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Slow adaptive ofdma systems through chance constrained programming


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Slow adaptive ofdma systems through chance constrained programming

  1. 1. IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. X, NO. X, XXX 2010 1 Slow Adaptive OFDMA Systems Through Chance Constrained Programming William Wei-Liang Li, Student Member, IEEE, Ying Jun (Angela) Zhang, Member, IEEE, Anthony Man-Cho So, and Moe Z. Win, Fellow, IEEE Abstract—Adaptive OFDMA has recently been recognized as I. I NTRODUCTION a promising technique for providing high spectral efficiency inarXiv:1006.4406v1 [cs.NI] 23 Jun 2010 future broadband wireless systems. The research over the last decade on adaptive OFDMA systems has focused on adapting the allocation of radio resources, such as subcarriers and power, F UTURE wireless systems will face a growing demand for broadband and multimedia services. Orthogonal fre- quency division multiplexing (OFDM) is a leading technology to the instantaneous channel conditions of all users. However, such “fast” adaptation requires high computational complexity to meet this demand due to its ability to mitigate wire- and excessive signaling overhead. This hinders the deployment of less channel impairments. The inherent multicarrier nature of adaptive OFDMA systems worldwide. This paper proposes a slow OFDM facilitates flexible use of subcarriers to significantly adaptive OFDMA scheme, in which the subcarrier allocation is enhance system capacity. Adaptive subcarrier allocation, re- updated on a much slower timescale than that of the fluctuation of instantaneous channel conditions. Meanwhile, the data rate cently referred to as adaptive orthogonal frequency division requirements of individual users are accommodated on the fast multiple access (OFDMA) [1], [2], has been considered as a timescale with high probability, thereby meeting the requirements primary contender in next-generation wireless standards, such except occasional outage. Such an objective has a natural as IEEE802.16 WiMAX [3] and 3GPP-LTE [4]. chance constrained programming formulation, which is known In the existing literature, adaptive OFDMA exploits time, to be intractable. To circumvent this difficulty, we formulate safe tractable constraints for the problem based on recent frequency, and multiuser diversity by quickly adapting sub- advances in chance constrained programming. We then develop carrier allocation (SCA) to the instantaneous channel state a polynomial-time algorithm for computing an optimal solution information (CSI) of all users. Such “fast” adaptation suffers to the reformulated problem. Our results show that the proposed from high computational complexity, since an optimization slow adaptation scheme drastically reduces both computational problem required for adaptation has to be solved by the base cost and control signaling overhead when compared with the conventional fast adaptive OFDMA. Our work can be viewed as station (BS) every time the channel changes. Considering the an initial attempt to apply the chance constrained programming fact that wireless channel fading can vary quickly (e.g., at methodology to wireless system designs. Given that most wireless the order of milli-seconds in wireless cellular system), the systems can tolerate an occasional dip in the quality of service, we implementation of fast adaptive OFDMA becomes infeasible hope that the proposed methodology will find further applications for practical systems, even when the number of users is small. in wireless communications. Recent work on reducing complexity of fast adaptive OFDMA Index Terms—Dynamic Resource Allocation, Adaptive includes [5], [6], etc. Moreover, fast adaptive OFDMA requires OFDMA, Stochastic Programming, Chance Constrained Programming frequent signaling between the BS and mobile users in order to inform the users of their latest allocation decisions. The Copyright c 2010 IEEE. Personal use of this material is permitted. overhead thus incurred is likely to negate the performance However, permission to use this material for any other purposes must be gain obtained by the fast adaptation schemes. To date, high obtained from the IEEE by sending a request to Manuscript received July 01, 2009; revised October 28, 2009 and February computational cost and high control signaling overhead are 09, 2010; accepted February 15, 2010. This research was supported, in part, the major hurdles that prevent adaptive OFDMA from being by the Competitive Earmarked Research Grant (Project numbers 418707 and deployed in practical systems. 419509) established under the University Grant Committee of Hong Kong, Project #MMT-p2-09 of the Shun Hing Institute of Advanced Engineering, We consider a slow adaptive OFDMA scheme, which is the Chinese University of Hong Kong, the National Science Foundation under motivated by [7], to address the aforementioned problem. Grants ECCS-0636519 and ECCS-0901034, the Office of Naval Research In contrast to the common belief that radio resource allo- Presidential Early Career Award for Scientists and Engineers (PECASE) N00014-09-1-0435, and the MIT Institute for Soldier Nanotechnologies. The cation should be readapted once the instantaneous channel associate editor coordinating the review of this manuscript and approving it conditions change, the proposed scheme updates the SCA for publication was Dr. Walid Hachem. on a much slower timescale than that of channel fluctuation. W. W.-L. Li is with the Department of Information Engineering, the Chinese University of Hong Kong, Hong Kong ( Specifically, the allocation decisions are fixed for the duration Y. J. Zhang is with the Department of Information Engineering and the of an adaptation window, which spans the length of many Shun Hing Institute of Advanced Engineering, the Chinese University of Hong coherence times. By doing so, computational cost and control Kong, Hong Kong ( A. M.-C. So is with the Department of Systems Engineering and Engineer- signaling overhead can be dramatically reduced. However, this ing Management and the Shun Hing Institute of Advanced Engineering, the implies that channel conditions over the adaptation window are Chinese University of Hong Kong, Hong Kong ( uncertain at the decision time, thus presenting a new challenge M. Z. Win is with the Laboratory for Information & Decision Systems (LIDS), Massachusetts Institute of Technology, MA, USA ( in the design of slow adaptive OFDMA schemes. An important Digital Object Identifier XXX/XXX question is how to find a valid allocation decision that remains
