1
Power Allocation in Multirate DS/CDMA Systems Based on Verhulst Equilibrium Lucas Dias H. Sampaio, Mois´es Fernando Lima, Bruno Bogaz Zarpel˜ao, Mario Lemes Proenc¸a Jr., and Taufik Abr˜ao, Member IEEE Abstract—This paper extends the discrete Verhulst power equilibrium method, previously suggested in [1], to the powerrate optimal allocation problem. Multirate users associated with different types of traffic are aggregated to distinct user’ classes, with the assurance of target rate allocation per user and QoS. Therein, single-rate Verhulst power allocation algorithm was adapted to the multirate DS/CDMA power control problem. The analysis was carried out taking into account the convergence time (number of iterations), quality of solution in terms of the normalized square error (NSE), and computational complexity when compared to the analytical solution based on interference matrix inverse and the Foschini solution as well. Index Terms—Power-rate allocation control; SISO multirate DS/CDMA; discrete Verhulst equilibrium equation; QoS.
I. I NTRODUCTION In the current telecommunications scenario, the multiple access networks are essential to comply with the new quadriplay services, composed by voice, video and data traffic associated with mobility [2]. In these networks the efficiently resources use, mainly spectrum and power1 allocation, is of great importance, both for telecommunications companies profits and for consumer satisfaction. Many efforts have been spent in the last years trying to find a better resource allocation algorithm that could be easy applied to direct sequence code division multiple access (DS/CDMA) communications systems in order to satisfy the new communications services needs in terms of capacity, availability and mobility. Aiming these requirements many distributed power control algorithms (DPCA) have been proposed in the last decades, e.g., Foschini and Miljanic [3] studies, which can be considerate as foundation of many well-known DPCAs. More recently, many researchers have been proposed new algorithms to solve the resource allocation problem in wireless networks. A distributed power control algorithm for the downlink of multiclass wireless networks was proposed in [4]. Also, under specific scenario of interference-limited networks, [5] proposed a power allocation algorithm to achieve global optimality in weighted throughput maximization based on multiplicative linear fractional programming (MLFP). Moreover, in [6] the goal is to maximize the fairness between users in the uplink of CDMA systems, satisfying different QoS requirements. In [1] the Verhulst mathematical model, initially designed to describe population growth of biological species with food and physical space restriction, was adapted to a single rate Computer Department, State University of Londrina, PR, 86051-990, Brazil. E-mails:
[email protected];
[email protected];
[email protected];
[email protected];
[email protected]. 1 Power allocation procedures imply in battery autonomy improvement.
DS/CDMA system DPCA. That work was the first to propose a Verhulst equilibrium equation adaptation to resource allocation problems in DS/CDMA networks. So, in this paper, the DPCA proposed in [1] was extended to the resource allocation problem in multirate DS/CDMA systems achieving different quality of service requirements (QoS), related to different user classes. II. P OWER AND R ATE A LLOCATION P ROBLEM In a multiple access system, such as DS/CDMA, the power control problem is of great importance in order to achieve relevant system capacity and throughput2 . The power allocation issue can be solved by finding the best vector that contains the minimum power to be assigned in the next time slot to each active user, in order to achieve the minimum quality of service (QoS) through the minimum carrier to interference ratio (CIR). In multirate multiple access wireless communications networks, the bit error rate (BER) is often used as a QoS measure and, since the BER is directly linked to the signal to interference plus noise ratio (SNIR), we are able to use the SNIR parameter as the QoS measurement. Hence, associating the SNIR to the CIR at time slot n results: δi [n] =
Rc × Γi [n], Ri [n]
n = 0, 1, . . . N
(1)
where δi [n] is the SNIR of user i at the nth iteration, Rc is the chip rate, Ri [n] is the data rate for user i, Γi [n] is the CIR for user i at iteration n, and N is the maximal number of iterations. In multirate DS/CDMA systems with multiple processing gains (MPG), where each user class has a different processing gain G > 1, defined as a function of the chip rate by: G` =
Rc , Ri`
` = 1, 2, . . . , L,
(2)
where L is the number of total user’s classes defined in the system (voice, data, video). Hence, in MPG-DS/CDMA multiple access systems, the SNIR and CIR for the `th user’s class are related to the processing gain of that service class: δi` = G` × Γi . From (1) we are able to calculate the data rate for user i at iteration n: Ri [n] =
Rc × Γi [n], δi [n]
n = 0, 1, . . . , N
(3)
2 In this paper the therm capacity is employed to express the total number of active users in the system.
2
The CIR for the ith user can be calculated as [1], [7]: Γi [n] =
pi [n]gii [n] K P j=1 j6=i
pj [n]gij [n] +
, i = 1, . . . , K
(4)
` = 1···L
(5)
k = 1 · · · K` ;
` = 1 · · · L.
