Novel Power and Time Allocation Algorithm for a Dynamic

TDMA Slot Assignment Multiuser Transmission SchemesYosra Abbes, Fatma Abdelkefi and Hichem Besbes

Research Unit TECHTRA Sup’Com

University of the 7th November at Carthage, Tunisia

Emails: abbes.yosra@yahoo.fr, fatma.abdelkefi@supcom.rnu.tn and hichem.besbes@supcom.rnu.tn

Abstract—In the literature, many design parameters have been utilizedin network optimization but till now little attention has been paid onthe buffer management in the base station. In this paper, we propose

a new power allocation algorithm suited for Dynamic Time DivisionMultiple Access (TDMA) scheme, that is based on the minimization ofthe total transmitted power under the constraint of ensuring the bufferstability since the buffer overflow is one of the important causes of trafficoutage and so the throughput degradation. We show that this optimization

problem can be efficiently solved througth using the convex optimizationtechniques. The global optimal minimization solution can be computednumerically since the objective and the constraint functions are non

linear. We can ensure the power control by adding an upper bound onthe transmitted power. The new formulated problem can be used foradmission control. Since the optimization problem is no longer convex,

we propose a simple heuristic algorithm to solve it. Simulation results aregiven to support our claims and to illustrate the good performance at the

level of power allocation obtained by the proposed allocation schemes.

Index Terms—Power allocation scheme, time allocation, buffer stability,

power control and admission control.

I. INTRODUCTION

High data rates, high spectral efficiency, flexibility and low power

consumption will be important features in the next generation of

wireless communications systems. An efficiently resource allocation

scheme is needed to distribute the available radio resources among

multiple users in the presence of limited resources (buffer’s length,

bandwidth and power).

The problem of resource management was investigated in many

previous works [1], [2], [3], [4]. In [1], a multiuser OFDM sub-

carrier, bit and power allocation algorithm was proposed in order to

minimize the total transmitted power using Lagrangian relaxation.

Despite its significant gains over the fixed allocation scheme, the

algorithm is computational intensive and is difficult to implement.

Linear programming was used in [2] to solve the subcarrier allocation

by linearizing the function of rate in terms of power. However, the

linearization can not generally be applied to all types of modulations

and it still needs further iterations to solve the linear programming

problem. In [3], an iterative water-filling algorithm was suggested

to solve a weighted sum power minimization problem. Authors in

[4] proposed a set of problem formulations that allocate network

resources to optimize the Signal to Interference Ratio (SIR), to

maximize throughput and to minimize delays. Such formulations

belong to non linear optimization problems, but can be efficiently

solved through the usage of convex optimization methods.

When the transmission demand is increased, buffers will eventually

start to become overflowed. This results in QoS degradation in the

network system. In [5], authors focus on the buffer management in the

base station. The resource allocation scheme aimed to maximize the

total packet throughput subject to individual user’s outage probability.

This strategy permits to allocate traffic channels according to uses

channel conditions as well as traffic patterns. In [6], authors suggest

This work was supported in part by the Qatar National Research Fundunder the project NPRP 08 - 577 - 2 - 241.

resource allocation algorithm based on queue-balancing and ensuring

stability of the system and establishing threshold rate of arrival

traffic.Notably, most of the existing works, in the field of power

allocation, doesn’t consider the problem of buffer stability as a

constraint of the power allocation scheme. This paper investigates

power allocation scheme under the constraint of maintaining buffer

stability in a dynamic Time Division Multiple Access (TDMA) mode.

We propose an efficient power allocation strategy that minimizes

the total power consumption and assigns to the different users the

required number of time-slots in a dynamic way.

We start by formulating the minimization problem, then showing

that this problem is convex and we propose to solve it under

the Karush-Khun-Tucker (KKT) conditions. The optimal solution

is computed numerically using an appropriate initialization and an

algorithm deduced from Fixed Point Method in the simulations part

of this paper.

Due to the limitation of power resource, minimum power require-

ments for all users may not be satisfied simultaneously. This motivates

the investigation of admission control where users are not automati-

cally admitted into the network. The constraint of power control will

be taken in account by reformulating the initial optimization problem

through the addition of an upper bound on the power allowed to

each user. However, the minimization problem is no longer convex.

In order to solve it, we suggest an heuristic suboptimal algorithm that

utilizes the outcome of the initial problem.

