be seen that there is some degree of performance degradation when 1/e is relatively small (high feedback efficiency), whereas the performance approaches to the full feedback as 1/e increases (low feedback efficiency). At 1/e = 6, the performance ratio is always higher than 99%, and there is no remarkable gain above 1/e = 10. Also, it should be noted that the feedback
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efficiency factor can be chosen using the property discussed in Section III-C to meet the maximum allowable feedback load by the system, e.g., the system which allocates a dedicated channel for the CQI feedback.
IV. S IMULATION R ESULTS To validate the effectiveness of our proposed algorithm, the IEEE 802.16e 1024-FFT OFDMA AMC (Adaptive Modulation & Coding) mode [18] was adopted, where the data subcarriers were grouped into 24 subchannels. The AMC operation is possible only when the channel coherence time is much longer than the lag between the time the channel is measured on the mobile and the time when the packet is actually transmitted from the BS [3], [18]. We thus adopted the ITU pedestrian B model [19]. For high mobility users, who are usually served by the Diversity mode, it is sufficient for them to predict the average CQI of the overall subchannels, because the fading channel follows the stationary distribution [3]. In the simulation, we further made the following assumptions: 1) The radius of the cell was set to 1km, and the distance, dk , between user k and the BS was generated as a 2-D uniformly distributed random variable. 2) We adopted P L(dk ) = 16.62 + 37.6log10 (dk )[dB] for the path loss model with log-normal shadowing with standard deviation σs = 8dB. 3) The ITU pedestrian B parameters was applied to the standard delay-spread model for frequency-selective fast fading [20]. 4) In making a fair comparison, we set the threshold of each scheme so as to exhibit the same feedback load at 30 number of users: M = 5 (best-M scheme), γth = 11.86 in dB scale (absolute SNR thresholding scheme), γk (t)/γ k ≥ 1.6 in linear scale (normalized SNR thresholding scheme), and e = 1/6.25 (efficiency-based feedback scheme). 5) The scheduling was performed every time slot (5 msec) for the multiple subchannels. In Fig. 4, we compare the performance of the schemes that have been used previously and our own under scheduling policies of different fairness criterion. It can be seen that the total throughput and Jain’s fairness index [21] of our efficiency-based feedback scheme always keep track of the full feedback for varying α. In the case of the best-M and normalized SNR thresholding schemes, there exists throughput degradation near α = 0 (MRS). However, as α approaches to 1 (PFS), the total throughput and fairness
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converge to the full feedback. On the other hand, in the case of the absolute SNR thresholding scheme, the total throughput and fairness performances diverge from the full feedback as α increases (MFS). Fig. 5 depicts the total throughput, Jain’s fairness index and total feedback load per channel of the schemes with PFS. In Fig. 5(a)-(b), the total throughput and Jain’s fairness index of our proposed scheme are very close to the full feedback for various number of users. However, in the case of the best-M and normalized SNR thresholding schemes, we can see considerable throughput degradation when there are small number of users. On the other hand, it can be seen that the total throughput of the absolute SNR thresholding scheme outperforms the full feedback, whereas the fairness performance is worse than that of the full feedback. This is because the weak users have no chance of being scheduling at all due to the lack of the CQI information, and consequently the resulting throughput vector is quite distant from the proportionally fair throughput allocation under the absolute SNR thresholding scheme (see Fig. 4 at α=1). From Fig. 5(c), it can be seen that the total feedback load per channel of our proposed scheme is maintained below 1/e, whereas the total feedback load per channel of existing schemes increase in proportion to the number of users.
V. C ONCLUDING R EMARKS In this paper, we have proposed an innovative feedback reduction scheme which uses a feedback efficiency factor as the feedback decision metric instead of received SNR. The key idea is to give more feedback opportunity to users who are more often scheduled. We have shown that the proposed scheme reduces total feedback load substantially while maintaining almost the same scheduling performance as in the case of full feedback. We have studied the interrelationship between feedback algorithms and scheduling policies under a gradient-based scheduling framework. The simulation results showed that our proposed scheme works well with a broad class of scheduling policies ranging from MRS (Maximum Rate Scheduler) aiming at extreme efficiency to MFS (Maximum Fairness Scheduler) aiming at extreme fairness. We have shown that our proposed scheme prevents both the underutilization of downlink resources due to insufficient feedback and the extravagant use of uplink resources due to excessive feedback by
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controlling the total feedback load to a target level regardless of the number of users in a cell. This is possible because each user adjusts their feedback load adaptively where the feedback efficiency factor acts as a coordinator. All these advantages are achieved without additional overhead. A PPENDIX A P ROOF OF O BSERVATION 1 We basically assume that the relative fluctuations of the achievable rates for various users around their respective time-average values are statistically identical. Denote by θ~0 = [θk0 , ∀k ∈ K] the long-term average ratio of time slots allocated to users by the scheduling policy (6), where 0 ≤ θk0 ≤ 1, ∀k ∈ K and P k∈K
~ 0 = [θ0 Rmax , ∀k ∈ K]. θk0 = 1. Then, we can express the resulting long-term throughput vector as R k k
~ = [θk Rmax , ∀k ∈ K], where Similarly, an arbitrary long-term throughput vector can be expressed as R k 0 ≤ θk ≤ 1, ∀k ∈ K and
P k∈K
θk = 1. Because the utility functions in (4) are strictly concave, it is easy
~ ∗ is the optimal solution of sum utility maximization if and only if (see [9] and references to show that R therein): ~ ∗ )T · (R ~ −R ~ ∗ ) ≤ 0, ∀R ~ ∈ Γ. ∇U (R
(15)
~ ∗ is the unique Under the assumption that the long-term feasible region is strictly convex, it follows that R ~ ~ ∗ as t → ∞ starting with any initial state R(0) ~ sum utility maximizer and R(t) →R ∈ Γ [9]. Therefore, ~ 0 should satisfy (15) for all possible throughput vector R ~ ∈ Γ, because the long-term throughput vector R ~ 0 is the unique optimal long-term throughput vector obtained by the gradient-based scheduler in (6) R which maximizes the sum utility. For the two-user case, (15) can be rewritten as follows: θ1 R1max − θ10 R1max (1 − θ1 )R2max − (1 − θ10 )R2max + ≤ 0, ∀θk ∈ [0, 1]. (θ10 R1max )α ((1 − θ10 )R2max )α
(16)
After simple manipulation, (16) reduces to: θ1 A − θ10 A ≤ 0,
∀θk ∈ [0, 1],
(17)
where A = (1 − θ10 )α (R2max )α−1 − θ10 α (R1max )α−1 . The inequality in (17) is always satisfied for ∀θk ∈ [0, 1] if and only if A = 0, and this leads to the result of Observation 1.
