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Minimizing the Transaction Time Difference for

NOMA-Based Mobile Edge Computing

Anam Yasir Kiani, Syed Ali Hassan, Binbin Su, Haris Pervaiz and Qiang Ni

Abstract—Non orthogonal multiple access (NOMA) and mo-

bile edge computing (MEC) are evolving as key enablers for

ﬁfth generation (5G) networks as this combination can provide

high spectral efﬁciency, improved quality-of-service (QoS), and

lower latency. This letter aims to minimize the transaction

time difference of two NOMA paired users ofﬂoading data

to MEC servers by optimizing their transmission powers and

computational resources of severs using a successive convex

approximation method. The equalization of transaction time

for paired users reduces the wastage of both frequency and

computational resources, and improves effective throughput of

the system to 19% on average.

I. INTRODUCTION

Some of the important concerns in today’s wireless net-

works are limited resources and media arbitration. The

medium used for the transmissions is being shared by mil-

lions of devices with heavy trafﬁc, which is expected to

increase by 1000 folds in the next decade. This would result

in services requiring high connectivity, reliability, ultra-low

latency, improved fairness and high throughput, etc. Non-

orthogonal multiple access (NOMA) has been introduced to

muddle through the demands of the epoch. One of main

purposes of NOMA is to serve multiple users by utilizing

the same resource block. NOMA provides a balanced trade-

off between the system throughput and user fairness [1] and

is being envisioned as a key technology in 5G networks. 5G

enabled devices are also expected to have latency constraint

and have computationally complex applications running on

them. For such applications, limited power and computational

capacity of mobile devices pose a problem, which can be

solved by using mobile edge computing (MEC) [2]. MEC

ofﬂoads computationally intensive data to base stations (BSs)

and access points (APs) that are equipped with powerful

servers. Servers being available at the edges result in reduction

of delay and improvement of computational efﬁciency [3]. The

advantages of both techniques (i.e., NOMA and MEC) have

drawn considerable attention of the researchers recently. In

NOMA-MEC, paired users ofﬂoad their data to MEC servers

by using the underlying NOMA principle.

A lot of work is being done in this context. For in-

stance, [4] formulated delay minimization for NOMA-MEC

data ofﬂoading as a form of fractional programming. Pure

NOMA is also compared with hybrid NOMA and orthogonal

multiple access (OMA) for data ofﬂoading purpose. In [5],

energy consumption of MEC users utilizing uplink NOMA is

reduced by optimizing user clustering, power, frequency and

computational resource allocation. [6] proved that the total

energy minimization is a convex problem and the authors

solved it by an iterative algorithm. [7] reduced the total

system energy by optimizing allocated power, transmission

time and ofﬂoaded task portions. In [8], energy consumption

is reduced by jointly optimizing power and time allocation by

formulating the problem to a form of geometric programming.

[9] studied energy harvesting for full duplex NOMA-MEC,

where total energy consumption is minimized by efﬁcient

power allocation, time scheduling and computing resources

allocation.

The time taken to process data (ofﬂoaded to MEC servers) is

not equal for each NOMA paired user because it is dependent

upon the amount of ofﬂoaded data and the channel conditions

of the paired users, etc. This inequality leads to under-

utilization of resources and reduced spectral efﬁciency. In this

letter, we propose a scheme to optimize the transaction time

of paired users and to reduce the transaction time difference

between them to improve spectral efﬁciency and to conserve

both frequency and computational resources. Transaction time

is the sum of transmission time and computational time. The

difference between transaction times is reduced by equalizing

transmission time and computational time separately, which is

achieved by optimizing a) power allocation and b) computa-

tional resource allocation, respectively. When the transaction

time of paired users becomes closer, the difference between

the transaction times reduces and hence the wasteful resources.

In the sequel, we describe the NOMA-MEC model followed

by optimization of transaction time difference. The results and

conclusions are presented towards the end.

