Near-Optimal Power Allocation and Multiuser ... - IEEE Xplore
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Near-Optimal Power Allocation and Multiuser Scheduling with Outage Capacity Constraints Exploiting Only Channel Statistics Kai-Kit Wong, Member, IEEE, and Jia Chen
Abstract— In this letter, we consider a delay-constrained wireless communication system where a delay constraint is characterized by the information outage probability in a transmission of K-block Rayleigh flat-fading channels. With multiple users, each given an individual delay requirement, this letter aims to minimize the total transmitted power by optimizing jointly the power distribution and the time sharing of a time-division (TD) multiuser system where the transmitter knows only the channel statistics (CST) but the mobile receivers (or users) have perfect knowledge of the channels. A suboptimal solution which utilizes Gaussian approximation with the mean and variance derived on the instantaneous mutual information and convex optimization, is proposed. Numerical results show that the proposed scheme performs nearly the same as the global optimum. Index Terms— Outage, power allocation, scheduling.
I. I NTRODUCTION
D
UE to the instability nature of wireless channels, it has long been the challenge of communicating reliably and efficiently (in terms of both power and bandwidth) over air [1]. In the past, most efforts focused on at what rate a particular wireless channel can support. In particular, in an additive white Gaussian noise (AWGN) channel, practical coding techniques with finite (but long) code length are available to approach the Shannon capacity within a fraction of decibel [2], [3]. Later in [4], Goldsmith et al. derived the ergodic capacity of a fading channel and showed that ergodic capacity can be achieved without the channel state information at the transmitter (CSIT) if we permit to have a very long codeword. The result of this sort is important to optimize the system performance if the aim is to maximize the rate over wireless channels. However, for delay-sensitive applications, the rate is usually preset and the preferred aim would be to minimize the transmission cost for a given outage probability constraint. For delay-sensitive communications, it is required to ascertain that the information arrives at the receiver before it expires. To model this, it is customary to consider a block-fading channel in which the fade is assumed to occur identically and independently from one block to another but remains static (or time-invariant) within a block1 of symbols [5]. In light of this, a delay constraint can be considered in terms of whether
Manuscript received September 27, 2006; revised February 13, 2007, April 27, 2007, and June 15, 2007; accepted July 4, 2007. The associate editor coordinating the review of this paper and approving it for publication was R. Fantacci. This work was supported by the Engineering and Physical Science Research Council (EPSRC) under grant EP/D053129/1. The authors are with the Adastral Park Research Postgraduate Campus, University College London, Martlesham Heath, Suffolk, IP5 3RE, United Kingdom (e-mail: {k.wong, j.chen}@adastral.ucl.ac.uk). Digital Object Identifier 10.1109/TWC.2008.060762. 1 A packet of information data for communications may be regarded as a block. In the context of this letter, we are interested in the case that the information is sent in K packets.
a target rate is obtained within a finite number of blocks, K [6], [7]. A delay constraint can be described as the probability of the outage event. Most related works can be found in [8]–[11]. In [8], Negi et al. considered exploiting the causal CSIT to optimize the power allocation over the blocks for minimizing the outage probability using a dynamic programming approach. Similar methodology was also proposed in [9] for a two-user downlink channel for expected capacity maximization with a short-term power constraint given the causal CSIT. Furthermore, in [10], Berry and Gallager looked into the delay-constrained problem taking into account of the size of buffer. More recently in [11], an algorithm that adapts the power allocation over the blocks to minimize the average transmit power while constraining an upper bound of the outage probability constraint was proposed. Unfortunately, the assumption of having causal CSIT may be unrealistic, and the required amount of channel feedback may not justify the advantage gained from the power control. The scope of this letter is fundamentally different from the previous works. First, the transmitter knows only the channel statistics (CST) though the receivers have perfect channel state information (CSIR). Secondly, a time-division (TD) multiuser downlink system is also considered. In this setting, each user is given an individual information outage probability constraint and only one user is allowed to access the channel for each block. Our objective is to optimize the power allocation among the users and to schedule the users smartly so that the overall transmit power is minimized while achieving the probabilistic delay constraints of the users. Having the assumption that all users are subjected to a delay tolerance of K blocks (or time slots),2 the exact order on how the users are scheduled within the blocks is irrelevant. As such, our aim boils down to finding the optimal power allocation and the optimal time-sharing (i.e., the number of blocks assigned) among the users. II. P ROBLEM F ORMULATION AND A SSUMPTIONS A. Single-User Channel Model We assume a block flat-fading noisy channel as in [8], [11]. Every KT0 data symbols are encoded as a single codeword and transmitted as a sequence of K blocks. The channel is assumed to fade identically and independently from one block to another, but the fade can be considered static within a block of T0 (which is assumed to be large enough to average out the effect of noise) symbols. We shall use ck to denote the channel power gain in block k and assume that the channel√ amplitude, ck , is in Rayleigh fading so that ck is exponential 2 The result of this letter can be easily extended to the case where users have different K. However, this assumption greatly simplifies the presentation of this letter.
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distributed and has the following probability density function (pdf) [1] F (ck ) =
c
k − 1 C0 d−γ C0 d−γ e
ck ≥ 0, ck < 0,
0
(1)
where E[ck ] = C0 d−γ ∀k in which d denotes the distance between the transmitter and the receiver, γ is the power loss exponent, and C0 is the distance-independent mean channel power gain. The assumption of this letter is that the transmitter knows only (1) (i.e., CST) but the receiver knows {cn }n≤k at time k (i.e., CSIR). For a given block, say k, the Gaussian codebook is used with an assigned power P , and the rate allows the following expression rk = log2
P ck 1+ N0
in bps/Hz
(2)
where K N0 denotes the noise power. An outage is said to occur if k=1 rk ≤ KR0 R for some target rate R. In a TD multiuser environment, the model will be exactly the same except that each block is given to only one of the users by some scheduling mechanism. More details will be given in Section II-C.
B. Single-User Problem: Power Allocation With only one user, the problem is to find the minimum required power (Q = KP ) [as the blocks are independent and identically distributed (i.i.d.), the optimum can always be attained with an equal-power policy] that assures an acceptable level of outage probability for a given R: min Q s.t. P Q≥0
K
rk ≤ R
≤ ε,
(3)
813
The problem of interest is, as a consequence, given by ⎧ N ⎪ ⎪ ⎪ min Qu ⎪ ⎪ {Qu },{nu } ⎪ ⎪ u=1 ⎪ ⎪ n ⎪ u ⎪ ⎪ ⎪ (u) ⎪ s.t. P rk ≤ Ru ≤ εu ∀u, ⎪ ⎨ k=1 M → ⎪ Qu ≥ 0 ∀u, ⎪ ⎪ ⎪ ⎪ ⎪ N ⎪ ⎪ ⎪ ⎪ nu ≤ K, ⎪ ⎪ ⎪ ⎪ u=1 ⎪ ⎩ nu ∈ {1, 2, . . . , K − N + 1} ∀u,
(4)
where Qu nu
(u)
rk N
the total power allocated to user u; the number of blocks allocated to user u;
(5) (6)
the rate achieved at the kth block of user u; (7) the total number of users; (8)
K Ru
the number of blocks; the target rate for user u;
(9) (10)
εu
the probability requirement for user u.
(11) (u)
Likewise, users have the mean channel power gains {C0 = C0 d−γ u }. The challenge of M is that it is a mixed integer problem which has no known method achieving the global optimum [14]. The remainder of the letter will be devoted to finding the suboptimal solution of (4). In particular, Section III will look into obtaining the optimal {Qu } for a given {nu }. Section IV will focus on finding the suboptimal time-sharing parameters {nu } using relaxation followed by convex optimization. Section V combines the two approaches to suboptimally solve (4). Numerical results in Section VI will, however, show that the proposed suboptimal method performs nearly the same as the global optimum with inappreciable difference.
k=1
where P(A) denotes the probability of an event A, and ε is the maximum allowable outage probability. If (3) is solved, then the optimal power allocation for each block is simply Popt = Qopt K . Solving the above problem is in general challenging because getting the pdf of the sum-rate ω = r1 +r2 +· · · +rK is difficult. This will be dealt with in Section III.
