Max-min fairness in downlink non-orthogonal multiple access with short packet communications

Journal Pre-proofs Regular paper Max-min Fairness in Downlink Non-Orthogonal Multiple Access with Short Packet Communications Fateme Salehi, Naaser Neda, Mohammad-Hassan Majidi PII: DOI: Reference:

S1434-8411(19)32157-0 https://doi.org/10.1016/j.aeue.2019.153028 AEUE 153028

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International Journal of Electronics and Communications

Received Date: Accepted Date:

28 August 2019 4 December 2019

Please cite this article as: F. Salehi, N. Neda, M-H. Majidi, Max-min Fairness in Downlink Non-Orthogonal Multiple Access with Short Packet Communications, International Journal of Electronics and Communications (2019), doi: https://doi.org/10.1016/j.aeue.2019.153028

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Max-min Fairness in Downlink Non-Orthogonal Multiple Access with Short Packet Communications Fateme Salehi, Naaser Neda, Mohammad-Hassan Majidi Faculty of Electrical and Computer Engineering, University of Birjand, Birjand, Iran

Abstract—In this paper, the performance of non-orthogonal multiple access (NOMA) technique for communication based on finite blocklength (FBL) codes is studied. NOMA, by co-serving multi users has great potential for achieving low-latency and better spectral efficiency rather than conventional orthogonal multiple access (OMA) techniques. Our goal is to have optimal power allocation to attain fair throughput in a DL-NOMA scenario. The optimization problem is formulated and an analytical approach using optimal solution search algorithm is introduced. Performance of the proposed NOMA system is compared with an OMA scheme which is benefiting from optimal power and blocklength allocation as well. Numerical results show that NOMA significantly improves the users’ fair throughput in FBL regime. Keywords – URLLC, finite blocklength, short packet communication, NOMA, max-min fairness I.

INTRODUCTION

The fifth generation (5G) of mobile communication systems, like its predecessor, supports human type communication for enhanced mobile broadband (eMBB) services such as telephony, multimedia, and mobile internet. Furthermore, two types of machine communications are considered in the 5G. The first one is massive machine type communications (mMTC) to connect everything that is benefiting from being connected and the next one is ultra-reliable low-latency communications (URLLC) for enabling mission-critical services [1,2]. Use cases of such services are industrial automation, smart transportation system, smart grid, e-health, virtual/augmented reality, and tactile internet [3]. Each of these applications has a specific latency and reliability requirement which was compared together in [3]. In some applications such as industrial automation, latency requirement is even less than 1 ms and reliability is about 99.9999999% (i.e., block error rate (BLER) 10-9) [3]. To realize low-latency communications, short packets with finite block length codes should be used to decrease the transmission delay. This will result in a fundamentally different system design and performance analysis. In fact the conventional Shannon’s information capacity, which assumes infinite blocklength of data, is no longer applicable. This is because in the FBL regime, decoding error probability in the receiver can’t be negligible due to short blocklength [4]. In [5], Polyanskiy and et.al calculated an exact approximation of transmission rate in the FBL regime for the AWGN channel. Triggered by that, research in this context developed to multiple-input multiple-output quasi-static fading channels [6], and quasi-static fading channels with retransmission [7] as well. Also, the effect of short packets in spectrum sharing and relaying networks were examined in [8] and [9] respectively. Scheduling of latency-critical packets was addressed in [10]. Furthermore, radio resource management in URLLC networks was investigated in [11-16]. 

Corresponding author. E-mail addresses: {f.salehi, nneda, m.majidi}@birjand.ac.ir

