1 Introduction

Increasing EE plays a key role in making mobile networks green communication networks [1, 2]. In addition, with the fast development of massive Machine Type Communication and massive amount of Ultra Dense Network technologies, network traffic is rapidly growing and many studies have been widely carried out to provide high QoS and EE for UEs. In particular, enhancing EE while providing the QoS in the uplink of LTE/LTE-A systems is important in reducing UE power consumption and increasing UE battery usage time.

In general, EE is defined as data rate per unit energy consumption [3]

$$ \varepsilon = { }\frac{R}{P} $$
(1)

where P represents power consumption and R denotes the data rate.

Many studies carried out to increase EE in LTE/LTE-A networks.

In [3], a downlink power allocation EE optimization scheme was proposed and verified to provide the minimum rate guaranteed services in Orthogonal Frequency Division Multiplexing (OFDM)-Distributed Antenna Systems. The authors proposed solution to the mathematical model as a non-convex NP-Hard problem. An EE improvement scheme combining channel status and power allocation was proposed in [4]. What is noted here is the power and resource allocation based on resource auction. The authors in [5] proposed energy efficient scheduling and power allocation method in downlink OFDM networks. They defined three kinds of EE: global EE, the weighted sum of EE achieved on each resource slot, the geometric weighted product of EE achieved on each resource slot. Then they studied energy efficient scheduling and power allocation method aimed at maximizing EE in the downlink, considering the constraints on the maximum transmission power of the base station. In [6], the researchers studied the problem of increasing the EE of UEs with limited battery life in OFDMA networks. In this work, they proposed a solution to solve the NP-Hard problem with nonlinear constraints.

As in any energy efficient resource allocation problems in OFDMA systems, the problems are proved to be NP-Hard, therefore many papers proposed algorithms to solve the problems. Unlike LTE networks, the energy efficient resource allocation problems in LTE-A networks, which using carrier aggregation (CA) and multiple carrier bands, have their own features.

In [7], the energy efficient resource allocation problem in LTE-A networks is considered. The authors proposed a resource block allocation algorithm that can maximize system throughput as well as minimizing power consumption in LTE-A networks for UEs which support multiple component carriers. They also proposed a solution algorithm for integer linear programming. The authors of [8] proposed a scheduling algorithm to reduce power consumption and enhance EE for high-speed packet data transmission in LTE-A which support Coordinated Multi-Point.

In addition, the EE scheduling based on deep neural network is considered. In [9], we used deep reinforcement learning models to solve non-convex dynamic optimization problems to enhance EE and showed the results of applying deep learning to Multiple Input Multiple Output-Non-Orthogonal Multiple Access systems. In [10, 11], the authors also proposed resource allocation methods based on deep neural networks, which have been widely used in studies due to their high solvency for nonlinear programming. But there are still some problems in the practical point of view due to the preparation time for train and the high computational cost.

On the other hand, the energy efficient scheduling methods that studied in the previous section did not consider the user priority. The user priority should be determined considering the buffer state of UE, QCI value, and so on, which is the QoS parameters. Priority metric to select users is investigated in [12, 13]. Authors of [12] proposed an EXP/PF method that is applied to both GBR and Non-GBR bearers, which includes packet delay and packet loss rate.

In general, services for users and the QoS requirements of each service are different. For example, user can browse a home page using a web browser at the same time as using File Transfer Protocol (FTP), and at the same time enable VoIP phones. The service associated with file download have a large tolerance to packet delay, but the need for low bit error rate. Conversely, for high speed data services such as VOIP, online gaming, video conferencing, multimedia streaming, and many others, delays have crucial effects on the performance.

The main information that are associated with the QoS profile include QCI [14]. QCI is a scalar that represents the specifications of the specific bearer (e.g. bearer priority, packet delay budget and packet loss rate), and that have been preconfigured by the operator owning the eNB. Table 1 shows different QCIs and their typical features reported in [14].