C. Multiuser Problem: Power Allocation and Scheduling With multiple users in TD, only one user is permitted to gain access to the channel for each block. As a consequence, optimization needs to be performed for both power allocation and scheduling. By scheduling, we mean that the transmitter requires to decide which user occupies the channel at any given time (or block). However, since the delay constraint specifies only that the rate is achieved in K blocks with some probability, the order of which the users are scheduled within the K blocks is unimportant, and scheduling boils down to sorting out the number of blocks being allocated to each user.
III. M INIMUM P OWER E QUATION Here, we shall derive an equation to determine the minimum power allocation for attaining a given outage probability if the number of blocks is fixed. For a TD system, each block is occupied by one user only and if {nu } are fixed, then the optimization for the users is completely uncoupled and will be equivalent to multiple individual users’ power minimizations. Therefore, it suffices to focus on a single-user system for a given number of blocks, n, i.e., (3) with K = n, or min Q s.t. P ({ω ≤ R}) ≤ ε. Q≥0
(12)
To solve the above problem, we argue that from the central limit theorem (CLT),3 if n is large enough, ω has normal 3 The validity of Gaussian approximation will be checked in Section VI. Results in Figure 1 will show that for a wide range of outage probabilities (e.g., ε ≥ 10−2 ), the approximation is good even if n is as small as 3. For more stringent outage probability requirements such as ε < 10−2 , the approximation tends to provide an upper bound in the constraint, so the constraint is guaranteed at the expense of a slight power penalty.
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distribution with the parameters μ = E[ω] and σ 2 = VAR[ω]. The constraint of (12) can then be simplified to 1 1 R − μ(Q) √ + erf ≤ε (13) 2 2 2σ(Q) which leads to g(Q) μ(Q) − where
√ 2erf −1 (1 − 2ε) σ(Q) − R ≥ 0
2 erf(x) = √ π
x
2
e−t dt.
(14) (15)
0
In Appendix A, we have derived the mean and variance of the sum-rate, ω, as ⎧ 1 ne a 1 ⎪ ⎪ ⎪ μ = E , 1 ⎪ ⎪ ln 2 a ⎪ ⎪ ⎪ 2 1 ⎨ 2 a e π 1 (16) σ2 = n a ln − γEM + ⎪ a 6 a ln2 2 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎩ − 23 F3 [1, 1, 1]; [2, 2, 2]; − − μ2 , a −γ
0d , E1 (·) denotes the exponential integral, where a QCnN 0 p Fq denotes the generalized hypergeometric function, and γEM is the Euler-Mascheroni constant [12]. Intuitively, g should be a strictly increasing function of Q because more transmit power leads to less chance of being in an outage. Accordingly, the minimum value of Q occurs when the equality of (14) holds, or the constraint becomes active, i.e., g(Qmin ) = 0. Throughout this letter, we shall refer to this equation as the minimum power equation (MPE). To find the solution of MPE efficiently, one possible method is to use the DIviding RECTangle (DIRECT) algorithm [13], which samples points in the domain, and uses the information it has obtained to decide where to search next. To apply DIRECT, we construct the problem as follows: (17) S → Qmin = arg min |g(Q)|.
0≤Q≤UQ
The upper bound UQ can be set to be a large enough number that makes sure Qopt ∈ (0, UQ ). We shall omit the details of DIRECT, but refer the interested readers to [13]. IV. O PTIMIZED T IME -S HARING BY C ONVEXIZATION In this section, we aim to optimize the time-sharing parameters of the users {nu } by joint consideration of the power consumption and the probability constraints of the users. Ideally, it requires to solve M, i.e., (4), which is unfortunately not known. Here, we propose to mimic M by considering a simpler problem with the probability constraints replaced by some upper bounds (as in [11]) n u (u) rk ≤ Ru P n u
k=1
≤P
k=1
ck ≤
nu N0 Qu
nu
2Ru
UB [u]. Pout
(18)
The original probability constraint in (4) can thus be ascertained by constraining the upper bound of the outage
UB [u] ≤ εu }. The advantage by doing so probability {Pout is substantial because first, the optimizing variable Qu can be separated from the random variable, and secondly, the distribution of = k ck can be approximated by CLT as log-normal with ⎧ (u) ⎪ ⎨ E[ln ] = nu ln C0 − γEM , (19) 2 ⎪ ⎩ VAR[ln ] = nu π . 6 UB The constraint {Pout [u] ≤ εu } can therefore be simplified to 1 nu 2Ru nu N0 eγEM · (20) Qu ≥ √n1 ςu . (u) u − √π erf −1 (1−2εu ) C0 e 3
In other words, the single-user problem (3) can be solved suboptimally through minQ≥ς Q, which suggests the optimal power allocation be simply Qsubopt = ς. Using the upper bound constraints in the multiuser problem (4), we then have
˜ → M
N
min
{Qu },{nu }
Qu
u=1
N0 eγEM
s.t. Qu ≥
(u)
C0
·
nu 2Ru
1 nu
π erf −1 (1−2ε ) −√ u
e
3
√1 nu
∀u,
N
nu ≤ K,
u=1
nu ∈ {1, 2, . . . , K − N + 1} ∀u,
(21)
where the constraints are now written in closed-form. There is no doubt that the power allocation from the modified problem (21) is very conservative, i.e., Qsubopt Qopt because of the upper bound constraint. However, our conjecture is that the problem structure of M on {nu } would be accurately ˜ Accordingly, we may be able to obtain nearimitated by M. ˜ although accurate optimal solution for {nu } by solving M ˜ power consumption cannot be estimated from M. ˜ is that the optimization A remaining difficulty of solving M is mixed with combinatorial search over the space of {nu } because they are integer-valued [14]. To tackle this, we relax ˜ becomes {nu } to positive real numbers {x2u } so that M ⎧ 1 N ⎪ x2u (au ) x2u ⎪ γEM ⎪ min N0 e ⎪ 1 ⎪ (u) ⎪ {x } x ⎪ u=1 C0 (bu ) u ⎨ u N ˜ r → (22) M ⎪ ⎪ s.t. x2u ≤ K, ⎪ ⎪ ⎪ u=1 ⎪ ⎪ √ ⎩ 1 ≤ xu ≤ K − N + 1, − √π erf −1 (1−2ε )
u where au 2Ru and bu e 3 . Apparently, both constraints in (22) are convex, and if the cost is also convex, the problem can be solved using known convex programming routines [14]. Now, let us turn our attention on a function of the form 1
f (x) = x2 ·
a x2 1
bx
≡ x2 h(x), for a, b, x > 0,
(23)
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1
where h(x) a x2 /b x . Our interest is to examine if f (x) is convex, or equivalently whether f (x) > 0. To show this, we first obtain ⎧ −2 ln a ln b ⎪ ⎪ h (x) = h(x) + , ⎪ ⎨ x3 x2 2 ⎪ −2 ln a ln b 6 ln a 2 ln b ⎪ ⎪ + 2 + h(x) − 3 . ⎩ h (x) = h(x) x3 x x4 x (24) Applying these results, f (x) can be found as 2 −2 ln a ln b −2 ln a 2 ln b f (x) = + + + + 2. h(x) x2 x x2 x (25) Letting α = 2 xln2 a and β = − xln b , we have
f (x) = (α + β)2 − α − 2β + 2 h(x) 2 3 1 + (β − 1)2 + + 2αβ > 0 = α− 2 4
(26)
since α, β > 0, which can be seen from the definition of (a, b) that α > 0 and β > 0 for εu < 0.5.4 Together with the fact that h(x) > 0 for all x > 0, we can conclude that f (x) > 0 and therefore, f (x) is convex. As the cost function in (22) is a summation of the functions of the form f (x), it is convex ˜ r. and hence the problem (22), or M ˜ r being convex, we can find the globally optimal With M {xu }opt at polynomial-time complexity. In particular, the complexity grows like O(N 3 ), which is scalable with the number of users. The remaining task, however, is to derive the integer-valued {nu } from {xu }. Simply setting nu = x2u would result in non-integer solutions while rounding them off could lead to violation of the outage probability constraints. In this letter, a greedy-approach is presented to obtain a feasible solution of {nu } from {xu }, which will be described in the next section. V. T HE P ROPOSED S CHEME Thus far, we have proposed two approaches: one that determines the optimal power allocation {Qu } based on MPE (see Section III); another one that finds the suboptimal (relaxed) time-sharing parameters {nu } by constraining the upper bound probability (see Section IV). In this section, we shall present an algorithm that combines the two approaches to jointly optimize the power allocation and time-sharing of the users. Our idea is to first map the optimal solution {xu }opt from ˜ r to a proper {nu } in M by rounding the optimal results M to the nearest integers, and then to step-by-step allocate one more block to the user who can minimize the overall required power using MPE. The proposed algorithm is described as follows: ˜ r (22) using convex optimization 1) Solve {xu } in M routines such as interior-point method. 2) Initialize nu = x2u ∀u where y returns the greatest integer that is smaller than y. Note that at this point, 4 It should be emphasized that the convexity of f is subjected to the condition that εu < 0.5. However, in practice, it would not make sense to have a system operating with outage probability greater than 50%.