  2. 2. 2 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. X, NO. X, XXX 2010optimal and feasible for the entire adaptation window. Such constrained programming in the context of resource allo-a problem can be formulated as a stochastic programming cation in wireless systems.problem, where the channel coefficients are random rather than • We exploit the special structure of the probabilisticdeterministic. constraints in our problem to construct safe tractable Slow adaptation schemes have recently been studied in other constraints (STC) based on recent advances in the chancecontexts such as slow rate adaptation [7], [8] and slow power constrained programming literature.allocation [9]. Therein, adaptation decisions are made solely • We design an interior-point algorithm that is tailored forbased on the long-term average channel conditions instead of the slow adaptive OFDMA problem, since the formu-fast channel fading. Specifically, random channel parameters lation with STC, although convex, cannot be triviallyare replaced by their mean values, resulting in a deterministic solved using off-the-shelf optimization software. Ourrather than stochastic optimization problem. By doing so, algorithm can efficiently compute an optimal solution toquality-of-service (QoS) can only be guaranteed in a long-term the problem with STC in polynomial time.average sense, since the short-term fluctuation of the channel is The rest of the paper is organized as follows. In Sectionnot considered in the problem formulation. With the increasing II, we discuss the system model and problem formulation. Anpopularity of wireless multimedia applications, however, there STC is introduced in Section III to solve the original chancewill be more and more inelastic traffic that require a guarantee constrained program. An efficient tailor-made algorithm foron the minimum short-term data rate. As such, slow adaptation solving the approximate problem is then proposed in Sectionschemes based on average channel conditions cannot provide IV. In Section V, we reduce the problem size based on somea satisfactory QoS. practical assumptions, and show that the revised problem can On another front, robust optimization methodology can be be solved by the proposed algorithm with much lower com-applied to meet the short-term QoS. For example, robust opti- plexity. In Section VI, the performance of the slow adaptivemization method was applied in [9]–[11] to find a solution that OFDMA system is investigated through extensive feasible for the entire uncertainty set of channel conditions, Finally, the paper is concluded in Section VII.i.e., to guarantee the instantaneous data rate requirementsregardless of the channel realization. Needless to say, theresource allocation solutions obtained via such an approach II. S YSTEM M ODEL AND P ROBLEM F ORMULATIONare overly conservative. In practice, the worst-case channel This paper considers a single-cell multiuser OFDM systemgain can approach zero in deep fading, and thus the resource with K users and N subcarriers. We assume that the instan-allocation problem can easily become infeasible. Even if the taneous channel coefficients of user k and subcarrier n areproblem is feasible, the resource utilization is inefficient as (t) described by complex Gaussian1 random variables hk,n ∼most system resources must be dedicated to provide guarantees 2 2 CN (0, σk ), independent in both n and k. The parameter σkfor the worst-case scenarios. can be used to model the long-term average channel gain as Fortunately, most inelastic traffic such as that from mul- −γ σk = dk ·sk , where dk is the distance between the BS andtimedia applications can tolerate an occasional dip in the d0instantaneous data rate without compromising QoS. This subscriber k, d0 is the reference distance, γ is the amplitudepresents an opportunity to enhance the system performance. path-loss exponent and sk characterizes the shadowing effect. (t) (t) 2In particular, we employ chance constrained programming Hence, the channel gain gk,n = hk,n is an exponentialtechniques by imposing probabilistic constraints on user QoS. random variable with probability density function (PDF) givenAlthough this formulation captures the essence of the problem, by 1 ξchance constrained programs are known to be computationally fgk,n (ξ) = exp − . (1)intractable except for a few special cases [12]. In general, such σk σkprograms are difficult to solve as their feasible sets are often The transmission rate of user k on subcarrier n at time t isnon-convex. In fact, finding feasible solutions to a generic given by (t)chance constrained program is itself a challenging research (t) pt gk,nproblem in the Operations Research community. It is partly rk,n = W log2 1 + , ΓN0due to this reason that the chance constrained programmingmethodology is seldom pursued in the design of wireless (t) where pt is the transmission power of a subcarrier, gk,n is thesystems. channel gain at time t, W is the bandwidth of a subcarrier, N0 In this paper, we propose a slow adaptive OFDMA scheme is the power spectral density of Gaussian noise, and Γ is thethat aims at maximizing the long-term system throughput capacity gap that is related to the target bit error rate (BER)while satisfying with high probability the short-term data and coding-modulation schemes.rate requirements. The key contributions of this paper are as In traditional fast adaptive OFDMA systems, SCA decisionsfollows: are made based on instantaneous channel conditions in order • We design the slow adaptive OFDMA system based on 1 Although the techniques used in this paper are applicable to any fading chance constrained programming techniques. Our formu- lation guarantees the short-term data rate requirements distribution, we shall prescribe to a particular distribution of fading channels for illustrative purposes. of individual users except in rare occasions. To the best 2 The case when frequency correlations exist among subcarriers will be of our knowledge, this is the first work that uses chance discussed in Section VI.
