1
Energy Efficient Non-Orthogonal Multiple Access for Machine-to-Machine Communications Zhaohui Yang, Wei Xu, Hao Xu, Jianfeng Shi, and Ming Chen Abstract—This letter investigates an uplink energy minimization problem for machine-to-machine (M2M) communications with non-orthogonal multiple access (NOMA). To solve this nonconvex problem, we first prove that transmitting with minimum rate and full time is optimal. Then, the original problem can be transformed into an equivalent convex problem, which can be effectively solved by the proposed optimal power control and time scheduling scheme. Numerical results show that the proposed scheme achieves the optimal energy consumption.
BS
UE 1 U UE N
ĂĂ MTCD 1 MTCD 3
Index Terms—M2M, resource allocation, NOMA, energy minimization
I. I NTRODUCTION Machine-to-machine (M2M) communication has been deemed as one of the next frontiers for dense wireless applications, such as smart grid and intelligent transport systems [1]. Different form conventional human type communications, the machine type communication devices (MTCDs) in a M2M enabled cellular network have their own unique features: large number of devices, time-controlled, small data transmission, extra low power consumption, etc [2]. Massive access control is a fundamental challenge for M2M communications. To tackle this challenge, an effective way is to deploy machine type communication gateways (MTCGs), which can serve as relays for MTCDs [2]. Considering that user equipments (UEs) have more power and storage space than MTCDs, a radio resource allocation scheme where UEs are configured as MTCGs was proposed in [3]. In [3], multiple MTCDs transmitted data to the UE by using orthogonal time division multiple access (TDMA). Recently, non-orthogonal multiple access (NOMA) was introduced in downlink cellular networks with randomly deployed users [4]. Different from TDMA, NOMA simultaneously serves multiple users in the same degrees of freedom by splitting them in the power domain. Therefore, NOMA appears attractive for M2M communications with the ability of simultaneously serving a large number of users [5]. Existing works about NOMA mainly focused on sum rate maximization and fair scheduling [6], [7]. However, energy minimization problem is important for M2M communications due to the fact that MTCDs are always configured with low power consumption [2]. In this letter, we consider an uplink energy minimization problem for an M2M enabled cellular network using NOMA. This work was supported in part by the National Science and Technology Major Project under Grant 2016ZX03001016-003, the National Nature Science Foundation of China under Grants 61372106, 61223001, 61471114, the Fundamental Research Funds for the Central Universities, and the Scientific Research Foundation of Graduate School of Southeast University under Grant YBJJ1650. The authors are with the National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China. (Email:
[email protected] [email protected],
[email protected],
[email protected],
[email protected]).
MTCD M
MTCD 2
Fig. 1.
MTCD M-1
The considered uplink M2M enabled cellular network.
ĂĂ
t1 MTCDs
MTCDs
MTCDs
UEs
T
Fig. 2.
Time sharing scheme during one uplink transmission period.
In a typical setup, a UE acting as an MTCG can decode and forward both the information of MTCDs and its own data to the base station (BS) directly. We aim to optimize the energy efficiency of the M2M enabled network which results in an energy consumption minimization problem for NOMA with individual MTCD and UE rate constraints. In order to tackle the nonconvex problem, we first analyze the problem and obtain some parametric conditions for the optimal solution. These conditions are insightful in helping transform the original problem into a convex one. Then, we accordingly propose an optimal power control and time scheduling scheme. We also analyze the complexity of the proposed algorithm. Numerical results show that the proposed algorithm outperforms existing representative scheme in terms of energy consumption. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION Consider an uplink M2M enabled cellular network with N UEs and M MTCDs, as shown in Fig. 1. Denote the sets of UEs and MTCDs as N = {1, · · · , N } and M = {1, · · · , M }, respectively. Every UE is regarded as a MTCG, which can serve as a relay for some MTCDs. Denote Ji∪as the specific set of MTCDs served by UE i ∈ N , and then i∈N Ji = M. Each UE i ∈ N or MTCD j ∈ Ji has a payload to transmit within time constraint T . As depicted in Fig. 2, time T consists of N + 1 uplink transmission phases for MTCDs and UEs. The amount of time allocated to MTCDs in Ji is denoted by ti , ∀i ∈ N . Assume that the decode-and-forward protocol [8] is adopted at each UE. Then, the last tN +1 amount of time is assigned to all UEs to transmit their own data and the decoded data from the served MTCDs. Obviously, the overall transmission time constraint, N +1 ∑ i=1
should be satisfied.