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˜ r may not give a feasible solu{nu } and {Qu } from M tion to M, and some outage rate probability constraints may not be satisfied. 3) For each user u, compute the minimal required power to ensure the outage probability constraint by solving MPE (27) Qu = arg min |gu (Q|nu )| , Q≥0
where the function gu (Q|nu ) is defined similarly as in (14), and the notation (Q|nu ) emphasizes the fact that nu is given as a fixed constant. N 4) Then, initialize m = K − u=1 nu . 5) Compute the power reduction metrics
Qu = Qu − arg min |gu (Q|nu + 1)| Q≥0
6) Find u∗ = arg maxu Qu and update ⎧ ⎪ ⎨ nu∗ := nu∗ + 1, Qu∗ := Qu∗ − Qu∗ , ⎪ ⎩ m := m − 1.
∀u.
(28)
(29)
If m ≥ 1, go back to Step 5. Otherwise, go to Step 7. 7) Optimization is completed and the solutions for both {nu } and {Qu } have been found. VI. N UMERICAL R ESULTS A. Setup and Benchmarks Computer simulations are conducted to evaluate the performance of the proposed algorithm for the power-minimization problem with outage rate probability constraints. Only CST has been assumed, and capacity-achieving codec is used so that the expression log2 (1 + SNR) can be used to express the rateachieved for each block. The totaltransmit SNR, defined N (u) N 1 as N1 u=1 C0 Q u=1 u N0 , is considered as the performance metric. To compute the required SNR for a given set of simulation parameters such as the numbers of users and blocks, the users’ target rates and outage probabilities, the algorithm presented in Section V, which iteratively solves the MPE, is used. Note that the MPE itself has already taken into account the randomness of the channel for outage evaluation. Results for the proposed algorithm will be compared with the following benchmarks: • Global-Optimal Time and Power Allocation from Exhaustive Searching—With MPE, the global optimum can be found from solving M by an exhaustive search over the space of {nu } at the expense of complexity, i.e., ⎧ N ⎪ ⎪ ⎪ arg min |gu (Qu |nu )| ⎪ min ⎪ ⎪ {n } {Qu ≥0} ⎪ u=1 ⎨ u N M → (30) ⎪ ⎪ nu ≤ K, s.t. ⎪ ⎪ ⎪ u=1 ⎪ ⎪ ⎩ nu ∈ {1, 2, . . . , K − N + 1} ∀u. The inner optimization in (30) can be performed using DIRECT for a given {nu }. The globally optimal solution for {nu , Qu } can therefore be obtained from a complete search over the space of {nu }. However, note that if N or K is large, the required complexity of the search can be
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TABLE I T HE NUMBER OF COMBINATIONS NEEDED FOR THE OPTIMIZATION OF M
K
2
20 30 40 20 30 40 20 30 40 20 30 40
3
4
5
Number of combinations
K N
Cumulative Density Function
N
K AND N .
190 435 780 1140 4060 9880 4845 27405 91390 15504 142510 658008
enormous. For instance, if N = 4 and K = 30, there are totally 27405 combinations for {nu } to be tested. More numerical examples are shown in Table I, which gives some idea of the complexity requirements for solving M from an exhaustive optimization approach. • Equal-Time Sharing with Optimized Power Allocation— An interesting benchmark is the system where each user is allocated equal number of blocks so that nu = K N ∀u, but the power allocation for each user is obtained by solving MPE. This system shows how important time-sharing optimization is if the system has highly heterogeneous users and channels. ˜ r —We can obtain the • Relaxed Optimal Solution from M ˜ r if suboptimal time and power allocation directly from M relaxation is allowed. It is noted, however, that this is an unrealizable solution unless fractional time allocation is permitted. Additionally, despite the relaxation, this does not give a lower bound for M because the upper bounds of the outage probability constraints are considered, and this can show the importance to have MPE. The simulations are conducted using MATLAB and the ˜ r is solved by the function “fminrelaxed convex problem M con”in MATLAB. A Rayleigh block-fading channel is consid(u) ered with the mean channel power C0 for user u. Results in Figures 1–4 are given for various simulation parameters. Each figure presents results with different sets of parameters and will be individually described. B. Results Figure 1 shows the cumulative distribution function (cdf) of the actual sum-rate for various number of blocks n and target rates and this cdf is compared with that of Gaussian approximation. As we can see, for a wide range of outage probabilities (e.g., ε ≥ 10−2 ), the approximation is good even if n is as small as 3. For more stringent outage probability requirements such as ε < 10−2 , the approximation tends to provide an upper bound in the constraint, but the constraint is always guaranteed at the expense of a slight power penalty. Figure 2 shows the SNR results against outage probability requirements for a 3-user system with 20 blocks (i.e., N = 3 and K = 20). Users are assumed to have the same target
1 n=
2 n= 3 n=
-2
10
5 n=
7
n=
10
n=
-4
10
-6
10
Simulation Gaussian approximation -8
10
-10
10
0
5
10
15
20
25
30
Transmission Ra te R Transmission Rate R (bps/Hz) (bps/Hz)
Fig. 1. Comparison between the actual and approximated distributions for an n-block Rayleigh fading channel when Q = 50.
