Energy Efﬁcient Resource Allocation for Multinumerology Enabled Hybrid Services in B5G Wireless Mobile Networks

—Multi-numerology (MN) providing a ﬂexible transmission frame structure has attracted a considerable attention for supporting abundant services for beyond ﬁfth- generation (B5G) networks. However, mobility induces severe performance degradation under different numerologies including temporal and spectral ﬂuctuation, which is not well-investigated in existing literature. We have conceived an MN-enabled energy efﬁciency (EE) problem aiming for alleviating mobility- and MN-induced interferences through moderate power and sub- carrier assignment, while considering quality-of-service (QoS) and latency requirements for different services. We propose a multi-numerology based power and resource block allocation (MNPRA) scheme considering time-/frequency-division (TD/MD) based MN leveraging temporal and spectral features among numerologies. The original non-solvable problem is theoretically transformed into a convex one by employing Dinkelbach process, Taylor approximation and difference of two concave functions (D.C.). Convergence of proposed MNPRA scheme is analyzed and veriﬁed by simulations. In simulation results, we have evaluated MNPRA under different service demands, user velocities and MN types. Our proposed scheme outperforms the conventional single-numerology framing and existing methods in open literature, which results in performances of higher EE as well as of lower throughput/delay outage probability.


I. INTRODUCTION
I N RECENT decades, there emerge diverse service types and increasing traffic demands for beyond fifth-generation (B5G) wireless networks such as latency-aware vehicular communications, reliability demands for smart applications, massive connectivity for internet-of-everything, and throughput-oriented ultra-high definition [1]. As elaborated in 5G specification, the 3rd generation partnership project (3GPP) has classified abundant scenarios into three primary use cases including enhanced mobile broadband (eMBB), massive machine-type communications (mMTC), and ultra-reliable and low-latency communications (URLLC) [2]. Furthermore, B5G promises unprecedented performance enhancement for various applications of further-eMBB (FeMBB), extremely-URLLC (EURLLC) and ultra-mMTC (umMTC) [3] providing over ten times spectrum efficiency and one tenth latency compared to existing 5G communications.
Accordingly, single-numerology (SN) system, i.e., a single transmission frame applied for a service type is utilized in traditional networks, which is potentially nonresilient due to various service demands. As a prospect mechanism specified in 5G new radio (5G-NR), it introduces multi-numerology (MN) that allows flexible transmission framing to supply ample service types [4]. The term numerology refers to as the parameterization of physical transmit waveform [5]. Different numerology types possess different sub-carrier spacings (SCSs) and symbol durations in spectral and temporal domains of resource blocks (RBs). For the latency-critical requirement, it is intuitive to assign RBs in a numerology type with larger SCS and shorter symbol duration, and vice versa for rate-oriented applications [6], which however induces potential tradeoff among numerologies. For instance, short symbol duration with low-latency provokes higher control overhead and lower signal gain [7], [8], whereas a larger SCS with high-rate declines the resource utilization. In papers [9], [10], [11], [12], they consider power-RB allocation under conventional SN mechanism while guaranteeing quality-of-service (QoS) and latency restriction. Moreover, papers [13], [14], [15] support high-speed applications under single-numerology based scheduling. The work in [13] has conducted a velocity-adaptive power allocation considering the QoS and fairness among users, whilst papers [14], [15] aim at maximizing system rate considering channel conditions of both mobile and static users. The authors of [15] optimize energy efficiency (EE) for versatile users under high-speed environments. However, system rate-latency performance cannot be simultaneously taken into account under plentiful types of services, which is confined by unavailability of different numerologies as in [9], [10], [13], [14], and [15].
To conquer the dilemma of SN mechanism mentioned above, allocation of numerologies for different services becomes substantially important as investigated in [16], [17], This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ [18], [19], [20], and [21]. The authors of [16] have addressed a scheduling problem by selecting different numerologies for each user with diversified service demands. The paper [17] has issued a delay-aware problem of URLLC users while maximizing the overall system EE. A joint power and RB allocation problem is investigated in [18] whose target is to minimize downlink power consumption under a heterogeneous network. However, the papers of [16], [17], [18] do not consider the inter-numerology interference (INI) caused by the overlapped numerologies [19], [20]. As analyzed in [19] and [20], INI cannot be neglected due to its serious interfered power induced by non-orthogonal transmission frames. Note that [19], [20] have analyzed INI and designed mitigation mechanism on waveforms from the physical layer perspective under the fixed numerology configuration. To mitigate INI, the work in [21] has employed multiplexing in time-domain based numerology rather than in frequency-domain, i.e., a single numerology type is configured at each timeslot to fulfill specific service requirement. Nonetheless, it has potentially lower flexibility compared to the system using frequency-domain based numerology in [16], [17] and [18]. Therefore, papers [5], [22], [23] have studied the architecture of multi-numerology with consideration of INI. The authors of [22] have designed a heuristic method to minimize INI by adjusting the power difference of RBs. While, the works of [5], [23] assign appropriate power to maximize INI-aware throughput. Furthermore, it is noticed that current INI-related works consider fixed numerology bandwidth ratio (NBR) among different numerologies, i.e., the number of RBs utilized by a specific numerology is predefined [19]. Nevertheless, INI can be possibly leveraged according to different utilized bandwidths compromising service requirements. Different number of available RBs for the respective numerologies should be jointly considered with power and RB allocation to mitigate total INI, which is not considered in most of existing papers.
Moreover, high mobility wireless networks such as vehicle, drone and high-speed railway communications have attracted compelling attention [24], [25], [26]. Accordingly, the next-generation wireless network is prospected to flexibly accommodate mobility-aware channels. Under high-speed scenario, it has to overcome the performance degradation induced by inter-carrier interference (ICI) from Doppler spread effect [27]. While, it also requires to conquer inter-symbol interference (ISI) caused by delay spread [27]. As for multinumerology, the system becomes more robust to interference where flexible RB with large SCS can reduce ICI, whereas small SCS can be applied against the ISI effect. Additionally, latency limitation is not studied in papers of [5], [13], [14], [15], [21], [22], which is specifically critical in most of high-speed wireless networks. The papers [4], [28], [29] have exploited the scalable numerology structure to meet the requirement of multi-services. The authors of [28] apply convex optimization to solve the RB allocation problem for a MN system with random ISI and ICI effects. In paper [29], the authors have formulated a two-dimensional (2D) resource allocation problem, i.e., time and frequency domains aiming at maximizing the achievable rate while restricting latency-aware demands. The paper [4] resolves a throughput maximization problem considering latency demands, which exhibits the random INI, ISI and ICI impacts on MN. However, actual channel is not well modeled but utilizing a random variance to characterize interferences. Furthermore, NBR and appropriate power assignment are not discussed which potentially generate higher interference resulting in a lower throughput and EE performance.
Note that RB allocation represents time-frequency domain constituted resource grids (RGs), whilst NBR allocation indicates total utilized bandwidth ratios of different numerologies. However, based on the investigation of state-of-the-art works, they consider simple models of the MN system without practical terms of interferences. Moreover, most of existing works target on the rate optimization, which however may waste power resource to satisfy the demands. We also extend from our previous works in [11], [12] and [23], which are focused on SN-based resource allocation for single and multi-services as well as for the fixed MN scenario, respectively. To achieve higher EE, conceiving a flexible MN mechanism becomes compellingly imperative under comprehensive resource utilizations and interference considerations for supporting different service requirements. Motivated by existing works, we aim at optimizing EE performance with joint power, RB and NBR assignments under interference factors of INI, ISI and ICI in a mobility-aware MN-based hybrid services. The main contributions of this work are summarized as follows. •  of eMBB/URLLC users. As a benefit of flexible mobility-aware MN network, the proposed MNPRA algorithm outperforms the existing benchmarks in open literature in terms of the lowest power consumption and the highest throughput/EE performance. The rest of this article is organized as follows. In Section II, we present the system model and problem formulation of considered MN system. The proposed MNPRA scheme is introduced in Section III. Analysis of MNPRA, including problem transformation and approximation is demonstrated in Section III-C. Section IV evaluates the results and benchmark comparisons, whilst the conclusion is drawn in Section V.