Table 1 Standardized QCI characteristics
Full size table

In [13], the whole system EE optimization problem is formulated and a user priority metric, which includes the EE of individual UEs, and sub-optimal algorithm are proposed. In [15], a spectral efficiency and energy efficiency optimization combination algorithm (H-PSSS) is proposed to improve the system EE, simultaneously guarantee the QoS for each UE. In this paper, the user priority is calculated by considering QoS parameters such as packet delay for UEs with GBR bearers or buffer length for UEs with Non-GBR bearers, and EE is considered by maximizing the modulation and coding schemes (MCS) in frequency domain resource allocation.

In [16], the researchers aim to reduce the complexity as well as improve the EE performance and analyze the inherent relation between EE and the number of subcarriers of a user and then give the closed-form expressions to calculate the EE gain with an additional subcarrier. Based on the closed-form expressions, they also propose a low complexity subcarrier assignment algorithm. In [17], the authors introduced a method of providing ad hoc treatment at a particular time interval based on the average waiting time for each user class in the case of low-priority services with the aim of achieving fairness among users in LTE-A and 5G networks. In [18], the researchers proposed algorithm to improve the QoS provision for the UEs with high delay and poor channel condition at the edge of cells and calculate the number of pre-assigned RBs by considering packet size and delay.

As can be seen the above literatures, in many scheduling algorithms, the UE priority is considered only in one aspect of QoS or EE. In this paper, we propose an energy efficient resource allocation method in LTE-A uplink, which support CA, and a priority metric considering EE as well as QoS. In this method, new framework for setting EE value as optimized ones is proposed. In the paper, high priority of UE is determined by its high EE approached possible EE optimization value when it is updated to decide resource allocation priority. Especially in order to guarantee real time scheduling, we propose new mechanism to refer to the optimization value for hysteresis updating mean EE, while the optimization process based on new scheme to solve the NP-Hard problem will run in parallel, because the time it takes for estimating optimum can be relatively long. By doing so, we will use the real time scheduling because the process for calculating mean EE is updating mean EE considering hysteresis, the while the optimization process is solving the optimization problem.

To demonstrate the effectiveness and availability of the proposed method, it is simulated in different situations with various number of UE like femto cell, Pico cell and micro cell, the simulation result is compared to the existing schemes. The paper also contributes to the analysis of the effect of the required time of NP-Hard optimization problem on the proposed method.

The rest of this paper is organized as follows.

In Sect. 2, the system model and problem formulation for scheduling on the LTE-A uplink are presented. In Sect. 3, we introduce a priority metric and resource allocation scheme to enhance QoS and EE in LTE-A uplink. Performance of the proposed algorithm is presented in Sect. 4. We conclude this work in Sect. 5.

2 System model and problem formulation

In this section, LTE/LTE-A uplink system model and a framework related to resource allocation are presented. The resource allocation problem is converted into an optimization problem, then the optimization methods are utilized to achieve the optimal solution.

2.1 LTE/LTE-A uplink system model

LTE/LTE-A uses Single-carrier Frequency Division Multiple Access (SC-FDMA) in the uplink [19, 20]. SC-FDMA has lower the peak-to-average power ratio (PAPR) value compared with OFDMA, which is employed in the downlink.

Figure 1 shows the resource allocation flowchart in LTE-A uplink system. As shown in the figure, the base station allocates resources considering channel status and buffer status based on the buffer status report of each UEs and broadcasts resource allocation information. Recently, QoS requirements have increased as the real-time interactive services are grown, and it means delay and jitter would be regarded as a network performance indicator. Thus, base station generally schedules in terms of EE, if the user priority is not determined according to the imperfect buffer status of UEs and packet loss due to excessive delay for GBR UEs and due to buffer overflow for Non-GBR UEs. VoIP can represent GBR service and thoughtfully-used service. Therefore, VoIP is created as a GBR service in the paper. Hence, we take into account the buffer status and packet delays in the LTE-A uplink, proposed to determine user priority and to allocate resources by combining EE aware scheduling and user priority. The number of non-empty buffers defines the buffer status report format, in the case that a user has only one bearer, the short format is used to conserve channel resources because the short format report requires fewer bits. However, when a user has multiple bearers, the long format is used.