28 26 24 Transmit SNR in dB
FOR VARIOUS
0
10
22 20 18 16 14 12 −5 10
Proposed method Equal−time with optimized power Relaxed solution of (34) Global optimum 10
−4
−3
10 Outage probability εout
10
−2
10
−1
Fig. 2. Results of transmit SNR against the outage probability requirement (u) when N = 3, K = 20, Ru = 4 ∀u and C0 = 1 ∀u.
rate R = 4 bps/Hz, the same mean channel power gain C0 d−γ = 1, and the same outage rate probability requirements ε ranging from 10−1 to 10−5 . Results illustrate that as ε decreases, the required SNR increases, and in particular, about 2–2.5 dB more in SNR is needed for a reduction of ε by an order of magnitude. In addition, results also indicate that the relaxed solution of (22) is significantly inferior than the other methods with about 6 dB difference. However, the proposed method achieves the same SNR performance as the global optimum. In this system, obviously, the optimal solution should be n1 = 6, n2 = 7, n3 = 7 with the power optimized from MPEs. Therefore, the equal-time allocation method with n1 = n2 = n3 = 6 is nearly optimal with difference less than 0.2 dB. Overall, from the results in this figure, it has been shown that MPE is important for power minimization, and using the upper bound probability to optimize power is likely to have a considerable power penalty. The performance for various number of blocks K is inves-
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26
817
32 Proposed method Equal−time with optimized power Relaxed solution of (34) Global optimum
30
24
28 Transmit SNR in dB
Transmit SNR in dB
22 Proposed method Equal−time with optimized power Relaxed solution of (34) Global optimum
20
18
26
24
22 16
14 10
20
15
20
25 30 35 Number of blocks K
40
45
50
Fig. 3. Results of transmit SNR against the number of blocks when N = 3, (u) εu = 10−3 ∀u, Ru = 4 ∀u and C0 = 1 ∀u.
tigated by the results in Figure 3. A close observation from the results indicates that the relaxed solution will converge to an irreducible SNR at about 24 dB after K = 30 whereas the other methods utilizing the MPE can further minimize the power by increasing K. This can be explained by the fact that the relaxed solution is based on an upper bound probability which is getting looser and looser if K increases. Thus far, we have looked at scenarios in which users are homogeneous and that the global optimum would be more or less the same as the equal-time allocation. Now, we consider the settings which have highly non-homogeneous (or highlyskewed) users, and we are interested to see whether the proposed method and the equal-time allocation can still give near-optimal results. The configurations that we have tested are listed in Table II and results provided in Figure 4. As we can see, the performance gap between the equal-time allocation and the global optimum does get larger (about 2 dB for Configuration 4 where there are high and low demand users). On the other hand, results also demonstrate that the proposed method works very well even for extreme scenarios and achieves the same SNR results as the optimum except for Configuration 4 where a performance gap of less than 0.1 dB is observed. Some details regarding how different methods work are provided in Table II. Results also reveals that the proposed method always arrives at the global optimal solution except for the very extreme case. This shows that the modified problem (22) can effectively mimic the structure of the original problem (4) for time-sharing optimization, making possible the overall power minimization using MPE. By contrast, regardless of the users’ channel statistics and requirements, the equal-time allocation always assigns 6 blocks (out of a total of 20 blocks) for each of the three users. Nonetheless, the results indicate that the SNR penalty for not optimizing {nu } is less than what we would have expected. The reason turns out to be that though a high-demand poor-channel-condition user should get more blocks for power minimization, the power saving is not remarkable because assigning more blocks means not only to increase the mean rate (which helps minimize power) but also
18
Fig. 4.
1
2 3 Configuration listed in Table 2
4
Results of transmit SNR for the configurations listed in Table 2.
to increase the variance of the sum-rate (which on the other hand requires more power to attain a given outage probability).
VII. C ONCLUSION This letter has addressed the optimization problem of power allocation and scheduling for a multiuser TD system where the transmitter knows only the channel statistics of the users, and the users are given individual outage probability constraints. By Gaussian approximation, we have derived a so-called MPE to determine the minimum power for attaining a given outage rate probability constraint if the number of blocks for a user is fixed. On the other hand, we have also proposed a convex programming approach to obtain the suboptimal number of blocks allocated to the users. The two main techniques have later been combined to obtain a joint solution for both power and time allocation for the users. Results have demonstrated that the proposed method achieves near-optimal performance.
A PPENDIX A: D ERIVATION OF M EAN AND VARIANCE OF log2 (1 + ax) WHEN x IS A STANDARD EXPONENTIAL RANDOM VARIABLE
The mean of r log2 (1 + ax), can be obtained by E[r]
= E[log2 (1 + ax)] ∞ = log2 (1 + ax)e−x dx 0 ∞ 1 = ln(1 + ax)e−x dx ln 2 0 ∞ 1 t ea = ln te− a dt a ln 2 1 1 ∞ t ea e− a = dt ln 2 1 t 1 1 ea = E1 ln 2 a
(31) (32) (33) (34) (35) (36)
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TABLE II VARIOUS CONFIGURATIONS TESTED IN F IGURE 4 WITH K = 20 WHERE THE SUPERSCRIPTS HIGHLIGHT THE VARIABLES THAT ARE NOT THE SAME AS THE OPTIMUMS .
Configuration 1
2
3
4
u 1 2 3 1 2 3 1 2 3 1 2 3
Ru (bps/Hz) 4 4 4 4 6 8 4 6 8 1 4 8
εu 10−3 10−3 10−3 10−3 10−3 10−3 10−5 10−5 10−5 10−5 10−5 10−5
where E1 (·) is the exponential integral. To find the variance, we find it useful to first evaluate the following integral
∞
e− a (ln t)2 dt = t
1
0
∞
e− a (ln t)2 dt − t
1
e− a (ln t)2 dt t
0
(37) where the first integrand can be recognized as a variant of the Euler-Mascheroni integral [12], so 2 ∞ 1 π2 − at 2 (38) e (ln t) dt = a ln − γEM + a 6 0 where γEM is the Euler-Mascheroni constant, and the second integrand can be evaluated by 1 1 ∞ t n −a − at 2 (ln t)2 dt e (ln t) dt = (39) n! 0 0 n=0 ∞ 1 n 1 −a = tn (ln t)2 dt (40) n! 0 n=0 ∞ 1 n −a (−1)2 · 2! = · (41) n! (n + 1)3 n=0 ∞ 1 n −a 1 · = 2 (42) n! (n + 1)3 n=0 1 = 23 F3 [1, 1, 1]; [2, 2, 2]; − (43) a in which p Fq denotes the generalized hypergeometric function. As a result, we have
∞
− at
e 1
2 2 1 π (ln t)2 dt = a ln − γEM + a 6 1 − 23 F3 [1, 1, 1]; [2, 2, 2]; − . a
(44)
(u)
C0 0.5 1 1.5 1.5 1 0.5 1.5 1 0.5 5 3 0.5
(nu )opt 9 6 5 4 6 10 4 6 10 2 4 14
(nu )proposed 9 6 5 4 6 10 4 6 10 3 5 12
(nu )equal−time 6 6 6 6 6 6 6 6 6 6 6 6
Finally, the variance of the rate can be obtained as 2 1 π2 1 ea a ln − γEM + VAR[r] = a 6 a ln2 2 1 −23 F3 [1, 1, 1]; [2, 2, 2]; − − (E[r])2 . a R EFERENCES
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[1] T. S. Rappaport, Wireless Communications: Principles and Practice. Prentice Hall, 1996. [2] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon limit error-correcting coding and decoding: turbo-codes,” in Proc. Int. Conf. Commun., pp. 1064–1070, May 1993. [3] J. Hagenauer, E. Offer, and L. Papke, “Iterative decoding of binary block and convolutional codes,” IEEE Trans. Inf. Theory, vol. 42, no. 2, pp. 429–445, Mar. 1996. [4] A. J. Goldsmith and P. P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Trans. Inf. Theory, vol. 43, pp. 1986– 1992, Nov. 1997. [5] R. J. McEliece and W. E. Stark, “Channels with block interference,” IEEE Trans. Inf. Theory, vol. 30, pp. 44–53, Jan. 1984. [6] G. Caire, G. Taricco, and E. Biglieri, “Optimum power control over fading channels,” IEEE Trans. Inf. Theory, vol. 45, pp. 1468–1489, July 1999. [7] S. Hanly and D. N. C. Tse, “Multi-access fading channels: delay-limited capacities,” in Proc. Int. Sym. Info. Theory, Cambridge, MA, p. 397, Aug. 1998. [8] R. Negi and J. M. Cioffi, “Delay-constrained capacity with causal feedback,” IEEE Trans. Inf. Theory, vol. 48, no. 9, pp. 2478–2494, Sept. 2002. [9] X. Liu and A. J. Goldsmith, “Optimal power allocation over fading channels with stringent delay constraints,” in Proc. Int. Conf. Commun., pp. 1413–1418, New York, 2002. [10] R. Berry and R. G. Gallager, “Communication over fading channels with delay constraints,” IEEE Trans. Inf. Theory, vol. 48. no. 5, pp. 1135–1149, May 2002. [11] K. K. Wong, “Optimizing the power allocation for Rayleigh blockfading channels with outage capacity constraints,” to appear in IEEE Trans. Wireless Commun., vol. 6, no. 9, pp. 3163–3169, Sept. 2007 [12] J. H. Conway and R. K. Guy, “The Euler-Mascheroni number,” in The Book of Numbers. New York: Springer-Verlag, pp. 260–261, 1996. [13] C. D. Perttunen, D. R. Jones, and B. E. Stuckman, “Lipschitzian optimization without the Lipschitz constant,” J. Optim. Theory and App., vol. 79, no. 1, pp. 157–181, Oct. 1993. [14] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004.