II. SY ST EM MO DE L

A single cell is considered with 2Nnumber of users, which

are served by a single BS. The BS is equipped with MEC

server having CTnumber of cores each with a computational

capability of fcycles/sec and total system bandwidth is BT.

Hybrid NOMA technique is used to pair the users into N

NOMA clusters, where each cluster has two users. A single

cluster with users u1and u2is considered for study. The user

u1is located at a distance dist1from BS whereas u2is located

at a distance dist2from BS, such that dist1<dist2. Without

the loss in generality, we assume that u1is a strong user with

allocated power p1and effective channel gain h1, however u2

is a weak user with allocated power p2and effective channel

gain h2, such that p1h1> p2h2. Let p1,max and p2,max are the

maximum transmission powers that can be allocated to u1and

u2, respectively. We assume that d1bits are ofﬂoaded by u1

and d2bits are ofﬂoaded by u2to the MEC server. Complete

ofﬂoading scheme is considered, where no local computation

is being performed. Each bit ofﬂoaded by u1requires c1

cycles and that of u2requires c2cycles for computation at

MEC server. The computational complexity of ofﬂoaded data

is dependent upon ofﬂoaded data type (i.e, video data requires

more CPU cycles as compared to text data). The total system

bandwidth is divided into Nnumber of frequency resource

blocks. A single frequency resource block with bandwidth

Bw=BT/N is allocated to a NOMA cluster and shared by

paired users, similarly the cores at MEC servers are divided

into Nnumber of computational resource blocks and are

allocated to NOMA clusters. The allocated computational

resources of a cluster (i.e., Ct=CT/N) are divided among

the paired users, depending upon the complexity and amount

of data being ofﬂoaded by them. Let u1and u2are allocated

with n1and n2cores, respectively.

The transaction time of the ith user is Ti=Ttxi+Tci, i ∈

{1,2},where Ttxiis the transmission time and Tciis the com-

putational time of the ith user, respectively. The transmission

time for the ith user is Ttxi=di

Ri,where Riis the data rate

of the ith user. In this work, the 2-user uplink NOMA cluster

is considered in which u1and u2experience channel gains

of f

h1and f

h2such that the user u1signal will be decoded

ﬁrst at BS. The achievable data rate of user u1will include

the interference from the user u2whereas the achievable data

rate of user u2will include noise only. The data rates are

dependent upon the effective channel gains (i.e., h1, h2) and

the allocated powers (i.e., p1, p2), such that [10]

R1=Bwlog21 + p1h1

p2h2+σ2,(1)

R2=Bwlog21 + p2h2

σ2,(2)

where σ2=Bw×σ2,σ2is the power spectral density of

noise and the effective channel gain for ith user is hi=

e

hi

distρ

i

, where e

hiis the exponential channel gain (corresponding to

Rayleigh fading) of ith user and ρis the path loss exponent.

The amount of data ofﬂoaded and the effective channel gains

are associated with the paired users, however, the powers

are optimized to reduce their transaction time difference.

Similarly, the computational time for the ith user is given by

Tci=dici

nif,where niis the number of cores allocated to the

user iand fis the computational capacity of each MEC core.

For a given paired users, di,ciand fare ﬁxed. The number of

computational resources allocated to the ith user is optimized

to balance the load across the cores in order to reduce the

difference between transaction time. By manipulating, it is

inferred that T1is equal to T2if

d1

R1

+d1c1

n1f=d2

R2

+d2c2

n2f

n1fd1+d1c1R1

n1fR1

=d2n2f+d2c2R2

n2fR2

d1(n1f+c1R1)

n1R1

=d2(n2f+c2R2)

n2R2

d1

d2

=R1

R2n1n2f+n1c2R2

n1n2f+n2c1R1(3)

Fig. 1: Unequal Transmission Time and Wasteful Resources

From (3), we can divide the original formulated problem into

two independent sub-problems and reformulate it as T1is

equal to T2, if Ttx1is equal to Ttx2as well as Tc1is equal

to Tc2. It is evident from Fig. 1 that unequal transmission

time results in wastage of allocated frequency resources. It

can be seen that for δtx =|Ttx1−Ttx2|amount of time, the

resources are under-utilized, i.e., a new NOMA signal cannot

be initiated. As this difference increases, the spectral efﬁciency

of the network decreases. The transmission time, Ttx, is equal

for both the users, if d1

d2=R1

R2,where R1and R2are the

data rates of both users u1and u2, respectively. Similarly,

the disparity in amount and computational complexity of data

ofﬂoaded by paired users (i.e., allocated with equal number of

cores) results in wastage of allocated computational resources.