1

On the other hand, it’s well known that NOMA, as an efficient radio access technique, by serving multiple users simultaneously, has a great capability of supporting low-latency communications and in comparison with conventional OMA, significantly improves the spectral efficiency. Performance of NOMA along with techniques like beamforming and cognitive radio was addressed in [17] and [18], respectively. A multiple-input single-output NOMA cognitive radio network relying on simultaneous wireless information and power transfer were studied in [19] and [20]. In [19], in order to improve the security of the primary network, an artificial-noise-aided cooperative jamming scheme was proposed. In [20], robust beamforming and power splitting ratio were jointly designed for minimizing the transmission power of the cognitive base station and maximizing the total harvested energy of the secondary users. In [21], NOMA capability in a decode-and-forward relaying system with energy harvesting nodes was considered. Furthermore OFDM-NOMA was recognized as a promising technique for next generation wireless communication in [22], where it suggests that combining NOMA and short packets, is a promising approach for low latency systems. In [23], the performance of NOMA in short-packet communications was investigated and the optimal power allocation and blocklength were obtained asymptotically for high SNR scenarios. Also, a closed form of achievable transmission latency reduction of NOMA compared to conventional OMA was characterized in [23]. In [24], the transmission rates and power allocation of NOMA scheme were optimized to make a trade-off between system throughput and user fairness. In that study, the effective throughput of the user with a higher channel gain was maximized while a predefined effective throughput target for the other user was guaranteed. In [25], the energy of the transmitter was minimized subject to heterogeneous reliability and latency constraints at receivers. It was shown that in contrast to conventional NOMA with homogeneous constraints, successive interference cancellation (SIC) maybe unfeasible and therefore different interference management techniques were introduced accordingly. In this paper, we merge short packet communications with NOMA and investigate its advantage in achieving higher fair throughput over its OMA counterpart subject to the same reliability, blocklength, and total transmit energy constraints. In contrast to the previous works, here we target to guarantee the QoS requirements of URLLC, i.e., achieving ultra-reliability and low latency and providing fair throughput among users at the same time. Fair throughput is an important criterion that, to the best of our knowledge, has not been considered in the short packet communication, so far. It should be noted that the blocklength is directly related to the latency. We consider a typical MTC scenario in local area communications [26] with two users and one BS which is assumed has access to perfect channel state information (CSI). The two users are assumed to be in one cluster and share one resource block. To fully exploit the benefit of NOMA, the channel gains of the users in the same cluster must be significantly different. Unlike the conventional communication with infinite blocklength where SIC could be assumed errorless, in the FBL regime, it is not possible to have a perfect SIC. This leads to challenges in optimal design of the NOMA scheme which is addressed in this paper. Since the successful connection of each user can be very critical, providing fairness between the users will be an important metric in URLLC and 5G systems [27]. In this work, like related works in [23-25], we consider a two-user single antenna scheme which serves as the basic scenario of more general cases, like [28] and [29]. The researchers in [28] and [29] considered more general multiple antenna two-way relay networks to design beamforming and energyefficient optimization respectively. However they assumed infinite blocklength codes which are not

2

applicable in URLLC. Therefore, extending our study to more general cases remains as a subject for future research. Our main contributions in this work are summarized as follows:  We specify the optimal design of the NOMA scheme by power allocation between two users to achieve fairness. Maximizing the minimum throughput is considered as the fairness criterion. To the best of our knowledge, max-min fairness problem in the FBL regime has not been addressed yet. A moderate approach to solve the complicated formulation of achievable rate in presence of imperfect SIC is introduced.  To show the advantages of the NOMA scheme in the context of FBL regime, its optimal OMA counterpart is considered. To this end, users’ power and blocklength are jointly optimized and optimal transmission rate is obtained.  The computational complexity of max-min fairness design algorithms in NOMA and OMA schemes are determined.  For comparison purposes, we consider a two user URLLC scenario with different reliability but the same delay requirements. It demonstrates that the NOMA scheme compared to OMA with same constraints, achieves higher fair throughput. The rest of the paper is organized as follows. In section II, the system model and transmission strategies with NOMA are provided. Problem formulation in two forms of NOMA and OMA schemes is presented in section III. Section IV solves the optimization problems. In section V, numerical results are provided. Finally, section VI concludes this work. II.

SYSTEM MODEL AND TRANSMISSION STRATEGIES WITH NOMA

As shown in Fig. 1(a), a DL communication system with two URLLC users is considered. Since the FBL regime consists of short packets transmission, it’s assumed that the channel is constant during one frame and it varies independently from one frame to the next one. According to [5], the achievable data rate Ri at user i, i 1, 2 , for a finite blocklength mi symbols ( mi

100 ), and an acceptable BLER  i ,

has an exact approximation as 1 Vi Q   i  log 2 mi  mi ln 2 mi

Ri  Ci 

where Ci  log 2 1   i  is the Shannon capacity, the Gaussian Q-function i.e., Q  x  





x

(1)

 i is the SNR/SINR ratio of user i, Q1  is inverse of

1 t2 2 exp( )dt , and Vi  1  1   i  is the channel 2 2

dispersion of a quasi-static Rayleigh fading channel. Therefore, the decoding error probability at user i is given by

  Ci  Ri  log 2 mi mi  ln 2   Q  i , Ri , mi   Vi mi  

εi  Q 

(2)

It’s assumed that the BS and each of the users have one antenna, and two users are grouped into one cluster. The NOMA scheme allows the BS to simultaneously serve the users in one cluster by using the entire blocklength via a superposition coding. It means that user multiplexing is performed in the power domain. In the receiver side, the user with stronger channel decodes and removes the signal of the user with weaker channel by SIC technique and extracts its own data. In this way, user i can cancel the 3

interference of the weaker users and its achievable rate related to the decoding of xk is as follows,

i, k 1, 2 , 1 Vi ,k Q   i ,k 

Ri ,k  Ci ,k 

mk

ln 2



log 2 mk mk

(3)

where  i , k is the decoding error of xk at user i. According to power domain NOMA principle, in the two users scenario, BS transmits