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Near-Optimal Power Allocation and Multiuser Scheduling with Outage Capacity Constraints Exploiting Only Channel Statistics Kai-Kit Wong, Member, IEEE, and Jia Chen
Abstract— In this letter, we consider a delay-constrained wireless communication system where a delay constraint is characterized by the information outage probability in a transmission of K-block Rayleigh flat-fading channels. With multiple users, each given an individual delay requirement, this letter aims to minimize the total transmitted power by optimizing jointly the power distribution and the time sharing of a time-division (TD) multiuser system where the transmitter knows only the channel statistics (CST) but the mobile receivers (or users) have perfect knowledge of the channels. A suboptimal solution which utilizes Gaussian approximation with the mean and variance derived on the instantaneous mutual information and convex optimization, is proposed. Numerical results show that the proposed scheme performs nearly the same as the global optimum. Index Terms— Outage, power allocation, scheduling.
I. I NTRODUCTION
D
UE to the instability nature of wireless channels, it has long been the challenge of communicating reliably and efficiently (in terms of both power and bandwidth) over air [1]. In the past, most efforts focused on at what rate a particular wireless channel can support. In particular, in an additive white Gaussian noise (AWGN) channel, practical coding techniques with finite (but long) code length are available to approach the Shannon capacity within a fraction of decibel [2], [3]. Later in [4], Goldsmith et al. derived the ergodic capacity of a fading channel and showed that ergodic capacity can be achieved without the channel state information at the transmitter (CSIT) if we permit to have a very long codeword. The result of this sort is important to optimize the system performance if the aim is to maximize the rate over wireless channels. However, for delay-sensitive applications, the rate is usually preset and the preferred aim would be to minimize the transmission cost for a given outage probability constraint. For delay-sensitive communications, it is required to ascertain that the information arrives at the receiver before it expires. To model this, it is customary to consider a block-fading channel in which the fade is assumed to occur identically and independently from one block to another but remains static (or time-invariant) within a block1 of symbols [5]. In light of this, a delay constraint can be considered in terms of whether
Manuscript received September 27, 2006; revised February 13, 2007, April 27, 2007, and June 15, 2007; accepted July 4, 2007. The associate editor coordinating the review of this paper and approving it for publication was R. Fantacci. This work was supported by the Engineering and Physical Science Research Council (EPSRC) under grant EP/D053129/1. The authors are with the Adastral Park Research Postgraduate Campus, University College London, Martlesham Heath, Suffolk, IP5 3RE, United Kingdom (e-mail: {k.wong, j.chen}@adastral.ucl.ac.uk). Digital Object Identifier 10.1109/TWC.2008.060762. 1 A packet of information data for communications may be regarded as a block. In the context of this letter, we are interested in the case that the information is sent in K packets.
a target rate is obtained within a finite number of blocks, K [6], [7]. A delay constraint can be described as the probability of the outage event. Most related works can be found in [8]–[11]. In [8], Negi et al. considered exploiting the causal CSIT to optimize the power allocation over the blocks for minimizing the outage probability using a dynamic programming approach. Similar methodology was also proposed in [9] for a two-user downlink channel for expected capacity maximization with a short-term power constraint given the causal CSIT. Furthermore, in [10], Berry and Gallager looked into the delay-constrained problem taking into account of the size of buffer. More recently in [11], an algorithm that adapts the power allocation over the blocks to minimize the average transmit power while constraining an upper bound of the outage probability constraint was proposed. Unfortunately, the assumption of having causal CSIT may be unrealistic, and the required amount of channel feedback may not justify the advantage gained from the power control. The scope of this letter is fundamentally different from the previous works. First, the transmitter knows only the channel statistics (CST) though the receivers have perfect channel state information (CSIR). Secondly, a time-division (TD) multiuser downlink system is also considered. In this setting, each user is given an individual information outage probability constraint and only one user is allowed to access the channel for each block. Our objective is to optimize the power allocation among the users and to schedule the users smartly so that the overall transmit power is minimized while achieving the probabilistic delay constraints of the users. Having the assumption that all users are subjected to a delay tolerance of K blocks (or time slots),2 the exact order on how the users are scheduled within the blocks is irrelevant. As such, our aim boils down to finding the optimal power allocation and the optimal time-sharing (i.e., the number of blocks assigned) among the users. II. P ROBLEM F ORMULATION AND A SSUMPTIONS A. Single-User Channel Model We assume a block flat-fading noisy channel as in [8], [11]. Every KT0 data symbols are encoded as a single codeword and transmitted as a sequence of K blocks. The channel is assumed to fade identically and independently from one block to another, but the fade can be considered static within a block of T0 (which is assumed to be large enough to average out the effect of noise) symbols. We shall use ck to denote the channel power gain in block k and assume that the channel√ amplitude, ck , is in Rayleigh fading so that ck is exponential 2 The result of this letter can be easily extended to the case where users have different K. However, this assumption greatly simplifies the presentation of this letter.
c 2008 IEEE 1536-1276/08$25.00
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distributed and has the following probability density function (pdf) [1] F (ck ) =
c
k − 1 C0 d−γ C0 d−γ e
ck ≥ 0, ck < 0,
0
(1)
where E[ck ] = C0 d−γ ∀k in which d denotes the distance between the transmitter and the receiver, γ is the power loss exponent, and C0 is the distance-independent mean channel power gain. The assumption of this letter is that the transmitter knows only (1) (i.e., CST) but the receiver knows {cn }n≤k at time k (i.e., CSIR). For a given block, say k, the Gaussian codebook is used with an assigned power P , and the rate allows the following expression rk = log2
P ck 1+ N0
in bps/Hz
(2)
where K N0 denotes the noise power. An outage is said to occur if k=1 rk ≤ KR0 R for some target rate R. In a TD multiuser environment, the model will be exactly the same except that each block is given to only one of the users by some scheduling mechanism. More details will be given in Section II-C.
B. Single-User Problem: Power Allocation With only one user, the problem is to find the minimum required power (Q = KP ) [as the blocks are independent and identically distributed (i.i.d.), the optimum can always be attained with an equal-power policy] that assures an acceptable level of outage probability for a given R: min Q s.t. P Q≥0
K
rk ≤ R
≤ ε,
(3)
813
The problem of interest is, as a consequence, given by ⎧ N ⎪ ⎪ ⎪ min Qu ⎪ ⎪ {Qu },{nu } ⎪ ⎪ u=1 ⎪ ⎪ n ⎪ u ⎪ ⎪ ⎪ (u) ⎪ s.t. P rk ≤ Ru ≤ εu ∀u, ⎪ ⎨ k=1 M → ⎪ Qu ≥ 0 ∀u, ⎪ ⎪ ⎪ ⎪ ⎪ N ⎪ ⎪ ⎪ ⎪ nu ≤ K, ⎪ ⎪ ⎪ ⎪ u=1 ⎪ ⎩ nu ∈ {1, 2, . . . , K − N + 1} ∀u,
(4)
where Qu nu
(u)
rk N
the total power allocated to user u; the number of blocks allocated to user u;
(5) (6)
the rate achieved at the kth block of user u; (7) the total number of users; (8)
K Ru
the number of blocks; the target rate for user u;
(9) (10)
εu
the probability requirement for user u.
