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Energy Efficient Resource Allocation in SC-FDMA

Uplink with Synchronous HARQ ConstraintsDan J. Dechene, Student Member, IEEE and Abdallah Shami, Member, IEEE

Abstract—In this paper we propose framework for energyefficient (margin adaptive, MA) resource allocation in multiuserSC-FDMA with synchronous HARQ constraints. Resource al-location is formulated as a two-stage problem where resourcesare allocated in both time and frequency. To limit the impactof retransmissions on the time-frequency problem segmentation,we propose use of a block scheduling interval that ensuresuplink users do not experience ARQ blocking. We formulate theoptimal MA allocation problem under this framework and basedon its structure, we propose a variable complexity sub-optimalapproach to minimize the energy in resource allocation. Resultsare presented for the energy performance relative to complexityand datarate.

Index Terms—SC-FDMA, Energy Efficiency, Margin Adaptive,

Resource AllocationI. INTRODUCTION

Energy efficiency is an on-going issue in modern wireless

communication systems as radio resources account for a large

portion of mobile device energy consumption. As the prolif-

eration of smaller and faster devices increases, efficient use of

limited battery resources becomes ever more paramount. The

radio resource allocation problem formulation of minimizing

transmission energy is known as the margin adaptive (MA)

problem [1]. MA problems are well-studied for a general

OFDM transmission systems [1], [2] however systems such as

uplink in 3GPP-LTE, employ localized SC-FDMA at the phys-

ical layer. Implementation of localized SC-FDMA suffers from

restrictions such as contiguous frequency block allocation and

limited flexibility in modulation and coding scheme (MCS)

assignment ultimately limiting the applicability of traditional

MA resource allocation framework to the SC-FDMA problem.

In this work, we propose an uplink resource allocation

scheme for localized SC-FDMA with a focus on energy

efficiency. The resource allocation problem is complicated by

restrictions imposed by synchronous hybrid automatic repeat

request (HARQ) such as transmission at the same rate for

retransmissions and by the fact that each UE can only transmit

a single transport block (TB) during a given scheduling frame.

Our proposed resource allocation method exploits periodicity

of the HARQ process in selection of a scheduling epoch toefficiently schedule uplink traffic.

The remainder of this paper is divided as follows. In

Section II we overview the details of the employed uplink

system model. In Section III and IV we formulate the energy-

optimal and suboptimal uplink resource allocation problem

with HARQ constraints respectively. In Section V simulation

results are provided. Finally in Section VI, conclusions are

drawn on this work.

D. J. Dechene and A. Shami are with the Department of Electrical andComputer Engineering, The University of Western Ontario, London, ON,Canada (e-mail: [email protected], [email protected]).

I I . SYSTEM MODEL

In this paper we focus on multi-user SC-FDMA used for

uplink in LTE, [3]. There are K users (UEs) within a single

cell, communicating with a single base station (eNB). Since

we focus on resource allocation within a single cell, for the

purpose of this paper, it is assumed that intercell interference is

negligible. The cell spectrum is divided into subcarriers which

are grouped into M resource blocks (RBs) consisting of 12subcarriers. Without loss of generality we assume there is an

Integer number (M ) of RBs.

The minimum duration of time that can be allocated to a

single UE is a scheduling block (SB). The nth SB falls in

the time duration [nT f , (n + 1)T f ) where T f is the subframe

duration (equal to 1ms) and is one RB in frequency. Resource

allocation decisions are made at each scheduling epoch. We

define the mth scheduling epoch as the time duration in

[mT e, (m+1)T e] and each epoch consists of T e/T f subframes

in time. In this paper, T e is chosen to be equal to 4T f . The

justification of this selection T e is discussed later. Figure 1

shows an example of the uplink frame layout with relevant

time durations shown. Successful transmission are shown in

gray, while failures are red. As shown retransmissions occur

exactly 8 subframes following the initial transmission using

the same number of resource blocks.

Localized SC-FDMA transmissions are limited to contiguityin frequency and each UE can transmit one TB using a com-

mon MCS mode per subframe1. Retransmissions employing

HARQ must be transmitted over the same number of resource

blocks using the same MCS after exactly 8 subframes have

passed from original transmission (non-adaptive HARQ).

In each subcarrier, there are N sym symbols in per subframe.