  3. 3. LI et al.: SLOW ADAPTIVE OFDMA SYSTEMS THROUGH CHANCE CONSTRAINED PROGRAMMING 3 SCA SCA SCA SCA SCA SCA SCA window3. Then, the time-average throughput of user k during the window becomes N slot slot slot time ¯k = b xk,n rk,n , ¯ (a) fast adaptive OFDMA n=1 where SCA SCA SCA 1 (t) rk,n = ¯ rk,n dt window window window T T ... ... ... is the time-average data rate of user k on subcarrier n during slot slot slot the adaptation window. The time-average system throughput time (b) slow adaptive OFDMA is given by K N ¯= b xk,n rk,n . ¯Fig. 1. Adaptation timescales of fast and slow adaptive OFDMA system(SCA = SubCarrier Allocation). k=1 n=1 Now, suppose that each user has a short-term data rate require- N (t) ment qk defined on each time slot. If n=1 xk,n rk,n < qk , then we say that a rate outage occurs for user k at time slot t,to maximize the system throughput. As depicted in Fig. 1a, and the probability of rate outage for user k during the windowSCA is performed at the beginning of each time slot, where [t0 , t0 + T ] is defined asthe duration of the slot is no larger than the coherence time of (t) Nthe channel. Denoting by xk,n the fraction of airtime assigned out (t)to user k on subcarrier n, fast adaptive OFDMA solves at each Pk Pr xk,n rk,n < qk , ∀t ∈ [t0 , t0 + T ], n=1time slot t the following linear programming problem: K N where t0 is the beginning time of the window. (t) (t) Pfast : max xk,n rk,n (2) Inelastic applications, such as voice and multimedia, that (t) xk,n k=1 n=1 are concerned with short-term QoS can often tolerate an N occasional dip in the instantaneous data rate. In fact, most (t) (t) s.t. xk,n rk,n ≥ qk , ∀k (3) applications can run smoothly as long as the short-term data n=1 rate requirement is satisfied with sufficiently high probability. K With the above considerations, we formulate the slow adaptive (t) xk,n ≤ 1, ∀n OFDMA problem as follows: k=1 K N (t) xk,n ≥ 0, ∀k, n, Pslow : max xk,n E rk,n (t) (4) xk,n k=1 n=1where the objective function in (2) represents the total system Nthroughput at time t, and (3) represents the data rate constraint (t) s.t. Pr xk,n rk,n ≥ qk ≥ 1 − ǫk , ∀k (5)of user k at time t with qk denoting the minimum required n=1data rate. We assume that qk is known by the BS and can be K (t) (t)different for each user k. Since gk,n (and hence rk,n ) varies on xk,n ≤ 1, ∀nthe order of coherence time, one has to solve the Problem Pfast k=1at the beginning of every time slot t to obtain SCA decisions. xk,n ≥ 0, ∀k, n,Thus, the above fast adaptive OFDMA scheme is extremelycostly in practice. where the expectation4 in (4) is taken over the random channel (t) In contrast to fast adaptation schemes, we propose a slow process g = {gk,n } for t ∈ [t0 , t0 + T ], and ǫk ∈ [0, 1] in (5)adaptation scheme in which SCA is updated only every adap- is the maximum outage probability user k can tolerate. In thetation window of length T . More precisely, SCA decision is above formulation, we seek the optimal SCA that maximizesmade at the beginning of each adaptation window as depicted the expected system throughout while satisfying each user’sin Fig. 1b, and the allocation remains unchanged till the next short-term QoS requirement, i.e., the instantaneous data ratewindow. We consider the duration T of a window to be of user k is higher than qk with probability at least 1 − ǫk .large compared with that of fast fading fluctuation so that The above formulation is a chance constrained program sincethe channel fading process over the window is ergodic; but a probabilistic constraint (5) has been imposed.small compared with the large-scale channel variation so thatpath-loss and shadowing are considered to be fixed in eachwindow. Unlike fast adaptive systems that require the exact 3 It is practical to assume x k,n as a real number in slow adaptive OFDMA.CSI to perform SCA, slow adaptive OFDMA systems rely Since the data transmitted during each window consists of a large mount of OFDM symbols, the time-sharing factor xk,n can be mapped into the ratioonly on the distributional information of channel fading and of OFDM symbols assigned to user k for transmission on subcarrier n.make an SCA decision for each window. 4 In (4), we replace the time-average data rate r ¯k,n by its ensemble average (t) Let xk,n ∈ [0, 1] denote the SCA for a given adaptation E rk,n due to the ergodicity of channel fading over the window.