Fig. 1
figure 1

Resource allocation flowchart in LTE-A uplink system

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Figure 2 shows the resource allocation framework.

Fig. 2
figure 2

Proposed resource allocation framework

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In the figure, \(\overline{{\varepsilon }_{u}^{his}\left(t-1\right)}\) is a mean EE calculated based on the mean EE Update (MeanEE) Process, refer to Eqs. (34), (34.1) for a moment of t − 1(or (t − 1)-th Transmission Time Interval (TTI), considering hysteresis.\(\overline{{\varepsilon }_{u}^{opt}\left(t\right)}\) is a optimum EE calculated by using EE Optimization (OptEE) Process at a moment of t (or t-th TTI).

As shown in the figure, we first determine whether the new connection exist after initializing the parameters. If new connection exists, NewConnectionFlag in OptEE Process is set to 1 or 0 in else case. Optimization process which always watch the flag state stops the previous calculation as soon as the flag is set to 1, the OptEE process, considering new connection, restarts to solve EE optimization. In the result, the synchronization between MeanEE process and OptEE process is obtained in terms of UE number. In other words, the synchronization means the coincidence of UE set for which both processes deal with. In this meaning, it will be called UE matching below.

Now, top (TTI) represents the time consumed by OptEE process to obtain the result, and MeanEE process will use the optimal EE at the time t ranging \(k \cdot \frac{{t}_{op}}{TTI}\le t<(k+1)\cdot \frac{{t}_{op}}{TTI}\), where the optimal EE is estimated by OptEE process under initializing parameters at the moment of \(\left\lfloor {\left( {k - 1} \right) \cdot \frac{{t_{op} }}{TTI}} \right\rfloor\). The operator ⌊ ⌋ rounds it to nearest integer less than or equal to it. OptEE process iterates this process. In the MeanEE process, the updating equation will utilize the resulting EE obtained according Eq. (34.1), comparing mean EE \(\overline{{\varepsilon }_{u}^{his}\left(t-1\right)}\) of MeanEE process for a moment (t − 1) and the latest optimization result of the OptEE process \(\overline{{\varepsilon_{u}^{opt} \left( {\left\lfloor {\left( {k - 1} \right) \cdot \frac{{t_{op} }}{TTI}} \right\rfloor } \right)}}\) in the buffer at every mean EE update. Here, the important thing is that the unit of time is not second but 1TTI = 1 ms.

After calculating EE, the priority of UE is evaluated according to the method proposed in Sect. 3 and then resources are allocated.

The scheduling related to resource allocation has 1 ms allocation period, which was introduced in 3GPP. This means that the updating EE would take more than 1 ms. The calculation time typically relies on the number of UEs.

However, OptEE process and MeanEE process are worked in parallel, and EE update value uses the latest value in the buffer. This assumed that the connection of the UE is not disconnected in several ms because the movement speed of UEs in small cell is very slow and they travel in public buses which are relatively slow transportations in macro cell. Therefore, even if the previous optimum EE used by MeanEE process is the value estimated for tens of TTI ago, we can suppose that there is no big change in energy efficiency, it similar to the present optimum EE and will be available to update mean EE. This will help to improve EE of the system. The effect of the time of optimized solution calculation is analyzed and evaluated quantitatively in the later section of the paper. Here, the upper layer determines if there is a new connection, thus it is not mentioned due to outside the scope of our treatment.

2.2 Formulation of initial energy efficient resource allocation

In this subsection, we propose the solution to optimize initial EE.