A. System Model
We consider a downlink orthogonal frequency division multiplexing (OFDM) based base station (BS) serving two different types of N = {1, 2, . . . , N} users with eMBB N e = {1, 2, . . . , N e } and URLLC N u = {1, 2, . . . , N u } users. We define numerology set of I = {1, 2, . . . , I}, where two numerologies 1 are considered as I = 2 with each having SCS of Δf i and corresponding symbol time duration of T i . Based on [30], Δf i · T i is a constant for all i to guarantee equivalent data amount. We define Δf i = 2 si × 15 (kHz) and s i as available SCS levels specified in the new radio (NR) protocol [30]. Note that we consider Δf 1 < Δf 2 and thus T 1 > T 2 without loss of generality. We also denote the system bandwidth as BW and total frame duration as T total .
1) Two-Dimensional (2D) Frame Structure: To support multiple service requirements by utilizing flexible numerologies, we introduce a 2D frame structure containing different size of resource blocks (RBs). As illustrated in Fig. 1, we construct a resource grid (RG) via the greatest common divisor, i.e., the RG is spanned by T 2 in time-domain and by Δf 1 in frequency-domain. The total bandwidth is further divided into sub-carrier sets of numerology-1 as F 1 = {1, 2, . . . , F 1 } and of numerology-2 as F 2 = {1, 2, . . . , F 2 }. Note that several RGs can generate an RB. Therefore, two types of RBs are introduced in our system with one having a larger SCS while the other possessing longer symbol duration. It can be observed that it becomes difficult to align symbols in an MN system. Accordingly, we partition all RBs into L RB groups (RBG) by aligning T 1 as the least common multiplier of symbol duration [31], and we denote the RBG set as L = {1, 2, . . . , L}, which is shown on the left-hand side of Fig. 1. Moreover, RBs within an RBG are configured in the order from f = 1 to f = F 1 and from left t = 1 to right t = T . The frequency indexes of two numerologies are different due to distinct SCSs. We denote the RB set in an RBG as 1 Here, it would be observable by considering 2 numerologies to know its nature of interferences and see how they impact on the system performance.
Note that the designed model can be extended to more than 3 numerologies with more complicate modeling. The key concept is the calculation of indexing via greatest common divisor (gcd) among different numerologies. For general case extension, we can adjust the ordering equations of (1)-(3) with i ≥ 3, as well as the INI model in (6) and (7) with additional summation index of i ∈ I.
2) TD-/FD-MN: We consider two kinds of MN including TD and FD based configurations. TD-MN allows a single numerology type within a single time duration, and different services are multiplexed in time-domain. While, FD-MN allows co-existence of numerologies at each time instant, i.e., partial RBs are for numerology-1 and the others are for numerology-2. It becomes a compelling tradeoff leveraging INI alleviation under TD-MN and high flexibility of FD-MN for throughput-oriented or latency-aware demands [32]. As depicted in Fig. 1, we define Q = Δf2 Δf1 as the ratio of two SCSs from the selected numerologies. Note that Q = 1 indicates that only one numerology is selected which is deemed to be TD-MN, whereas it is FD-MN when Q = 1.
Note that M is the total number of possible NBR values with the total set of M = {1, 2, . . . , M}. Let us take F 2 = 6 for example, we can derive φ m ∈ {0, 1 6 , 2 6 , . . . , 1} with M = 8. As exemplified on the right-hand side of Fig. 1 as well as in Fig. 2, φ 1 = 0 and φ 2 = 1 stands for TD-MN with SCSs Δf 1 and Δf 2 , respectively. As for FD-MN, φ 3 = 1 6 indicates two RBs for SCS Δf 1 and the remains for SCS Δf 2 . To adjust NBR, we consider a binary indicator α l,m ∈ {0, 1} which α l,m = 1 implies that we select φ m as the NBR in the l-th RBG. We notice that in each RBG we can only choose a single φ m , i.e., m∈M α l,m = 1, ∀l ∈ L. Accordingly, based on α l,m and φ m , we can obtain 0 ≤ η l = m∈M α l,m φ m ≤ 1 as the candidate NBR value in the l-th RBG, that is, numerology-1 will occupy a ratio of η l of bandwidth, whilst numerology-2 takes the rest of a ratio of (1 − η l ). Therefore, deciding a discrete policy of η l is equivalent to determining the indicator α l,m . We denote ω m i,k as the activation indicator 2 to represent whether the k-th RB under numerology i is applying the m-th NBR value φ m , which can be derived by 2 Note that we could define a compound variable combining both α l,m and ω m i,k . However, we keep α l,m simple and neat and let the non-solvable expression emerge in ω m i,k in the interference model, which benefits the problem transformation and derivation by depending only on the binary policy of α l,m . Fig. 2, we take φ m = 1 6 , F 1 = 12 and F 2 = 6 for example with its outcome of K m 1 's index set of {1, 2} and K m 2 's index set of {1, F 2 + 1}. We can know from Fig. 2 that top two RBs are occupied by numerology-1 with K m 1 's and the remaining ones are allocated to K m 2 , which follows the activation condition in the first criterion of (3). According to the above definition, α l,m ω m i,k = 1 indicates that we firstly select the m-th NBR value φ m in the l-th sub-RG as α l,m = 1, whereas the k-th RB of numerology i in RBG l is available to be allocated to users as ω m i,k = 1.