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R EFERENCES [1] R. Knopp and P. Humblet, “Information capacity and power control in single-cell multiuser communications,” in Proc. IEEE ICC’95, Seattle, Jun. 1995. [2] ”CQI report and scheduling procedure,” 3GPP Tdoc R1-051045, 2005. [3] D. Tse and P. Viswanath, Fundamentals of wireless communication, Cambridge University Press, 2005. [4] J.G. Andrews, A. Ghosh and R. Muhamed, Fundamentals of WiMAX: Understanding broadband wireless networking, Prentice Hall, 2007. [5] S. Shakkottai and A. L. Stolyar, ”A study of scheduling algorithms for a mixture of real and non-real time data in hdr”, Bell Laboratories, Lucent Technologies, Oct. 2000. [6] D. Gesbert and M. -S. Alouini, “How much feedback is multi-user diversity really worth?,” in Proc. IEEE ICC’04, Paris, Jun. 2004. [7] L. Yang, M. -S. Alouini and D. Gesbert, “Further results on selective multiuser diversity,” in Proc. ACM MSWiM’04, Itary, Oct. 2004. [8] T. Tang and R. W. Heath, Jr., “Opportunistic feedback for downlink multiuser diversity,” IEEE Communications Letters, vol. 9, no. 10, Oct. 2005. [9] R. Agrawal and V. Subramanian, ”Optimality of certain channel aware scheduling policies,” in Proc. 2002 Allerton Conference on Communication, Otc. 2002. [10] J. Huang, V. Subramanian, R. Agrawal, and R. Berry, ”Downlink scheduling and resource allocation for ofdm systems,” in Proc. Conference on Information Sceiences and Systems, New Jersey, Mar. 2006. [11] H. J. Kushner and P. A. Whiting, ”Convergence of proportional-fair sharing algorithms under general conditions,” IEEE Trans. Wireless Communications, vol. 3, no. 4, pp. 1250–1259, Jul. 2004. [12] R. Agrawal, A. Bedekar, R. La, and V. Subramanian, ”A class and channel-condition based weighted proportional fair scheduler,” in Proc. ITC 2001, Salvador, Brazil, Sep. 2001. [13] J. Mo and J. Walrand, “Fair end-to-end window-based congestion control,” IEEE/ACM Trans. Networking, vol. 8, no. 5, pp. 556–567, Oct. 2000. [14] F. Kelly, “Charging and rate control for elastic traffic,” Eur. Trans. Telecommun., vol. 8, pp. 33–37, Oct. 1996. [15] S. Borst, “User-level performance of channel-aware scheduling algorithm in wireless data networks,” in Proc. IEEE INFOCOM’03, San Francisco, Mar. 2003. [16] ”Project 802.16m evaluation methodology document (EMD),” IEEE 802.16m-08/004, Mar. 2008. [17] Q. Zhang and S. A. Kassam, ”Finite-state markov model for rayleigh fading channels,” IEEE Trans. Communications, vol. 47, no. 11, pp. 1688–1692, Nov. 1999. [18] IEEE Std 802.16e-2005 and IEEE 802.16-2004/Cor 1-2005. Part 16: Air interface for fixed and mobile broadband wireless access systems. Dec. 2005. [19] ”Guidelines for the evaluation of radio transmission technologies for IMT-2000,” Recommendation ITU-R M.1225, 1997. [20] M. P¨atzold, Mobile fading channels. Baffins Lane, England: John Wiley and Sons, 2002
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[21] R. Jain, D. Chiu, and W. Hawe, “A quantitative measure of fairness and discrimination for resource allocation in shared system,” DEC TR-301.
17
1 θ1’
0.9
θ2’
0.8 PFS
Time sharing ratio
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Fig. 1.
0
5
α
10
15
Time sharing ratio in the two-user case, R1max /R2max = 2
80 Full feedback EFR (simulation, e=1/15) EFR (analysis: e=1/15) ABS (simulation, γth=−2.2dB)
Total feedback load per channel
70 60
ABS (analysis, γth=−2.2dB) Best−M, M=15 NOR (simulation, A=0.83) NOR (analysis, A=0.83)
50 40 30 20 10 0
Fig. 2.
10
20
30 40 Number of users
Total feedback load comparison of proposed and existing schemes
50
60
70
18
1 0.995
Performance ratio
0.99 Number of users = 10 Number of users = 20 Number of users = 30 Number of users = 40 Number of users = 50 Number of users = 60 Number of users = 70
0.985 0.98 0.975 0.97 0.965 0.96 0.955 0.95
Fig. 3.