2 i 1

pi xi , where xi and pi are the message and the allocated power of

user i respectively. The received signal at user i is given by

yi  hi





p1 x1  p2 x2  ni

(4)

2 where ni is the complex additive white Gaussian noise with variance  i , and hi shows the channel

coefficient between BS and user i. Without loss of generality, it’s assumed that h1

2

 12  h2

2

 22 .

User 2 directly detects x2 by considering x1 as interference, and the received SINR of x2 at user 2 is given by

 2,2 

p2 h2

2

(5)

p1 h2   22 2

The decoding error probability of x2 at user 2 is denoted by  2,2 , where based on (2) is approximated by



ε 2,2  Q  2,2 , R2,2 , m2



(6)

Since x2 is directly detected,  2,2 is the overall decoding error probability at user 2, i.e.,  2   2,2 . On the other hand, user 1 performs SIC. It first decodes x2 while x1 treats as interference and after that,

x1 is detected normally. So, the received SINR of x2 at user 1 is given by

 1,2 

p2 h1

2

p1 h1   12 2

(7)

Similarly, the decoding error probability of x2 at user 1 is approximated by

1,2  Q   1,2 , R1,2 , m2 

(8)

It’s a reasonable assumption that x1 can’t be detected correctly if x2 is in error. However, if user 1 successfully decode and remove x2 , following (4), the received SNR of x1 is given by

 1,1 

p1 h1

2

 12

(9)

Accordingly, the decoding error probability of x1 at user 1, i.e., 1,1 , is denoted by

1,1  Q   1,1 , R1,1 , m1 

(10)

Based on the above discussion, it’s clear that the overall decoding error probability at user 1 can be approximated as

1 = 1,2  1  1,2  1,1  1,2  1,1

4

(11)

where we took into account the fact that in URLLC services  i , k is in order of 10-5  10-9. III.

PROBLEM FORMULATION

In the considered scenario, BS should serve two users within Dmax symbols period to provide fairness subject to the total transmit energy. If a channel feedback is available at the transmitter side, users’ rates can be set according to their instantaneous channel conditions. In this case, a suitable criterion is the max-min fairness. The throughput of user i, Ti , is defined as the average bits per each complex symbol (or channel use), which is decoded correctly at the receiver;

Ti

mi Ri 1   i  Dmax

(12)

where Ri  Ri ,i and 1   i is defined as the reliability of user i, and it’s acceptable amount is defined by URLLC application.

(a) System model

P2 P1

P1

P2

m

m1

m2

(b) NOMA frame structure

(c) OMA frame structure

Fig. 1. System model and frame structures.

A. Optimization Problem in NOMA In the NOMA scheme, the superposition coding is performed at the transmitter such that the BS enables to transmit users’ signals with different power simultaneously, so m1  m2  m . In Fig. 1(b) a frame of NOMA is presented. Therefore, the desired optimization problem is formulated as follows

max min T1 , T2 

(13a)

s.t. p1  p2  Pave

(13b)

p1 , p2

pi  0, i  1, 2

(13c)

 i   i th , i  1, 2

(13d)

m  Dmax

(13e)

5

From (13a) it’s clear that the optimization variables are the allocated powers of two users subject to some constraints. Constraints (13b) and (13c) are to guarantee the practical power allocations, where

Pave shows the upper limit of average power of transmitter. (13d) guarantees that the preferred th reliability of the user i, doesn’t violate than  i . Moreover, finite blocklength of the two users is stated

by (13e). B. Optimization Problem in OMA As a benchmark for comparison, we formulate the optimization problem for an equivalent OMA scheme as well, where the two users are served in orthogonal (i.e., different) channel uses. Fig. 1(c) demonstrates the OMA frame structure. Now, the received signal at user i, i 1, 2 , is given by

yi  hi pi xi  ni

)14(

In (14), because of orthogonality, the users have no interference on each other, so, the SNR for each user is given by

i 

pi hi

2

)15(

 i2

As a result, following (2), the decoding error probability of user i is approximated by

 i  Q  i , Ri , mi 

)16(

Finally, the optimization problem of the OMA transmission scheme is formulated as

max min T1 , T2 

(17a)

s.t. m1 p1  m2 p2  Dmax Pave

(17b)

mi , pi i1,2

0  pi  Pave , i  1, 2

(17c)

 i   i th , i  1, 2

(17d)

m1  m2  Dmax

(17e)

where Ri in (12) defined by (1). Now the optimization parameters are the blocklength and power allocated to the two users. Constraints (17b)-(17d) are similar to (13b)-(13d) respectively. The coefficient in (17c) shows the peak to average power ratio (PAPR) factor. Also, (17e) is to guarantee that the two scenarios have the same overall latency. IV.