3) Signal Model:
We define P n,l i,k as the power allocated to the n-th user on the k-th RB of numerology i in RBG l. The channel loss is denoted as h n, indicates the distance based large-scale fading with distance d n and pathloss exponent . The RB assignment indicator is defined as ρ n,l i,k ∈ {0, 1}. Note that ρ n,l i,k = 1 means that the k-th RB in numerology i of RBG l is allocated to user n, and vice versa for ρ n,l i,k = 0. Owing to different mobility-aware channel conditions and the non-orthogonality induced from MN, interferences from ICI, ISI, and INI should be jointly alleviated and modelled in the problem, which are not considered in most of existing open literature. Inspired by Jakes' power spectral density model, we can obtain the ICI function [33] for the k-th RB in RBG l under numerology i as where v n , f c and c are velocity of user n, operating carrier frequency and the speed of light, respectively. The subscript f = F i k Fi − 1 + f is the RB index for sub-carrier f under numerology i. It can be seen that higher velocity and frequency bands will intrinsically provoke large ICI, which can be compromised by assigning proper power P n,l i,f , RB policy ρ n,l i,f and NBR policy α l,m . Based on paper [34], the ISI function in an MN system can be derived as where ξ RMS is the root mean square (RMS) signal delay spread, γ is the cyclic prefix ratio in OFDM, and Ξ(ξ) is the function of the channel power delay profile which can be asymptotically approximated as an exponential distribution based on [35]. According to [36], there arise asymmetric interference impacts from two different numerology types which should be also taken into account in INI. We can write out the MN-based INI function as the received interference of numerology-2 from numerology-1 as and interference of numerology-1 from numerology-2 as where the pertinent parameters in INI functions are expressed as (8)-(10), shown at the bottom of the next page. We can observe from INI models that when Q = 1, the values of sine functions become zero in numerators of both (6) and (7), i.e., no interference IN I l i →i,k = 0 is induced due to orthogonality property in either TD-MN or single-numerology system.
Considering all interference factors as defined above, the received signal-to-interference-plus-noise ratio (SINR) of user n on RB k in RBG l under numerology i is formulated as where IN I l i →i,k is INI from other numerologies with i = i, and N 0 is the noise power spectral density. By determining the policy set of power P = {P n,l i,k |∀n ∈ N , l ∈ L, i ∈ I, k ∈ K}, RB assignment ρ = {ρ n,l i,k |∀n ∈ N , l ∈ L, i ∈ I, k ∈ K} and NBR indicator α = {α l,m |∀l ∈ L, m ∈ M}, the throughput of user n can thus be derived by Note that total transmit data amount is considered for both eMBB and URLLC users in (12) by leveraging the Shannon throughput as well as time duration, which is different from the existing works of [17] and [4] considering only Shannon throughput. The power consumption of user n in the l-th RBG is expressed as where η PA is the power amplifier efficiency, and P circuit is the constant circuit power consumption averaged over each RBG.
To characterize the joint performance of throughput and power, we are able to acquire the system EE according to (12) and (13) as