4
6
8
10 1/e
Performance ratio of the efficiency-based feedback scheme with PFS
12
14
16
19
30
Total throughput (Mbps)
28
26 Full feedback EFR (e=1/6.25) Best−M (M=5) NOR (A=1.6) ABS (γth=11.86 dB)
24
22
20
18
16
0
0.5
1 α
1.5
2
(a) Total throughput (when the number of users = 30) 1 0.9
Jain’s fairness index
0.8 0.7 0.6 0.5 0.4 Full feedback EFR (e=1/6.25) Best−M (M=5) NOR (A=1.6) ABS (γth=11.86 dB)
0.3 0.2 0.1
0
0.5
1 α
1.5
2
(b) Jain’s fairness index (when the number of users = 30) Fig. 4.
Comparison of total throughput and Jain’s fairness index with scheduling policies of different fairness criterion
20
28
Total throughput (Mbps)
26 Full feedback EFR (e=1/6.25) Best−M (M=5) NOR (A=1.6) ABS (γ =11.86 dB)
24
22
th
20
18
16 10
20
30
40 Number of users
50
60
70
(a) Total throughput with PFS (i.e., α=1) 0.9 0.85 0.8 Full feedback EFR (e=1/6.25) Best−M (M=5) NOR (A=1.6) ABS (γth=11.86 dB)
Jain’s fairness index
0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 10
20
30
40 Number of users
50
60
(b) Jain’s fairness index with PFS (i.e., α=1)
70
21
70
Total feedback load per channel
60
Full feedback EFR (e=1/6.25) Best−M (M=5) NOR (A=1.6) ABS (γth=11.86 dB)
50
40
30
20
10
0
10
20
30 40 50 Number of users
60
70
(c) Total feedback load per channel with PFS (i.e., α=1) Fig. 5.
Comparison of total throughput, Jain’s fairness index and total feedback load per channel for various number of users
Feedback Reduction for Multiuser OFDM Systems Jeongho Jeon, Student Member, IEEE, Kyuho Son, Student Member, IEEE, Hyang-Won Lee, Member, IEEE, and Song Chong, Member, IEEE
Abstract
Feedback reduction in multiuser OFDM systems becomes an important issue due to the excessive amount of feedback required to run opportunistic scheduling, especially when the number of users and carriers in the system increases. In this paper we propose a novel feedback reduction scheme for efficient downlink scheduling. In the proposed scheme, each user determines the amount of feedback based on so called feedback efficiency factor in a distributed manner. The key idea is to give more feedback opportunity to users who are more often scheduled. Simulation results demonstrate that the proposed scheme can substantially decrease feedback load while achieving almost the same scheduling performance as in the case of full feedback. In addition, the proposed scheme offers unique advantages over existing ones. First, it is not tailored to a specific scheduling policy; thus, its performance is insensitive to underlying scheduling policy. Secondly, total feedback load can be maintained below a target level regardless of the number of users in the system.
Index Terms
OFDM system, opportunistic scheduling, CQI feedback reduction.
This research was supported by the Ministry of Knowledge Economy, Korea, under the ITRC(Information Technology Research Center) support program supervised by the IITA(Institute of Information Technology Advancement), (IITA-2008-C1090-0801-0037). Jeongho Jeon, Kyuho Son and Song Chong are with the School of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea (e-mail: {jjeon, skio}@netsys.kaist.ac.kr, [email protected]). Hyang-Won Lee is with the Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail:[email protected]).
2
I. I NTRODUCTION
O
PPORTUNISTIC scheduling has been introduced as an effective means of increasing system throughput in modern wireless systems [1]. It exploits independently-fading wireless channels
of users in order to take advantage of multiuser diversity gain, but CQI (Channel Quality Indicator) information of all users must be known to the scheduler residing at base station (BS) every time slot. The amount of CQI feedback from users to the BS increases as the number of users increases, consuming non-negligible amount of radio resource. The CQI feedback load becomes a more serious issue in a multicarrier system such as up-to-date OFDM-based IEEE 802.11/16/20 systems since every time slot the CQI information must be fed back to the BS for all users and for all carrier frequencies (interchangeably, subchannels). Therefore, a proper CQI feedback reduction scheme must be incorporated into a system to prevent extravagant amount of channel feedback. There have been some works on CQI feedback reduction. In [2], best-M feedback scheme was proposed for multi-carrier systems. In this scheme, each user selects M (≤ N ) best subchannels where N is the number of subchannels in the system and feeds back the CQIs of these subchannels to the BS. Since this scheme always gives an equal number of feedback opportunity M to all users irrespective of whether they are in deep fading or not, it has potential to work reasonably well with scheduling policies concerning fairness such as PFS (Proportional Fairness Scheduler) [3] and MFS (Maximum Fairness Scheduler)1 [4]. However, it will not work well with MRS (Maximum Rate Scheduler) [4], [5], since some subchannels can be scheduled to a user in deep fading thereby yielding throughput loss unless M is closer to the number of subchannels N . In [6], selective multiuser diversity concept was proposed for a single-channel system, in which each user sends the CQI feedback based on the absolute SNR threshold scheme. More specifically, user k sends the feedback every time slot t only if γk (t) ≥ γth where γk (t) denotes the instantaneous received SNR of user k at time slot t and γth is the threshold. This scheme can be easily extended to multi-channel systems
1
The MFS aims to allocate resources such that the minimum user’s data rate is maximized [4].