We assume a constant amount of symbols N ctrl are reserved

for control while the rest are for data transmission. We assume

for simplicity that valid TB sizes are those defined by CQI

values given in Table 7.2.3-1 of [4]. Consequently, the overall

number of data bits that can be transmitted per subframe over

a set N of RBs in frequency is given as

η(k, N ) = 12(N sym − N ctrl)k · |N| (1)

where k is the spectral efficiency from [4], N is a set of RBs

1For single layer (non-MIMO) transmissions. [4]

ARQ subframe = ARQw = 8T F

ARQ slot (m)=T e=4T F

F r e q u e n c y

( R B s )

LTE

subframe (n)=T F

Time

(subframes, n)

8 subframes

Fig. 1: Frame Layout

978-1-61284-231-8/11/$26.00 ©2011 IEEE

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

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in frequency and | · | is the size of a set. For N sym = 14(regular cyclic prefix, [4]) and assuming N ctrl = 3, the above

reduces to η(k, N ) = 132k · |N|.

The SNR of the channel between each UE and the eNB

and from RB to RB is independent. A block fading model is

assumed, where the channel is static for the duration of the

decision epoch and independent from epoch to epoch. Channel

estimation is assumed to be error-free and available at the eNB.

The block error rate (BLER) is the average failure rate of

TBs. In general this will depend on γ i,eff (the effective SNR

of a transmission), T i (the TB size), and ki (the MCS em-

ployed). There exists no analytical solution to obtain the error

rate of a coded transmission. Practically, estimates of BLER

are based on receiver training/calibration. For the remainder

of this paper, the measure of Information Outage probability

(IOP) derived in [5] is used as a measure for BLER. The

IOP was shown to closely model BLER for moderate block

lengths. Extensions are trivial when system BLER can be more

accurately estimated and stored for a particular implementation

(via proper training). From [5], we obtain

BLER(γ, N i, T i) ≈ Q⎛⎝ log(1 + γ − log(2)T i

132|N i| 2

132|N i|γ

1+γ

⎞⎠ (2)

where N i is set of RBs in use, Q(·) is the well-known Q-

function and γ i,eff is the effective SNR. 132|N i| and T i132|N i|

corresponds to the number of modulated symbols and effective

spectral efficiency per allocation respectively.

The measured effective SNR of a transmission is related

to the SNR of the subchannels allocated for that particular

transmission. As in [6], the effective SNR of an SC-FDMA

symbol cannot be approximated using EESM or MIESM (as

in OFDM), but rather it can be computed as the average SNR

over the set of subchannels or as

γ (0)i,eff (m, N i) =

1

|N i|

k∈N i

γ i,k(m)

|N i|(3)

where γ i,k(m) is the instantaneous SNR in resource block kseen from UE i relative to a reference power level γ 0 and

N i is the set of RBs in the computation. The superscript

(x) is used to denote the xth retransmission number where

0 is the initial transmission. The additional |N i| in the above

equation describes that power is equally allocated across the

all assigned resource blocks. Further, we can model the effect

of retransmissions as an increase in the measured SNR [7]. We

employ this model with coefficients for SNR gains given in Ta-

ble 2 of [7]. We note however that for a given implementation,these gain factors can be obtained through receiver calibration

measurements. The effective SNR of retransmission z is then

given as (in dB)

γ (z)i,eff,(dB)(m, N i, ki) = γ

(0)i,eff,(dB)(m, N i)

+ μ(z, ki) · Rcode + (z, ki) (4)

where Rcode is the code rate ×1024 which is given in [4]

(i.e., 78, 120, . . .), μ(z, ki) and (z, ki) are given from [7]

for each retransmission and modulation scheme and where

γ (0)i,eff (m, N i) is the effective SNR for UE i given in (3).

The required power to obtain BLERtgt is then simply

P i =γ i,eff ( N i, T i)

γ (z)i,eff (m, N i)

(5)

where γ i,eff ( N i, T i) is the γ argument in (2) when (2) is set

equal to BLERtgt for given arguments N i and T i.

III . OPTIMAL ALLOCATION FRAMEWORK

It is well-known [2] that OFDM MA problems are NP-

hard due to Integer constraints on the subcarrier allocation

variables. The problem is further complicated in localized SC-

FDMA due to restrictions on contiguity of allocated resources.