  4. 4. 4 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. X, NO. X, XXX 2010 III. S AFE T RACTABLE C ONSTRAINTS Then, the allocation decision x satisfies ˆ N (t) Pr xk,n rk,n ≥ qk ˆ ≥ 1 − ǫk , ∀k. (12) Despite its utility and relevance to real applications, the n=1chance constraint (5) imposed in Pslow makes the optimization Proof: Our argument will use the Bernstein approxi-highly intractable. The main reason is that the convexity of the mation theorem proposed in [13].6 Suppose there exists anfeasible set defined by (5) is difficult to verify. Indeed, given x ∈ RN K such that Gk (ˆ) ≤ 0, i.e., ˆ xa generic chance constraint Pr {F (x, r) > 0} ≤ ǫ where r is arandom vector, x is the vector of decision variable, and F is a Nreal-valued function, its feasible set is often non-convex except inf qk + ̺ Λk (−̺−1 xk,n ) − ̺ log ǫk ˆ ≤ 0. (13) ̺>0for very few special cases [12], [13]. Moreover, even with the n=1 N (t)nice function in (5), i.e., F (x, r) = qk − n=1 xk,n rk,n is The function inside the inf ̺>0 {·} is equal to (t)bilinear in x and r, with independent entries rk,n in r whose N (t)distribution is known, it is still unclear how to compute the qk + ̺ log E exp − ̺−1 xk,n rk,n ˆ − ̺ log ǫk (14)probability in (5) efficiently. n=1 N To circumvent the above hurdles, we propose the following (t) ˜formulation Pslow by replacing the chance constraints (5) with = qk + ̺ log E exp ̺−1 − xk,n rk,n ˆ − ̺ log ǫk n=1a system of constraints H such that (i) x is feasible for (5) (15)whenever it is feasible for H, and (ii) the constraints in H are Nconvex and efficiently computable5. The new formulation is (t) = ̺ log E exp ̺−1 qk − xk,n rk,n ˆ − ̺ log ǫk ,given as follows: n=1 K N (16) ˜ (t) Pslow : max xk,n E rk,n (6) xk,n where the expectation E {·} can be computed using the distri- k=1 n=1 (t) N butional information of gk,n in (1), and (15) follows from the (t) s.t. inf qk + ̺ Λk (−̺−1 xk,n ) independence of random variable rk,n over n. ̺>0 (t) n=1 Let Fk (x, r) = qk − N xk,n rk,n . Then, (13) is equiva- n=1 − ̺ log ǫk ≤ 0, ∀k (7) lent to K inf ̺E exp ̺−1 Fk (ˆ, r) x − ̺ǫk ≤ 0. (17) ̺>0 xk,n ≤ 1, ∀n (8) According to Theorem 2 in Appendix A, the chance con- k=1 straints (12) hold if there exists a ̺ > 0 satisfying (17). Thus, xk,n ≥ 0, ∀k, n, (9) the validity of (12) is guaranteed by the validity of (11). (t) Now, we prove the convexity of (7) in the followingwhere Λk (·) is the cumulant generating function of rk,n , proposition. ˆ W xk,n ∞ − pt ξ ̺ ln 2 Proposition 2. The constraints imposed in (7) are convex in Λk (−̺−1 xk,n ) = log ˆ 1+ 0 ΓN0 x = [x1,1 , · · · , xN,1 , . . . , x1,K , · · · , xN,K ]T ∈ RN K . 1 ξ · exp − dξ . (10) Proof: The convexity of (7) can be verified as follows. σk σk We define the function inside the inf ̺>0 {·} in (11) asIn the following, we first prove that any solution x that is N ˜feasible for the STC (7) in Pslow is also feasible for the chance Hk (x, ̺) qk + ̺ Λk (−̺−1 xk,n ) − ̺ log ǫk , ∀k. (18) ˜constraints (5). Then, we prove that Pslow is convex. n=1Proposition 1. Suppose that gk,n (and hence rk,n ) (t) (t) It is easy to verify the convexity of Hk (x, ̺) in (x, ̺), sinceare independent random variables for different n and the cumulant generating function is convex. Hence, Gk (x) in (t)k, where the PDF of gk,n follows (1). Furthermore, (11) is convex in x due to the preservation of convexity bygiven ǫk > 0, suppose that there exists an x = ˆ minimization over ̺ > 0.[ˆ1,1 , · · · , xN,1 , . . . , x1,K , · · · , xN,K ]T ∈ RN K such that x ˆ ˆ ˆ N IV. A LGORITHM −1Gk (ˆ) x inf qk +̺ Λk (−̺ xk,n )−̺ log ǫk ≤ 0, ˆ ∀k. In this section, we propose an algorithm for solving Problem ̺>0 ˜ ˜ n=1 Pslow . In Pslow , the STC (7) arises as a subproblem, which (11) by itself requires a minimization over ̺. Hence, despite its ˜ convexity, the entire problem Pslow cannot be trivially solved 5 Condition (i) is referred to as “safe” condition, and condition (ii) is referred 6 For the reader’s convenience, both the theorem and a rough proof areto as “tractable” condition. provided in Appendix A.