The remainder of this paper is organized as follows. In Section

II, we describe the system model and we formulate the optimization

problem. In Section III, we prove the uniqueness and existence of

the global minimum solution and we formulate the KKT conditions

used in the derived results. In Section IV, we represent the power

allocation algorithm with admission control. In Section V, we states in

a detailed way the considered simulation parameters and we represent

the obtained results. Finally, conclusions are given in Section VI.

II. SYSTEM MODEL AND OPTIMIZATION PROBLEM

A. Dynamic TDMA system

The TDMA scheme is a technique for resource management based

on the basic idea of time-sharing. In this paper, we denote the total

signal occupancy period as T = N Ts, where Ts is the time-slot

duration and N is the total number of time-slots. The time of channel

occupancy of each user is equal to:

Tk = tk T,

where tk denotes the percentage of spectral occupancy of the user

“k” such that 0 ≤ tk ≤ 1 and

K∑

k=1

tk ≤ 1.

We consider a single cell downlink Dynamic TDMA system with

K users. The Channel State Information (CSI) is assumed accurately

known at the transmitter and the K receivers.

IEEE Globecom 2010 Workshop on Broadband Wireless Access

978-1-4244-8865-0/10/$26.00 ©2010 IEEE 824

In this paper, we assume that the channel is “slow flat fading

channels”. In fact, it is not straightforward to consider a more realistic

channel model and we point out that the frame length is smaller than

the channel coherence time. The received signal at the user “k” side

is given by:

yk = hk.s+ nk

where s is the transmitted signal, hk is the channel gain, assumed

constant during the period T and nk is an Additive White Gaussian

Noise (AWGN) having a variance equal to σ2k.

We consider that the channel gain to noise ratio of a user “k” has

the following expression:

γk =|hk|

2

σ2k

. (1)

Remind that the channel capacity for the kth user is given by

ck = B log2(1 + Pkγk)

where Pk denotes the amount of power allocated to the user “k”.

As we consider in our work a dynamic TDMA systems, it results

that the rate offered to the user “k” is equal to:

rk = tkck. (2)

We assume that the stability condition is verified by each user [7],

this is equivalent to say that when it is supposed a finite time horizon

n, the average offered traffic tends to the arrival data rate λk. This

means that rk = λk. Also, we must emphasize that λk is known

at the receiver side. Therefore: λk = tkck and as

K∑

k=1

tk ≤ 1, we

deduce the following stability condition[6]:

K∑

k=1

λk

ck≤ 1. (3)

However considering the stability approach in such a system, we

believe that it is a new approach which implies one of the main

contributions of this paper.

B. Problem formulation

Our main purpose consists in the minimization of the total

transmitted power when the buffer is maintained stable. Therefore,

we formulate our optimization problem by applying the stability

condition already presented in (3). Since our goal is to minimize

the total transmitted power, the transmitted rate rk must be closed

to the arrival data rate λk in order to reduce the waste of power. To

define the optimum power allocation for each user “k”, we propose

in (4) to minimize the total energy consumed during the period T ,

given by ET =∑K

k=1 PkTk =

K∑

k=1

λkPk

ckT . Since T is constant,

our optimization problem is formulated by:

minPk

K∑

k=1

λkPk

B log2 (1 + Pk γk)

subject to

K∑

k=1

λk

B log2 (1 + Pk γk)≤ 1 (C1)

(4)

where we assume that Pk ≥ 0, since Pk < 0 is not a feasible

solution.

Although the problem (4) is a non- linear optimization problem,

it will be shown in the next section that it has a unique and global

solution.

III. UNIQUENESS OF THE SOLUTION

Lemma 1: The non-linear optimization problem introduced in (4)

has a unique optimal solution.

Proof: The proof will be presented in three steps.

Step1: Introduction of variables change

In order to simplify the resolution of the minimization problem (4),

we introduce new variables xk and Ak given by xk = ln (1 + Pkγk)and Ak = λk ln(2)

B. The objective function is given by

f(X) =

K∑

k=1

Ak (exp (xk)− 1)

γkxk

(5)

subject to

g(X) =

K∑

k=1

Akγk

xk

− 1 (6)

where X = [x1, . . . , xK ]T .