Nevertheless, for the interest of the reader we formulate the

MA resource allocation optimization problem. Based on the

characteristics of this problem formulation, we propose an

online suboptimal scheme that tries to minimize the average

transmission power of each new transmission in Section IV.

A. Time Domain, Time-Frequency Domain Packet Scheduler

Recent research, particularly for LTE systems, proposes

segmenting the packet scheduling problem into a time-

domain packet scheduling (TDPS) and frequency-domain

packet scheduling (FDPS) problem. Such methods [8] choose

a set of users at each scheduling interval to schedule (TDPS)

and allocate them in frequency (FDPS) to maximize cell

throughput (rate adaptive, RA).

In the energy limited regime, the goal is to minimize

transmission energy while meeting throughput requirements.

In this regime, the optimal allocation for MA differs from

that of an RA problem. As a result, RA methods are not

directly applicable in solving the MA problem. Further, due

to limitations imposed by synchronous HARQ, the abovedescribed approaches may experience ARQ blocking since the

TDPS is limited in knowledge to a single subframe in time.

A simple method to eliminate ARQ blocking is to limit

the number of new transmissions a station can initiate in

each ARQ window. In synchronous HARQ, retransmissions

are limited to exactly ARQw subframes following the initial

transmission (where ARQw = 8 in LTE uplink) and the

maximum number of times a packet can be transmitted is

4. Based on these observations, the maximum number of

new transmissions that any station should initiate during any

ARQw subframes is ARQw/4 = 2 to eliminate any ARQ

blocking.

Based on this, we define an ARQ slot as the duration of time such that each UE can be allotted at most one new TB for

transmission. Consequently, each ARQ slot m has a duration

of 4 subframes or 1/2 of an ARQw (i.e., T e = 4T f ).

We formulate the resource allocation problem by segment-

ing it into a TDPS and Time-Frequency Domain packet

scheduler (TFDPS). At the beginning of scheduling epoch

(ARQ slot) m, the TDPS chooses which UEs can access the

channel (and with how much data, T i(m)) based on the buffers

and priority of each UE over ARQ slot m. The TFDPS then

allocates the UEs within the Time-Frequency grid of a ARQ

slot based on channel conditions and on-going retransmission.


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B. TDPS Formulation

During ARQ slot m, the TDPS analyzes the number of

resources available for transmission. Based on this information

combined with buffer and quality of service information,

the TDPS chooses the set of UEs to transmit with TB size

T i(m). For the remainder of this paper, we assume all UEs

transmit T i(m) = T̄ i bits per frame. The detailed design of

an energy efficient TDPS will be presented elsewhere.

C. TFDPS Formulation

The TFDPS is formulated as follows. Given a set of

UEs {1, 2, . . . , K } and a set of TB sizes for each UE

{T 1(m), T 2(m), . . . , T K (m)} for all m, the TFDPS en-

ergy minimization problem can be formulated to solve for

resource assignment (S i,j,n(m) and S i,j,n(m), ∀i,j,n,m),

power assignment (P i,n(m), ∀i,n,m), and rate assignment

(ki,n(m), ∀i,n,m). Here S i,j,n(m) and S i,j,n(m) are binary

indicators that denote whether a given RB j is allocated to

a given UE i for both new transmissions and retransmissions

respectively, P i,n(m) denotes the power allocated to UE i and

ki,n(m) denotes the spectral efficiency of the MCS chosen for

UE i all during subframe n of ARQ slot m.

The energy-optimal objective function can be given as

min

⎛⎝ lim

t→∞

1

t

tm=0

K i=1

M j=1

3n=0

αi

(S i,j,n(m) + S i,j,n(m))P i,n(m)

(6)

where αi is the relative importance parameter of power mini-

mization for UE i and subframe n ∈ {0, 1, 2, 3} within ARQ

slot m.