PROBLEM SOLVING

In this section we are going to introduce a step by step approach to solve the optimization problems in (13) and (17). To simplify the approach, we have to first analyze the constraints and determine their optimal status. Let’s first consider the constraint (13d) and (17d) on acceptable BLER of two users. Since each URLLC application needs a specific reliability, allocating more resources to get BLER lower than the th th required  i , wastes the resources. Therefore,  i   i is an optimal selection. In addition, it’s easy to





show that the throughput presented in (12), that is Ti  1   ith Ri mi Dmax , is a monotonic function of SNR/SINR  i . This is stated by proposition 1, as follows:

6

th Proposition 1: The acceptable data rate (i.e., Ri  0 ) in (1), where  i   i , is a monotonically increasing

function of SNR/SINR  i . (See Appendix A for proof.) Based on the Proposition 1, the inequality in (13b) or (17b) can be changed to equality for the optimal solution using the following corollary. Corollary 1: To maximize min T1 , T2  , equality in the energy constraint, i.e., m1 p1  m2 p2  Dmax Pave , is always guaranteed. (See Appendix B for proof.) Corollary 1 significantly simplifies the power allocation at the BS, since p2 is now related to p1 directly. Finally, Proposition 2 is essential for simplifying the solutions which is expressed here. Proposition 2: The optimum resource allocation in (13) and (17) should always get T1  T2 . (See Appendix C for proof.) To obtain the optimal solution of problems (13) and (17), we propose an approach according to the above propositions and corollary. A. Design of Max-Min Fairness in NOMA Based on the discussion presented above, one should note the following points: 

According to Corollary 1;  2,2 ,  1,2 , and  1,1 presented in (5), (7), and (9) respectively are just functions of p1 .



Based on Proposition 2, the optimal solution only can be achieved while T1  T2 . Using (12) and

 1   2th th  1  1

the fact that in the NOMA scheme m1  m2  Dmax , this leads to R1,1   

  R2,2 . 

A certain amount of achievable bit rate for user 2 is considered at the receiver 1 or 2. This means that R1,2  R2,2 .

Therefore, the optimization problem for NOMA in (13), can be rearranged as follows

max R2,2

(18a)

p1

s.t. Q   1,2 , R1,2 , Dmax   Q   1,1 , R1,1 , Dmax    Q   2,2 , R2,2 , Dmax   

th 1

(18b)

th 2

(18c)

 1   R1,2  R2,2 , R1,1    R2,2  1   (5), (7), and (9) th 2 th 1

(18d) (18e)

Although it is not possible to obtain p1 in a closed-form, however by searching algorithms like bisection method [30], problem (18) and equivalently the optimization problem (13) can be solved. Proposition 3:

 p1  which is defined as  p1  Q  1,2 , R1,2 , Dmax   Q  1,1 , R1,1 , Dmax 

)19(

is a convex function with respect to p1 . (See Appendix D for proof.) Proposition 3 indicates that there are at most two p1 values that guarantee value which satisfies

 p1   1th . The smaller

 p1   0 , is the optimal solution p1* that assures T1  T2 . Note that in NOMA 7

scenario, it is always assumed that the two users which share the same resources, experience different channel gains and since we suppose that h1

2

 12  h2

2

 22 , then we should have p1*  p2* . The

power allocation algorithm in the NOMA scheme is demonstrated in table 1. TABLE 1 POWER ALLOCATION ALGORITHM IN THE NOMA SCHEME th Input: total blocklength Dmax , overall BLER of user i  i , BS average power Pave , required accuracy . * * * Output: optimum power p1 , p2 , and fair throughput T1  T2  T0 .

1. Set p1  0 . 2. while

 1th do

3.

Set p1  min  p1  p, Pave  .

4.

Calculate  2,2 ,  1,2 , and  1,1 presented in (5), (7), and (9) respectively.

5.

Calculate R2,2 , R1,2 , and R1,1 using (18c) and (18d) respectively.

6. end while lb ub 7. Set p1  p1  p , and p1  p1 . 8. Find p1   p1lb , p1ub  that satisfies

 1th via bisection method with required accuracy .





* th * * * 9. return p1  p1 , p2  Pave  p1 and T0  1   2 R2,2 .