In an LTE-A cell of u users, multiple component carriers (CCs) can be allocated to each UE. Denote downlink bandwidth allocated to a user u by \({B}_{u,c}\), then according to Shannon’s theorem the data rate is expressed as

$${R}_{u,c}={B}_{u,c}\cdot {log}_{2}(1+{SNR}_{u,c})$$
(2)

In this case the relationship between resource block and bandwidth allocated to the UE u is

$$ B_{u,c} = b_{RB} \mathop \sum \limits_{r = 1}^{R} \mathop \sum \limits_{i = 1}^{I} a_{c,r,u}^{i} , $$
(3)

where \(b_{RB}\) is a bandwidth of a resource block, \(r=\stackrel{-}{1..R}\) is resource block number, \(i=\stackrel{-}{1..I}\) is CQI number, and \({a}_{c,r,u}^{i}\in \{\mathrm{0,1}\}\) is an integer that indicates whether the resource block r of the CC c is allocated to the user u with MCS index i or not. So, if allocated, then \({a}_{c,r,u}^{i}=1\), or \({a}_{c,r,u}^{i}=0\) if not allocated.

In general, \({SNR}_{u,c}\) is proportional to the transmit power and channel gain and inversely proportional to the bandwidth and power spectral density of the additive white Gaussian noise (AWGN). So, it is

$$ SNR_{u,c} = \frac{{P_{u,c} \cdot G_{u,c} }}{{B_{u,c} \cdot N_{g} }}, $$
(4)

where \({P}_{u,c}\) is a transmit power allocated to the user u on the CC c, \({N}_{g}\) is a power spectral density of AWGN, \({L}_{c}\) is a path loss on the CC c, \({F}_{u,c}\) is a random variable for fading between the user u and base station, \({G}_{u,c}\) and \({D}_{u}\) are a channel gain and distance on the CC c between the UE u and the base station, respectively, denote \(\alpha_{c}\) as the path loss exponent of CC c, \(G_{u,c}\) can be formulated as

$$ G_{u,c} = L_{c} \cdot F_{u,c} \cdot D_{u}^{{ - \alpha_{c} }} . $$
(5)

The amount of path loss \({L}_{c}\) depends on the carrier frequency

$${L}_{c}\cong {(\frac{{\lambda }_{c}}{4\pi })}^{2}$$
(6)

where \({\mu }_{c}\) is a wavelength of the CC c.

Uplink data rate for the user u can be written as

$${R}_{u}=\sum_{c=1}^{C}{R}_{u,c}=\sum_{c=1}^{C}{B}_{u,c}{\mathit{log}}_{2}\left(1+\frac{{P}_{u,c}\cdot {G}_{u,c}}{{B}_{u,c}\cdot {N}_{g}}\right).$$
(7)

\({p}_{c}\) is transmit power per resource block allocated to the user u on the CC c and number of allocated resource block is \(\sum_{r=1}^{R}\sum_{i=1}^{I}{a}_{c,r,u}^{i}\), so power consumption \({P}_{u}\) for user u is

$${P}_{u}=\sum_{c=1}^{C}{P}_{u,c}=\sum_{c=1}^{C}{p}_{c}\cdot \sum_{r=1}^{R}\sum_{i=1}^{I}{a}_{c,r,u}^{i}.$$
(8)

In this paper, power consumption that does not depend on carrier frequency, for example, due to the heat energy are not considered.

Therefore, EE of user u and LTE-A network are as follows;

$${\varepsilon }_{u}=\frac{{R}_{u}}{{P}_{u}}$$
(9)
$$\varepsilon =\sum_{u=1}^{U}{w}_{u}\cdot {\varepsilon }_{u}$$
(10)

where \({\varepsilon }_{u}\) is EE of user u of uplink system, \(\varepsilon \) is EE of LTE-A network and weight factor \({w}_{u}\) determined by which UE of uplink system has higher priority. By adjusting weight factor, we balance resource allocation of users according to their services.