(11) (u)
Likewise, users have the mean channel power gains {C0 = C0 d−γ u }. The challenge of M is that it is a mixed integer problem which has no known method achieving the global optimum [14]. The remainder of the letter will be devoted to finding the suboptimal solution of (4). In particular, Section III will look into obtaining the optimal {Qu } for a given {nu }. Section IV will focus on finding the suboptimal time-sharing parameters {nu } using relaxation followed by convex optimization. Section V combines the two approaches to suboptimally solve (4). Numerical results in Section VI will, however, show that the proposed suboptimal method performs nearly the same as the global optimum with inappreciable difference.
k=1
where P(A) denotes the probability of an event A, and ε is the maximum allowable outage probability. If (3) is solved, then the optimal power allocation for each block is simply Popt = Qopt K . Solving the above problem is in general challenging because getting the pdf of the sum-rate ω = r1 +r2 +· · · +rK is difficult. This will be dealt with in Section III.
C. Multiuser Problem: Power Allocation and Scheduling With multiple users in TD, only one user is permitted to gain access to the channel for each block. As a consequence, optimization needs to be performed for both power allocation and scheduling. By scheduling, we mean that the transmitter requires to decide which user occupies the channel at any given time (or block). However, since the delay constraint specifies only that the rate is achieved in K blocks with some probability, the order of which the users are scheduled within the K blocks is unimportant, and scheduling boils down to sorting out the number of blocks being allocated to each user.
III. M INIMUM P OWER E QUATION Here, we shall derive an equation to determine the minimum power allocation for attaining a given outage probability if the number of blocks is fixed. For a TD system, each block is occupied by one user only and if {nu } are fixed, then the optimization for the users is completely uncoupled and will be equivalent to multiple individual users’ power minimizations. Therefore, it suffices to focus on a single-user system for a given number of blocks, n, i.e., (3) with K = n, or min Q s.t. P ({ω ≤ R}) ≤ ε. Q≥0
(12)
To solve the above problem, we argue that from the central limit theorem (CLT),3 if n is large enough, ω has normal 3 The validity of Gaussian approximation will be checked in Section VI. Results in Figure 1 will show that for a wide range of outage probabilities (e.g., ε ≥ 10−2 ), the approximation is good even if n is as small as 3. For more stringent outage probability requirements such as ε < 10−2 , the approximation tends to provide an upper bound in the constraint, so the constraint is guaranteed at the expense of a slight power penalty.
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distribution with the parameters μ = E[ω] and σ 2 = VAR[ω]. The constraint of (12) can then be simplified to 1 1 R − μ(Q) √ + erf ≤ε (13) 2 2 2σ(Q) which leads to g(Q) μ(Q) − where
√ 2erf −1 (1 − 2ε) σ(Q) − R ≥ 0
2 erf(x) = √ π
x
2
e−t dt.
(14) (15)
0
In Appendix A, we have derived the mean and variance of the sum-rate, ω, as ⎧ 1 ne a 1 ⎪ ⎪ ⎪ μ = E , 1 ⎪ ⎪ ln 2 a ⎪ ⎪ ⎪ 2 1 ⎨ 2 a e π 1 (16) σ2 = n a ln − γEM + ⎪ a 6 a ln2 2 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎩ − 23 F3 [1, 1, 1]; [2, 2, 2]; − − μ2 , a −γ
0d , E1 (·) denotes the exponential integral, where a QCnN 0 p Fq denotes the generalized hypergeometric function, and γEM is the Euler-Mascheroni constant [12]. Intuitively, g should be a strictly increasing function of Q because more transmit power leads to less chance of being in an outage. Accordingly, the minimum value of Q occurs when the equality of (14) holds, or the constraint becomes active, i.e., g(Qmin ) = 0. Throughout this letter, we shall refer to this equation as the minimum power equation (MPE). To find the solution of MPE efficiently, one possible method is to use the DIviding RECTangle (DIRECT) algorithm [13], which samples points in the domain, and uses the information it has obtained to decide where to search next. To apply DIRECT, we construct the problem as follows: (17) S → Qmin = arg min |g(Q)|.
0≤Q≤UQ
The upper bound UQ can be set to be a large enough number that makes sure Qopt ∈ (0, UQ ). We shall omit the details of DIRECT, but refer the interested readers to [13]. IV. O PTIMIZED T IME -S HARING BY C ONVEXIZATION In this section, we aim to optimize the time-sharing parameters of the users {nu } by joint consideration of the power consumption and the probability constraints of the users. Ideally, it requires to solve M, i.e., (4), which is unfortunately not known. Here, we propose to mimic M by considering a simpler problem with the probability constraints replaced by some upper bounds (as in [11]) n u (u) rk ≤ Ru P n u
k=1
≤P
k=1
ck ≤
nu N0 Qu
nu
2Ru
UB [u]. Pout
(18)
The original probability constraint in (4) can thus be ascertained by constraining the upper bound of the outage
UB [u] ≤ εu }. The advantage by doing so probability {Pout is substantial because first, the optimizing variable Qu can be separated from the random variable, and secondly, the distribution of = k ck can be approximated by CLT as log-normal with ⎧ (u) ⎪ ⎨ E[ln ] = nu ln C0 − γEM , (19) 2 ⎪ ⎩ VAR[ln ] = nu π . 6 UB The constraint {Pout [u] ≤ εu } can therefore be simplified to 1 nu 2Ru nu N0 eγEM · (20) Qu ≥ √n1 ςu . (u) u − √π erf −1 (1−2εu ) C0 e 3
In other words, the single-user problem (3) can be solved suboptimally through minQ≥ς Q, which suggests the optimal power allocation be simply Qsubopt = ς. Using the upper bound constraints in the multiuser problem (4), we then have
˜ → M
N
min
{Qu },{nu }
Qu
u=1
N0 eγEM
s.t. Qu ≥
(u)
C0
·
nu 2Ru
1 nu
π erf −1 (1−2ε ) −√ u
e
3
√1 nu
∀u,
N
nu ≤ K,
u=1
nu ∈ {1, 2, . . . , K − N + 1} ∀u,
(21)
where the constraints are now written in closed-form. There is no doubt that the power allocation from the modified problem (21) is very conservative, i.e., Qsubopt Qopt because of the upper bound constraint. However, our conjecture is that the problem structure of M on {nu } would be accurately ˜ Accordingly, we may be able to obtain nearimitated by M. ˜ although accurate optimal solution for {nu } by solving M ˜ power consumption cannot be estimated from M. ˜ is that the optimization A remaining difficulty of solving M is mixed with combinatorial search over the space of {nu } because they are integer-valued [14]. To tackle this, we relax ˜ becomes {nu } to positive real numbers {x2u } so that M ⎧ 1 N ⎪ x2u (au ) x2u ⎪ γEM ⎪ min N0 e ⎪ 1 ⎪ (u) ⎪ {x } x ⎪ u=1 C0 (bu ) u ⎨ u N ˜ r → (22) M ⎪ ⎪ s.t. x2u ≤ K, ⎪ ⎪ ⎪ u=1 ⎪ ⎪ √ ⎩ 1 ≤ xu ≤ K − N + 1, − √π erf −1 (1−2ε )
u where au 2Ru and bu e 3 . Apparently, both constraints in (22) are convex, and if the cost is also convex, the problem can be solved using known convex programming routines [14]. Now, let us turn our attention on a function of the form 1
f (x) = x2 ·
a x2 1
bx
≡ x2 h(x), for a, b, x > 0,
(23)
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 3, MARCH 2008 1
1
where h(x) a x2 /b x . Our interest is to examine if f (x) is convex, or equivalently whether f (x) > 0. To show this, we first obtain ⎧ −2 ln a ln b ⎪ ⎪ h (x) = h(x) + , ⎪ ⎨ x3 x2 2 ⎪ −2 ln a ln b 6 ln a 2 ln b ⎪ ⎪ + 2 + h(x) − 3 . ⎩ h (x) = h(x) x3 x x4 x (24) Applying these results, f (x) can be found as 2 −2 ln a ln b −2 ln a 2 ln b f (x) = + + + + 2. h(x) x2 x x2 x (25) Letting α = 2 xln2 a and β = − xln b , we have