B. Design of Max-Min Fairness in OMA In the OMA scheme, based on the aforementioned discussion, the following points are considered 

According to Corollary 1 and (17e);  1 and

 2 determined in (15) are functions of only p1 and

m1 . 

 1   2th  m2  Based on Proposition 2 and (12), one can claim in the optimal design R1    R2 . th   1  1  m1 

Therefore, the optimization problem for OMA in (17), can be rearranged as follows

max R2

(20a)

s.t. Q   1 , R1 , m1   1th

(20b)

Q   2 , R2 , m2    2th

(20c)

m1 , p1

 1   2th  m2  R1    R2 th   1  1  m1  m1  m2  Dmax , and (15)

 m1, p1 

We consider the decoding error of user 1, as

(20d) (20e)

Q  1 , R1 , m1  . The above optimization

problem consists of two independent variables p1 and m1 . Because of nonlinear property of constraints, the feasible solutions which guarantee Q  1 , R1 , m1   1 are introduced using a searching th

algorithm via taking m1 as a constant in each iteration and attaining p1 .

8



* * Remark: It’s clear that m2  Dmax  m1 and p2*  Dmax Pave  m1* p1*

*



m2* . However, after determining

Pave , we choose the nearest solution to p2* among other feasible

the optimal solution, if p2

solutions, which doesn’t violet the power bound. In table 2, the power and blocklength allocation algorithm in the OMA scheme is demonstrated. TABLE 2 POWER AND BLOCKLENGTH ALLOCATION ALGORITHM IN THE OMA SCHEME th Input: total blocklength Dmax , overall BLER of user i  i , BS average power Pave , required accuracy . * * * Output: optimum power pi and blocklength mi allocated to user i, and fair throughput T1  T2  T0 .

1. Set p1  Pave . 2. for m1  1: Dmax

 1 do nothing if else  1th do while th

3. 4. 5. 6.

Set p1  max  p1  p,0  .

7.

Calculate

 1 , and  2 presented in (15). Calculate R2 , and R1 using (20c) and (20d) respectively.

8. 9. 10.

end while lb ub Set p1  p1 and p1  p1  p .

11.

Find p1   p1lb , p1ub  that satisfies

12. end if 13. end for



  m

 1th via bisection method with required accuracy .



* * 14. return m1* , p1*  arg max R2 , m2  Dmax  m1 , p2*  Dmax Pave  m1* p1*

T0*  1   2th

* 2

Dmax  R2 .



m2* , and

C. Computational complexity The complexity of a searching algorithm depends on the stopping criterion. For both algorithms presented in Table 1 and Table 2, the iterations stop, if the condition where

( j)

( j)



( j 1)



is satisfied,

is the result obtained after j iterations. According to [11,27], power allocation for two

NOMA users through a bisection procedure constraint to desired accuracy  (Table 1), has





computational complexity of O log 2 1th

 . On the other hand, power and blocklength allocation

among two OMA users through the algorithm in table 2 subject to desired accuracy , has computational





complexity of O Dmax log 2 1th

 . It reveals that the computational complexity of the OMA scheme

is Dmax times of those with the NOMA scheme. V.

NUMERICAL RESULTS

9

In this section, performance of the proposed NOMA scheme and its OMA counterpart are evaluated th 7 through the numerical results. A heterogeneous network is supposed with users BLER 1  10 and

 2th  105 . PAPR factor is considered as

 1.2 . Furthermore, the noise power at each receiver is

2 2 2 assumed as  1   2   . Numerical results are provided based on fixed channel gains and required

accuracy   10 15 . In order to realize the relation between the objective function and parameters of interest we consider the normalized channel gains of two users, i.e., hi and h2

2

 2 , set to be fixed. For instance h1

2

2

 2  0.8

 2  0.1 . Also, it is assumed that Pave  10 W and Dmax  200 channel uses, unless

otherwise stated.

 p1 

In Fig. 2, the plot of user 1 decoding error, i.e.,

in (19), is indicated versus p1 . It’s observed

th 7 there are two p1 values that satisfy 1  10 , where according to the outcome of Proposition 3, the

smaller value is optimal. It should be noted that due to the channel gains of the two users and limited power, the reliability of user 1 maybe never satisfied. In that case the solution does not exist.

Fig. 2. User 1 decoding error,

 p1  , versus

p1

in the NOMA scheme.

In Fig. 3, the two users’ throughput achieved by the NOMA scheme, defined in (12) and (3), is plotted versus p1 . It is clear that T1 and T2 are monotonically increasing and decreasing functions, respectively in the reasonable range (i.e., Ti > 0). Their intersection is the solution of the max-min fairness optimization problem. This means that with power allocation in about p1 * users have equal throughput as T0  0.52 bits ch.use .

