I Introduction
Nonorthogonal multiple access (NOMA) is a powerdomain multiplexing technology that allows users to transmit signals on the same timefrequency resources [1]. This technology supports scenarios with massive connectivity, therefore, it is a 5G candidate technology for uplink massive machinetype communications (MTC) [2]. MTC devices are usually power limited but have applications with diverse quality of service (QoS) requirements, e.g., public monitoring, realtime localization, industrial automation. Therefore, energy efficient transmission is a crucial requirement in most cases. Therefore, it is important to design an energyefficient power control scheme in uplink NOMA systems with delay QoS constraint.
Extensive research has been done on power control for NOMA systems by taking linklayer QoS requirements into consideration. Yang et al. [3] and Cai et al. [4] proposed a power allocation scheme for NOMA systems under minimum rate constraints. When modeling a wireless communication system as a queueing system, the delay QoS requirements can be approximated by effective capacity model [5]. Yu et.al [6] used effective model to analyze the performance of a twouser downlink NOMA network. Furthermore, Choi [7], Liu et.al [8] and Chen et al. [9] used the effective capacity model and proposed a crosslayer power control policy to guarantee a certain delay QoS requirement in NOMA systems. However, in these work, the transmitter is always in transmission mode, which will over estimate transmit power consumption in their power control schemes. In 2016, Sinaie et al. [10] proposed a crosslayer power consumption model by considering a twomode circuitry (a circuitry works in the transmission mode or sleep mode). Xu et al. [11], [12] followed Sinaie’s work and proposed a new energy efficiency analytical model by using a twomode circuitry. The model is simple but only apply to the pointtopoint wireless communication systems, which is not suitable for NOMA systems.
The aim of this paper is to design an energyefficient power control scheme with delay QoS constraint in uplink NOMA systems. By continuing Xu’s work, we develop Xu’s energy efficiency analytical model in uplink NOMA systems and formulate an optimization problem to maximize the energy efficiency of NOMA systems under delay QoS requirements and peak power constraints. We further use the Dinkelbach method and develop a new power control scheme that can solve the above optimization problem.
The rest of this paper is organized as follows. Section II describes the uplink NOMA system model as well as the effective bandwidth and the effective capacity model. In section III, we analyze the effective capacity in NOMA systems. In section IV, we will formulate our power control problem under target delayoutage constraints. In section V, the problem will be solved by Dinkelbach algorithm. Performance of the proposed algorithm is evaluated in Section VI by simulations. Section VII summarizes our work.
Ii System Model
Iia NOMA System Mode
As shown in Fig. 1, we consider an uplink NOMA system, in which a NOMA base station serves MTC users or user equipments (UEs) (in this paper, we use the terms MTC users and UE interchangeably). A UE works either in a transmission mode if there is data to transmit or in a sleep mode otherwise. Transitions between two modes can be modeled as a state machine, which is illustrated in Fig. 2. All the UEs utilize the same time and frequency resources but different power domain to transmit data. The received signal at the base station is given by
(1) 
where
is the Rayleigh fading coefficient with unit variance between the base station and user
, denotes the transmit power for user , denotes the transmit signal for user , and is the additive white Gaussian noise (AWGN) with mean and variance .The successive interference cancelation (SIC) technique is used at base station to eliminate multiuser interference. The SIC decoding order is based on the channel gain information. The highest channel gain user’s signal is decoded at first since it is strongest at the base station. Then removing the strongest signal from the received signal to decode the second highest channel gain user’s signal. That is, to decode the user’s signal first decodes the user’s signal and removes this signal from received signal, in the order ; the signals from the users are treated as noise.
Let denote the channel gain of the user, where is the absolute value of a complex number.
has an independent and identically distributed exponential distribution with unit mean. For simplicity, we assume that and the channel gains are sorted in a descending order, i.e.,
. Therefore, the signaltointerferenceplusnoise ratio (SINR) experienced when decoding user’s signal is(2) 
where is the received user’s signal and represents other users interference. Specially, the user’s SINR is
(3) 
We assume that the instantaneous channel gain information is perfectly known at each UE. Based on the Shannon theory, the achievable rate of the user is given by
(4) 
where is the total bandwidth utilized by all the users.
IiB The preliminaries of effective bandwidth and the effective capacity model
All the NOMA UEs have its own buffer with infinite buffer size. The buffer model is shown at Fig. 3. The UE arrival data from the data source at slot is . Furthermore, we follow the work [11] and assume that

and the arrival is exponentially distributed with a mean data length .
Based on the above assumptions, the UE arrivals
are identically distributed (IID) random variables (RVs) identical to a RV
; the probability density function (PDF) of the arrival
is(5) 
the average data arrival rate for UE k is
(6) 
where denotes the duration of a slot.