The computational time difference δtc =|Ttc1−Ttc2|for both

the users is zero, if d1c1

n1f=d2c2

n2f,which can be written in

simpliﬁed form as d1

d2=n1c2

n2c1.

III. PROB LE M FOR MU LATI ON

The original problem described in the previous section is

given by

(P) min λ, (4a)

s.t. d1

R1

+d1c1

n1−d2

R2

+d2c2

n2≤λ,

(4b)

d2

R2

+d2c2

n2−d1

R1

+d1c1

n1≤λ,

(4c)

λ≥0(4d)

0≤pi≤pi,max i∈ {1,2}(4e)

0≤ni≤Cti∈ {1,2}(4f)

n1+n2≤Ct(4g)

where the problem (P) is subjected to constraints (4e), (4f)

and (4g) i.e., the power allocated to the individual user is

positive and less than respective maximum, the number of

cores allocated to the individual user is positive and less

than total number of allocated cores, moreover, sum of cores

allocated to both the users is less than or equal to the total

number of allocated cores. As can be seen from the formation

of problem (P), it can be decomposed into two independent op-

timization sub-problems. The results for original optimization

problem and sub-problems are equivalent. The objective of

the ﬁrst optimization problem is to minimize the transmission

time difference of given paired users with known di’s, by

optimizing the power allocation. From equations (1) and (2),

we have

R1+R2=Bwlog21 + p1h1+p2h2

σ2(5a)

R1≤Bwlog21 + p1h1

p2h2+σ2

=Bwlog21 + p1h1+p2h2

σ2−R2,

(5b)

R2≤Bwlog21 + p2h2

σ2,(5c)

For the objective with power allocation, we introduce a new

variable µand hence the sub-problem of minimizing the trans-

mission time difference of paired users can be reformulated

as

(P1) min µ, (6a)

s.t. d1

R1

−d2

R2

≤µ, (6b)

d2

R2

−d1

R1

≤µ, (6c)

µ≥0(6d)

0≤pi≤pi,max i∈ {1,2}(6e)

where the objective function (6a) is subjected to data rate (5b,

5c) and power ( 6e) constraints. By manipulating (6b), we get

µR1R2≥µα1≥α2

2≥d1R2−d2R1,(7)

where α1and α2are real valued variables, having values such

that inequality holds. The equation (7) is equivalent to

R1R2≥α1,(8a)

µ α2

α2α10,(8b)

α2

2≥d1R2−d2R1,(8c)

where (8b) is a convex linear matrix inequality (LMI), and (8c)

is non-convex. The non-convex parts in left side of (8c) can

be approximated using the Taylor series expansion to get the

approximated lower bound. By applying the ﬁrst Order Taylor

Approximation, the left side of (8c) can be approximated as

α22≥α(j)

22+ 2α(j)

2α2−α(j)

2

α22≥α(j)

22+ 2α(j)

2α2−2α(j)

22

α22≥2α(j)

2α2−α(j)

22

(9)

The right side of Eq. (9) is the ﬁrst order approximation around

the point α(j)

2. By substituting Eq. (9) into left side of (8c),

the (8c) can be rewritten as follows:

2α(j)

2α2−(α(j)

2)2≥d1R2−d2R1,(10)

where jshows the number of iteration, α(j)

2denotes the value

of α2during the j−th iteration. The equation (8a) is rewritten

as

R1R2≥β2,(11)

where β2≥α1. The problem (P1) deﬁned in Eq. (6a) subject

to the constraints deﬁned in Eq. (8b), Eq. (10) and Eq. (11) is

a convex optimization problem and can be efﬁciently solved

using standard convex optimization tool such as CVX [11]. It

will provide a lower bound approximation solution [12], [13]

of (P1) due to the ﬁrst order Taylor approximation in Eq. (10).