10

1 W and p2 = 9 W , both

Fig. 3. The two users’ throughput achieved by the NOMA scheme versus

p1 .

The two users’ throughput defined in (12) and (1), are also plotted versus p1 and m1 in Figs 4 and 5 respectively. From these figures, it is clear that in the reasonable range, T1 and T2 are respectively monotonically increasing and decreasing functions in both p1 and m1 . The intersection of the two * curves in Fig. 4 with fixed and optimal blocklength value, shows the p1 . Similarly, the intersection of the * two curves in Fig. 5 with fixed and optimal power value, shows the m1 .

11

Fig. 4. The two users’ throughput achieved by the OMA scheme versus

p1 .

Fig. 5. The two users’ throughput achieved by the OMA scheme versus

m1 .

In Fig. 6, we plot the feasible solution set which guarantees the reliability requirement of user 1 in the OMA scheme (i.e.,

 m1, p1   1th ). Since the blocklength could only be an integer within the range

of 1 to Dmax , the feasible solutions are limited. Among the feasible solutions, that pair that results to maximum throughput and simultaneously does not lead to power limit violation is optimal, which is 12

* demonstrated by a red point in Fig. 6. We notice that this point which is equal to p1  5.5 W and

m1*  62 is the same cross point of the curves in Figs 4 and 5.

Fig. 6. The feasible solution set achieved by the OMA scheme.

In Figs 7 and 8, the throughput 𝑇1 = 𝑇2 = 𝑇0 achieved by the OMA scheme versus p1 and m1 is * illustrated respectively. It’s seen that both are concave and have one global maximum, T0  0.5 , which * * * * determines the optimal values of p1 and m1 . These optimal values, i.e., p1 and m1 , are determined

by red points in Fig. 7 and Fig. 8 respectively.

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Fig. 7. Throughput

T0

achieved by the OMA scheme versus

p1 .

Fig. 8. Throughput

T0

achieved by the OMA scheme versus

m1 .

In Fig. 9, the effect of maximum blocklength, Dmax , on the achievable fair throughput in the NOMA scheme is assessed and compared with the results of optimal OMA scheme. It’s observed the proposed NOMA scheme effectively outperforms the OMA, regardless of blocklength.

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Fig. 9. Maximum achievable fair throughput by the NOMA and OMA schemes versus

Dmax .

In Fig. 10, the effect of average power, Pave , on the achievable fair throughput is investigated. The superiority of the NOMA scheme with respect to the OMA is notable in this graph as well.

Fig. 10. Maximum achievable fair throughput by the NOMA and OMA schemes versus

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Pave .

VI. CONCLUSION In this paper, the combination of NOMA with short packets was considered in critical-IoT, where low latency is a requirement. On the other hand, achieving fairness in URLLC is essential to guarantee the QoS for all users of critical services. For a DL-NOMA system with two users, an optimization problem was formulated to address the fairness by optimizing power and rate subject to total energy, reliability and latency constraints. To simplify the design of optimal power allocation and users transmission rate, optimal states of inequality constraints were determined first and then, an analytical approach with an optimal solution search algorithm was proposed. The OMA was considered as an evaluation scheme of the proposed system. An analytical approach for optimal power and blocklength allocation was introduced for OMA, as well. Numerical results show that the proposed NOMA scheme improves the users’ fair throughput compared to the OMA with the same constraints, significantly. In addition, the computational complexity of the proposed algorithms was studied. It was indicated that design of NOMA with respect to OMA has lower complexity, because unlike the OMA scheme it doesn’t need the blocklength optimization. However, some concepts like jointly design of users clustering and transmission strategy in a multi-user scenario as well as extension of the proposed approach to a relaying enabled system remain for the future studies. APPENDIX A: PROOF OF PROPOSITION 1 Proof: To prove that the reasonable Ri in (1) is monotonically increasing function of the corresponding SNR/SINR, i.e.,  i , i 1, 2 , the partial derivative of Ri with respect to  i is derived as

  Ri log e2     1 2    i 1   i  1   i  1   i   1  where   Q1   i 

)A.1(





1/ 2

1  mi . From Ri  i  0 , the answer is given by ˆi   1  1  4 2  2 

 1 . In

the range of  i  ˆi , we have Ri  i  0 , so Ri is monotonically increasing function, and in the range of 0   i  ˆi , we have Ri  i  0 , i.e., Ri is monotonically decreasing function. On the other hand, for

 i  0 , we have Ri

log2 mi mi

0 for mi

100 . As a result, in the range of 0   i  ˆi , we

would have Ri  0 which is unacceptable. So one can suppose that Ri is a monotonically increasing function of

 i . It should be noted that in general Ri can be Ri ,k , and also  i can be  i , k . ■