ti ≤ T,
(1)
2
During the uplink transmission phase for MTCDs in Ji , all the MTCDs in Ji simultaneously transmit data to UE i. According to the NOMA principle, the received signal of UE i is Ji ∑ √ yi = hij pj sj + ni , (2) j=Ji−1 +1
∑i
where J0 = 0, Ji = l=1 |Jl |, | · | is the cardinality of a set, hij is the channel between MTCD j ∈ Ji and UE i, pj and sj denote the transmission power and message of MTCD j, respectively, and ni represents the additive zero-mean Gaussian noise with variance σ 2 . Without loss of generality, the channels are sorted as |hi(Ji−1 +1) |2 ≥ · · · ≥ |hiJi |2 . According to [9] and [10], it is always recommended that the transmission time for each MTCD could be as long as possible. Under this consideration, we assume that each MTCD in Ji must use all the transmission time ti to upload the required payload for the benefit of energy minimization. Applying the successive interference cancellation in NOMA [6], the achievable data throughput for MTCD j ∈ Ji in an uplink transmission period is given by ( ) |hij |2 pj rij = Bti log2 1 + ∑Ji , (3) 2 2 l=j+1 |hil | pl + σ where ∑Ji B is the bandwidth of the system, and we define l=Ji +1 pl = 0 for j = Ji . After the UEs successfully decode messages from the MTCDs, all UEs simultaneously transmit data to the BS based on the NOMA principle. Denote hi as the channel between UE i and BS. These channels are also sorted in decreasing order, i.e., |h1 |2 ≥ · · · ≥ |hN |2 . As a result, the achievable data throughput of UE i can be written as ( ) |hi |2 qi ri = BtN +1 log2 1 + ∑N , (4) 2 2 l=i+1 |hl | ql + σ where qi is the transmission power of UE i. Now it is ready to formulate the energy minimization problem for the M2M enabled cellular network as min p,qq ,tt
N ∑
ti
i=1
Ji ∑
pj + tN +1
qi
∀i ∈ N , j ∈ Ji Ji ∑
ri ≥ Ei +
(5a)
i=1
j=Ji−1 +1
s.t. rij ≥ Dj ,
N ∑
Dj ,
∀i ∈ N
(5b) (5c)
j=Ji−1 +1 N +1 ∑
ti ≤ T
(5d)
i=1
0 ≤ pj ≤ Pj , 0 ≤ qi ≤ Qi , ∀i ∈ N , j ∈ Ji 0 ≤ ti , 0 ≤ tN +1 , ∀i ∈ N ,
(5e) (5f)
where p = (p1 , · · · , pM )T , q = (q1 , · · · , qN )T , t = (t1 , · · · , tN , tN +1 )T , Dj is the payload that MTCD j has to upload within time constraint T , Ei is the data generated by UE i itself in time T , Pj is the maximal transmission power of MTCD j, and Qi is the maximal transmission ∑ power of UE i. ¯ i = E i + Ji For notational convenience, we set E j=Ji−1 +1 Dj .