3
by applying the same threshold to all subchannels, but it can cause serious fairness problem when PFS or MFS is employed because users in deep fading have no chance of being scheduled at all due to their low received SNR at all carrier frequencies. In order to overcome this problem, [7] proposed a modified SNR thresholding scheme, referred to as the normalized SNR thresholding scheme. In this scheme, user k sends the CQI feedback when the normalized SNR value, γk (t)/γ k , exceeds a certain threshold A where γ k is the average of γk (t) from time slot 0 to t. In the average sense, however, the behavioral characteristic of the normalized SNR thresholding scheme is similar with that of the best-M scheme and consequently it may work well with PFS or MFS but not with MRS. A common drawback of all existing schemes mentioned above is that the total amount of feedback required at each time slot tends to increase in proportion to the number of users in the system, which significantly undermines their practicality for a large system. In order to overcome this problem, a random access based feedback protocol, named as opportunistic feedback, was proposed in [8]. Instead of giving feedback opportunity to all the users whose received SNR is greater than a threshold, it limits the opportunity to the users who become winners in the underlying random access process among them. Thereby it can explicitly control the total number of feedback per time slot regardless of number of users in the system. However, it inherently possesses the same drawbacks as the absolute SNR thresholding scheme, and feedback delay will be much longer due to the random access process. In contrast to the previous works, our proposed scheme uses a new metric called feedback efficiency factor as the feedback decision metric instead of received SNR. The key idea behind the proposed scheme is to give more feedback opportunity to users who are more often scheduled. The rest of the paper is organized as follows. In Section II, we describe the system model and define some notations. Section III proposes and analyzes the efficiency-based feedback algorithm in detail. In Section IV, we show through simulation that the proposed scheme reduces total feedback load substantially while maintaining almost the same scheduling performance as in the case of full feedback. In addition, we also demonstrate that the proposed scheme offers two unique advantages over the existing ones. First, its performance is insensitive to underlying scheduling policy and hence it works well with a broad class of scheduling policies ranging
4
between two extreme policies: MRS aiming at extreme efficiency and MFS aiming at extreme fairness. Secondly, it explicitly keeps total feedback load below a desired level regardless of the number of users in the system. We conclude the paper in Section V.
II. S YSTEM M ODEL We consider the downlink of a single cell in an OFDM system with K users and N subchannels. Denote by K and N the set of all users and subchannels respectively, i.e., K = {1, 2, ..., K} and N = {1, 2, ..., N }. A time-slotted system is considered, where channels are assumed to hold their state for the duration of a time slot. Let ~π (t) = [~πk , ∀k ∈ K] be the channel state vector at time slot t. In a single-channel system (N = 1), each element of ~π (t) can be replaced with a parameter which reflecting thermal noise, path-loss, shadowing and multi-path fading in a single value. Let Γ~π (t) be the feasible region of the transmission rate vector ~r(t) = [rk (t), ∀k ∈ K] given channel state vector ~π (t) at time slot t, i.e., ~r(t) ∈ Γ~π (t). Note that Γ~π (t) can be differed according to the underlying physical layer model. The transmission rate of user k at time slot t is determined by a scheduler as rk (t) =
PN n=1
xnk (t)rkn (t), where rkn (t) is the achievable
rate of user k over subchannel n during time slot t, and xnk (t) is the scheduling indicator function such that xnk (t) = 1 if user k is chosen to be served over subchannel n during time slot t and xnk (t) = 0 otherwise. Because at most one user can be assigned to each subchannel at each time slot, the scheduling indicator function should satisfy
PK k=1
xnk (t) ≤ 1, ∀n, ∀t.
Let Rk (t) be the average throughput of user k up to time slot t, i.e., Rk (t) =
1 t
Pt
τ =1 rk (t),
and
~ let R(t) = [Rk (t), ∀k ∈ K]. Each user is associated with a utility function Uk (Rk (t)) of their average P ~ throughput, and let U (R(t)) = K k=1 Uk (Rk (t)). We assume that the utility function Uk (·), ∀k ∈ K is an increasing, strictly concave, and continuously differentiable function on R+ . Define the long-term average ~ ~ = limt→∞ R(t), and let Γ be the long-term feasible region of the long-term throughput vector as R ~ considering all the possible scheduling policies. It can be easily shown that average throughput vector R, the long-term feasible region Γ is convex, coordinate convex, and compact set. We finally denote by Rkmax the intercept of the long-term feasible region Γ along the k-th dimension. This is simply the long-term
5
average throughput user k would get if user k was the only one scheduled for all subchannels and for all time slots.
III. E FFICIENCY- BASED F EEDBACK R EDUCTION A. Proposed Algorithm As mentioned, our feedback scheme gives more feedback opportunity to users who are more often scheduled. To apply such a concept in determining each user’s feedback amount, we have introduced feedback efficiency defined as the ratio of the average number of allocated subchannels, sk (t), to the amount of feedback, fk (t), for an arbitrary user k at time slot t. More specifically, fk (t) denotes the number of subchannels of which the CQI information was sent back to the BS. A proper target efficiency factor will be predetermined from the system aspect, and all mobile users are expected to maintain the same efficiency by adjusting their feedback amount. The efficiency-based feedback reduction (EFR) algorithm is summarized in Algorithm 1. For the given target efficiency factor e, each user computes their feedback success probability pk (t) using (1) at time slot t. We denote by [·]+ 1 the orthogonal projection onto the interval [0, 1] since a probability is expressed on a linear scale from 0 (impossibility) to 1 (certainty). Intuitively, if one increases the target efficiency factor, then each user will decrease their feedback success probability in order to balance their own efficiency to the target value. With the success probability pk (t), each user performs totally N independent Bernoulli trials, and sends the CQIs of the best subchannels up to the number of successful trials at time slot t + 1. After that, according to the scheduling results sk (t), each user updates the average number of allocated subchannels. In this paper, we assume that each user can estimate their average number of allocated subchannels, which is a reasonable assumption; in the IEEE 802.16e system, the DL-MAP (downlink map) message contains such resource allocation information. We also assume the full queue condition that the outgoing queues residing at the BS for each user are full. It should be noted that each user determines their amount of feedback not based on the direction from the BS, but distributively based on the feedback efficiency factor.
6
Algorithm 1 Efficiency-based Feedback Reduction Algorithm 1: At time slot t, – compute the feedback success probability: ·
1 sk (t) pk (t) = · N e
¸+ .
(1)
1
– generate a finite Bernoulli sequence {Xki : i = 1, ..., N }, where each Xki is a binary random variable with the success probability pk (t), and the total number of successes is Xk = 2:
PN i=1
Xki .