Due to the fact that the service rate in any timevarying wireless channels fluctuates, we assume that the user services are IID RVs identical to a RV
(7) 
In order to guarantee a QoS requirement, the effective bandwidth is defined as the minimum constant service rate and the effective capacity is defined as maximum constant arrival rate specified by exponent [5]. Based on the effective bandwidth model and the effective capacity model, the UE’s data arrival rate and service rate can be characterized by their own effective bandwidth and effective capacity :
(8) 
(9) 
where is UE’s QoS exponent. If the assumptions of the GartnerEllis theorem hold, there is a unique QoS exponent that satisfies
(10) 
then the complementary cumulative distribution function (CCDF) of backlog size can be approximated by [5]:
(11) 
where is the nonempty buffer probability for UE. The parameter plays a critically important role for statistical QoS guarantees, which indicates the exponential decay rate of the QoS violation probability. A smaller corresponds to a slower decay rate, which implies that the user can only provide a looser QoS guarantee, while a larger leads to a faster decay rate, which means that a more stringent QoS requirement can be supported. When the arrival is exponentially distributed with a mean data length , the value of the nonempty buffer probability can be calculated from Xu’ s work [11] as
(12) 
Denote the UE circuits power consumption and transmission probability by and respectively. The UE total power consumption is
(13) 
The probability of a UE being in transmission mode is equivalent to the probability that traffic arrives from the upper layer or the buffer storage is nonempty. Denote the event of traffic arrival by and the event of buffer is nonempty by . Since two events are mutually independent, therefore, can be expressed as
(14) 
Iii Effective capacity analysis in NOMA
Iiia Effective capacity analysis for twouser NOMA
We first consider a twouser NOMA system, where two users utilize the same timefrequency resources. This scenario has been extensively studied in many papers [7], [13], which is called multiuser superposition transmission or paired NOMA. In a twouser NOMA system, the corresponding distribution of SINR for the first user is given by:
Case 1: If second user is in sleep mode, the PDF of the SINR is
(15) 
Case 2: If second user is in transmission mode, the PDF of the SINR is [14]
(16) 
Therefore, the effective capacity for the first user is given by
(17)  
Since the second user’s signal is only interfered by the noise, therefore, the effective capacity for the second user is
(18)  
IiiB Effective capacity analysis for Kuser NOMA
The user effective capacity in a NOMA systems depends on the distribution of , which is difficult to derive the closeform expression when the number of user is larger than three. Gu et al. [14] derive a simple method to reduce the computational complexity to calculate the effective capacity of user in a fullinterference scenario. But in NOMA systems, due to the SIC mechanism, only the first user’s signal interfered by other users’ signal. The user’s signal only interfered with user’s signal. Based on the above facts and integrated with Gu’s work, can be calculated as
(19) 
where denotes the set of transmission mode users and
(20) 
Iv Problem Formulation of the Power control
In this work, we use bits per Joule to measure the system’s energy efficiency, which is the ratio of the system’s sum effective capacity to the total power consumption:
(21) 
Now let us consider the delayQoS requirement. The total delay experienced by user consists of queueing delay and transmission delay that equals :
(22) 
When the maximum delay bound and a tolerance are specified by a typical MTC application, the system is in delayoutage if it cannot guarantee the following inequality:
(23) 
where denotes the probability of a random event.
Let and =[
] denote the vector of transmit power and QoS exponent, respectively. The optimal power control problem can be formulated as
P1:(24) 
(25) 
(26) 
where (25) is the delayoutage constraint and (26) is peak power constraint for each UE.
V Power control Strategy
In this section, we will solve P1 to obtain the optimal power control strategy for maximizing uplink NOMA energy efficiency.
According to [12], the CCDF of queueing delay can be approximated by
(27) 
By substituting into (27) with
(28)  
The constraint (25) can be rewritten as
(29)  
where is the equivalent sign and . Result (29) indicates that constraint (25) give a lower bound of QoS exponent . A large value of indicates a stringent delay QoS requirement and thus requires more power consumption. Based on the above observation, we first have the following result:
Result 1: When the average data arrival rate is , the energy efficiency of the uplink NOMA system is a decreasing function of QoS exponent .
For a proof of Result 1, see Appendix A. Based on Result 1, the optimal QoS exponent is the boundary value
(30) 
By substituting in (30) into (24) in P1, the problem P1 becomes P2:
(31) 
(32) 
where is the optimal QoS exponent vector. The problem P2 is to find such a that satisfy the constraint (32). We have the following result to formally characterize this problem:
Result 2: In an uplink NOMA system, for given QoS exponent the vector of QoS exponents of different UEs, the sum of effective capacity is concave of the transmission power .