Obviously, problem (5) is nonconvex. In the following, we first provide the necessary condition for the optimal solution of problem (5) and then obtain the optimal solution. III. O PTIMAL S OLUTION A. Optimal Condition By analyzing problem (5), we have the following lemma. Lemma 1: The optimal (pp∗ , q ∗ , t ∗ ) of problem (5) satisfies ∗ ¯i , and ∑N +1 ti = T , ∀i ∈ N , j ∈ Ji . rij = Dj , ri∗ =E i=1 Obviously, we know that serving the minimum required throughput for both MTCDs and UEs is optimal. This observation is intuitively reasonable as less resources are used and hence less energy is consumed. Moreover, we can also observe that energy saving can always benefit from longer transmission time from [9] and [10]. As a result, Lemma 1 can be easily proved according to the above two observations. B. Optimal Power Control and Time Scheduling Theorem 2: The original problem in (5) can be equivalently transformed into the following convex problem as ) Ji ∏ Ji Ji ( C N ∑ ∑ ijk σ 2 ti ∑ ti −1 min e t |hij |2 m=j i=1 j=Ji−1 +1
k=m
) N N N ( F ∑ ik σ 2 tN +1 ∑ ∏ tN +1 + −1 e |hi |2 m=i i=1
(6a)
k=m
s.t.
N +1 ∑
ti ≤ T
(6b)
i=1
Ti ≤ ti , TN +1 ≤ tN +1 ,
∀i ∈ N ,
(6c)
¯N +i−k /B, where Cijk = (ln 2)DJi +j−k /B, Fik = (ln 2)E −1 −1 Ti = maxj∈Ji {Tij }, Tij = vij (Pj ), vij (x) is the inverse function of vij (x) defined in (9), TN +1 = maxi∈N {Ti(N +1) }, −1 Ti(N +1) = u−1 i (Qi ), and ui (x) is the inverse function of ui (x) defined in (10). Proof: By applying Lemma 1, the inequality constraints (5b) and (5c) are always active and we can accordingly further simplify problem (5) in the following. ∗ = Dj . First, we consider the optimal condition as rij ∗ Plugging rij = Dj into (3), we have ( (ln 2)D ) ∑ Ji j |hij |2 pj = e Bti − 1 pl |hil |2 + σ 2 . (7) l=j+1
Using the recursion method, we have ( (ln 2)D ) Ji |hiJi |2 pJi = e Bti − 1 σ 2 , Ji ∑
|hi(Ji −1) |2 pJi −1 = σ 2
) Ji ( (ln 2)D ∏ 2Ji−1−k Bti e −1 ,
(8a) (8b)
m=Ji −1 k=m
.. . |hi(Ji−1+1) |2 pJi−1+1 = σ 2
Ji ∑
) Ji ( (ln 2)DJ +J ∏ i i−1 +1−k Bti e −1 .
m=Ji−1 +1 k=m
(8c)
3
pj =
) Ji ( C Ji ∏ ijk σ2 ∑ ti e − 1 , vij (ti ), |hij |2 m=j
) Ji ∏ Ji ( C ijk σ2 ∑ ti e − 1 +λ− |hij |2 m=j j=Ji−1 +1 k=m ( C ) ijk ∏J i 2 ti σ C e −1 Ji Ji ∑ Ji ijl k=m ∑ ∑ ( C ) = 0, ∀i ∈ N , (13) ijl j=Ji−1+1m=j l=m |hij |2 ti e ti −1 ∂L = ∂ti
Based on (8), we get ∀j ∈ Ji . (9)
k=m
According to (9), we find that the transmission power of MTCD j ∈ Ji is a function of the transmission time ti . Obviously, pj in (9) is a decreasing function with ti . Then, from (5e), we can equivalently transform the constraint 0 < pj ≤ Pj −1 into ti ≥ vij (Pj ). Further, with the same method in (8) we can obtain power qi as ) N N ( F ik σ2 ∑ ∏ tN +1 qi = e −1 , ui (tN +1 ), ∀i ∈ N , (10) |hi |2 m=i
Ji ∑
) N N ( F N ∑ ik ∂L σ2 ∑ ∏ tN +1 = e −1 +λ− ∂tN +1 |hi |2 m=i i=1 k=m ( F ) ik ∏Ji 2 tN +1 σ F e −1 N ∑ N ∑ N il k=m ∑ ( F ) = 0. il i=1 m=i l=m |hi |2 tN +1 e tN +1 −1
(14)
k=m
ri∗
¯i from Lemma 1. Obviously, qi is a by inserting = E decreasing function with tN +1 and 0 < qi ≤ Qi is equivalent to tN +1 ≥ u−1 i (Qi ). Substituting (9) and (10) into problem (5), we can obtain the equivalent problem in (6). Since the constraints of problem (6) are linear, we only need to prove that the objective function (6a) is convex. To show this, we first define g(x) =
Ji ∏ (
Cijk x
e
) −1 ,
x > 0.