(6)
Hence, the total number of active users in the system is given by K = K1 ∪. . .∪K` ∪. . .∪KL . Note that indexes associated to the K users are obtained by concatenation of ascending rates from different user’s classes. Thus, K1 identifies the lowest user’s rate class, and KL the highest. The K × K channel gain matrix, considering path loss, shadowing and fading effects, between user j and user i (or base station) is given by: g11 g12 · · · g1K g21 g22 · · · g2K G= . .. .. , .. .. . . . gK1 gK2 · · · gKK which could be assumed static or even dynamically changing over the optimization window (N time slots). Assuming multirate user classes, we are able to adapt the classical power control problem to achieve the target rates for each user, simply using the Shannon capacity relation between minimum CIR and target rate in each user class, resulting: `
Γ`min = 2Rmin − 1
(7)
Now, considering a K × K interference matrix B, with 0, i = j; Γi,min gji Bij = (8) , i 6= j; gii where Γi,min can be obtained from (5), taking into account each rate class requirement and the column vector u = [u1 , u2 , . . . , uK ]T , with elements: ui =
Γi,min σi2 , gii
(9)
u
(10)
if and only if the maximum eigenvalue of B is smaller than 1 [9]; I is the K × K identity matrix. In this situation, the power control problem shows a feasible solution. A. Multirate Power Allocation Criterion The classical power allocation problem is extended to incorporate multirate criterion in order to guarantee the target data rate per user class. Mathematically, we want to solve the following optimization problem: £ ¤ L min p = p11 . . . p1K1 , . . . , p`1 . . . p`K` , . . . , pL 1 . . . pKL s.t.
R` δ ∗ = min , Rc
` where Γ`min and Rmin is the minimum CIR and target user rate associated to the `th user class, respectively, δ ∗ is the minimum (or target) signal to noise ratio (SNR) to achieve minimum acceptable BER (or QoS). Besides, the power allocated to the kth user belonging to the `th class at nth iteration is:
p`k [n],
−1
p∗ = (I − B)
σ2
where pi [n] is the power allocated to the ith user at iteration n and is bounded by [Pmin ; Pmax ], the channel gain (including path loss, fading and shadowing effects) between user j and user (or base station) i is identified by gij , K is the number of active users in the system (considering all user’s classes), and σi2 = σj2 = σ 2 is the average power of the additive white Gaussian noise (AWGN) at the input of ith receiver, admitted identical for all users. Therefore, in DS/CDMA multirate systems the CIR relation to achieve the target rate can be calculated to each user class as follows [8]: Γ`min
we can obtain the analytical optimal power vector allocation p∗ = [p1 , p2 , . . . , pK ]T simply by matrix inversion as:
` ` (11) Pmin ≤ p`k ≤ Pmax ` ` R = Rmin , ∀k ∈ K` , and ∀ ` = 1, 2, · · · L
III. V ERHULST P OWER -R ATE O PTIMIZATION A PPROACH The Verhulst mathematical model was first idealized to describe population dynamics based on food and space limitation. In [1] that model was adapted to single-rate DS/CDMA distributed power control using a discrete iterative convergent equation as follows: · ¸ δi [n] pi [n + 1] = (1 + α) pi [n] − α pi [n], i = 1, · · · , K δi∗ (12) where pi [n + 1] is the user i power at the n + 1 iteration and is bounded by Pmin ≤ pi [n + 1] ≤ Pmax , α ∈ (0; 1] is the Verhulst convergence factor, δi [n] is the ith user’ SNIR at iteration n, δi∗ is the minimum SNR for the ith user that guarantee a minimum QoS in terms of performance (BER). The recursion (12) can be effectively implemented in the ith mobile unit since all necessary parameters α, the QoS level given by δi∗ , the transmitted power pi [n], except δi [n], can be considered known in the mobile unit i. The SINR δi [n] can be obtained only at the correspondent base station that demodulates the signal from user i. In this way, the BS estimates δi [n], quantizes it in a convenient number of bits, and transmits this information to the ith user through the direct channel. Thus, (12) depends on local parameters just allowing that the power control works in a distributed manner, i.e., each one of the K links (mobile terminals to base station) carries out separately the respective power control mechanism, justifying the name distributed power control algorithm. The proposed DPCA has a remarkable characteristic: the estimated SINR, inside the brackets in (12), is in the numerator of the division. This property makes that the algorithm sensitivity to estimation errors is independent of the real value of SINR. It is known that a function like ϕ = κδ −1 (SINR in the denominator is inherent to all known DPCAs proposed in the literature, as the classical [3] and sigmoidal [10]) has a differential increment dϕ = −κδ −2 dδ i.e., the amount of ϕ variation for small δ deviations depends on the real value of the SNIR, δ. For a function like ϕ = κδ as in (12), we have dϕ = κdδ, which is in fact independent of the