B. Problem Formulation
For the optimization problem in a mobility-aware MN system, we aim at maximizing the system EE by optimally determining the policy of power, RB and NBR assignments constrained by available resources and service requirements, which is formulated as (15a)-(15k), shown at the bottom of the next page. Constraints (15b), (15c) guarantee the QoS requirements of R e min and R u min for throughput-oriented eMBB and latency-aware URLLC users. Note that without loss of generality, R e min has a much higher value than that of R u min . Moreover, constraints (15d), (15e) confine the latency restrictions of both eMBB and URLLC services with τ e and τ u , respectively. Similarly, more stringent service of smaller τ u for URLLC should be guaranteed, i.e., in general we have τ u < τ e . The limitation of per-RB transmit power and total BS power are respectively defined as P max in constraint (15f) and as P tot max in (15g). Note that (15f) confines per-RB power to prevent greedy assignment consuming full power from a certain RB leading to inappropriate candidate solution set. In (15i) and (15h), we represent the discrete policy of RB allocation ρ n,l i,k and NBR adjustment α l,m . Constraint (15j) implies that only one NBR value of φ m can be selected in an RBG. While, constraint (15k) indicates that the activation of RB k in RBG l for numerology i is activated, that is, means that the RB is activated and can be allocated to a single user, and vice versa. We can observe that the optimization problem P1 in (15) is complex and non-solvable due to non-linearity and non-convexity properties owing to the fractional objective, the coupled discrete and continuous variables, and the non-concave throughput function, which is regarded as a nonconvex mixed-integer non-linear problem (MINLP) and hence an NP-hard problem [37]. Therefore, we will transform the original problem into a solvable one by proposing the MNPRA scheme which is elaborated in the followings.

A. Problem Transformation
To convert the considered complicated problem into a tractable one, we have to first tackle the fractional objective, coupled continuous and discrete variables and the non-convexity caused by interferences of ICI, ISI, and INI. In the followings, we will introduce some mathematical transformations and variable relaxing methods to handle these problems. The flowchart of problem transformation for proposed MNPRA algorithm is summarized in Fig. 3.
1) Transformation of Fractional Objective: Based on the fractional form in our objective function, we adopt the Dinkelbach method to transform a fractional function into a subtractive and linear expression [38]. We define D as the feasible set of policy in problem P1 consisting of all possible constraints in (15b)-(15k). Therefore, under constrained set of D, there exists an optimal EE λ * which can be expressed as where the set of {P * , ρ * , α * } indicates the optimal allocation policy of power, RB assignment, and NBR adjustment, respectively. We can infer that the optimal EE in (16) is obtained if and only if the following problem holds Note that in P2, the solution set of D exists as a non-empty feasible set when min Such condition indicates that under the case at least one RB is assigned to each user with maximum power, we can acquire a feasible set by guaranteeing the worst throughput/latency being above/below the predefined thresholds, respectively. The maximum of EE with a fractional objective function is transformed into problem P2 in (17) with a subtractive objective function [39], [40]. We notice that optimal EE λ * can be obtained iteratively until convergence. However, it still leads to non-linearity and non-convexity in the first term of throughput function which is addressed as belows.
2) Auxiliary of Coupled Variables: We can observe from SINR in (11) that the coupled term regarding all policies take place as P n,l i,k ρ n,l i,k m∈M α l,m ω m i,k containing both continuous and discrete variables. To deal with this, we define an auxiliary parameter denoted by P n,l i,k as P n,l Note that (19) implies that we transmit data with power P n,l i,k once the specific RB and NBR is selected. To ensure the relationship stated above, a new constraint related to power constraint (15f) should be equivalently introduced as where (a) holds according to (19) and (b) follows the fact that ρ n,l i,k m∈M α l,m ω m i,k ≤ 1. Note that we neglect the similar procedure for total power constraint in (15g). Therefore, we can deduce the following lemma regarding the upper bound of INI.
Lemma 1: According to the defined auxiliary parameter of power P n,l i,k , the MN-based INI function in (6) and (7) can be derived to be upper bounded by and where σ is a positive constant indicating the tightness compared to the original INI functions. The parameters of δ 1→2,m , δ 2→1,m and δ where σ > 1 satisfying 1 The corresponding throughput function R n ( P ) and power consumption P n,l ( P ) can also be acquired in a similar manner by replacing auxiliary parameters in (19) which are given by Subsequently, the problem P2 in (17) is transformed into the optimization problem formulated as Note that λ is a predefined penalty parameter derived from P2. We can have the following proposition regarding the problem in (27).  (P , ρ, α) ≥ R n ( P ). Furthermore, P n,l ( P ) is equivalent to the original power consumption of P n,l (P , ρ, α). As a result, the objective function (27a) is the lower bound of that in (17). Based on a similar derivation, we can prove that QoS constraints (27c) and (27d) are also the corresponding subsets of original ones in (15b) and (15c), respectively. While, the power constraints (27e) and (27f) possess the same feasible region as (15f) and (15g), respectively. We can thus prove that the maximization of problem P2 in (17) is equivalent to optimizing its lower bound in problem P3 in (27), which completes the proof.
3) Integer Relaxation of Binary Variables: Motivated by [41], [42], to deal with the binary variables problem, we relax the RB and NBR assignments within the region between 0 and 1, i.e., the policies of variable constraints in (15h) and (15i) are alternatively expressed as respectively. Inspired by [43], in order to approximate the continuous relaxed variables to zero and one, we design two penalty functions as auxiliary constraints which are given by where problem P3 in (27) will stay unchanged as proven in the following proposition. Leveraging the solution of (30) and 0 ≤ ρ n,l i,k ≤ 1 in (28) yields ρ n,l i,k ∈ {0, 1}, which sustains the property of the original discrete variable ρ n,l i,k . By employing the same derivation stated above, we can deduce that α n,l i,k ∈ {0, 1} considering (29) and (31). As a result, we can conclude that problem P3 is unchanged by using integer relaxation of (28) and (29) with additional auxiliary constraints (30) and (31), respectively. This completes the proof.
We define that D denotes a set consisting of all constraints (27b)-(27f), and integer relaxation in (28) and (29). Benefited by relaxing discrete variables, the optimization problem in (27) is equivalent to the following one Note that the constraints (32b) and (32c) are non-convex, which can be resolved with the employment of abstract Lagrangian duality [44]. We transform both constraints into the respective penalty terms, and the abstract Lagrangian objective is given by where β ρ ≥ 0 and β α ≥ 0 are denoted as penalty constants. The transformed problem can be formulated as a primal problem as ρ, α, β ρ , β α ). (34) Note that the primal problem P4 is in a form of max-min with β ρ and β α existing in the minimization operator. This is because minimization of the negative part of the induced penalties in Lagrangian objective in (33) aims for sustaining the constraints in (28) Proof: We can learn from Proposition 1 that the problem in (32) is unchanged by introducing integer relaxation with its primal problem of P4 in (34). While, the corresponding dual problem is a min-max optimization as We define the optimum of primal problem in (34) as p * , whereas the optimum for its dual one (35) is denoted as d * .
According to the weak duality theorem, we can obtain the following relationship as where the inner maximization problem in the dual problem of (35) is defined as ν(β ρ , β α ) max { P , ρ, α}∈D Υ( P , ρ, α, β ρ , β α ). Therefore, we have four possible cases to be discussed in the followings.