APPENDIX B: PROOF OF COROLLARY 1 (ACCORDING TO [16, APPENDIX B]) † † † † Proof: We first suppose that optimal power allocation p1 and p2 , satisfying m1 p1  m2 p2  Dmax Pave , † † † gives the maximum of min T1 , T2  which is denoted by T0 . We increase p1 and p2 by multiplying in a

scalar value   Dmax Pave

m p

† 1 1

 m2 p2†  to attain p1*   p1† and p2*   p2† which satisfies

m1 p1*  m2 p2*  Dmax Pave . We note that since   1 , so p1*  p1† and p2*  p2† . According to (9) and (5), * † we have respectively  1   1 and

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 

p2* h2

2

p2† h2

2

p2† h2

2

 † 2   2† 2 )B.1( 2  p h2   p1 h2   2 p1† h2  2  † † * * This shows that as p1 and p2 increase to p1 and p2 respectively,  i increases which causes to Ri * 2

* 1

2

2 2



2

† * increase according to the Proposition 1. As a result, the answer increases from T0 to T0 , where † contradicts with the claim of optimality T0 that is the maximum of min T1 , T2  . So we conclude that

m1 p1*  m2 p2*  Dmax Pave is always guaranteed in the optimal solution of problems (13) and (17). ■ APPENDIX C: PROOF OF PROPOSITION 2 * Proof: To prove in the optimal solution T1  T2  T0 , we first prove in the allowed SNR/SINR which

causes to 𝑅𝑖 > 0, T1 and T2 are monotonically increasing and decreasing functions of p1 respectively. As per the chain rule, T1 p1   T1  1   1 p1  , which the partial derivative of

 1 with respect to

p1 is always positive according to (8). As well, pursuant to Proposition 1 one can say that Ti  i  1   ith   mi Dmax  Ri  i  0 and following T1 p1  0 . Similarly we have T2 p1   T2  2   2 p1  . As for (5) and also p2  Pave  p1 , one can write



 

2  2  Pave h2   2 h2  0 2 2 p1 p1 h2   22 2



2

)C.1(

According to this and former result it can be concluded T2 p1  0 . Now the amount of T1 and T2 in

p1  0 is calculated. If T2  T1 , this two functions have one cross point that would be the optimal

1 th 1 Q  2  log 2 m2  solution. In p1  0 we have R1  log 2 m1 m1 and R2  log 2 1   2   m2 ln 2 m2

where  2  Pave h2

2

 22 . Since in the NOMA scheme m1  m2 , clearly R2  R1 . Finally, due to

1th   2th it can be derived that T2  T1 . Therefore, min T1 , T2  is maximized when T1  T2  T0* . This would be provable for the OMA scheme too, where due to similarity and page limit has been omitted. ■ APPENDIX D: PROOF OF PROPOSITION 3 Proof:

 p1 

is sum of the two functions 1,1 and  1,2 . Asymptotically when p1  0 , the decoding

error probability of x1 data stream by user 1 is at most, i.e.,  1,1  1 . In contrast to this, the decoding error probability of x2 data stream by user 1 is at least, i.e.,  1,2  0 , because the most portion of Ptot is allocated to user 2. In other word, 1,1

1,2 and can be said

 p1   1,1 . By increasing

p1 , 1,1

decreases but  1,2 increases, till 1,1  1,2 . Finally at high p1 (i.e., when p2  0 ), with the same argument  1,2

1,1 and can be said

 p1   1,2 . As we know,

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1,1 and  1,2 are monotonically

decreasing and increasing functions of p1 respectively in range of (0,1] . As a result,