3
real value of δ. For the sigmoidal case, not only SINR is in the denominator but, additionally, the division is argument of a hyperbolic tangent function. Obviously, the hyperbolic function influences, but it does not cancel, the relation between δ and the algorithm behavior with imperfect measures of SINR. Equation (12) gives a recursive power update, close to the optimal power solution after N iteration. However, originally it does not consider the rate requirements in a multirate environment. In order to achieve the QoS to each user class, (12) must be adapted to reach the equilibrium lim pi [n] = p∗i n→∞ when the power allocated to each user satisfies the target rate constraint given in (11). Hence, the recursive equation must be rewritten considering SNIR per user class, via (1), in order to incorporate multirate scenario. The minimum CIR per user class is obtained directly by (5). In this way, considering the relation between CIR and SNIR in a multirate DS/CDMA context, we propose the following equation in order to iteratively solve optimization problem in (11): δi [n]
= F × Γi [n] = F×
(13) pi [n]gii [n]
K P j=1 j6=i
pj [n]gij [n] +
, i = 1, . . . , K σ2
where F is the spreading factor per user class, given by: F =
Rc ` Rmin
(14)
Note that the CIR of ith user at the nth iteration is weighted by spreading factor; so the corresponding SNIR is inversely proportional to the actual (minimum) rate of the ith user of the `th class. A. Quality of Solution × Convergence Speed The quality of solution achieved by iterative Verhulst equation (12) is measured by how close to the optimum solution is p[n], and can be quantified by means of the normalized squared error (NSE) when equilibrium is reached. The NSE definition is given by: " # 2 kp[n] − p∗ k N SE[n] = E , 2 kp∗ k where k · k2 denotes the squared Euclidean distance to the origin, and E[·] the expectation operator. On the other hand, the convergence speed in Verhulst equation is dictated by the parameter α. Hence, for small values of convergence factor, i.e., α → 0, the convergence is slow, but the NSE is very small after N iteration, when compared with the opposite configuration: the convergence is fast when α → 1, but the NSE is a concern. So, in order to accelerate convergence, we propose two adaptive criteria for convergence factor α when iterations evolve, based on a) SNIR to target SNR difference, and b) tanh mapping for this difference, as following: ¾ ½ |δi [n − 1] − δi∗ | + αmin , (15) a) αi [n] = min αmax ; δi∗
b)
αi [n] = max {αmin ; tanh(|δi [n − 1] − δi∗ |)} ,
(16)
with αmin = 0.1 and αmax = 0.95. IV. N UMERICAL R ESULTS System parameters are indicated in Table I. In all simulation it was assumed a rectangular multicell geometry with a number of base station (BS) equal to 4 and mobile terminals (mt) uniformly distributed. Besides, the initial rate assignment for all multirate users was admitted discretely and uniformly distributed over three submultiple rates of chip rate, Rmin = 1 1 1 [ 128 ; 32 ; 16 ]Rc [bps]. TABLE I M ULTIRATE DS/CDMA SYSTEM PARAMETERS Parameters Adopted Values DS/CDMA Power-Rate Allocation System Noise Power Pn = −63 [dBm] Chip rate Rc = 3.84 × 106 Min. Signal-noise ratio SN Rmin = 4 dB Max. power per user Pmax = 20 [dBm] Min. Power per user Pmin = SN Rmin + Pn [dBm] Time slot duration Tslot = 666.7µs or Rslot = 1500 slots/s # mobile terminals K ∈ {5; 30} # base station BS = 4 cell geometry rectangular, with xcell = ycell = 5 Km mobile term. distrib. ∼ U [xcell , ycell ] Channel Gain path loss ∝ d−2 shadowing uncorrelated log-normal, σ 2 = 6 dB fading Rice: [0.6; 0.4] Max. Doppler freq. fD max = 11.1 Hz b = (1 + ε)G, where ε ∼ U [±δ] Error estimates G δ = 0 : 0.02 : 0.2 User Types # user classes L = 3 (voice, data, video) 1 1 1 User classes Rates Rmin = [ 128 ; 32 ; 16 ]Rc [bps] Verhulst Power-Rate algorithm Type partially distributed α range [0.10; 0.95] Optimization window N ∈ [100; 1000] iterations Performance parameters Trials number, TR 100 samples