Hence, the optimization problem (4) can be equivalently rewritten

in a standard form as:

Pbeq :

minX∈R

n

+

f (X)

subject to g (X) ≤ 0(7)

It is interesting to note that f which was given in (5) and g that

was given in (6) are convex and continuous functions, so Pbeqis a

convex programming problem.

Step 2: existence and uniqueness of the solution

Let L = {X ∈ Rn+; g (X) ≤ 0} a set of the feasible solution of

Pbeq . Since f and g are convex, L is a convex and closed set of

Rn+. Furthermore, lim

‖X‖→+∞f (X) = +∞ ∀X ∈ L and X > 0 .

We deduce that it exists a unique and global minimum, solution of

Pbeq [8].

Using the fact that there exists x̄ solution of Pbeq and noting that f

et g are differentiable at x̄, we can deduce that Karush-Khun-Tucker

(KKT) conditions are satisfied [8]. So, there exist a scalar α ≥ 0 such

the following conditions hold while considering a single constraint:

∇f (x̄) + α∇g (x̄) = 0

(α∇g (x̄)) = 0

g (x̄) ≤ 0

α ≥ 0

Consequently, we obtain the following system:

{

exp (xk) (xk − 1) + 1− αγk = 0, k = 1, ..,K

α(

∑K

k=1Ak

xk

− 1)

= 0 α ≥ 0(8)

Note that g is an active constraint if g (X) = 0. Assuming that g

is inactive, it means that g (X) 6= 0 and α = 0, then (8) becomes

exp (xk) (xk − 1) + 1 = 0 and it results that xk = 0 , k = 1, ..,K.

This is impossible because xk> 0 .

Hence, g is an active constraint, g (X) = 0 and α > 0.

As a result, (8) leads to the following system{

∑K

k=1Ak

xk

− 1 = 0

α = exp(xk)(xk−1)+1γk

> 0 , k = 1, ..,K.(9)

Finally, solving the system (9), we finds that

xk = W

(

α.γk − 1

e

)

+ 1 (10)

where W is the LambertW function 1.

1

825

Step 3: The obtained solution

Let us define the function

h (α) =

K∑

k=1

Ak

W(

α.γk−1e

)

+ 1− 1. (11)

It is important to mention that h is a decreasing function of α.

Furthermore, the Lambert W function is positive for α > 0 and

is strictly concave, then limα→+∞

h (α) = 0 and limα→0

h (α) = +∞. We

can conclude that it exists a unique α solution of h (α) = 0.

We deduce from the system (9) that

P∗k =

exp(

W(

α.γk−1e

)

+ 1)

− 1

γk. (12)

We recognize the expression of the capacity and the percentage of

the channel spectral occupancy (see Section II.1):

ck = B ln 2

(

W

((

α.γk − 1

e

)

+ 1

))

,

tk =Ak

(

W(

α.γk−1e

)

+ 1) .

It is clear that an explicit expression of α cannot be readily

obtained. However, the equation (11) can be solved numerically.

For finding the root of (11), we consider an iterative algorithm

deduced from the Fixed Point Method algorithm (see Algorithm 1)

with an appropriate initialization α0 defined as:

α0 =eAeqexp(Aeq) + 1

gb(13)

where Aeq = W (α0gb−1e

) =∑K

k=1 Ak and gb =∏K

k=1(γk)1K

.

We start the iterative process by this initial value (13) and we

consider the function defined as

en =K∑

k=1

Ak(

W(

αn γk−1e

)

+ 1) − 1. (14)

This iterative process stops once en = 0.

Algorithm 1 alpha search algorithm

Initialize α0,eps,e0and µ

en = e0;

αn = α0;while (en > eps)αn+1 = αn(1 + µen);calculate en+1 using (14);

en = en+1;

end

The appropriate initial value α0 further reduces the number of

iterations and the convergence is mostly obtained less than 10iterations.

IV. POWER ALLOCATION WITH ADMISSION CONTROL

As it was mentioned before, since the power resource at the

Base Station (BS) is limited, all user’s power requirement may not

be satisfied. An admission control mechanism should be employed

to determine which users to be admitted into the network. This

scenario is important for real-time multimedia application in order

to achieve the required QoS performance.In [9] authors proposed a

The Lambert W function is the inverse function of f (w) = wew whereew is the exponential function and w is any complex number.

cross-layer model based on adaptive resource allocation scheme by

developing different admission control and power time-slot allocation

algorithm. The joint of power allocation and admission control has

been considered in [10]. The global problem is posed as a two-

stage optimization problem. In the first admission control stage, we

define the set of admitted users. Then, we determine the amount of

power needed by these admitted users in the power allocation stage.