(11)
k=m
Then, the second derivative of g(x) follows ( )2 Ji Ji 2 Cijk x Cijk x ∑ ∑ C e C e ijk ijk g ′′ (x) = g(x) − Cijk x − 1)2 eCijk x − 1 (e k=m k=m [ J ] Ji 2 2 i ∑ ∑ Cijk e2Cijk x Cijk eCijk x ≥ g(x) − Cijk x − 1)2 Cijk x − 1)2 k=m (e k=m (e ( ) Ji 2 ∑ Cijk eCijk x eCijk x − 1 = g(x) (12) > 0. 2 (eCijk x − 1) k=m From (12), we find that g(x) is convex for x > 0. According ( ) to [11, Page 89], the prospective function s(x, t) = tg xt is convex with respect to (x, t) for x, t > 0. Thus, function s(1, t) is convex with t, which indicates that the objective function (6a) is convex and problem (6) is consequently convex. Since problem (6) has been proven convex, it is ready to obtain the globally optimal solution via the well-known interior point method. However, the complexity of solving problem (6) by the interior point method is O(N 3 M 5 ), which is in general high. Here, we further obtain the optimal solution with low complexity by analyzing the Karush-Kuhn-Tucker (KKT) conditions of problem (6). The Lagrangian function of problem (6) can be written as ) Ji Ji ∏ Ji ( C N ∑ ∑ ijk σ 2 ti ∑ t i L(tt, λ) = e −1 |hij |2 m=j i=1 j=Ji−1 +1 k=m (N +1 ) ) N N ( F N ∑ ∑ ik σ 2 tN +1 ∑ ∏ e tN +1 −1 +λ + ti −T , 2 |h | i m=i i=1 i=1 k=m
where λ ≥ 0 is the Lagrange multiplier associated with constraint (6b). According to [11], the optimal solution should satisfy the KKT conditions of problem (6):
∂L strictly increasSince the objective function (6a) is convex, ∂t i es with ti , ∀i ∈ N ∪ {N + 1}. Using the bisection method, we can obtain the unique solution of ti from (13) and (14) given λ. Denote the unique solution of ti as wi (λ) for all i ∈ N ∪ {N + 1}. Obviously, wi (λ) decreases with λ. Further combining constraints (6c), we have
ti = max{wi (λ), Ti },
∀i ∈ N ∪ {N + 1}.
(15)
From Lemma 1, constraint (6b) holds with equality. Thus, applying (15) into (6b), we have N +1 ∑
max{wi (λ), Ti } = T.
(16)
i=1
Because the left term of (16) decreases with λ, we can use the bisection method to obtain the unique solution of λ. Having obtained Lagrange multiplier λ from (16), we can get the optimal t from (15). Then, we obtain the optimal p and q according to (9) and (10), respectively. C. Further Discussion Assume that the payload for each MTCD is the same, i.e., D1 = · · · = DM = D. According to our derived closed-form solution in (9), the transmission power of MTCD j ∈ Ji can be expressed as pj =
Ji ∏ Ji ( ) σ 2 d(1 − dJi −j+1 ) C σ2 ∑ ti e −1 = , |hij |2m=j |hij |2 (1 − d)
(17)
k=m
where C = (ln 2)D/B, and d = eC/ti − 1. Since (1 + d)(1 − dJi −j ) ≥ (1 − dJi −j+1 ) for 0 < d ≤ 1 and (1 + d)(dJi −j − 1) > (dJi −j+1 − 1) for d > 1, we have the power allocation relationship as: |hij |2 1 − dJi −j |hij |2 pj+1 = ≥ D . 2 J −j+1 i pj |hi(j+1) | 1 − d 2 Bti |hi(j+1) |2
(18)
Considering the small load case [2] where the device payload D D makes |hij |2 ≥ 2 Bti |hi(j+1) |2 satisfied, we can obtain pj+1 ≥ 1, which indicates that MTCDs with poor channel pj conditions obtain more transmission power. This power allocation strategy ensures that minimal payload requirements for MTCDs with poor channel conditions can be satisfied. Noth that this observation shows different behavior characteristics of power control from the traditional water-filling feature.