4) Approximation of Non-Concavity:
However, we can know from problem P4 in (34) that the objective function Υ( P , ρ, α, β ρ , β α ) performs a non-concave form. Hence, we employ D.C. approximation [45] to convert non-convex problem into a solvable convex one. First of all, we rewrite the throughput function into a subtractive form as R n ( P ) = R n R n 2 ( P ) = l∈L i∈I k∈K Therefore, according to (38) and (39), we partition terms in (33) by defining an alternative D.C. objective function as We can infer that concavity property does not exactly hold in F ( P , ρ, α) due to two subtractive concave functions. Therefore, we employ first-order Taylor approximation [46] on F 2 ( P , ρ, α) to transform concave functions into affine ones. We first approximate the term R n 2 ( P ) which can be derived by where ∇ T P R n 2 ( P ) indicates the first-order partial derivative of R n 2 ( P ) with respect to P . Note that T is transpose operation, whilst index j means the iteration count. The element of partial derivative of ∇ T P R n 2 ( P ) is further expressed by (43), shown at the bottom of the next page, where the related parameters are given by the equation can be derived, shown at the bottom of next page. Similarly, we can derive the remaining terms as where the elements of partial derivatives in (44) and (45) (42), (44) and (45), we can approximate F 2 ( P , ρ, α) as (46) We can then define the new objective as F ( P , ρ, α) = F 1 ( P , ρ, α) − F 2 ( P , ρ, α). Accordingly, we can transform P4 into the following problem formulated as (15d), (15e), (15j), (15k), (27e), (27f), (28), (29), where R n ( P ) = R n 1 ( P ) − R n 2 ( P ). The following theorem demonstrates that P5 preserves concavity property.
Theorem 3: With the aid of D.C. and Taylor's first-order approximation, the transformed problem P5 is convex with a concave objective and convex constraints. Moreover, converted QoS constraints preserve the original throughput satisfaction.
Proof: Based on D.C. and Taylor's first-order approximation, we can infer from (42), (44) and (45) that the respective solutions of R n 2 ( P ), G 1 ( ρ) and G 2 ( α) are given at previous iteration j − 1, whereas the only variables are those to be determined at j-th iteration, which results in affine functions. Therefore, F ( P , ρ, α) is concave owing to the summation of concave function F 1 ( P ) and affine one F 2 ( P ). Moreover, concave function of R n ( P ) provokes convex constraints of (47c) and (47d). Therefore, with the aid of D.C. and Taylor's first-order approximation, the transformed problem P5 is convex with a concave objective and convex constraints. Due to the concave property of R n 2 ( P ) function in (39), we can know that the inequality R n 2 ( P ) ≤ R n 2 ( P ) holds. Accordingly, it can be deduced that By comparing (27c), (27d), (47c), (47d) and (48), we can x ∈ {e, u} indicates either eMBB or URLLC services. As a result, the new QoS constraints provide a smaller feasible solution set of the transformed problem P5, which preserves the original throughput satisfaction. This completes the proof.

B. Proposed MNPRA Algorithm
We can observe from Theorem 3 that problem P5 is a convex problem, which can be resolved through arbitrary convex optimization tools. Owing to the transformations stated above, we have to iteratively acquire the candidate solution of the power P , RB indicator ρ and NBR assignment α in P5 until convergence. We notice that the solution of P5 is equivalently applied in the original problem P1 owing to introduced auxiliary constraints. The overall MNPRA algorithm is demonstrated in Algorithm 1. First, we initialize the required inputs including channel gain h n,l i,k , ∀n ∈ N , l ∈ L, i ∈ I, k ∈ K, iteration index j = 1, temporary EE parameter λ (18). On the other hand, the inner loop conducts convex problem resolution to derive the optimal solution from the convex problem P5 in (47). Note that both iterations converge when the difference of the objective between two successive iterations is smaller than a given threshold. That is,