At time slot t + 1, – select fk (t + 1) = Xk best subchannels and send back the CQIs of these subchannels to the BS. – update the average number of allocated subchannels: t+1
1X sk (t + 1) = sk (τ ) t τ =1
(2)
– set t = t + 1 and go to 1.
B. Interrelationship between Feedback Algorithms and Scheduling Policies To have a better understanding of the interrelationship between feedback algorithms and scheduling policies, it is worthwhile knowing how users share resources with scheduling policies of different fairness criterion. This is because it is appropriate to give more feedback opportunities to users who occupy more subchannels. However, existing schemes were designed without thoroughly considering the underlying scheduling policies. Many of the scheduling policies considered—including the PFS—can be viewed as a gradient-based ~ + 1)) − U (R(t)) ~ algorithm [9], [10]. It aims to attain sum utility maximization by maximizing U (R(t at each time slot t, which is equivalent to choosing the transmission rate vector having the maximum projection onto the gradient of the sum utility: T ~ arg max ∇U (R(t)) · ~r(t). ~ r(t)∈Γ~π(t)
(3)
Under the strict concavity of the utility function, it has been proved by several researchers that such a
7
~ over the long-term feasible region Γ [9], [11]. gradient-based algorithm maximizes sum utility U (R) On the other hand, there was an attempt to develop utility functions to exploit a tradeoff between two different aims: total throughput maximization and fairness among users. For example, one class of utility functions applied for wireless packet schedulers was [12]2 : X (1 − α)−1 Rk (t)1−α , k∈K ~ U (R(t)) = X log(Rk (t)) ,
α > 0, α 6= 1, (4) α = 1,
k∈K
where α is a fairness exponent factor. With the utility functions in (4), (3) becomes: X
arg max
~ r(t)∈Γ~π(t)
(Rk (t))−α rk (t).
(5)
k∈K
For simplicity of the exposition, consider a single-channel system (N = 1) in which the resource allocation problem in (5) turns into a situation where a single user selection is needed who has the maximum weighted rate where the weights are the marginal utilities3 : arg max (Rk (t))−α rk (t). k∈K
(6)
As α → 0, the total throughput is maximized (MRS), and when α = 1, the proportionally fair throughput allocation is achieved (PFS). As α increases, fairness is improved at the cost of reduced total throughput, and especially as α → ∞, the throughput allocation becomes max-min fair (MFS). With the scheduler in (6), each user shares the single-channel as in Observation 1 below. Observation 1: For the two-user case, each user receives time slots in a long-term sense as in the following ratio: θ10 =
µ 1+
1 R1max R2max
¶(α−1)/α ,
θ20 = 1 − θ10 ,
(7)
where Rkmax is the long-term maximum achievable rate of user k. Proof: See Appendix A for the proof.
2
The utility functions in (4) were originally introduced in the context of wireline window-based congestion control [13].
3
The scheduler form in (6) is often referred to as Generic Proportional Fair Scheduler in the IEEE 802.16 Task Group m (TGm) [16].
8
In the single-channel system (N = 1), the long-term average number of allocated subchannels is equal to each user’s time sharing ratio on the channel, θk0 . Thus, the feedback success probability is given by h 0 i+ θk in the long-term sense, and the CQI information of the channel will be sent back to the BS if the e 1
outcome of a Bernoulli trial is successful. In this context, one can easily notice that the expected feedback opportunity of each user is proportional to their resource sharing ratio under our proposed scheme. This result can be readily extended to a multi-channel system without loss of generality. Fig. 1 plots (7) when the long-term maximum achievable rate of user 1 is twice than that of user 2, i.e., R1max /R2max = 2. Under our proposed scheme, each user’s feedback amount will be determined according to each sample instance of the resource sharing ratio at the given value of α from 0 to ∞. However, existing schemes have their single ideal operating point. For example, when α = 0 (MRS), best-M and normalized SNR thresholding schemes will reveal redundant feedback, because the strong user monopolizes all the resources, and the weak user does not need to send feedback at all. However, the absolute SNR thresholding scheme will be well-matched with MRS because the scheme lets the strong user send more feedback. On the other hand, when α = 1 (PFS), each user shares an equal fraction of time slots4 . Hence, the absolute SNR thresholding scheme can cause a serious fairness problem because the weak user has no chance of being scheduled at all due to the lack of the CQI information. Yet the bestM and normalized SNR thresholding schemes will be harmonious with PFS because these schemes give an equal or approximately equal number of feedback opportunities irrespective of their relative channel strengths between users. The case of α → ∞ (MFS) also can be explained analogously.
C. Feedback Load Estimation A common drawback of existing schemes is that the total feedback load increases in proportion to the number of users, which significantly undermines their practicality for a large system. However, under our proposed scheme, the total feedback load can be maintained below a target level regardless of the number
4
By simply extending the result in (7), if there are totally K users in the system, then each user will equally receive fraction 1/K of the
time slots referred to as the equal time sharing property of the PFS [15].