Similarly, the objective of the second optimization problem

is to minimize the computational time difference of given

paired users with known di’s and ci’s by optimizing the core

allocation. By introducing a new variable ζ, the sub-problem

of the computational resource allocation can be transformed

as

(P2) min ζ, (12a)

s.t. d1c1

n1

−d2c2

n2

≤ζ, (12b)

d2c2

n2

−d1c1

n1

≤ζ, (12c)

ζ≥0,(12d)

0< ni< Ct,(12e)

n1+n2≤Ct(12f)

where the objective function (12a) is subject to constraints

(12e), the number of cores allocated to individual user is

greater than zero and less than total cores allocated to the

cluster and (12f), the sum of cores allocated to both the users

is less than or equal to total cores allocated to the cluster. The

number of cores allocated to individual user must be greater

than zero to ensure the minimum requirement of the user. The

integer constraint is relaxed for ni. By manipulating (12b), we

get

ζn1n2≥ζ γ1≥γ2

2≥(d1c1)n2−(d2c2)n1,(13)

where γ1and γ2are variables with real values. The equation

(13) implies

n1n2≥γ1,(14a)

ζ γ2

γ2γ10,(14b)

γ2

2≥(d1c1)n2−(d2c2)n1,(14c)

where (14b) is a convex LMI. The left side of (14c) can

be approximated using the Taylor series expansion to get the

approximated lower bound. By applying the ﬁrst Order Taylor

Approximation, the left side of (14c) can be approximated as

γ22≥2γ(j)

2γ2−γ(j)

22

(15)

The right side of Eq. (15) is the ﬁrst order approximation

around the point γ(j)

2. By substituting Eq. (15) into left

side of (14c), the (14c) can be rewritten as follows:

2γ(j)

2γ2−γ(j)

22≥(d1c1)n2−(d2c2)n1,(16)

Fig. 2: Transaction Time difference for Benchmark (or No Optimization) Case, Approaches A, B and C

where γi’s are updated in each iteration and jshows the

number of iteration. From (14a), we have

n1n2≥η2,(17)

where η2≥γ1. The problem (P2) deﬁned in Eq. (12a) subject

to the constraints deﬁned in Eq. (14b), Eq. (16) and Eq. (17)

is a convex optimization problem and can be efﬁciently solved

using standard convex optimization tool such as CVX [11].

For a given pair of users, we obtain optimal values of power

and number of cores once the optimization is performed.

These parameters result in minimization of transaction time

difference, which is illustrated in next section.

IV. PERFORMANCE EVALUATION

The maximum power for u1,p1,max, is 2W and of u2,p2,max,

is 4W. Initially p1is 1W and p2is 2W. The dist1and dist2are

200 m and 600 m, respectively. The cluster bandwidth is 200

kHz and the path loss exponent is 3.8. The transaction time

difference is considered for three different approaches namely:

Power Optimization with Equal Core Allocation (A), Power

Optimization with Random Core Allocation (B) and proposed

Power Optimization with Optimal Core Allocation (C). The

power and core optimization is achieved by successive convex

approximation as discussed in the Section III. In equal core

allocation, the cores are equally divided between the paired

users, i.e., n1=n2. In random core allocation, the cores are

randomly divided between the paired users n1=κCtand

n2= (1−κ)Ct, where, κis from uniform random distribution

varying from 0 to 1. The ratio of ofﬂoaded data amount,

i.e., d2/d1and complexity, i.e., c2/c1is varied to study their

impact on the transaction time.