[j] and initial policy set {P [j], ρ[j], α[j]}. The EE approximation is performed at outer loop by calculating λ[j] in
is for EE approximation at outer loop, whereas holds for convergence of the inner loop of Algorithm 1.
We also note that the maximum allowable iterations are upper bounded by T in and T out for inner/outer loops, respectively. Note that as our initial feasible policy of Algorithm 1, we follow the random assignment satisfying the conditions when min R n (P max , ρ, α) ≥ R x min , max Θ(l, i, k)ρ n,l i,k ≤ τ x , and l∈L i∈I k∈K ρ n,l i,k ≥ 1, ∀n ∈ N x , x ∈ {e, u}, l ∈ L, i ∈ I, k ∈ K. Note that each user may be allocated at least one RB with the maximum allowable power for the worst throughput case.  (Pmax, ρ, α)

C. Convergence and Complexity Analysis
Since we convert the original fractional objective function to a subtractive D.C. form, we should prove that both iteration loops of Algorithm 1 will converge to its respective optimal candidate solutions.
Proposition 2: Improved solutions can be obtained from problem P2 in (17), which results in a convergence of outer loop of MNPRA in Algorithm 1.
Proof: First, we can infer from (17) that the optimal EE obtained at j-th outer iteration is greater than zero, i.e., The last inequality holds due to the derivation from (51  ( P , ρ, α), it can be inferred that the first-order Taylor approximation of F ( P , ρ, α) becomes the lower bound of its original function. Then, we have the following deduction in the inner loop of Algorithm 1, shown at the bottom of the next page. Note that D is defined as a feasible set containing all constraints in P5. Based on the relationship stated above, we conclude that the optimal value of P5 in (47) will be improved or stay unchanged. When j → ∞, we can acquire the optimum by . That is, the solution obtained by problem P5 in (47) in the inner loop of MNPRA in Algorithm 1 will converge to a local optimum. This completes the proof.
Moreover, we compare the computational complexity of proposed MNPRA with the global solution. For an MN system with I numerologies, K RBs, and N users, the computational complexity of obtaining the global optimum is O (|P | · 2) L·(IKN+M) , which is not implementable due to its exponential complexity. Note that |P | is the searching space over power assignment. Based on algorithm 1, the proposed MNPRA scheme has a complexity order of O(| P | · | ρ| · | α|) with the relaxed continuous solution set size of RB | ρ| and of NBR | α|. It can be inferred that | P | = |P |, | ρ| 2 LIKN , and | α| 2 LM based on convex optimization operation, which achieves a considerably lower and implementable complexity order than that of global optimum.

IV. PERFORMANCE EVALUATION
We evaluate the system performance of proposed MNPRA scheme via simulations. We consider the total bandwidth and time duration of MN configurations as 180 kHz and 2 ms, respectively. The two of three selected numerologies are adopted in our simulation, including numerology-1 with s 1 = 0, Δf 1 = 15 kHz, T 1 = 1 ms, F 1 = 12 sub-carriers, numerology-2 with s 2 = 1, Δf 2 = 30 kHz, T 2 = 0.5 ms, F 2 = 6 sub-carriers, and numerology-3 with s 3 = 2, Δf 2 = 60 kHz, T 3 = 0.25 ms, F 3 = 3 sub-carriers. Note that all numerologies possess the totally identical bandwidth for fair comparison. We consider that the channel experiences a small-scale Rayleigh fading with complex Gaussian distribution as h n,l i,k ∼ CN(0, 1). The CP ratio of ISI is set as γ = 0.07, whilst RMS delay is ξ RMS = 28 μs. We notice that throughput is calculated based on the multiplication of bandwidth, time duration and signal strength resulting in a unit of bits, whilst energy is in a unit of J, namely Joule. Note that all users are supposed to have the same velocity. To average the channel variations, the simulations are performed over around 100 Monte Carlo iterations with 100 timeslots per Monte Carlo run. We consider that the BS is capable of periodically collecting user velocity information from the receiver ends with second-level, which is much longer than channel coherence time with milliseconds. Accordingly, the velocity is regarded as a constant under the execution of resource allocation. Nevertheless, due to mechanical perturbation on device, we consider the velocity variance with 5% of its average value. For example, user with average speed of 60 km/hr possesses a velocity range of [57, 63] km/hr. For simplicity, we only label the average velocity values in the simulations. The remaining parameter setting referring [17], [47] is summarized in Table I.

A. Convergence of MNPRA
We firstly evaluate the convergence of proposed MNPRA scheme under TD-MN and FD-MN for outer and inner parts of EE optimization. Note that the Dinkelbach value in the outer iteration is referring (18). We can observe from Figs. 4(a) and 4(b) that the FD-MN requires more iterations for converging its optimal solution due to more flexible numerology structure simultaneously benefiting all services, whereas TD-MN having quicker convergence owing to a single transmission type at a single timeslot. Furthermore, it is observed that the proposed scheme asymptotically approaches the global optimum in Dinkelbach values with corresponding EE difference lower than 0.2 × 10 6 bits/J. As seen in Fig. 4(b), FD-MN requires more iteration rounds as well as higher computational complexity order to conduct resource allocation due to severe INI effect. Moreover, following the same reason, TD-MN may achieve higher EE performance than FD-MN, i.e., at an EE difference of around 10 6 bits/J, which strikes a compelling tradeoff between the flexibility of resource allocation and convergence performance.