  5. 5. LI et al.: SLOW ADAPTIVE OFDMA SYSTEMS THROUGH CHANCE CONSTRAINED PROGRAMMING 5Algorithm 1 Structure of the Proposed Algorithm later, such an algorithm will in fact terminate in a polynomialRequire: The feasible solution set of Problem Pslow is a˜ number of steps. compact set X defined by (7)-(9). We now give the structure of the algorithm. A detailed flow 1: Construct a polytope X 0 ⊃ X by (8)-(9). Set i ← 0. chart is shown in Fig. 2 for readers’ interest. 2: Choose a query point (Subsection IV. A-1) at the ith iteration as xi by computing the analytic center of X i . Initialize: The set X 0 : (A0 , b0 ) Initially, set x0 = e/K ∈ X 0 where e is an N -vector of and x0 = e/K. ones. 3: Query the separation oracle (Subsection IV. A-2) with xi : 4: if xi ∈ X then Oracle 5: generate a hyperplane (optimality cut) through xi to ̺∗ = arg inf [H(xi , ̺)] ̺>0 remove the part of X i that has lower objective values 6: else N Y 7: generate a hyperplane (feasibility cut) through xi to H(xi , ̺∗ ) ≤ 0 remove the part of X i that contains infeasible solutions. 8: end if Adding Adding 9: Set i ← i + 1, and update X i+1 by the separation feasibility cut optimality cut hyperplane. (23) (24)10: if termination criterion (Subsection IV. B) is satisfied then Update11: stop X i+1: (Ai+1,bi+1)12: else13: return to step 2. Query Point14: end if Compute the Generator analytical center of X i+1 . xi+1 , Ai+1, bi+1 xi+1using standard solvers of convex optimization. This is dueto the fact that the subproblem introduces difficulties, for (Optional)example, in defining the barrier function in path-following Atkinson and Vaidya N Termination?algorithms or providing the (sub-)gradient in primal-dual Modification [19] onmethods (see [14] for details of these algorithms). Fortunately, (Ai+1 , bi+1 ). Ywe can employ interior point cutting plane methods to solve ˜Problem Pslow (see [15] for a survey). Before we delve into N Has any optimalitythe details, let us briefly sketch the principles of the algorithm cut been generated?as follows. Y Suppose that we would like to find a point x that is feasible The problem is The problem is feasible.for (7)-(9) and is within a distance of δ > 0 to an optimal ˜ infeasible. The optimal solution x∗ = xi .solution x∗ of Pslow , where δ > 0 is an error toleranceparameter (i.e., x satisfies ||x − x∗ ||2 < δ). We maintain theinvariant that at the beginning of each iteration, the feasibleset is contained in some polytope (i.e., a bounded polyhedron). EndThen, we generate a query point inside the polytope and ask Fig. 2. ˜ Flow chart of the algorithm for solving Problem Pslow .a “separation oracle” whether the query point belongs to thefeasible set. If not, then the separation oracle will generatea so-called separating hyperplane through the query point tocut out the polytope, so that the remaining polytope contains A. The Cutting-Plane-Based Algorithmthe feasible set.7 Otherwise, the separation oracle will returna hyperplane through the query point to cut out the polytope 1) Query Point Generator: (Step 2 in Algorithm 1)towards the opposite direction of improving objective values. In each iteration, we need to generate a query point inside We can then proceed to the next iteration with the new the polytope X i . For algorithmic efficiency, we adopt the ana-polytope. To keep track of the progress, we can use the so- lytic center (AC) of the containing polytope as the query pointcalled potential value of the polytope. Roughly speaking, when [17]. The AC of the polytope X i = {x ∈ RN K : Ai x ≤ bi }the potential value becomes large, the polytope containing the at the ith iteration is the unique solution xi to the followingfeasible set has become small. Thus, if the potential value convex problem:exceeds a certain threshold, so that the polytope is negligibly Mismall, then we can terminate the algorithm. As will be shown max log si m (19) {xi ,si } m=1 7 Note that such a separating hyperplane exists due to the convexity of thefeasible set [16]. s.t. s = bi − Ai xi . i
  6. 6. 6 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. X, NO. X, XXX 2010We define the optimal value of the above problem as the called feasibility cut is generated at xi as follows:potential value of the polytope X i . Note that the uniqueness T ui,¯ κof the analytic center is guaranteed by the strong convexity (x − xi ) ≤ 0, κ ¯ ∀¯ ∈ K, (23) Miof the potential function si → − m=1 log si , assuming that ||u κ i,¯ || mX i is bounded and has a non-empty interior. The AC of a where || · || is the Euclidean norm, K = {k : ¯polytope can be viewed as an approximation to the geometric Hk (xi , t∗ ) > 0, k = 1, 2, · · · , K} is the set ofcenter of the polytope, and thus any hyperplane through the users whose chance constraints are violated, and ui,¯ = κ i,¯ κ i,¯ κ i,¯ κ i,¯ T κ NKAC will separate the polytope into two parts with roughly the [u1,k , · · · , uN,1 , . . . , u1,K , · · · , uN,K ] ∈ R is the gradi-same volume. ent of Gκ (x) with respect to x, i.e., ¯ Although it is computationally involved to directly solve i,¯ κ ∂Hκ (x, ̺∗ ) ¯(19) in each iteration, it is shown in [18] that an approximate uk,n = ∂xk,n xk,n =xiAC is sufficient for our purposes, and that an approximate AC k,nfor the (i + 1)st iteration can be obtained from an approximate W xi k,n − ̺∗ ln 2AC for the ith iteration by applying O(1) Newton steps. W − ln 2 ∞ 1+ ΓNξ0 pt ln 1+ ΓNξ0 pt 1 exp − σξκ dξ 0 σκ¯ ¯ 2) The Separation Oracle: (Steps 3-8 in Algorithm 1) = . W xi The oracle is a major component of the algorithm that plays − k,n ̺∗ ln 2 ∞ pt ξtwo roles: checking the feasibility of the query point, and 0 1+ ΓN0 1 σk exp − σξκ dξ ¯generating cutting planes to cut the current set. • Feasibility Check The reason we call (23) a feasibility cut(s) is that any x which ˜ does not satisfy (23) must be infeasible and can hence be We write the constraints of Pslow in a condensed form as dropped.follows: If the point xi is feasible, then an optimality cut is generated Gk (x) = inf {Hk (x, ̺)} ≤ 0, ∀k (20) as follows: T ̺>0 v A0 x ≤ b 0 (21) (x − xi ) ≤ 0, (24) ||v||where where v = (t) (t) − E{r1,1 }, · · · , −E{rN,1 }, . . . , −E{r1,K }, · · · , (t) IN