As can be seen in (2)–(10), optimization model is characterized by

$$\underset{a,e}{\mathit{max}}\varepsilon $$
(11)

The constraints on the above expressions are:

$$\forall u, \sum_{c=1}^{C}{P}_{u,c}\le {P}_{u}^{max}$$
(11_a)
$$\forall u, \sum_{c=1}^{C}{B}_{u,c}{\mathit{log}}_{2}(1+\frac{{G}_{u,c}\cdot {P}_{u,c}}{{B}_{u,c}\cdot {N}_{g}})\ge {R}_{u}^{min}$$
(11_b)
$$\forall c, \sum_{u=1}^{U}{B}_{u,c}\le {BW}_{c}$$
(11_c)
$$\forall u,c,r, {a}_{c, r,u}^{i}\le {e}_{u,c}^{i}$$
(11_d)
$$\forall c,\mathrm{r}, \sum_{u=1}^{U}\sum_{i=1}^{I}{a}_{c, r,u}^{i}\le 1$$
(11_e)
$$\forall u,c, \sum_{c=1}^{C}\sum_{i=1}^{I}{e}_{u,c}^{i}\le {\mu }_{u}$$
(11_f)
$$\forall u,c, \sum_{i=1}^{I}{e}_{u,c}^{i}\le 1$$
(11_g)
$$\forall u, c, {B}_{u,c}\ge 0, {P}_{u,c}\ge 0$$
(11_h)
$${\forall u,c,r, e}_{u,c}^{i},{a}_{c,r,u}^{i}\in \{\mathrm{0,1}\}$$
(11_i)

The constraint (11_a) presents the limitation of power for user u. In other words, the total sum of transmission power of each component carrier cannot exceed the maximum power \({P}_{u}^{max}\) of user u. The constraint (11_b) defines the lower bound of the uplink data rate for user u. The inequality (11_c) indicates the limitation of the allocated bandwidth for each user. As shown in (11_a–11_c), the total sum of the allocated bandwidth for each user in a component carrier cannot exceed the bandwidth of that. The constraint (11_d) shows that if the c-th CC has been allocated to user u with CQI index i (in this case, \({e}_{u,c}^{i}=1\)), then the RBs of the CC would be allocated to the user with CQI index which is larger than i (\({a}_{c, r,u}^{i}=1)\). The constraint (11_e) indicates that an RB can be allocated to only one user. Because to satisfy the inequality, only one term \({a}_{c, r,u}^{i}\) can be set 1. The inequality (11_f) indicates that the maximum number of the carrier aggregation allocated to each user is \(\mu \), the maximum of \(\mu \) is 5 according to 3GPP. The constraint (11_g) means that all the RBs of the CC allocated to user u have the same only one MCS.

In order to solve (11) taking into account (7), (8) we can divide it into two parts as follows.

$$\underset{{B}_{u,c},{P}_{u,c}}{\mathit{max}}\varepsilon $$
(12)
$$\forall u, \sum_{c=1}^{C}{P}_{u,c}\le {P}_{u}^{max}$$
(12_a)
$$\forall u, \sum_{c=1}^{C}{B}_{u,c}{\mathit{log}}_{2}(1+\frac{{G}_{u,c}\cdot {P}_{u,c}}{{B}_{u,c}\cdot {N}_{g}})\ge {R}_{u}^{min}$$
(12_b)
$$\forall c, \sum_{u=1}^{U}{B}_{u,c}\le {BW}_{c}$$
(12_c)