9
of users in a cell. Intuitively, if there are a large number of users in a cell, then each user will receive a relatively small number of subchannels. Consequently, each user will reduce their feedback amount in order to keep the target efficiency factor. Under our proposed scheme, the feedback success probability is given by pk =
£ s ¤+ k
Ne 1
, k ∈ K, where
we dropped the time index t for brevity. With the success probability pk , each user repeatedly performs totally N independent but identical Bernoulli trials, and the resulting number of successful trials, Xk , follows the binomial distribution with parameters N and pk , i.e., Xk ∼ B(N, pk ). Thus, the k-th user’s expected number of subchannels, of which the CQI information will be sent back to the BS, is given by: E[Xk ] = N pk ≤
sk , e
and the total feedback load is obtained by summing the feedback amount of all the individual users: X k∈K
E[Xk ] ≤
1X N sk = , e k∈K e
(8)
where the equality in (8) is because the available number of subchannels is always N regardless of the number of users in a cell. Finally, the total feedback load per channel of our efficiency-based feedback scheme is bounded by: 1 FEFR ≤ . e
(9)
Note that the estimated total feedback load per channel of our proposed scheme in (9) is not a function of the number of users in a cell, but a function of the target efficiency factor. We now look into the statistics of the feedback load of existing schemes by assuming that users experience independent identically distributed (i.i.d) fading. For a Rayleigh fading channel, the instantaneous received SNR γk , k ∈ K is distributed exponentially with a probability density function [17]: µ ¶ 1 γk g(γk ) = exp − , γk γk
γk ≥ 0,
where γ k is the average SNR. Under the absolute SNR thresholding scheme, each user sends the CQI of a channel if the instantaneous received SNR exceeds a certain threshold γth . Thus, the total feedback load
10
per channel of the absolute SNR thresholding scheme is: ¶ X µ ¶ Z γth K µ K X γth FABS = 1− g(γk )dγk = exp − . γk 0 k=1 k=1
(10)
On the other hand, under the normalized SNR thresholding scheme, each user sends the CQI of a channel when the normalized SNR value, γk /γ k , exceeds a certain threshold A. By simply replacing γth in (10) by γ k A for each user, the total feedback load per channel of the normalized SNR thresholding scheme is obtained by: FNOR =
K X
exp (−A) = K exp (−A) .
(11)
k=1
Admittedly, the total feedback load per channel of the best-M feedback scheme is deterministically given by: FBest-M =
KM . N
(12)
Both best-M and normalized SNR thresholding schemes let the users send equal or approximately equal amounts of feedback irrespective of their relative channel strengths to other users. By simply setting A = − ln M , each user’s feedback amount and the total feedback load of the normalized SNR thresholding N scheme becomes identical with those of the best-M feedback in the average sense. To verify the analysis, we simulated our proposed and existing schemes for the 24-subchannel system. In the case of the absolute SNR thresholding or normalized SNR thresholding scheme, the same threshold was applied to the multiple subchannels respectively, and resulting sum feedback load was averaged over 24 subchannels to obtain the per channel feedback load. For the efficiency-based feedback scheme, PFS was performed. Fig. 2 plots the total feedback load per channel of our proposed and existing schemes. It can be seen that the analytic results agree well with the simulation ones. Note that the slopes of existing schemes and the upper bound of our efficiency-based feedback scheme can be adjusted by tuning the feedback control parameters. To sum up, the total feedback loads of existing schemes are very sensitive to variation in the number of users in a cell, and thus there is a possibility that existing schemes can cause not only the underutilization of downlink resources due to insufficient feedback when the number of users is small, but also the extravagant use of uplink resources due to excessive feedback when the number
11
of users is large. However, the total feedback load of our proposed scheme is maintained below a target level regardless of the number of users in a cell by adjusting each user’s feedback load adaptively. Also, it should be noted that our proposed scheme does not require any underlying contention mechanisms, like random access based opportunistic feedback [8], to control the total feedback load.
D. Selection of the Feedback Efficiency Factor With imperfect knowledge of the channel state due to the partial CQI feedback, the scheduler selects ˆ ~π(t) at each time slot t as follows: the transmission rate vector ~r(t) over the reduced feasible region Γ T ~ˆ arg max ∇U (R(t)) · ~r(t), ˆ ~π(t) ~ r(t)∈Γ
(13)
~ˆ ~∗ where R(t) is the average throughput vector up to time slot t with imperfect channel knowledge. Let R ~ˆ ∗ be the long-term throughput vector obtained by the scheduler with perfect and imperfect channel and R knowledge respectively. Because the gradient-based scheduler aims to maximize sum utility, we define the performance ratio of the scheduling with imperfect channel knowledge to perfect channel knowledge as follows:
PK ρ = Pk=1 K
ˆ∗ ) Uk (R k
∗ k=1 Uk (Rk )
.
(14)
However, finding the closed-form of (14) under a certain feedback scheme is a daunting task. It requires to formulate the throughput achieved with a specific scheduling policy under a certain feedback scheme, but even formulating the throughput of the multi-channel systems with a specific scheduling policy under perfect channel knowledge itself is not an easy task. Fig. 3 plots the empirically obtained performance ratio of our efficiency-based feedback scheme for various number of users. PF scheduling was performed and all the other simulation environments are identical with that of Section IV. It can be seen that there is some degree of performance degradation when 1/e is relatively small (high feedback efficiency), whereas the performance approaches to the full feedback as 1/e increases (low feedback efficiency). At 1/e = 6, the performance ratio is always higher than 99%, and there is no remarkable gain above 1/e = 10. Also, it should be noted that the feedback
12
efficiency factor can be chosen using the property discussed in Section III-C to meet the maximum allowable feedback load by the system, e.g., the system which allocates a dedicated channel for the CQI feedback.