IN · · · IN (t) −E{rN,K } T ∈ RN K is the derivative of the objective of A0 = ∈ R(N +N K)×N K , −IN K ˜ Pslow in (6) with respect to x. The reason we call (24) an b0 = [eT , 0N K ]T ∈ RN +N K N T optimality cut is that any optimal solution x∗ must satisfy (24)with IN and eN denoting the N × N identity matrix and N - and hence any x which does not satisfy (24) can be dropped.vector of ones respectively, and (21) is the combination8 of (8) Once a cutting plane is generated according to (23) or (24),and (9). Now, we first use (21) to construct a relaxed feasible we use it to update the polytope X i at the ith iteration asset via follows X 0 = {x ∈ RN K : A0 x ≤ b0 }. (22) X i = {x ∈ RN K : Ai x ≤ bi }.Given a query point x ∈ X 0 , we can verify its feasibility to Here, Ai and bi are obtained by adding the cutting plane to the ˜Pslow by checking if it satisfies (20), i.e., if inf ̺>0 {Hk (x, ̺)} previous polytope X i−1 . Specifically, if the oracle provides ais no larger than 0. This requires solving a minimization feasibility cut as in (23), thenproblem over ̺ > 0. Due to the unimodality of Hk (x, ̺) in ̺, Ai−1 i−1 ¯ Ai = ∈ R(M +|K|)×N K ,we can simply take a line search procedure, e.g., using Golden- (ui /||ui ||)T k ksection search or Fibonacci search, to find the minimizer bi−1 i−1 ¯̺∗ . The line search is more efficient when compared with bi = ∈ RM +|K| (ui /||ui ||)T xi k kderivative-based algorithms, since only function evaluations9are needed during the search. where Mi−1 is the number of rows in Ai−1 , and | · | is the number of elements contained in the given set; if the oracle •Cutting Plane Generation provides an optimality cut as in (24), then In each iteration, we generate a cutting plane, i.e., a hyper-plane through the query point, and add it as an additional Ai−1 (M i−1 +1)×N K Ai = T ∈ R ,constraint to the current polytope X i . By adding cutting (v/||v||)plane(s) in each iteration, the size of the polytope keeps shrink- bi−1 i−1 bi = ∈ RM +1 .ing. There are two types of cutting planes in the algorithm (v/||v||)T xidepending on the feasibility of the query point. If the query point xi ∈ X i is infeasible, then a hyperplane B. Global Convergence & Complexity (Step 10 in Algorithm 1) In the following, we investigate the convergence properties 8 To reduce numerical errors in computation, we suggest normalizing each of the proposed algorithm. As mentioned earlier, when theconstraint in (21). 9 The cumulant generating function Λ (·) in (10) can be evaluated numer- k polytope is too small to contain a full-dimensional closed ballically, e.g., using rectangular rule, trapezoid rule, or Simpson’s rule, etc. of radius δ > 0, the potential value will exceeds a certain
  7. 7. LI et al.: SLOW ADAPTIVE OFDMA SYSTEMS THROUGH CHANCE CONSTRAINED PROGRAMMING 7 (t)threshold. Then, the algorithm can terminate since the query When gk,n for different n are identically distributed, dif-point is within a distance of δ > 0 to some optimal solution of ferent subcarriers become indistinguishable to a user k. In ˜Pslow . Such an idea is formalized in [18], where it was shown this case, the optimal solution, if exists, does not depend onthat the analytic center-based cutting plane method can be used ˜ n. Replacing xk,n by xk in Pslow , we obtain the followingto solve convex programming problems in polynomial time. formulation:Upon following the proof in [18], we obtain the following K N ˜′ (t)result: Pslow : max xk E rk,n xk k=1 n=1Theorem 1. (cf. [18]) Let δ > 0 be the error toleranceparameter, and let m be the number of variables. Then, s.t. inf qk +̺N Λk (−̺−1 xk )−̺ log ǫk ≤ 0, ∀k ̺>0Algorithm 1 terminates with a solution x that is feasible for K˜Pslow and satisfies x − x∗ 2 < δ for some optimal solution xk ≤ 1, ˜x∗ to Pslow after at most O((m/δ)2 ) iterations. k=1 Thus, the proposed algorithm can solve Problem Pslow ˜ xk ≥ 0, ∀k. 2within O((N K/δ) ) iterations. It turns out that the algo- ˜′ Note that the problem structure of Pslow is exactly therithm can be made considerably more efficient by dropping same as that of P ˜slow , except that the problem size is re-constraints that are deemed “unimportant” in [19]. By incor- duced from N K variables to K variables. Hence, the al-porating such a strategy in Algorithm 1, the total number gorithm developed in Section IV can also be applied toof iterations needed by the algorithm can be reduced to ˜′ solve Pslow , with the following vector/matrix size reductions:O(N K log2 (1/δ)). We refer the readers to [15], [19] for A0 = [eN , −IK ]T ∈ R(1+K)×K , b0 = [1, 0, · · · , 0]T ∈details. R1+K in (21), ui,¯ = [u1 κ , · · · , uKκ ]T ∈ RK in (23), and κ i,¯ i,¯ (t) (t) T v = − E{r1 }, · · · , −E{rK } ∈ RK in (24). ComparedC. Complexity Comparison between Slow and Fast Adaptive with P ˜slow , the iteration complexity of P ′ is now reduced ˜ slow 2OFDMA to O(K log (1/δ)). Indeed, this can even be lower than the √ complexity of solving one Pfast — O( N KL0 ), since K It is interesting to compare the complexity of slow and ˜ is typically much smaller than N in real systems. Thus, thefast adaptive OFDMA schemes formulated in Pslow and Pfast , overall complexity of slow adaptive OFDMA is significantlyrespectively. To obtain an optimal solution to Pfast√we need , lower than that of fast adaptation over T time solve a linear program (LP). This requires O( N KL0 ) Before leaving this section, we emphasize that the problemiterations, where L0 is number of bits to store the data defining ˜′ size reduction in Pslow does not compromise the optimality ofthe LP [20]. At first glance, the iteration complexity of solving ˜ ˜ the solution. On the other hand, Pslow is more general in thea fast adaptation Pfast can be lower than that of solving Pslow sense that it can be applied to systems in which the frequencywhen the number of users or subcarriers are large. However, it ˜ bands of parallel subchannels are far apart, so that the channelshould be noted that only one Pslow needs to be solved for each distributions are not identical across different subchannels.adaptation window, while Pfast has to be solved for each timeslot. Since the length of adaptation window