However, the admission control problem is combinatorial hard, which

introduces high complexity for practical implementation. Therefore,

a low-complexity solution approach for the joint admission control

and power allocation problem is highly desirable.

A. Problem formulation

We reformulate the initial optimization problem (4) by adding the

constraint of power control for each user,

minPk

K∑

k=1

λkPk

B log2 (1 + Pk γk)(C1)

subject to

K∑

k=1

λk

B log2 (1 + Pk γk)≤ 1

subject to 0 < Pk ≤ Pmax

(15)

where Pmax is the maximal power budget allowed to each user.

As we can see, the reformulated problem (15) is a no longer convex

and its resolution is not straightforward. Therefore, we resort to a

suboptimal heuristic algorithm to solve it.

B. Solution and proposed algorithm

In the following, we propose a suboptimal algorithm which over-

comes the main difficulty of finding a straightforward solution to the

considered power allocation problem (15). We propose a reduced-

complexity iterative algorithm that is constituted of three main steps.

At the beginning of this algorithm, we consider the solution obtained

in the initial optimization problem (12) We sort the initial power

allocated Pk in a decreasing order so P1 > P2 > · · · > PK . In some

cases, the second constraint is ignored, i.e Pk > Pmax. Therefore,

we propose to reallocate a new value of power to the corresponding

users. The power value that can be assigned can be equal to 0 or equal

to Pmax. In fact, we allocate to a given user, the amount Pmax only

if he can fulfill a number of bit bk = ⌊log2 (1 + Pmaxγk)⌋ 6= 0so tk is changed to tk = λk

B log2(1+Pmax γk), otherwise we assign

to him an amount of power equal to 0 and tk = 0. However, the

constraint of buffer stability cannot be respected since the value of

tk is changed. In order to solve this problem, we suggest a power

reallocation. In fact, after fixing the amount of power assigned to the

kth user and the time needed tk, we substitute this user from the

list of user K and we consider a number of user equal to K − 1.

The constraint is no longer equal to 1 but to 1 − tk. The proposed

algorithm is explained in Figure 1.

V. SIMULATION RESULTS

A. Parameters

We describe in this part the different parameters used in our simu-

lations which are given in Tab.I. We consider a single cell with 1kmradius. In the performed simulations, each user’s location is randomly

generated and evenly distributed over the cell. Users’data rate arrive

according to a Poisson process with rate λk. We consider the path

loss, shadowing and flat fading in the channel propagation model. For

path loss, we use the modified Hata urban propagation model. The

shadowing component follows a log-normal distribution with mean

826

Figure 1. Flowchart of power allocation with admission control

value of 0 and standard deviation of 8dB. We perform Monte-Carlo

simulations over 103 realizations. For each configuration, we consider

T = 104 time-slots to simulate the outcome of the proposed power

allocation schemes. Although we derived an optimal power allocation

scheme, it is not straightforward to apply this result in a more realistic

scenario. In order to make the optimization problem more tractable,

we use simplified simulations parameters.

Table ISIMULATION PARAMETERS

Bandwidth B = 5MhzCarrier frequency fc = 2Ghz

Noise power σ = −174 dBm/Hz

Average of arrival rate λ = 100kbit/sTime symbol duration Ts = 200 ns

B. Results

Illustrating the optimal scheme given in Section Section III, Figure

2 represents the time allocated to each user in terms of channel

conditions (4). We notice that the time allocated to each user is

decreasing when the radio channel conditions become better. In

Figure 3, we plotted the total throughput achieved and the arrival data

rate in terms of number of users. We remark that the total throughput

is closed to the total arrival data rate to ensure the buffer stability

and to minimize the total allocated power. We fix Pmax = 1dBm.