IV. N UMERICAL R ESULTS In this section, we evaluate the performance of the proposed scheme. We set N = 4, B = 180 KHz, σ 2 = −174 dBm/Hz, Ei = 20 Kbits, Pj = 14 dBm, Qi = 34 dBm and T = 1 s, ∀i ∈ N , j ∈ Ji . The number of uniformly distributed MTCDs, M , is tested from 55 to 100. Besides, the payload for each MTCD is the same, i.e., D1 = · · · = DM = 2 Kbits, Moreover, the path loss model is 128.1 + 37.6 log10 d (d is in km) and the standard deviation of shadow fading is 4 dB [12]. We compare our proposed OPT scheme for a M2M enabled system with NOMA (labeled as ‘OPT-NOMA’) with the following three schemes: the equal time sharing scheme, where ti = tN +1 = T /(1 + N ), ∀i ∈ N , and the transmission power of MTCDs and UEs can be respectively obtained from (9) and (10), for a M2M enabled system with NOMA (labeled as ‘ET-NOMA’), fair power allocation scheme for NOMA [7] (labeled as ‘Fair NOMA’), and the optimal power allocation scheme for a M2M enabled system with TDMA [3] (labeled as ‘OPA-TDMA’), where the MTCDs in Ji upload their data to UE i by time division and the formulated energy minimization problem can be directly proved to be convex. Fig. 3 shows the total energy consumption of all UEs and MTCDs in time T . It can be seen that OPT-NOMA, ETNOMA and Fair NOMA outperform the conventional OPTTDMA, especially when the number of MTCDs is large. This is because the MTCDs served by the same UE can simultaneously transmit data by using the NOMA scheme, and the transmission time of each MTCD is larger than that by using TDMA scheme, resulting in that the energy consumption with NOMA is less than TDMA. Compared to TDMA, there are several sources of additional signaling and processing overhead in NOMA. Since the OPA-NOMA can optimally allocate the transmission time according to the minimum required payloads and channel conditions, the energy consumption of OPANOMA is greatly reduced compared to ET-NOMA. The OPANOMA outperforms fair NOMA, because the transmission rates for all users are set as the same for fair NOMA [7]. Channel gains and power allocation for MTCDs served by the same UE are displayed in Fig. 4, where channel gains and power allocation are modified by subtracting the corresponding minimal channel gain and minimal allocated power. It can be observed that MTCDs with lower channel gains are allocated with higher power, which verifies the discussions in Section
0.8 Proposed OPT−NOMA Proposed ET−NOMA Fair NOMA [6] OPA−TDMA [3]
0.6 0.4 0.2 0 55
60
65
70 75 80 85 Number of MTCDs
90
95
100
Fig. 3. Comparison of total energy consumption with existing NOMA and TDMA schemes. 20 Normalized channel gains Normalized power allocation
15 dB
D. Complexity Analysis For the proposed optimal power control and time scheduling (OPT) scheme in Section III-B, the major complexity lies in the computation of ti in each iteration, ∀i ∈ N ∪ {N + 1}. Since there are three summation operators and one product operator in both (13) and (14), ti can be obtained with bisection method which has a complexity of O(M 4 log2 (1/ϵ1 )) for accuracy ϵ1 . Thus, the total complexity of the proposed algorithm is O(N M 4 log2 (1/ϵ1 ) log2 (1/ϵ2 )), where ϵ2 is the accuracy of the bisection method for obtaining λ. Considering that the dimension of the variables in problem (6) is N +1 and there are four summation operators and one product operator in the second derivative of (6a), the complexity of solving problem (6) by using the standard interior point method is O(N 3 M 5 ) [11, Pages 487, 569], which is in general high.