f (x) = (α + β)2 − α − 2β + 2 h(x) 2 3 1 + (β − 1)2 + + 2αβ > 0 = α− 2 4
(26)
since α, β > 0, which can be seen from the definition of (a, b) that α > 0 and β > 0 for εu < 0.5.4 Together with the fact that h(x) > 0 for all x > 0, we can conclude that f (x) > 0 and therefore, f (x) is convex. As the cost function in (22) is a summation of the functions of the form f (x), it is convex ˜ r. and hence the problem (22), or M ˜ r being convex, we can find the globally optimal With M {xu }opt at polynomial-time complexity. In particular, the complexity grows like O(N 3 ), which is scalable with the number of users. The remaining task, however, is to derive the integer-valued {nu } from {xu }. Simply setting nu = x2u would result in non-integer solutions while rounding them off could lead to violation of the outage probability constraints. In this letter, a greedy-approach is presented to obtain a feasible solution of {nu } from {xu }, which will be described in the next section. V. T HE P ROPOSED S CHEME Thus far, we have proposed two approaches: one that determines the optimal power allocation {Qu } based on MPE (see Section III); another one that finds the suboptimal (relaxed) time-sharing parameters {nu } by constraining the upper bound probability (see Section IV). In this section, we shall present an algorithm that combines the two approaches to jointly optimize the power allocation and time-sharing of the users. Our idea is to first map the optimal solution {xu }opt from ˜ r to a proper {nu } in M by rounding the optimal results M to the nearest integers, and then to step-by-step allocate one more block to the user who can minimize the overall required power using MPE. The proposed algorithm is described as follows: ˜ r (22) using convex optimization 1) Solve {xu } in M routines such as interior-point method. 2) Initialize nu = x2u ∀u where y returns the greatest integer that is smaller than y. Note that at this point, 4 It should be emphasized that the convexity of f is subjected to the condition that εu < 0.5. However, in practice, it would not make sense to have a system operating with outage probability greater than 50%.
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˜ r may not give a feasible solu{nu } and {Qu } from M tion to M, and some outage rate probability constraints may not be satisfied. 3) For each user u, compute the minimal required power to ensure the outage probability constraint by solving MPE (27) Qu = arg min |gu (Q|nu )| , Q≥0
where the function gu (Q|nu ) is defined similarly as in (14), and the notation (Q|nu ) emphasizes the fact that nu is given as a fixed constant. N 4) Then, initialize m = K − u=1 nu . 5) Compute the power reduction metrics
Qu = Qu − arg min |gu (Q|nu + 1)| Q≥0
6) Find u∗ = arg maxu Qu and update ⎧ ⎪ ⎨ nu∗ := nu∗ + 1, Qu∗ := Qu∗ − Qu∗ , ⎪ ⎩ m := m − 1.
∀u.
(28)
(29)
If m ≥ 1, go back to Step 5. Otherwise, go to Step 7. 7) Optimization is completed and the solutions for both {nu } and {Qu } have been found. VI. N UMERICAL R ESULTS A. Setup and Benchmarks Computer simulations are conducted to evaluate the performance of the proposed algorithm for the power-minimization problem with outage rate probability constraints. Only CST has been assumed, and capacity-achieving codec is used so that the expression log2 (1 + SNR) can be used to express the rateachieved for each block. The totaltransmit SNR, defined N (u) N 1 as N1 u=1 C0 Q u=1 u N0 , is considered as the performance metric. To compute the required SNR for a given set of simulation parameters such as the numbers of users and blocks, the users’ target rates and outage probabilities, the algorithm presented in Section V, which iteratively solves the MPE, is used. Note that the MPE itself has already taken into account the randomness of the channel for outage evaluation. Results for the proposed algorithm will be compared with the following benchmarks: • Global-Optimal Time and Power Allocation from Exhaustive Searching—With MPE, the global optimum can be found from solving M by an exhaustive search over the space of {nu } at the expense of complexity, i.e., ⎧ N ⎪ ⎪ ⎪ arg min |gu (Qu |nu )| ⎪ min ⎪ ⎪ {n } {Qu ≥0} ⎪ u=1 ⎨ u N M → (30) ⎪ ⎪ nu ≤ K, s.t. ⎪ ⎪ ⎪ u=1 ⎪ ⎪ ⎩ nu ∈ {1, 2, . . . , K − N + 1} ∀u. The inner optimization in (30) can be performed using DIRECT for a given {nu }. The globally optimal solution for {nu , Qu } can therefore be obtained from a complete search over the space of {nu }. However, note that if N or K is large, the required complexity of the search can be
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TABLE I T HE NUMBER OF COMBINATIONS NEEDED FOR THE OPTIMIZATION OF M
K
2
20 30 40 20 30 40 20 30 40 20 30 40
3
4
5
Number of combinations
K N
Cumulative Density Function
N
K AND N .
190 435 780 1140 4060 9880 4845 27405 91390 15504 142510 658008
enormous. For instance, if N = 4 and K = 30, there are totally 27405 combinations for {nu } to be tested. More numerical examples are shown in Table I, which gives some idea of the complexity requirements for solving M from an exhaustive optimization approach. • Equal-Time Sharing with Optimized Power Allocation— An interesting benchmark is the system where each user is allocated equal number of blocks so that nu = K N ∀u, but the power allocation for each user is obtained by solving MPE. This system shows how important time-sharing optimization is if the system has highly heterogeneous users and channels. ˜ r —We can obtain the • Relaxed Optimal Solution from M ˜ r if suboptimal time and power allocation directly from M relaxation is allowed. It is noted, however, that this is an unrealizable solution unless fractional time allocation is permitted. Additionally, despite the relaxation, this does not give a lower bound for M because the upper bounds of the outage probability constraints are considered, and this can show the importance to have MPE. The simulations are conducted using MATLAB and the ˜ r is solved by the function “fminrelaxed convex problem M con”in MATLAB. A Rayleigh block-fading channel is consid(u) ered with the mean channel power C0 for user u. Results in Figures 1–4 are given for various simulation parameters. Each figure presents results with different sets of parameters and will be individually described. B. Results Figure 1 shows the cumulative distribution function (cdf) of the actual sum-rate for various number of blocks n and target rates and this cdf is compared with that of Gaussian approximation. As we can see, for a wide range of outage probabilities (e.g., ε ≥ 10−2 ), the approximation is good even if n is as small as 3. For more stringent outage probability requirements such as ε < 10−2 , the approximation tends to provide an upper bound in the constraint, but the constraint is always guaranteed at the expense of a slight power penalty. Figure 2 shows the SNR results against outage probability requirements for a 3-user system with 20 blocks (i.e., N = 3 and K = 20). Users are assumed to have the same target
1 n=
2 n= 3 n=
-2
10
5 n=
7
n=
10
n=
-4
10
-6
10
Simulation Gaussian approximation -8
10
-10
10
0
5
10
15
20
25
30
Transmission Ra te R Transmission Rate R (bps/Hz) (bps/Hz)
Fig. 1. Comparison between the actual and approximated distributions for an n-block Rayleigh fading channel when Q = 50.