Figure 4 shows the power allocated with admission control in terms

of channel conditions. As it is shown, we can identify from this

figure “active” users which can be accepted by the admission control,

in order to guarantee the required system performance and buffer

stability. Figure 5 depicts the total power allocated in terms of the

average of the arrival data rate when the number of user is 15 and

30. It is clearly noticeable that when the number of users increases,

the total power allocated get higher. In fact, when the number of

users increases, the arrival traffic increases and the system becomes

more crowded and the demand in power will increase. Applying the

admission control mechanism (explained previously), we can remark

a minimization in the total consumed power (remain less then 1dBm)

since the user that has a bad channel conditions and who requires

the largest amount of power are dropped.

-20 -10 0 10 20 30 400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Normalized radio channel conditions (dBm)

pourc

enta

ge o

f tim

e a

llocate

d (

%)

Figure 2. The impact of gain channel to noise on the time allocation

5 10 15 20 25 300.5

1

1.5

2

2.5

3

Number of users

rate

(M

bit/s

)

total throughput

arrival data rate

Figure 3. total throughput versus number of users

-20 -15 -10 -5 0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

radio channel state (dB)

Pow

er

allo

cate

d (

dB

m)

Power allocated without admission control

Power allocated with admission control

Figure 4. The impact of admission control on power allocation

VI. CONCLUSIONS

In this paper, we have proposed a non linear power and time

allocation schemes in Dynamic TDMA system to minimize total

transmitted power while ensuring buffer stability. We had suggested

an optimal outcome for the initial optimization problem. As in any

827

10 20 30 40 50 60 70 80 90 1000.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5

mean of arrival data rate (kbits/s)

tota

l pow

er

consum

ed (

dB

m)

15 users

30 users

Figure 5. The impact of the variation of the arrival data rate on the totalpower consumed

practical transmission situation, we are constrained by the individual

user power budget, we reformulated the initial derived optimization

problem considering the inclusion of the constraint of power control.

The joint admission control and power control algorithm aims to

minimize the total consumed power and providing better QoS.

Finally, we proposed a reduced-complexity heuristic algorithm that

results in the efficient resolution of this optimization problem.

REFERENCES

[1] C. Y. Wong, R. S. Cheng, K. B. Letaief, and R. D. Murch, ”Mul-tiuser OFDM with adaptive subcarrier, bit and power allocation,” IEEE

J.Select. Areas Commun., vol. 17, pp.1747-1757, Oct. 1999.[2] Inhyoung Kim, Hae Leem Lee, Beomsup Kim and Lee, Y. H., ”On the

use of Linear Programming for Dynamic Subchannel and Bit allocationin Multiuser OFDM ,” IEEE Global Telecommunications Conference,vol. 6, pp. 3648 -3652, 2001.

[3] D. Yu, J. M. Chioffi, ”Iterative water-filling for optimal resource alloca-tion in OFDM Multiple-Access and broadcast channels,” in Proceedings

of IEEE GLOBECOM, Nov. 2006.[4] D. Julian, M. Chiang, D. O’Neill, and S. Boyd, ”QoS and fairness

constrained convex optimization of resource allocation for wirelesscellular and ad hoc networks,”. In Proceedings of IEEE INFOCOM 2002,vol. 2, pp. 477-486, June 2002.

[5] G. Li and H. Liu, ”Dynamic Resource Allocation with Finite BufferConstraints in Broadband OFDMA Networks,” in Proceedings of the

IEEE Wireless Communications and Networking Conference, New Or-

leans, vol. 2, pp. 1037-1042, March 2003.[6] S. Najeh and H. Besbes, ”Queue-Balancing Resource Allocation for

Rate-Stable OFDMA Systems,” in Proceedings of the IWCMC 2010,July 2010.

[7] S. Kittipiyakul and T. Javidi, ”A Fresh Look at Optimal SubcarrierAllocation in OFDMA Systems,” in Proceedings of the IEEE Conference

on Decision and Control, vol.3, pp. 3289- 3294, December, 2004.[8] R. Fletcher, Practical Methods of Optimization. John Wiley and sons,

1987.[9] J.Tang and X.Zhang, “Cross-Layer based modeling for quality of service

guarantees in mobile wireless networks,” IEEE Commun. Magazine,pp.100-106, Jan.2006.

[10] E. Matskani, N. D. Sidiropoulos, Z.-Q. Luo, and L. Tassiulas, “Convexapproximation techniques for joint multiuser downlink beamforming andadmission control,” IEEE Transactions on Wireless Communications,vol. 7, no. 7, pp. 2682–2693, 2008.

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