For a proof of Result 2, see Appendix B. Based on result 2, the numerator of in (21) is concave. Since the denominator of is an affine function, the fractional function is qusiconcave [15]. A fractional quasiconcave problem can be solved by Dinkelbach’s algorithm as a sequence of parameterized concave problems [16]. Let be the optimal value of original problem, can be expressed as
(33) 
Problem P2 can be transformed to the following parametric concave problem:
(34) 
The maximal value of in (34) is a root of the equation . The value of can be found by solving the parameterized problem in (34) according to the Dinkelbach method. For a given in (34), we solve the problem as P3:
(35) 
(36) 
The problem P3 is a concave optimization problem and constraint (36) satisfies Slater’s condition [15], one that can be efficiently solved by the Lagrange dual method. The Lagrangian function of problem P3 can be written as
(37) 
where is the nonnegative Lagrange multipliers. The equivalent dual problem can be decomposed into two parts: 1) the maximization problem solves the power control problem and 2) the minimization problem solves corresponding Lagrange multiplier, which is given by
(38) 
By using the Lagrange dual decomposition, the maximization problem can be solved by differentiating with . We denote the optimal power control by . The Karush–Kuhn–Tucker (KKT) conditions for P3 are given by
(39) 
(40) 
(41) 
(42) 
where represents the value of corresponding variable at the optimal point. Equation (39) and (40) are feasibility conditions, (41) is the complementary slackness condition, and (42) is the stationary condition. The optimal power can be solved as
(43) 
where denotes and is the optimal Lagrangian multiplier, which need to ensure the system to meet the peak power constraint of each UE.
As for minimization problem, the multiplier can be updated by the subgradient method [15] as follows:
(44) 
where is the iteration index, is the positive step sizes for the iteration (e.g., ). When the step sizes are chosen properly, the convergence to the optimal solution is guaranteed.
Parameters  Values 

Time duration of a slot,  1ms 
Noise spectral density,  174dBm/Hz 
System Bandwidth,  18kHz 
Circuit power,  10dBm 
Maximum transmit power,  46dBm 
Predefined threshold,  0.1 
Vi Result and discussion
In this section, we evaluate the performance of the proposed algorithm in Rayleigh fading environment [17]. The channel gain is generated by exponential distribution with parameter [18], where is the distance between the user and the base station, denotes the path loss exponent and we set . Three different QoS requirements UEs are in a NOMA cell. The distance between base station and three users are 300, 600 and 900 meters, respectively. The other parameters are listed in Table I.
Fig. 4 shows the simulated and approximated of average delay violation probability for different UEs when the all the UEs average date rates are 600 Kbps. The delay QoS requirements for three UEs are set as (10ms, 0.1), (20ms, 0.1) and (30ms, 0.1) respectively. As shown in Fig. 4, the approximation results are very close to the simulation results. This verifies the correctness of delayoutage probability and indicates that our power control scheme is capable of providing precise delayoutage probability guarantees.
Fig. 5 shows a comparison of energy efficiency between the twomode circuitry and singlemode circuitry. The target delay violation probability is 0.1. The figure shows the energy efficiency improvement when the twomode circuitry is used in NOMA systems. The reason is that the singlemode circuitry overestimates transmit power, thus it is apparently less energy efficient.
Vii Conclusion
It is important to design an energy efficient power control scheme in uplink NOMA systems under predefined delayoutage constraints for MTC. Previous work on power control in NOMA systems only consider singlemode UEs, which will over estimate transmit power consumption. In this paper, we consider twomode UEs in uplink NOMA systems, and propose an energyefficient power control scheme under predefined delayoutage constraints. Simulation results confirm our power control scheme is capable of providing precise delayoutage probability guarantees in uplink NOMA systems.
Acknowledgement
The work was supported in part by the National Nature Science Foundation of China Project under Grant 61471058, in part by the Hong Kong, Macao and Taiwan Science and Technology Cooperation Projects under Grant 2016YFE0122900, in part by the Beijing Science and Technology Commission Foundation under Grant 201702005 and in part by the 111 Project of China under Grant B16006.
Appendix A
Since effective capacity is a decreasing function of [5], therefore, the sum of effective capacity is also a decreasing function of . When the QoS exponent increases, the transmission power increases accordingly to meet a stringent QoS requirement. Thus the total power consumption is an increasing function of the QoS exponent . Because the numerator is a decreasing function and the denominator is an increasing function, therefore the system’s energy efficiency is a decreasing function of the QoS exponent .
Appendix B
Let , where . Since is concave for all , so is concave in the domain set. This implies that is logconvex, and is logconvex as well since logconvexity is preserved under sums [15]. Noting that is convex for logconvex . Therefore, is a concave function of for . Meanwhile, the sum effective capacity can be written as . Since concavity is preserved under sums, thus is a concave function of transmission power , completing the proof.
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