1) Rate Constraints: The amount of new information that

must be transmitted during a frame is constrained as

M j=1

3n=0

S i,j,n(m)ki,n(m) ≥ T i(m), ∀i, m (7)

2) HARQ Constraints: The ARQ constraints ensure that

resource allocation cannot occur in a subframe where a re-

transmission exists. This means S i,j,n(m) = 0, n ∈ S i(m −2), ∀i, m, where S i(m−2) is the set of subframes during ARQ

slot m − 2 whose transmissions were unsuccessful and have

not reached the maximum number of retransmissions. Further,

the number of RBs allocated to a user during a retransmission

must equal the original amount of RBs assigned, i.e.,

M j=1

S i,j,n(m) =M j=1

(S i,j,n(m − 2) + S i,j,n(m − 2)),

n ∈ S i(m − 2), ∀i, m (8)

3) New Transmission Limitation: In order to ensure ARQ

blocking does not occur, each station is limited to a new

transmission in a ARQ slot. This constraint is given as

n∈S C

i(m−2)

I

⎛⎝ M

j=1

S i,j,n(m)

⎞⎠ ≤ 1, ∀i, m (9)

where I (x) = 0 when x = 0 and 1 otherwise and S C i (m − 2)is the complementary set of S i(m − 2).

4) Allocation and Contiguity Constraints: Additionally, lo-

calized SC-FDMA is limited to RB allocations in contiguity,

this constraint can be given jointly as

K

i=1

(S i,j,n(m) +

S i,j,n(m)) ≤ 1,

S i,j,n(m), S i,j,n(m) ∈ {0, 1}, ∀ j, n, m (10)

ensuring only a single user can occupy an RB during an instant

of time and from [9] as

S i,j,n(m) − S i,j+1,n(m) + S i,x,n(m) ≤ 1,

x = j + 2, . . . , M , ∀i,j,n ∈ S C i (m − 2), m (11)

S i,j,n(m) − S i,j+1,n(m) + S i,x,n(m) ≤ 1,

x = j + 2, . . . , M , ∀i,j,n ∈ S i(m − 2), m (12)

5) Power Level Constraints: The applied power level con-

straints are expressed as

P i,n(m) =γ i,eff ( N i,n, T i(m))

γ (0)i,eff (m, N i,n)

, ∀i, n ∈ S C i (m−2), m (13)

P i,n(m) =γ i,eff ( ¯ N i,n, T i(m))

γ (zi,n)i,eff (m, ¯ N i,n)

, ∀i, n ∈ S i(m − 2), m (14)

where zi,n(m) is the retransmission number in subframe n for

UE i during ARQ slot m and N i,n = { j|S i,j,n(m) = 1} and¯ N i,n = { j|S i,j,n(m) = 1}

Combining (6)-(14), forms the optimal TFDPS allocation

for inputs T i(m), ∀m. In general, it is not computationallyfeasible to solve this problem for all time m. This is due to

the time dependance on m (i.e., scheduling for slot m depends

on the outcome of all previous slots m).

IV. SUB -O PTIMAL RESOURCE ALLOCATION SCHEME

As previously noted, it is not feasible to allocate resources

described in Section III for all time m in advance. Traditional

approaches in the literature [10] have done so by exhaus-

tively breaking the system into channel states and solving the

optimization problem as a function of these states. However

unlike the above methods, the SC-FDMA channel has a large

number of subchannels (RBs) resulting in a state space thatgrows exponentially with the number of RBs. This combined

with the Integer allocation constraints above, makes solving

the problem in this fashion infeasible.

A greedy, less complex approach is to solve the problem

online at each ARQ slot m. The resulting problem is still

rather complex, particularly when considering the contiguity

constraints. Recent work by Wong et al. [11] exploited these

contiguity constraints as a method of reducing the overall

problem search space to solve the SC-FDMA RA allocation

problem. This was accomplished by exploiting the property

that the number of exhaustive frequency contiguous allocations


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is significantly less than the exhaustive allocations for a gen-

eral OFDMA RA allocation problem. From the reduced search

space, allocations were solved for each UE to maximize the

cell uplink sum-rate. While the exhaustive set of allocations

reduces the search space, it is still quite large, particularly for

a moderate number of RBs in frequency and UEs.

In the case of an MA problem (where we look to minimize

energy used for transmission while meeting a target rate

rather than maximizing cell throughput), it is not necessary

to allocate all RBs as in RA problems (as with the approach

in [11]), but to simply allocate those necessary to minimize

transmission power while meeting rate constraints. Further,

Wong’s method suffers from needing to exhaustively consider

all possible allocations while in practice, only a small number

of allocations need to be considered in the problem.