B. Impacts of INI
We reveal an example for the impact of ICI and INI in frequency-domain as demonstrated in Fig. 5. Note that similar figure can be shown for ISI in time-domain, which is neglected here because of it compellingly smaller influence. We consider three exemplified signal sources, including signal with SCS of 15 kHz and that with Doppler effect as well as the one  of 30 kHz. As explained previously, Doppler effect will cause ICI, whilst different numerologies provoke INI between each other. As observed from the figure, the ICI-induced signal leads to a smaller degradation than INI. This is because that different numerologies have totally alternated from their original orthogonal waveforms to non-orthogonal ones.
Moreover, in Fig. 6 of σ ∈ {2, 10, 30}. Note that power is equally assigned to all sub-carriers for readily observing effect of INI before resource allocation. Note that we select numerology-1 occupying the RBs with sub-carriers indexed from 1 to ηF 1 , whilst numerology-2 utilizes the remaining sub-carriers. Accordingly, the INI at the left-hand side of each subfigure is termed as interference generated from numerology-2 and received by numerology-1, and vice versa for the other side. We can observe that the boundary sub-carriers suffer from considerable INI degradation, for example, the 2-nd as well as 4-th sub-carriers in Fig. 6(a) have the highest INI peaks in their respective numerology configuration. Furthermore, the remote sub-carriers will receive less INI due to adequately weak sidelobes. We can infer from Figs. 6(b) to 6(d) that INI generated by numerology-2 periodically fluctuates owing to discontinuous concatenation of its symbols with twice bandwidth compared to numerology-1. Therefore, we should assign appropriate power and RBs for mitigating such condition. With the increment of η, lower INI from numerology-2 can be acquired, whereas higher interference from numerology-1 occurs due to more allocated resources to that configuration. Moreover, with the aid of higher value of σ, a tighter bound of INI can be achieved and consequently σ = 30 is selected in our following simulation setting.
The resultant INI of proposed MNPRA scheme is shown in Fig. 7 by considering two numerologies and fixed NBR η = { 1 6 , 2 6 }, where Fig. 7(a) and Fig. 7(b) have demonstrated  the assigned power and the corresponding INI values, respectively. We can know that lower INI can be achieved with the benefit of lower power assigned. By contrast, more power can be allocated to those sub-carriers with little interference for improving rate performance. Moreover, when η = 2 6 , more power is required to mitigate interference, as observed by comparing Figs. 6(b) and 7(a), which results in comparably smaller INI in Fig. 7(b), i.e., the original non-optimized INI of around 600 mWatt is reduced to around 13 mWatt at sub-carrier 4 under MNPRA. However, fixed numerology is potentially unable to meet the requirement for complex use cases of multi-service optimization, which will be discussed in the following simulations.

C. Effect of Service Constraints
In Fig. 8, we have studied the EE performance of proposed MNPRA algorithm under TD-/FD-MN constrained by different service requirements. As shown in Fig. 8(a), we can observe an increased EE curve with looser delay constraint for URLLC services owing to more flexible numerology assignment. Moreover, FD-MN inducing severe INI leads to a comparably lower EE due to simultaneous satisfaction of both URLLC and eMBB services, especially when restricted by higher QoS demands, which can also be found in Fig. 8(b). Additionally, little performance difference can be reached under relaxed requirements of delay and QoS in both results, i.e., we have an EE difference from 5 × 10 5 to 10 5 bits/J in FD-MN from τ u = 0.5 to 2 ms, as seen in Fig. 8(a). To elaborate on a little further, we observe from Fig. 8(b) that higher EE is attained if more power is utilized to flexibly and properly assign radio resources for alleviating interferences of ICI, ISI and INI, i.e., it improves EE of around 5 × 10 5 bits/J by increasing power from 25 to 35 dBm.

D. Impact of User Velocity
We evaluate the performance of user velocity on EE performance of proposed MNPRA scheme considering different channel delay spread τ RMS ∈ {2, 28} μs under TD-/FD-MN in Fig. 9. Note that we fix Δf 1 = 15 kHz and adjust different Δf 2 ∈ {30, 60} kHz, i.e., Q = Δf2 Δf1 to observe the effect of different numerology's setting. Note that Q = 1 is for TD-MN adopting one type of the two numerologies {Δf 1 , Δf 2 } in each RBG. We can readily observe that EE performance monotonically decreases with the increment of velocity values owing to severe ICI caused by the Doppler shift. Moreover, we can see that the channel experiencing a longer delay spread leads to a degradation of EE performance due to the ISI. It has a better EE of Q = 4 than that of Q = 2 when it experiences low RMS delay spread of 2 μs, which is mainly caused by lower ICI from wider bandwidth as observed from (4). However, it performs reversely under high delay spread of 28 μs thanks to simultaneous mitigation of both ICI and ISI. That is, selection of a smaller SCS Q = 2 is capable of resolving comparably severe ISI problem in higher delay spread than that in a lower one. Furthermore, when increasing user speed, EE difference becomes remarkably larger from Q = 2 to Q = 4 mainly due to high ICI generated by large SCS. To elaborate a little further, we compare TD-MN and FD-MN under delay spread of 28 μs. It is revealed that lower EE is obtained when configured with FD-MN compared to that with TD-MN, which is substantially affected by relentless INI, i.e., we observe an EE difference of about 1.5×10 6 bits/J when SCS as {15, 30} kHz and around 2.8 × 10 6 bits/J when SCS as {15, 60} kHz under most of user velocities. Such high EE difference is mainly caused by high SCS since SCS with 60 kHz in SCS set of {15, 60} kHz causes larger INI than that with SCS set of {15, 30} kHz spacing.