Energy consumption (J)
4
10 5 0
Fig. 4.
1
2
3
4
5
6
7 8 9 10 11 12 13 14 15 MTCD ID
Channel gains and power allocation for MTCDs.
III-C. It should be noted that the discussions in Section IIIC do not hold for the case of unequal payload for MTCDs, because MTCDs with poor channel conditions can obtain less power when the payloads for MTCDs with poor channel conditions are small. V. C ONCLUSION In this letter, we have investigated an uplink energy minimization problem for M2M communications with NOMA. We propose an optimal power control and time scheduling scheme by solving the KKT conditions for optimality. Numerical results show that the proposed scheme for NOMA yields lower energy consumption than the existing NOMA and TDMA schemes. R EFERENCES [1] G. Wu, S. Talwar, K. Johnsson, and N. Himayat, “M2M: From mobile to embedded internet,” IEEE Commun. Maga., vol. 49, no. 4, pp. 36–43, Apr. 2011. [2] K. Zheng, F. Hu, W. Wang, and W. Xiang, “Radio resource allocation in LTE-advanced cellular networks with M2M communications,” IEEE Commun. Maga., vol. 50, no. 7, pp. 184–192, Jul. 2012. [3] G. Zhang, A. Li, K. Yang, L. Zhao, Y. Du, and D. Cheng, “Energyefficient power and time-slot allocation for cellular-enabled machine type communications,” IEEE Commun. Lett., vol. 20, no. 2, pp. 368–371, Feb. 2016. [4] Z. Ding, Z. Yang, P. Fan, and H. V. Poor, “On the performance of non-orthogonal multiple access in 5G systems with randomly deployed users,” IEEE Signal Process. Lett., vol. 21, no. 12, pp. 1501–1505, Jul. 2014. [5] Z. Ding, Y. Liu, J. Choi, Q. Sun, M. Elkashlan, and H. V. Poor, “Application of non-orthogonal multiple access in LTE and 5G networks,” IEEE Commun. Mag., 2017. [Online]. Available: http://arxiv.org/abs/1511.08610 [6] X. Chen, A. Benjebbour, A. Li, and A. Harada, “Multi-user proportional fair scheduling for uplink non-orthogonal multiple access (NOMA),” in Proc. IEEE Veh. Technol. Conf. Seoul, Korea, May. 2014, pp. 1–5. [7] S. Timotheou and I. Krikidis, “Fairness for non-orthogonal multiple access in 5G systems,” IEEE Signal Process. Lett., vol. 22, no. 10, pp. 1647–1651, Oct. 2015. [8] S. Karmakar and M. K. Varanasi, “The diversity-multiplexing tradeoff of the dynamic decode-and-forward protocol on a MIMO half-duplex relay channel,” IEEE Trans. Inf. Theory, vol. 57, no. 10, pp. 6569–6590, Oct. 2011. [9] V. Angelakis, A. Ephremides, Q. He, and D. Yuan, “Minimum-time link scheduling for emptying wireless systems: Solution characterization and algorithmic framework,” Trans. Inf. Theory, vol. 60, no. 2, pp. 1083– 1100, Feb. 2014. [10] C. K. Ho, D. Yuan, L. Lei, and S. Sun, “Power and load coupling in cellular networks for energy optimization,” IEEE Trans. Wireless Commun., vol. 14, no. 1, pp. 509–519, Jan. 2015. [11] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. [12] Access, Evolved Universal Terrestrial Radio, “Further advancements for E-UTRA physical layer aspects, 3GPP TS 36.814,” V9. 0.0, Mar. 2010.