A number of mobile terminals ranging from K = 5 to 30 was considered, which experiment slow fading channels, i.e., the following relation is always satisfied: Tslot < (∆t)c
(17)
−1 where Tslot = Rslot is the time slot duration, Rslot is the transmitted power vector update rate, and (∆t)c is the coherence time of the channel3 . This condition is part of the SINR estimation process, and it implies that each power updating accomplished by the DPCA happens with rate of Rslot , assumed here equal to 1500 updates per second. The recursion in (12) should converge to the optimum point before each channel gain gij experiments significant changing. Note that satisfying (17) the gain matrices remain approximately static during one convergence process interval, i.e., 666.7µs. In all of the simulations the entries values for the QoS targets were fixed in δ ∗ = 4 dB, the adopted receiver noise 3 Corresponds to the time interval in which the channel characteristics do not suffer expressive variations.
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a) static channel, situation where the channel coefficients remain constant during all the convergence process (N iterations), i.e., for a time interval equal or bigger than Tslot ; b) dynamic channel: the coefficients suffer modifications according to the channel coherence time, observing the slow fading condition of (17), considering the two adopted mobilities, i.e., pedestrian and vehicular (fDmax ≤ 90Hz). Under this condition the minimum number of iterations needed for the algorithm achieves convergence (under acceptable NSE values) was quantified. Hence, we were able to evaluate the Verhulst power allocation algorithm implementation viability in multirate systems with realistic channels and system operation conditions. Typical convergence behavior for fixed α, K = 7 multirate users, with rate assignment uniformly distributed, is shown in Fig. 1, for power allocation solution (inferior plots). Plateaux indicate convergence to the optimum power vector, p∗ ; hence dot lines (Popt in legend) indicates analytical solution given by (10). Note the fast convergence for all users when α = 0.9, i.e., N ≈ 25 iterations, against ≈ 150 for α = 0.1. Evidently, the quality of solution in this two situation is distinct, as discussed in the next subsection. A. Performance under Channel Gain Error Estimates In a real scenario, the SINR estimations at BS are not perfect. In order to incorporate this characteristic, a random error is added in each element of channel gain matrix, in each iteration basis. The ratio of the estimated and real channel gain values is given by: gc ij = (1 + ε)gij , where ε will be considered as a random variable with uniform distribution in the range [−δ; δ]. In the subsequent simulations the adopted range values for δ were 0 to 0.2, in steps of 0.02. −4
x 10
Verhulst; K= 7users; α = 0.1
−4
x 10
Power Allocation per user [W]
2.5
Verhulst; K= 7users; α = 0.9
2.5 Avg Power Pmin
Avg Power Pmin
Popt
2
2
1.5
1.5
1
1
0.5
0.5
0
0
50
100 Time slots
150
0
P
opt
0
50
100 Time slots
150
Fig. 1. Power convergence for K = 7; Rk,min [120, 120, 240, 120, 30, 120, 30][Kbps]. α = 0.1 and 0.9
=
Since we have some idea how fast the Verhulst algorithm reaches the equilibrium with different values of α, it is important to determine solution quality in terms of convergence time. For the same system configuration of Fig. 1, we have obtained in Fig. 2 the associated NSE ratio, defined as: N SER =
N SE(α = 0.9) N SE(fast converg.) = N SE(α = 0.1) N SE(slow converg.)
(18)
1
0.8 NSE Ratio
power for all users is Pn = −63 dBm, and the gain matrix G have intermediate values between those used in [10] and [8]. The simulations results discussed in the next section were obtained under two channel conditions:
0.6
0.4
0.2
0 0.2 0.1 G error estimate, δ
0
0
50
100
150
200
250
300
Iteration, N
Fig. 2. NSE Ratio considering fast (α = 0.9) and slow (α = 0.1) power convergence behavior of K = 7 mt, and δ channel gain estimation error values.