28 26 24 Transmit SNR in dB
FOR VARIOUS
0
10
22 20 18 16 14 12 −5 10
Proposed method Equal−time with optimized power Relaxed solution of (34) Global optimum 10
−4
−3
10 Outage probability εout
10
−2
10
−1
Fig. 2. Results of transmit SNR against the outage probability requirement (u) when N = 3, K = 20, Ru = 4 ∀u and C0 = 1 ∀u.
rate R = 4 bps/Hz, the same mean channel power gain C0 d−γ = 1, and the same outage rate probability requirements ε ranging from 10−1 to 10−5 . Results illustrate that as ε decreases, the required SNR increases, and in particular, about 2–2.5 dB more in SNR is needed for a reduction of ε by an order of magnitude. In addition, results also indicate that the relaxed solution of (22) is significantly inferior than the other methods with about 6 dB difference. However, the proposed method achieves the same SNR performance as the global optimum. In this system, obviously, the optimal solution should be n1 = 6, n2 = 7, n3 = 7 with the power optimized from MPEs. Therefore, the equal-time allocation method with n1 = n2 = n3 = 6 is nearly optimal with difference less than 0.2 dB. Overall, from the results in this figure, it has been shown that MPE is important for power minimization, and using the upper bound probability to optimize power is likely to have a considerable power penalty. The performance for various number of blocks K is inves-
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26
817
32 Proposed method Equal−time with optimized power Relaxed solution of (34) Global optimum
30
24
28 Transmit SNR in dB
Transmit SNR in dB
22 Proposed method Equal−time with optimized power Relaxed solution of (34) Global optimum
20
18
26
24
22 16
14 10
20
15
20
25 30 35 Number of blocks K
40
45
50
Fig. 3. Results of transmit SNR against the number of blocks when N = 3, (u) εu = 10−3 ∀u, Ru = 4 ∀u and C0 = 1 ∀u.
tigated by the results in Figure 3. A close observation from the results indicates that the relaxed solution will converge to an irreducible SNR at about 24 dB after K = 30 whereas the other methods utilizing the MPE can further minimize the power by increasing K. This can be explained by the fact that the relaxed solution is based on an upper bound probability which is getting looser and looser if K increases. Thus far, we have looked at scenarios in which users are homogeneous and that the global optimum would be more or less the same as the equal-time allocation. Now, we consider the settings which have highly non-homogeneous (or highlyskewed) users, and we are interested to see whether the proposed method and the equal-time allocation can still give near-optimal results. The configurations that we have tested are listed in Table II and results provided in Figure 4. As we can see, the performance gap between the equal-time allocation and the global optimum does get larger (about 2 dB for Configuration 4 where there are high and low demand users). On the other hand, results also demonstrate that the proposed method works very well even for extreme scenarios and achieves the same SNR results as the optimum except for Configuration 4 where a performance gap of less than 0.1 dB is observed. Some details regarding how different methods work are provided in Table II. Results also reveals that the proposed method always arrives at the global optimal solution except for the very extreme case. This shows that the modified problem (22) can effectively mimic the structure of the original problem (4) for time-sharing optimization, making possible the overall power minimization using MPE. By contrast, regardless of the users’ channel statistics and requirements, the equal-time allocation always assigns 6 blocks (out of a total of 20 blocks) for each of the three users. Nonetheless, the results indicate that the SNR penalty for not optimizing {nu } is less than what we would have expected. The reason turns out to be that though a high-demand poor-channel-condition user should get more blocks for power minimization, the power saving is not remarkable because assigning more blocks means not only to increase the mean rate (which helps minimize power) but also
18
Fig. 4.
1
2 3 Configuration listed in Table 2
4
Results of transmit SNR for the configurations listed in Table 2.
to increase the variance of the sum-rate (which on the other hand requires more power to attain a given outage probability).
VII. C ONCLUSION This letter has addressed the optimization problem of power allocation and scheduling for a multiuser TD system where the transmitter knows only the channel statistics of the users, and the users are given individual outage probability constraints. By Gaussian approximation, we have derived a so-called MPE to determine the minimum power for attaining a given outage rate probability constraint if the number of blocks for a user is fixed. On the other hand, we have also proposed a convex programming approach to obtain the suboptimal number of blocks allocated to the users. The two main techniques have later been combined to obtain a joint solution for both power and time allocation for the users. Results have demonstrated that the proposed method achieves near-optimal performance.
A PPENDIX A: D ERIVATION OF M EAN AND VARIANCE OF log2 (1 + ax) WHEN x IS A STANDARD EXPONENTIAL RANDOM VARIABLE
The mean of r log2 (1 + ax), can be obtained by E[r]
= E[log2 (1 + ax)] ∞ = log2 (1 + ax)e−x dx 0 ∞ 1 = ln(1 + ax)e−x dx ln 2 0 ∞ 1 t ea = ln te− a dt a ln 2 1 1 ∞ t ea e− a = dt ln 2 1 t 1 1 ea = E1 ln 2 a
(31) (32) (33) (34) (35) (36)
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TABLE II VARIOUS CONFIGURATIONS TESTED IN F IGURE 4 WITH K = 20 WHERE THE SUPERSCRIPTS HIGHLIGHT THE VARIABLES THAT ARE NOT THE SAME AS THE OPTIMUMS .
Configuration 1
2
3
4
u 1 2 3 1 2 3 1 2 3 1 2 3
Ru (bps/Hz) 4 4 4 4 6 8 4 6 8 1 4 8
εu 10−3 10−3 10−3 10−3 10−3 10−3 10−5 10−5 10−5 10−5 10−5 10−5
where E1 (·) is the exponential integral. To find the variance, we find it useful to first evaluate the following integral
∞
e− a (ln t)2 dt = t
1
0
∞
e− a (ln t)2 dt − t
1
e− a (ln t)2 dt t
0
(37) where the first integrand can be recognized as a variant of the Euler-Mascheroni integral [12], so 2 ∞ 1 π2 − at 2 (38) e (ln t) dt = a ln − γEM + a 6 0 where γEM is the Euler-Mascheroni constant, and the second integrand can be evaluated by 1 1 ∞ t n −a − at 2 (ln t)2 dt e (ln t) dt = (39) n! 0 0 n=0 ∞ 1 n 1 −a = tn (ln t)2 dt (40) n! 0 n=0 ∞ 1 n −a (−1)2 · 2! = · (41) n! (n + 1)3 n=0 ∞ 1 n −a 1 · = 2 (42) n! (n + 1)3 n=0 1 = 23 F3 [1, 1, 1]; [2, 2, 2]; − (43) a in which p Fq denotes the generalized hypergeometric function. As a result, we have
∞
− at
e 1
2 2 1 π (ln t)2 dt = a ln − γEM + a 6 1 − 23 F3 [1, 1, 1]; [2, 2, 2]; − . a
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(u)
C0 0.5 1 1.5 1.5 1 0.5 1.5 1 0.5 5 3 0.5
(nu )opt 9 6 5 4 6 10 4 6 10 2 4 14
(nu )proposed 9 6 5 4 6 10 4 6 10 3 5 12
(nu )equal−time 6 6 6 6 6 6 6 6 6 6 6 6
Finally, the variance of the rate can be obtained as 2 1 π2 1 ea a ln − γEM + VAR[r] = a 6 a ln2 2 1 −23 F3 [1, 1, 1]; [2, 2, 2]; − − (E[r])2 . a R EFERENCES
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