Fig. 2 depicts the transaction time difference without any

power optimization and equal number of core allocation, i.e.,

benchmark case, approach A, approach B and approach C

in contour plots from left to right. It can be observed in

all plots that for a ﬁxed value of d2/d1, different values

of c2/c1result in different transaction time differences. The

larger the transaction time difference, the more the under-

utilized resources. It can be observed that the transaction

time difference for second plot (i.e., approach A) is overall

Fig. 3: Comparison of Simulation and Optimization Results

lesser than the previous. For the same ratios of d2/d1and

c2/c1, the transaction time difference is reduced by optimizing

only the power allocations. The transaction time difference

for Approach B (i.e., third plot) is lesser than the transaction

time difference for ﬁrst plot, i.e., benchmark case. However,

this difference is comparable with Approach A, as the only

difference is in the core allocation. The transaction time

difference for proposed scheme (i.e., Approach C, fourth plot

in Fig. 2), where both the power and cores are optimized, is

minimum. It is also clear that the paired users have optimal

values of d2/d1and c2/c1for which the transaction time

difference is minimum. For instance in fourth plot Fig. 2, when

d2/d1= 1.2, the transaction time difference is 2 seconds for

c2/c1of 0.7. As d2/d1is increased to 1.7, the transaction time

gap jumps to 35 seconds for the same ratio of c2/c1. Similarly,

when d2/d1is decreased to 0.7, the transaction time difference

increases to 20 seconds.

To validate the proposed solution, the approaches A, B and

C are also solved heuristically by searching over the whole

solution space labelled as ”Simulation” and compared with the

results obtained for the approaches A, B and C using succes-

sive convex optimization method labelled as ”Optimization” in

Fig. 4: Effective System Throughput

Fig.3. The maximum number of iterations for the Optimization

results for the Approaches A, B and C is set to 100. In Fig.3

when d2/d1is 0.4, the transaction time difference without

optimization is 822 seconds, approximately 402 seconds for

both heuristic and optimized solutions of Approach A and 467

seconds for Approach B. The transaction time difference for

Approach C goes to 196.1 and 61.58 seconds for optimized

and heuristic solutions,respectively. The difference between

the simulation and optimization results is due to use of Taylor

series expansion for approximation.

We now illustrate the effect of reducing the transaction

time on the effective throughput of the system. The effective

throughput of the system is given by

Φeff =P2

i=1 Ri

max (T1, T2),(18)

where the numerator is the sum of achieved data rates by

the paired users while the denominator is the maximum of

transaction times of the paired users. As both the users are

paired, therefore, the resources allocated to them are free only

when both of them complete their transactions, hence the de-

nominator is characterized by the max(.) operator. A decrease

in effective transaction time increases the system’s effective

throughput. It is clear from Fig. 4 that for a ﬁxed value of

c2/c1and a range of d2/d1, the system effective throughput

for proposed Approach C is greater than the other approaches.

It is also evident that larger is the ofﬂoaded data disparity,

the larger is the difference between the system’s effective

throughput for the compared schemes. The reason behind

this trend is the optimal core allocation. When the ofﬂoaded

data is same in characteristic (i.e., amount and complexity is

same), the cores allocation for the schemes are same (i.e.,

equal number of cores for no optimization, Approach A and

Approach C) and the difference in the throughput appears only

because of the power allocation. However, as the ofﬂoaded

data disparity increases, the proposed scheme outperforms

others. The average increase in the system effective throughput

is 19% for the case shown in Fig. 4.

V. CONCLUSION

In this letter, it has been shown that the transaction time

plays an important role in improving the overall resource

utilization and the transaction time difference of two users is

minimized by optimizing both the transmission powers and

computational resources allocation independently. The pro-

posed optimization resulted in increased effective throughput

of the system. As a future direction to this work, the joint

problem can also be investigated while considering correlation

both communication and computation resources. The approach

can be extended to multiple users in a NOMA cluster. A

data aware NOMA clustering scheme can be used where

the users are paired considering both the power disparity

as well as their data ofﬂoading requirements, which can

contribute further towards improvement of spectral efﬁciency

and system’s effective throughput.

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