IV. S IMULATION R ESULTS To validate the effectiveness of our proposed algorithm, the IEEE 802.16e 1024-FFT OFDMA AMC (Adaptive Modulation & Coding) mode [18] was adopted, where the data subcarriers were grouped into 24 subchannels. The AMC operation is possible only when the channel coherence time is much longer than the lag between the time the channel is measured on the mobile and the time when the packet is actually transmitted from the BS [3], [18]. We thus adopted the ITU pedestrian B model [19]. For high mobility users, who are usually served by the Diversity mode, it is sufficient for them to predict the average CQI of the overall subchannels, because the fading channel follows the stationary distribution [3]. In the simulation, we further made the following assumptions: 1) The radius of the cell was set to 1km, and the distance, dk , between user k and the BS was generated as a 2-D uniformly distributed random variable. 2) We adopted P L(dk ) = 16.62 + 37.6log10 (dk )[dB] for the path loss model with log-normal shadowing with standard deviation σs = 8dB. 3) The ITU pedestrian B parameters was applied to the standard delay-spread model for frequency-selective fast fading [20]. 4) In making a fair comparison, we set the threshold of each scheme so as to exhibit the same feedback load at 30 number of users: M = 5 (best-M scheme), γth = 11.86 in dB scale (absolute SNR thresholding scheme), γk (t)/γ k ≥ 1.6 in linear scale (normalized SNR thresholding scheme), and e = 1/6.25 (efficiency-based feedback scheme). 5) The scheduling was performed every time slot (5 msec) for the multiple subchannels. In Fig. 4, we compare the performance of the schemes that have been used previously and our own under scheduling policies of different fairness criterion. It can be seen that the total throughput and Jain’s fairness index [21] of our efficiency-based feedback scheme always keep track of the full feedback for varying α. In the case of the best-M and normalized SNR thresholding schemes, there exists throughput degradation near α = 0 (MRS). However, as α approaches to 1 (PFS), the total throughput and fairness
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converge to the full feedback. On the other hand, in the case of the absolute SNR thresholding scheme, the total throughput and fairness performances diverge from the full feedback as α increases (MFS). Fig. 5 depicts the total throughput, Jain’s fairness index and total feedback load per channel of the schemes with PFS. In Fig. 5(a)-(b), the total throughput and Jain’s fairness index of our proposed scheme are very close to the full feedback for various number of users. However, in the case of the best-M and normalized SNR thresholding schemes, we can see considerable throughput degradation when there are small number of users. On the other hand, it can be seen that the total throughput of the absolute SNR thresholding scheme outperforms the full feedback, whereas the fairness performance is worse than that of the full feedback. This is because the weak users have no chance of being scheduling at all due to the lack of the CQI information, and consequently the resulting throughput vector is quite distant from the proportionally fair throughput allocation under the absolute SNR thresholding scheme (see Fig. 4 at α=1). From Fig. 5(c), it can be seen that the total feedback load per channel of our proposed scheme is maintained below 1/e, whereas the total feedback load per channel of existing schemes increase in proportion to the number of users.
V. C ONCLUDING R EMARKS In this paper, we have proposed an innovative feedback reduction scheme which uses a feedback efficiency factor as the feedback decision metric instead of received SNR. The key idea is to give more feedback opportunity to users who are more often scheduled. We have shown that the proposed scheme reduces total feedback load substantially while maintaining almost the same scheduling performance as in the case of full feedback. We have studied the interrelationship between feedback algorithms and scheduling policies under a gradient-based scheduling framework. The simulation results showed that our proposed scheme works well with a broad class of scheduling policies ranging from MRS (Maximum Rate Scheduler) aiming at extreme efficiency to MFS (Maximum Fairness Scheduler) aiming at extreme fairness. We have shown that our proposed scheme prevents both the underutilization of downlink resources due to insufficient feedback and the extravagant use of uplink resources due to excessive feedback by
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controlling the total feedback load to a target level regardless of the number of users in a cell. This is possible because each user adjusts their feedback load adaptively where the feedback efficiency factor acts as a coordinator. All these advantages are achieved without additional overhead. A PPENDIX A P ROOF OF O BSERVATION 1 We basically assume that the relative fluctuations of the achievable rates for various users around their respective time-average values are statistically identical. Denote by θ~0 = [θk0 , ∀k ∈ K] the long-term average ratio of time slots allocated to users by the scheduling policy (6), where 0 ≤ θk0 ≤ 1, ∀k ∈ K and P k∈K
~ 0 = [θ0 Rmax , ∀k ∈ K]. θk0 = 1. Then, we can express the resulting long-term throughput vector as R k k
~ = [θk Rmax , ∀k ∈ K], where Similarly, an arbitrary long-term throughput vector can be expressed as R k 0 ≤ θk ≤ 1, ∀k ∈ K and
P k∈K
θk = 1. Because the utility functions in (4) are strictly concave, it is easy
~ ∗ is the optimal solution of sum utility maximization if and only if (see [9] and references to show that R therein): ~ ∗ )T · (R ~ −R ~ ∗ ) ≤ 0, ∀R ~ ∈ Γ. ∇U (R
(15)
~ ∗ is the unique Under the assumption that the long-term feasible region is strictly convex, it follows that R ~ ~ ∗ as t → ∞ starting with any initial state R(0) ~ sum utility maximizer and R(t) →R ∈ Γ [9]. Therefore, ~ 0 should satisfy (15) for all possible throughput vector R ~ ∈ Γ, because the long-term throughput vector R ~ 0 is the unique optimal long-term throughput vector obtained by the gradient-based scheduler in (6) R which maximizes the sum utility. For the two-user case, (15) can be rewritten as follows: θ1 R1max − θ10 R1max (1 − θ1 )R2max − (1 − θ10 )R2max + ≤ 0, ∀θk ∈ [0, 1]. (θ10 R1max )α ((1 − θ10 )R2max )α
(16)
After simple manipulation, (16) reduces to: θ1 A − θ10 A ≤ 0,
∀θk ∈ [0, 1],
(17)
where A = (1 − θ10 )α (R2max )α−1 − θ10 α (R1max )α−1 . The inequality in (17) is always satisfied for ∀θk ∈ [0, 1] if and only if A = 0, and this leads to the result of Observation 1.