is equal to T timeslots, the overall complexity of the slow adaptive OFDMA VI. S IMULATION R ESULTScan be much lower than that of conventional fast adaptation In this section, we demonstrate the performance of ourschemes, especially when T is large. proposed slow adaptive OFDMA scheme through numerical Before leaving this section, we emphasize that the advantage simulations. We simulate an OFDMA system with 4 usersof slow adaptive OFDMA lies not only in computational cost and 64 subcarriers. Each user k has a requirement on itsreduction, but also in reducing control signaling overhead. We short-term data rate qk = 20bps. The 4 users are assumedwill investigate this in more detail in Section VI. to be uniformly distributed in a cell of radius R = 100m. That is, the distance dk between user k and the BS follows 2d the distribution10 f (d) = R2 . The path-loss exponent γ is V. P ROBLEM S IZE R EDUCTION equal to 4, and the shadowing effect sk follows a log-normal ˜ In this section, we show that the problem size of Pslow can be distribution, i.e., 10 log10 (sk ) ∼ N (0, 8dB). The small-scalereduced from N K variables to K variables under some mild channel fading is assumed to be Rayleigh distributed. Supposeassumptions. Consequently, the computational complexity of that the transmission power of the BS on each subcarrier isslow adaptive OFDMA can be markedly lower than that of 90dB measured at a reference point 1 meter away from thefast adaptive OFDMA. BS, which leads to an average received power of 10dB at the In practical multicarrier systems, the frequency intervals boundary of the cell11 . In addition, we set W = 1Hz and N0 =between any two subcarriers are much smaller than the carrier 1, and the capacity gap is Γ = − log(5BER)/1.5 = 5.0673,frequency. The reflection, refraction and diffusion of electro- 10 The 2dmagnetic waves behave the same across the subcarriers. This distribution of user’s distance from the BS f (d) = R2 is derived 1 (t) from the uniform distribution of user’s position f (x, y) = πR2 , where (x, y)implies that the channel gain gk,n is identically distributed is the Cartesian coordinate of the position.over n (subcarriers), although it is not needed in our algorithm 11 The average received power at the boundary is calculated by 90dB + 10 log10 100 −4derivations in the previous sections. 1 dB = 10dB due to the path-loss effect.
  8. 8. 8 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. X, NO. X, XXX 2010 120 40 100 35 80 30 Number of Iteration ∆¯ = ¯ i − ¯ i − 1 60 25 b 40 20 b b 20 15 0 10 −20 5 −40 0 0 5 10 15 20 25 30 10 20 30 40 50 60 Iteration Window NumberFig. 3. Trace of the difference of objective value ¯i between adjacent b Fig. 4. Number of iterations for convergence of all the feasible windowsiterations (ǫk = 0.2). (ǫk = 0.2).where the target BER is set to be 10−4 . Moreover, the length 30 feasibleof one slot, within which the channel gain remains unchanged, infeasibleis T0 = 1ms.12 The length of the adaptation window is chosen 25to be T = 1s, implying that each window contains 1000slots. Suppose that the path loss and shadowing do not change 20 Number of Iterationwithin a window, but varies independently from one window toanother. For each window, we solve the size-reduced problem 15 ˜′Pslow , and later Monte-Carlo simulation is conducted over 61independent windows that yield non-empty feasible sets of ˜′Pslow when ǫk = 0.1. 10 In Fig. 3 and Fig. 4, we investigate the fast convergenceof the proposed algorithm. The error tolerance parameter is 5chosen as δ = 10−2 . In Fig. 3, we record the trace of oneadaptation window13 and plot the improvement in the objective 0 10 20 30 40 50 60 70 80 90 100function value (i.e., system throughput) in each iteration, i.e., Window Number∆¯ = ¯i − ¯i−1 . When ∆¯ is positive, the objective value b b b bincreases with each iteration. It can be seen that ∆¯ quickly b Fig. 5. Number of iterations for feasibility check of all the windows (ǫk =converges to close to zero within only 27 iterations. We also 0.2).notice that fluctuation exists in ∆¯ within the first 11 iterations. bThis is mainly because during the search for an optimalsolution, it is possible for query points to become infeasible. iterations, where each iteration takes 1.467 seconds.14However, the feasibility cuts (23) then adopted will make sure Moreover, we plot the number of iterations needed forthat the query points in subsequent iterations will eventually ˜ checking the feasibility of Pslow . In Fig. 5, we conduct abecome feasible. The curve in Fig. 3 verifies the tendency. ˜ simulation over 100 windows, which consists of 61 feasibleAs Pslow is convex, this observation implies that the proposed ˜ windows (dots with cross) and 39 infeasible windows (dotsalgorithm can converge to an optimal solution of Pslow within ˜ with circle). On average, the algorithm can determine if Pslowa small number of iterations. In Fig. 4, we plot the number is feasible or not after 7 iterations. The quick feasibility checkof iterations needed for convergence for different application can help to deal with the admission of mobile users in thewindows. The result shows that the proposed algorithm can ˜ cell. Particularly, if there is a new user moving into the cell,in general converge to an optimal solution of Pslow within the BS can adopt the feasibility check to quickly determine35 iterations. On average, the algorithm converges after 22 if the radio resources can accommodate the new user without 12 The coherence time is given by T = 9c , where c is the speed of sacrificing the current users’ QoS requirements. 0 16πfc vlight, fc is the carrier frequency, and v is the velocity of mobile user. As an In Fig. 6, we compare the spectral efficiency of slow adap-example, we choose fc = 2.5GHz, and if the user is moving at 45 miles perhour, the coherence time is around 1ms. 14 We conduct a simulation on Matlab 7.0.1, where the system configu- 13 The simulation results show that all the feasible windows appear with rations are given as: Processor: Intel(R) Core(TM)2 CPU P8400@2.26GHzsimilar convergence behavior. 2.27GHz, Memory: 2.00GB, System Type: 32-bit Operating System.