One can see from Fig. 2 that regardless of channel error estimates δ, the quality of solution for both α = 0.9 (fast) and α = 0.1 (slow convergence) at the final section of iterations (N > 170) shows high similarity (N SER ≈ 1), but with a slight advantage in terms of convergence for α = 0.1. In that region, with both convergence factors, the algorithm approaches to the optimal solution at same speed; as a consequence the N SER → 1. Conversely, after a initial approaching convergence, i.e., after 23 and until ≈ 120 iterations, the Verhulst algorithm with α = 0.9 produces a much better solution, resulting in N SE(α = 0.9) 200 then N SE(αAdpt ) ≈ 2 · 10−2 N SE(α = 0.1). Fig. 5 c) shows the NSE simulation results for both methods considering K = 30 users and δ = 0. Note that the hyperbolic tangent mapping always results a lower NSE, although this difference is marginal. Therefore, for any number of iterations the convergence solution provided by the tanh method is better than the SNR to SNR target difference mapping at cost of more computational effort spent with tanh evaluation.
−3
x 10
5 4 3 2 1
0 10
0.2 15 Users, K
0.15 20
0.1 25
0
10
δ
a)
0.05 30
0
Fig. 3. NSE degradation as a function of K mobile terminals and δ channel gain estimation error, α = 0.2.
−2
10 NSE
Normalized Squared Error, NSE
6
−4
10
B. Adaptive Convergence Methods Performance
Adaptive Tanh α = 0.1 (fixed)
−6
10
8
Verhulst Fixed α; K= 30; α = 0.1 δ = 0 −4 x 10
8
6
6
5
Avg Power Pmin
4
Popt
3 2
Power Allocation per user [W]
7
Power Allocation per user [W]
7
1 0
200
400 Time slots
600
800
5
Avg Power Pmin
4
Popt
3 2
0
0
200
400 Time slots
600
100
150
200
250
300
350
400
NSER, eq. (18)
NSER
b)
−1
10
−2
10
0
50
100
150
200
250
300
350
400
10
SNR diff, eq. (15) Tanh map., eq (16)
c)
−1
10
−2
10
−3
10
−4
10
−5
10
50
100
150
200 Iteration, N
250
300
350
400
Fig. 5. a) NSE and b) NSER for the adaptive method using tanh function against fixed α = 0.1. c) NSE for both suggested adaptive methods. K = 30 users.
C. Performance and Convergence under Dynamic Channels
Verhulst − Hyper. Tan; K= 30; δ = 0 −4 x 10
1
0
50
0
10
NSE
In order to speed up the algorithm convergence, two adaptive criteria based on SNIR’s difference were suggested in section III-A. There are two important performance aspects to be analyzed, considering the suggested adaptive methods against fixed α: convergence time and solution quality. However, since for the three methods (both adaptive methods, and fixed α method as well) a lower convergence factor was adopted, bounded by α = 0.1, in this section we evaluate the results just in terms of convergence time (number of iterations, N ). An first approach to evaluate the convergence time reduction with the adoption of adaptive convergence factor is provided in Figure 4. Both plots were generated under the same channel conditions with K = 30 users and no channel gain error estimates. As expected, the convergence rate, mainly at the beginning (early iterations), is greatly speeded up. One can see that roughly the adaptive tanh−α procedure allows the proposed algorithm to achieve total convergence 50% early regards to the classical fixed α = 0.1.
800
Fig. 4. Convergence comparison between the adaptive α using the hyperbolic tangent function (right) and the classical fixed α method (left).