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R EFERENCES [1] R. Knopp and P. Humblet, “Information capacity and power control in single-cell multiuser communications,” in Proc. IEEE ICC’95, Seattle, Jun. 1995. [2] ”CQI report and scheduling procedure,” 3GPP Tdoc R1-051045, 2005. [3] D. Tse and P. Viswanath, Fundamentals of wireless communication, Cambridge University Press, 2005. [4] J.G. Andrews, A. Ghosh and R. Muhamed, Fundamentals of WiMAX: Understanding broadband wireless networking, Prentice Hall, 2007. [5] S. Shakkottai and A. L. Stolyar, ”A study of scheduling algorithms for a mixture of real and non-real time data in hdr”, Bell Laboratories, Lucent Technologies, Oct. 2000. [6] D. Gesbert and M. -S. Alouini, “How much feedback is multi-user diversity really worth?,” in Proc. IEEE ICC’04, Paris, Jun. 2004. [7] L. Yang, M. -S. Alouini and D. Gesbert, “Further results on selective multiuser diversity,” in Proc. ACM MSWiM’04, Itary, Oct. 2004. [8] T. Tang and R. W. Heath, Jr., “Opportunistic feedback for downlink multiuser diversity,” IEEE Communications Letters, vol. 9, no. 10, Oct. 2005. [9] R. Agrawal and V. Subramanian, ”Optimality of certain channel aware scheduling policies,” in Proc. 2002 Allerton Conference on Communication, Otc. 2002. [10] J. Huang, V. Subramanian, R. Agrawal, and R. Berry, ”Downlink scheduling and resource allocation for ofdm systems,” in Proc. Conference on Information Sceiences and Systems, New Jersey, Mar. 2006. [11] H. J. Kushner and P. A. Whiting, ”Convergence of proportional-fair sharing algorithms under general conditions,” IEEE Trans. Wireless Communications, vol. 3, no. 4, pp. 1250–1259, Jul. 2004. [12] R. Agrawal, A. Bedekar, R. La, and V. Subramanian, ”A class and channel-condition based weighted proportional fair scheduler,” in Proc. ITC 2001, Salvador, Brazil, Sep. 2001. [13] J. Mo and J. Walrand, “Fair end-to-end window-based congestion control,” IEEE/ACM Trans. Networking, vol. 8, no. 5, pp. 556–567, Oct. 2000. [14] F. Kelly, “Charging and rate control for elastic traffic,” Eur. Trans. Telecommun., vol. 8, pp. 33–37, Oct. 1996. [15] S. Borst, “User-level performance of channel-aware scheduling algorithm in wireless data networks,” in Proc. IEEE INFOCOM’03, San Francisco, Mar. 2003. [16] ”Project 802.16m evaluation methodology document (EMD),” IEEE 802.16m-08/004, Mar. 2008. [17] Q. Zhang and S. A. Kassam, ”Finite-state markov model for rayleigh fading channels,” IEEE Trans. Communications, vol. 47, no. 11, pp. 1688–1692, Nov. 1999. [18] IEEE Std 802.16e-2005 and IEEE 802.16-2004/Cor 1-2005. Part 16: Air interface for fixed and mobile broadband wireless access systems. Dec. 2005. [19] ”Guidelines for the evaluation of radio transmission technologies for IMT-2000,” Recommendation ITU-R M.1225, 1997. [20] M. P¨atzold, Mobile fading channels. Baffins Lane, England: John Wiley and Sons, 2002
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[21] R. Jain, D. Chiu, and W. Hawe, “A quantitative measure of fairness and discrimination for resource allocation in shared system,” DEC TR-301.
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1 θ1’
0.9
θ2’
0.8 PFS
Time sharing ratio
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Fig. 1.
0
5
α
10
15
Time sharing ratio in the two-user case, R1max /R2max = 2
80 Full feedback EFR (simulation, e=1/15) EFR (analysis: e=1/15) ABS (simulation, γth=−2.2dB)
Total feedback load per channel
70 60
ABS (analysis, γth=−2.2dB) Best−M, M=15 NOR (simulation, A=0.83) NOR (analysis, A=0.83)
50 40 30 20 10 0
Fig. 2.
10
20
30 40 Number of users
Total feedback load comparison of proposed and existing schemes
50
60
70
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1 0.995
Performance ratio
0.99 Number of users = 10 Number of users = 20 Number of users = 30 Number of users = 40 Number of users = 50 Number of users = 60 Number of users = 70
0.985 0.98 0.975 0.97 0.965 0.96 0.955 0.95
Fig. 3.
4
6
8
10 1/e
Performance ratio of the efficiency-based feedback scheme with PFS
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16
19
30
Total throughput (Mbps)
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26 Full feedback EFR (e=1/6.25) Best−M (M=5) NOR (A=1.6) ABS (γth=11.86 dB)
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1 α
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(a) Total throughput (when the number of users = 30) 1 0.9
Jain’s fairness index
0.8 0.7 0.6 0.5 0.4 Full feedback EFR (e=1/6.25) Best−M (M=5) NOR (A=1.6) ABS (γth=11.86 dB)
0.3 0.2 0.1
0
0.5
1 α
1.5
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(b) Jain’s fairness index (when the number of users = 30) Fig. 4.
Comparison of total throughput and Jain’s fairness index with scheduling policies of different fairness criterion
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Total throughput (Mbps)
26 Full feedback EFR (e=1/6.25) Best−M (M=5) NOR (A=1.6) ABS (γ =11.86 dB)
24
22
th
20
18
16 10
20
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40 Number of users
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(a) Total throughput with PFS (i.e., α=1) 0.9 0.85 0.8 Full feedback EFR (e=1/6.25) Best−M (M=5) NOR (A=1.6) ABS (γth=11.86 dB)
Jain’s fairness index
0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 10
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40 Number of users
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(b) Jain’s fairness index with PFS (i.e., α=1)
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Total feedback load per channel
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Full feedback EFR (e=1/6.25) Best−M (M=5) NOR (A=1.6) ABS (γth=11.86 dB)
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30 40 50 Number of users
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(c) Total feedback load per channel with PFS (i.e., α=1) Fig. 5.
Comparison of total throughput, Jain’s fairness index and total feedback load per channel for various number of users