  9. 9. LI et al.: SLOW ADAPTIVE OFDMA SYSTEMS THROUGH CHANCE CONSTRAINED PROGRAMMING 9 18 outage probability of user 1 0.1 fast adaptation 16 0.05 slow adaptation (ǫ k = 0. 1) 0 Spectral Efficiency (bps/Hz/subcarrier) 14 10 20 30 40 50 60 outage probability of user 2 0.1 12 0.05 10 0 10 20 30 40 50 60 outage probability of user 3 8 0.1 6 0.05 0 4 10 20 30 40 50 60 outage probability of user 4 0.1 2 0.05 0 0 10 20 30 40 50 60 10 20 30 40 50 60 Window Number ǫ k = 0. 1 ǫ k = 0. 3Fig. 6. Comparison of system spectral efficiency between fast adaptive Fig. 7. Outage probability of the 4 users over 61 independent feasibleOFDMA and slow adaptive OFDMA. windows.tive OFDMA with that of fast adaptive OFDMA15 , where zero 5.5outage of short-term data rate requirement is ensured for each 5.4user. In addition, we take into account the control overheads 5.3 Spectral Efficiency (bps/Hz/subcarrier)for subcarrier allocation, which will considerably affect thesystem throughput as well. Here, we assume that the control 5.2signaling overhead consumes a bandwidth equivalent to 10% 5.1of a slot length T0 every time SCA is updated [21]. Note that 5within each window that contains 1000 slots, the control sig-naling has to be transmitted 1000 times in the fast adaptation 4.9scheme, but once in the slow adaptation scheme. In Fig. 6, 4.8the line with circles represents the performance of the fast 4.7adaptive OFDMA scheme, while that with dots correspondsto the slow adaptive OFDMA. The figure shows that although 4.6slow adaptive OFDMA updates subcarrier allocation 1000 4.5times less frequently than fast adaptive OFDMA, it can achieve 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ǫkon average 71.88% of the spectral efficiency. Considering thesubstantially lower computational complexity and signaling Fig. 8. Spectral efficiency versus tolerance parameter ǫk . Calculated fromoverhead, slow adaptive OFDMA holds significant promise the average overall system throughput on one window, where the long-termfor deployment in real-world systems. average channel gain σk of the 4 users are −65.11dB, −56.28dB, −68.14dB ˜ As mentioned earlier, Pslow is more conservative than the and −81.96dB, respectively.original problem Pslow , implying that the outage probability isguaranteed to be satisfied if subcarriers are allocated according ˜to the optimal solution of Pslow . This is illustrated in Fig. 7, ˜ Pslow is enlarged. A question that immediately arises is howwhich shows that the outage probability is always lower than to choose the right ǫk , so that the actual outage probabilitythe desired threshold ǫk = 0.1. ˜ stays right below the desired value. Towards that end, we can Fig. 7 shows that the subcarrier allocation via Pslow could perform a binary search on ǫk to find the best parameter thatstill be quite conservative, as the actual outage probability is satisfies the requirement. Such a search, however, inevitablymuch lower than ǫk . One way to tackle the problem is to set involves high computational costs. On the other hand, Fig. 8ǫk to be larger than the actual desired value. For example, we shows that the gain in spectral efficiency by increasing ǫk iscould tune ǫk from 0.1 to 0.3. By doing so, one can potentially marginal. The gain is as little as 0.5 bps/Hz/subcarrier whenincrease the system spectral efficiency, as the feasible set of ǫk is increased drastically from 0.05 to 0.7. Hence, in practice, 15 For illustrative purpose, we have only considered P we can simply set ǫk to the desired outage probability value fast as one of thetypical formulations of fast adaptive OFDMA in our comparisons. However, to guarantee the QoS requirement of users.we should point out that there are some work on fast adaptive OFDMA which In the development of the STC (7), we considered that theimpose less restrictive constraints on user data rate requirement. For example,in [5], it considered average user data rate constraints which exploits time channel gain gk,n are independent for different n’s and k’s.diversity to achieve higher spectral efficiency. While it is true that channel fading is independent across