In our proposed method, we adapt the approach in [11] for

the MA problem and solve the problem for each ARQ slot m.

The proposed sub-optimal algorithm provides for a varying

degree of complexity. By including a limited number of the

most energy efficient allocations for a UE determined based

on channel measurements, the optimization framework canconsider a reduced search space. We denote this as the Best-

N method. The search space increases as N increases, and

in the limiting case when N is large, this method obtains

the optimal allocation at each ARQ slot m. The algorithm

operates as follows. At the beginning of each ARQ slot

m determine the subset of RBs available for transmission

as described in Section IV-A. For each UE, find the N most energy efficient allocations (power, rate and subcarriers)

subject to the restrictions and details described in Section IV-B

and formulate the reduced search-space optimization problem.

Finally, allocate the chosen subcarriers to each UE at a given

power level and using a given MCS.

A. Residual Resource Allocation

Due to the synchronous HARQ mechanism, any transmis-

sion that was erroneous, and has not exceeded the maximum

number of transmissions is rescheduled exactly 8 subframes

following its original transmission. These transmissions are

scheduled using the same set of RBs2 and using the same

MCS. To accommodate these restrictions, we denote the

resource restriction for each UE the vector given as Si(m)T =[ST i,1, ST i,2, ST i,3, ST i,4] where Si,n has elements given as

S i,n:j =

⎧⎨⎩

1, if block {n, j} is reserved for ReTx

1, if UE i ReTx during n0, otherwise j = 1, . . . , M

(15)

The resource restriction vector ensures that UEs are not able

to initiate a new transmission during subframes where they

have a retransmission, and cannot be allocated any resources

that are reserved for retransmission by any UE.

B. Best-N Allocation Computation

The Best-N allocations represent the N allocations (set

of RBs, MCS mode, and power allocated) that are the most

energy efficient for the UE use for transmission. We denote

2Retransmissions are restricted to the same RBs in frequency.

the tuple B i,n ≡ {ki, P i, N i} as the nth most energy efficient

allocation for a UE i. These allocations minimize the required

applied power to satisfy a target error rate BLERtgt. To obtain

these allocation, for each UE i, we perform the following

1) For known T i(m), obtain the number of RBs M (ki)needed for each MCS mode ki given from [4] and

using (1)

2) For each unique quantity of resource blocks M

(ki),choose the minimum effective coding rate that can

transmit T i(m) bits of data during that block

3) For each such MCS scheme, determine the power level

needed to maintain BLERtgt for all possible allocations

of M (ki) contiguous resource blocks using (2). As the

Q-function is monotonic in its arguments, this can easy

be found through efficient search techniques.

4) Store the Best-N allocations, including MCS, power

level required and resource block set in B i,n ranked by

required power allocation level.

Once efficient allocations have been computed, the optimiza-

tion problem is a general set-packing problem and can be

formulated using binary programming as

minx

cT x (16)

s.t. Ax ≤ 1M , Aeqx = 1K , xj ∈ {0, 1}, ∀ j ∈ x

where c is a real-valued vector, x is the vector of allocation

selections, Aeq is the equality constraint matrix of K rows

and A is the inequality constraint matrix. Each non-zero

entry of the solution vector x corresponds to selecting the

corresponding column allocation in A. Consequently, the

matrices A and Aeq are given as

A= [

A1, · · · ,

AK ] (17)

Aeq =

⎡⎢⎣1T C 1

· · · 0T C K.

..

.

. .

.

..0T C 1

· · · 1T C K

⎤⎥⎦ (18)

where the columns of matrices A1, . . . ,AK each contain a set

of potential RB allocations for each UE based on the efficient

allocations found previously and C i is the number of columns

in Ai. The inequality ensures that at most 1 UE occupies any

RB in frequency during any subframe while the equality is

used to ensure exactly 1 allocation is chosen for each UE.

The submatrix Ai is obtained as follows. Let N (i, n) be

the set of RB allocations for the nth allocation of i (stored in

B i,n). Ai is then comprised of small submatrices for each of

the n allocations and for each subframe of the ARQ slot or as

Ai =

⎡⎢⎣ Ai,1 · · · 0M · · · · · · Ai,n · · · 0M

.... . .