 p1 

in terms

of p1 is first decreasing and then increasing, therefore it’s a convex function. ■ REFERENCES [1] J. Sachs, G. Wikström, T. Dudda, R. Baldemair, and K. Kittichokechai, “5G Radio Network Design for Ultra-Reliable Low-Latency Communication,” IEEE Network, vol. 32, no. 2, pp. 24-31, Mar. 2018. [2] P. Bhoyar, P. Sahare, S. B. Dhok, and R. B. Deshmukh, “Communication technologies and security challenges for Internet of things: A comprehensive review,” AEU – International Journal of Electronics and Communications, vol. 99, pp. 81—99, Feb. 2019. [3] G. J. Sutton, J. Zeng, R. P. Liu, W. Ni, D. N. Nguyen, B. A. Jayawickrama, X. Huang, M. Aboihasan, Z. Zhang, E. Dutkiewicz, and T. Lv, “Enabling Technologies for Ultra-Reliable and Low Latency Communications: From PHY and MAC Layer Perspectives,” IEEE Communications Surveys & Tutorials, pp. 1—1, 2019. [4] G. Durisi, T. Koch, and P. Popovski, “Toward Massive, Ultrareliable, and Low-Latency Wireless Communication with Short Packets,” Proc. IEEE, vol. 104, no. 9, pp. 1711–26, Aug. 2016. [5] Y. Polyanskiy, H. V. Poor, and S. Verdu´, “Channel coding rate in the finite blocklength regime,” IEEE Trans. Inf. Theory, vol. 56, no. 5, pp. 2307–2359, May 2010. [6] W. Yang, G. Durisi, T. Koch, and Y. Polyanskiy “Quasi-static multiple antenna fading channels at finite blocklength,” IEEE Trans. Inf. Theory, vol. 60, no. 7, pp. 4232–4265, Jul. 2014. [7] P. Wu and N. Jindal, “Coding versus ARQ in fading channels: How reliable should the PHY be?,” IEEE Trans. Commun., vol. 59, no. 12, pp. 3363–3374, Dec. 2011. [8] B. Makki, T. Svensson, and M. Zorzi, “Finite block-length analysis of the incremental redundancy HARQ,” IEEE Wireless Commun. Lett., vol. 3, no. 5, pp. 529–532, Oct. 2014. [9] Y. Hu, A. Schmeink, and J. Gross, “Blocklength-Limited Performance of Relaying under Quasi-Static Rayleigh Channels,” IEEE Transactions on Wireless Communications, pp. 1—1, 2016. [10] S. Xu, T.-H. Chang, S.-C. Lin, C. Shen, and G. Zhu, “Energy-efficient packet scheduling with finite blocklength codes: Convexity analysis and efficient algorithms,” IEEE Trans. Wireless Commun., vol. 15, no. 8, pp. 5527–5540, Aug. 2016. [11] C. She, C. Yang, and T. Q. S. Quek, “Cross-layer optimization for ultrareliable and low-latency radio access networks,” IEEE Trans. Wireless Commun., vol. 17, no. 1, pp. 127–141, Jan. 2018. [12] C. Sun, C. She, C. Yang, T. Q. S. Quek, Y. Li, and B. Vucetic, “Optimizing Resource Allocation in the Short Blocklength Regime for Ultra-Reliable and Low-Latency Communications,” IEEE Trans. Wireless Commun., pp. 1—1, 2018. [13] C. She, C. Yang, and T. Q. S. Quek, “Joint uplink and downlink resource configuration for ultrareliable and low-latency communications,” IEEE Trans. Commun., vol. 66, no. 5, pp. 2266-2280, May 2018. [14] Z. Hou, C. She, Y. Li, T. Q. S. Quek, and B. Vuectic, “Burstiness aware bandwidth reservation for Ultra-reliable and Low-latency Communications in Tactile Internet,” IEEE Journal on Selected Areas in Commun., vol. 36, no. 11, pp. 2401—2410, Nov. 2018. [15] C. She, Z. Chen, C. Yang, T. Q. S. Quek, Y. Li, and B. Vucetic, “lmproving Network Availability of Ultra-Reliable and Low-Latency Communications With Multi-Connectivity,” IEEE Trans. Commun., vol. 66, no. II, pp. 5482—5496, Nov. 2018.

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Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fateme Salehi, Naaser Neda, Mohammad-Hassan Majidi (corresponding author) Faculty of Electrical and Computer Engineering, University of Birjand, Birjand, Iran

Max-min Fairness in Downlink Non-Orthogonal Multiple Access with Short Packet Communications

Fateme Salehi, Naaser Neda, Mohammad-Hassan Majidi (corresponding author) Faculty of Electrical and Computer Engineering, University of Birjand, Birjand, Iran {f.salehi, nneda, m.majidi}@birjand.ac.ir

Abstract—In this paper, the performance of non-orthogonal multiple access (NOMA) technique for communication based on finite blocklength (FBL) codes is studied. NOMA, by co-serving multi users has great potential for achieving low-latency and better spectral efficiency rather than conventional orthogonal multiple access (OMA) techniques. Our goal is to have optimal power allocation to attain fair throughput in a DL-NOMA scenario. The optimization problem is formulated and an analytical approach using optimal solution search algorithm is introduced. Performance of the proposed NOMA system is compared with an OMA scheme which is benefiting from optimal power and blocklength allocation as well. Numerical results show that NOMA significantly improves the users’ fair throughput in FBL regime.

Keywords – URLLC, finite blocklength, short packet communication, max-min fairness, NOMA.

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