In order to quantify the α-adaptation effect over the normalized square error, Fig. 5 a) and b) shows the NSE and NSER, respectively, for each number of iterations in the range [0; 400], and considering the same scenario and data generated on Fig. 4. Note that for any iteration after the initial iterations (N > 100) the adaptive convergence factor provides at least one and half order better performance in terms of NSE. Specifically,
The Verhulst power allocation algorithm behavior in terms of NSE and the ability to follow the channel changes are analyzed in this subsection. There were evaluated two channel profiles with Rayleigh fading: pedestrian channels, which the maximum Doppler frequency around fDmax = 11Hz for a carrier frequency of fc = 1.8Ghz; and vehicular channels, typically with a fDmax = 90Hz. Hence, in terms of power update, under a Tslot = 666.7µs, the channel may be considered slow. The adaptive factor α based on tanh were adopted. As reference, the classical algorithm of Foschini [3] were considered. The allocated power vector evolution was analyzed considering K = 10 users distributed uniformly in three different classes (rates), over 500 · Tslot . Results in Fig. 6 are presented for the allocated average power and respective average NSE over K = 10 users, considering a) pedestrian and b) vehicular mobilities. As expected, the power allocation follows the opposite way from instantaneous channel fading realization, in an attempt to compensate such fades. To reduce complexity, the minimum number of iterations per time slot was obtained, i.e., N = 10 for pedestrian scenario and N = 45 for vehicular scenario. Those number of iteration is the minimum needed for the
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Verhulst algorithm achieves a marginally lower NSE (averaged over 500 · Tslot ) when compared to the Foschini classical algorithm [3], as indicated in Table II. Since Verhulst multirate algorithm needed a low number of iterations per time slot to achieve a desired power allocation solution quality, even under vehicular channels, it is easy to extend the algorithm implementation to fast fading channels, obtaining the similar solution quality for the allocated power. −5
8.5
a) f
x 10
Dmax
8 Avarage Power Allocation [W]
−5
= 11Hz
3 Opt Power Foschini Verhulst
7.5
b) fDmax = 90Hz
x 10
Opt Power Foschini Verhulst
2.5
7 2 6.5 6 1.5 5.5 5
1 0
−2
50
100
150
200
100
150
200
Foschini
Foschini
10
50
Verhulst
Verhulst −5
10 −3
NSE
10
−6
10
−4
10
−5
10
NSE Avg: Fosch: 4.38e−4 Verh.: 3.542e−4 0
50
100 Time Slots
150
−7
10
200
So, comparing the proposed and the classical Foschini [3] algorithms, both result in same complexity. Besides, computational complexity of the proposed algorithm is N (K + 10), where N is the number of iterations necessary for convergence. Comparing it to the best case complexity of the matrix inversion operation, which is given by O[K 2 · log(K)] [11], our algorithm could achieve a considerable complexity reduction when the number of mobile terminals is large and the NSE requirement is not excessively tight, once in the proposed method, the complexity could be controlled simply specifying the maximal admissible NSE. V. C ONCLUSIONS This paper proposed an extension of Verhulst power equilibrium approach with adaptive convergence factor for DS/CDMA multirate systems. Simulations showed that the proposed algorithm has advantages over the analytical solution. Besides, the use of adaptive convergence factor result in a significantly reduction in the number of iterations needed to achieve convergence. The proposed DPCA can be used to solve practical resource allocation problems in advanced DS/CDMA networks. The algorithm has demonstrated efficiency even in dynamic channels, which just few iterations per time slot were needed to achieve convergence. Furthermore, the Verhulst algorithm presents a better NSE merit figure over a wide range of mobility (Doppler frequencies) when compared to the classical Foschini algorithm. This feature enlarges the field of applicability of the proposed DPCA.
NSE Avg: Fosch: 4.254e−6 Verh.: 3.95e−6 0
50
100 Time Slots
150
200
Fig. 6. Power Allocation and NSE average for α−tanh Verhulst: mean over K = 10 users values, considering a) fDmax = 11 Hz and N = 10 iterations per Tslot ; b) fDmax = 90 Hz and N = 45 iterations per Tslot .
From Table II and further simulations results (not shown here) it is possible to conclude that given a number of iteration N > 10 for pedestrian channels, or N > 45 for vehicular channels case, the difference between the solution quality given by Verhulst algorithm and the Foschini algorithm tends to grow in favor to the first one, since Verhulst gets equilibrium very near to the optimal solution (given by the interference matrix inversion), more quickly than the rate observed with Foschini algorithm. TABLE II NSE AVERAGE OVER 500 · Tslot . N SEAvg Pedestrian Vehicular fD max = 11 Hz fD max = 90 Hz Foschini 4, 38 · 10−4 4, 25 · 10−6 Verhulst 3, 54 · 10−4 3, 95 · 10−6 N Iter./Tslot 10 45
D. Complexity As in a distributed method, each link mobile terminal-BS performs separately their updating, i.e., as a whole the power control is performed by K processors in parallel and each one performs only scalar operations. Hence, the analysis is reduced to the study of one iteration on each mobile terminal.
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