... · · · · · ·...

. . ....

0M · · · Ai,1 · · · · · · 0M · · · Ai,n

⎤⎥⎦ (19)

where 0M is an M length column vector and Ai,n is a M length column vector given as

ai,n:j =

1, j ∈ N (i, n)0, otherwise

, j = 1, 2, . . . , M (20)

Finally, Ai is obtained by removing any columns from Ai

that share any non-zero row elements with the column vector

Si(m) (to account for pre-allocated retransmissions).


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2 2.5 3 3.5 4 4.5 5 5.5 6

9

10

11

Best−N Parameter (N)

A v e r a g e P o w e r ( d B )

2 2.5 3 3.5 4 4.5 5 5.5 6

0

500

1000

C o m p l e x i t y I n c r e a s e ( % )

Average Power, K=2

Average Power, K=6

Average Power, K=10

Complexity Time, K=2



Fig. 2: Complexity Comparison

200 400 600 800 1000 1200 1400 1600 1800 2000−8

−6

−4

−2

0

2

4

6

8

10

TB Size (bits)

A v e r a g e P o w e r p e r U E ( d B )

Average Power, K=2

Average Power, K=6

Average Power, K=10

Fig. 3: Datarate Performance

The objective function cT = [cT 1 , . . . ,cT K ] is simply the cost

of choosing the corresponding allocation. The problem objec-

tive vector for each ci is ci:j = αiP i(N ( j)), j = 1, 2, . . . , C iwhere P i(N ( j)) is the power level required for the nth

transmission scheme, αi is the priority of UE i and N ( j)

denotes the nth allocation corresponding to column j in Ai.

The above problem can be solved using binary programming

techniques such as branch and bound at each ARQ slot m.

The proposed optimization framework allows for solving

the resource allocation problem efficiently at each scheduling

epoch. While the proposed approach dramatically reduces

the problem complexity, in rare cases the solution may be

infeasible during a given scheduling epoch as we do not

exhaustively allow selection of every possible allocation. To

minimize the probability of this occurrence the designer may

take a number of steps including limiting the maximum

number of resource blocks that can be allocated to a single

user as a function of available, enforcing overlap restrictions

on the Best-N selection scheme or incrementing N and re-

solving the problem when a solution is found infeasible.

V. RESULTS

To evaluate the performance of our proposed method, we

implement the above framework and enforce the overlap

restriction described above. The number of UEs and RBs

is configured to 10 and 24 respectively with T f = 1ms,

T e = 4ms, the Best-N depth set to 3 and all UEs have a

priority (αi) equal to 1. The reference SNR is set to γ 0 = 10dB

and BLERtgt = 10%. Each subchannel is assumed to follow

the Rayleigh SNR distribution p(γ ) = 1γ 0

exp

− γ γ 0

where

γ 0 is the mean SNR for a reference power level. Results are

averaged over 40000 subframes. Due to length restrictions,limited results are shown for complexity and datarate.

To observe the complexity tradeoff, Figure 2 shows the

average power applied per UE versus the number of Best-

N allocations. We see that beyond 3, there is little reduction

in average power, while a large complexity increase (measured

via relative computation time) to the base case (2 UEs,

N = 2). Further, by increasing the number of UEs, there is an

increase in average power as result of each UE is less likely

to be scheduled using its locally optimal allocation. While we

note the effect of complexity and number of UEs on average

power, the dependance on per UE TB size has a much larger

effect on the average power performance (shown in Figure 3).

This confirms that proper design of the TDPS (i.e., minimizing

T i(m)) is vital in minimizing average transmission power.

VI . CONCLUSION

In this work we have proposed framework for energy

efficient resource allocation in the SC-FDMA multi-user up-link. Firstly, we proposed an alternative method of selecting

an appropriate scheduling epoch based on the impact of

synchronous HARQ. By utilizing the proposed method, we

can reduce the number of allocation procedures in time and

ensure users can always initiate a new transmission during

any frame (i.e.., do not experience ARQ blocking). Secondly,

we proposed a sub-optimal energy efficient resource allocation

method with a tunable complexity parameter to trade-off

efficiency and complexity. In our future work, we will present

the design of the energy efficient TDPS as well as look at

more efficient allocation methods.

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