E. Comparison Between SN/MN Under Different Scenarios
As depicted in Fig. 10, we evaluate the performance of proposed MNPRA scheme under TD-/FD-MN by comparing SN and fixed MN. Different scenarios are considered, including low, medium and high as well as hybrid speed levels, which represent user velocity of 20, 60, 200 and a hybrid case of {20, 200} km/hr, respectively. Note that one of {15, 30} kHz is adopted as SCS in SN, whilst Δf 1 = 15 kHz is set in MN. We know that large SCS is applicable for ICI-dominated channels, whereas small SCS benefits ISI-based environments. Therefore, similar observation can be readily inferred from both Figs. 9 and 10 that TD-MN outperforms the other mechanisms in support of flexible assignment of different numerologies. Nevertheless, FD-MN is compromising its EE for providing hybrid services, which is mainly degraded by INI effect. Moreover, FD-MN with fixed NBR η = 1 2 admits the worst performance due to inflexibility of RB assignment for different services, which induces even higher INI than flexible NBR selection, i.e., around 10 6 bits/J degradation compared to dynamic NBR allocation. Additionally, under a hybrid-speed scenario, MNPRA under TD-/FD-MN is able to deal with all interferences by striking a tradeoff between performance and different service requirements.
In order to reveal its benefits of FD-MN in our proposed MNPRA, we further evaluate the scenario with different ratios of URLLC users to all serving users as {25, 50, 75}%. Note that outage is calculated if the outcome does not satisfy either constraints (15b) and (15d) for eMBB service or (15c) and (15e) for URLLC users. We can observe that under fewer URLLC users, i.e., at a ratio of {25, 50}%, MNPRA in TD-MN attains the highest EE in Fig. 11(a) with an acceptably lower outage probability in Figs. 11(b) and 11(c), which is benefited by moderate assignment of RB and power. Nonetheless, EE of SN configuration becomes inadequate by considerably downgrading the outage performance owing to little degree of freedom for flexible allocation, i.e., SN with SCS of 15 kHz leads to around 0.25 throughput outage and 0.09 delay outage under a ratio of 25% URLLC users. We now turn to the results of FD-MN, which outperforms the other mechanisms when 75% URLLC users are served. This is because FD-MN allows the co-existence of multiple numerologies within a single timeslot, supporting simultaneous throughput-and delay-oriented applications while mitigating induced interferences. FD-MN offering resilient services performs a even lower outage below 0.05 than the other configurations with respect to both throughput and delay outage probabilities.

F. Benchmark Comparison of MNPRA
In Fig. 12, we compare the proposed MNPRA scheme to the existing four benchmarks (BMs) in open literature and the baseline as elaborated as follows.
• BM1: Sliding Window [17] utilizes a fixed-size temporal sliding window to gradually assign MN-based RBs to mainly fulfill latency-aware applications. The power is assigned to meet the essential requirement of rateoriented users. Note that the channel fluctuation is not  considered, that is, all RBs are considered to possess the same channel condition. • BM2: Priority Greedy [21] performs MN-enabled resource allocation according to a priority-based greedy method and queuing conditions. This method assigns low-delay based numerology's resources to the timecritical tasks. The power is allocated to meet the QoS constraint. • BM3: SN-PRA [47] conducts EE-based power-RB allocation under conventional SN structure. The problem is resolved based on D.C. transformation as well as integer relaxation for tackling binary variables. However, it induces quantization errors when converting from the continuous relaxed RB indicator to the discrete one. • BM4: MNPRA-P employs solely power allocation scheme in Algorithm 1 under MN, whereas RB is arbitrarily assigned. Note that we would like to observe the improvement of either sub-carrier or power assignment. • Baseline adopts a round robin method to randomly allocate RBs in an MN-enabled system, whilst the power is equally assigned constrained by its allowable maximum transmit power.
We can know from Figs. 12(a) and 12(b) that BM3 using SN-based system experiences a stringent service under inflexible frame structure, which results in a comparably lower throughput and higher consumed energy than BMs 1 and 2 in an MN-enabled network. Following the same reason, the difference of the required energy in SN-PRA is higher than BMs 1 and 2 owing to its fixed framing. Moreover, BM1 reaches a better performance than BM2. This is because BM1 performs local optimization by sliding with a window size, whilst priority based method will relentlessly sacrifice the other user demands for supporting a more urgent task. To elaborate a little further, as shown in Fig. 12(c), it reveals a compelling performance gap in EE if improper RB policy is conducted in an MN system, i.e., BM4 has an EE degradation of around 30% and 45% compared to BMs 1 and 2 as well as the proposed MNPRA scheme, respectively. Moreover, we can achieve at least 15% improvement under the optimal power allocation, i.e., MNPRA attains an enhanced EE compared to BMs 1 and 2 from 6 × 10 6 to 7 × 10 6 bits/J under the lowest velocity. In this context, we can summarize from Figs. 12(a) and 12(b) that the proposed MNPRA achieves the highest throughput and consumes the lowest energy since it appropriately assigns RBs and power to conquer serious impacts of INI, ISI, and ICI. As illustrated in Fig. 12(c), MNPRA also results in the highest EE comparing to other existing methods in open literature.

V. CONCLUSION
In this paper, we have conceived an MNPRA scheme to resolve the designed EE optimization under a mobility-aware MN network. We have also characterized the temporal and spectral interferences of ICI, ISI and INI in MN. The non-solvable problem is transformed into a convex and solvable one with the benefit of Dinkelbach process, D.C. approximation and advanced integer relaxation. The local convergence of MNPRA is proven and validated through theory and simulation. We can observe that the proposed MNPRA can strike a compelling tradeoff between EE improvement and throughput/delay satisfaction under different parameter setting. Benefited by flexible numerology and resource allocation, the proposed MNPRA achieves the highest EE and throughput performance as well as the lowest consumed energy compared to the other existing benchmarks in open literature.