and

$${B}_{u,c}={b}_{RB}\sum_{r=1}^{R}\sum_{i=1}^{I}{a}_{c,r,u}^{i}$$
(13-1)
$${P}_{u,c}={p}_{c}\cdot \sum_{r=1}^{R}\sum_{i=1}^{I}{a}_{c,r,u}^{i}$$
(13-2)
$$\forall u,c,r, {a}_{c, r,u}^{i}\le {e}_{u,c}^{i}$$
(13_a)
$$\forall c,r, \sum_{u=1}^{U}\sum_{i=1}^{I}{a}_{c, r,u}^{i}\le 1$$
(13_b)
$$\forall u,c, \sum_{c=1}^{C}\sum_{i=1}^{I}{e}_{u,c}^{i}\le {\mu }_{u}$$
(13_c)
$$\forall u,c, \sum_{i=1}^{I}{e}_{u,c}^{i}\le 1$$
(13_d)
$$\forall u, c, {B}_{u,c}\ge 0, {P}_{u,c}\ge 0$$
(13_e)
$${\forall u,c,r, e}_{u,c}^{i},{a}_{c,r,u}^{i}\in \{\mathrm{0,1}\}$$
(13_f)

Thus, (11) is divided into (12) and (13). Now, we solve the (12) considering \((\sum_{r=1}^{R}\sum_{i=1}^{I}{a}_{c,r,u}^{i})\in {I}^{+}\) (where \({I}^{+}\) is a positive integer including zero) in (13-1), (13-2). We can transform (13-1), (13-2) into as follows considering solution \({B}_{u,c},{P}_{u,c}\) could be indivisible by \({b}_{RB}, {p}_{0}\).

$$\underset{{a}_{c,r,u}^{i},{e}_{u,c}^{i}}{\mathit{min}}\Vert {B}_{u,c}-{b}_{RB}\sum_{r=1}^{R}\sum_{i=1}^{I}{a}_{c,r,u}^{i}\Vert .$$
(14-1)
$$\underset{{a}_{c,r,u}^{i},{e}_{u,c}^{i}}{\mathit{min}}\Vert {P}_{u,c}-{p}_{c}\cdot \sum_{r=1}^{R}\sum_{i=1}^{I}{a}_{c,r,u}^{i}\Vert .$$
(14-2)

As a result, (13) is reduced to (14-1), (14-2) and constraints (13_a), (13_f). (12) is sum-of-ratios problems that maximize the sum of several fractional functions, resulting in non-convex optimization problems due to the non-convexity of the objective function and is hard to solve.

2.3 Transformation and solution of equations

Now, given that

$$ w_{u} \cdot \varepsilon_{u} = \varphi_{u} . $$
(15)

Equation (12) is transformed as follows.

$$ \mathop {\max }\limits_{{B_{u,c} ,P_{u,c} }} \mathop \sum \limits_{u = 1}^{U} \varphi_{u} . $$
(16)

Taking into account (9) and (15), we can rewrite as:

$$ w_{u} \cdot R_{u} = \varphi_{u} \cdot P_{u} . $$
(17)

\({\varphi }_{u}\) is a monotonic function for \({B}_{u,c},{P}_{u,c}\) and the constraint (12_a–12_c), we can extend Eqs. (17) and (18) as follows:

$${w}_{u}\cdot {R}_{u}-{\varphi }_{u}\cdot {P}_{u}\ge 0$$
(18)

which is another constraint of Eq. (16).

To solve the optimization problem considering (16) and the constraint (18), from (12_a–12_c), we define the Lagrangian \(\Lambda ({P}_{u,c},{B}_{u,c},\varphi ,\kappa ,\tau ,\zeta ,\epsilon )\):

$$\Lambda \left(P,B,\varphi ,\kappa ,\tau ,\nu ,\zeta ,\epsilon ,\pi \right)=\sum_{u=1}^{U}{\varphi }_{u}+\sum_{u=1}^{U}{\kappa }_{u}\cdot ({w}_{u}\cdot {R}_{u}-{P}_{u}\cdot {\varphi }_{u})-\sum_{u=1}^{U}{\tau }_{u}\cdot \left(\sum_{c=1}^{C}{P}_{u,c}-{P}_{u}^{max}\right)-\sum_{c=1}^{C}{\zeta }_{c}\cdot \left[\sum_{u=1}^{U}{B}_{u,c}-{BW}_{c}\right]+\sum_{u=1}^{U}{\epsilon }_{u}\cdot \left[\sum_{c=1}^{C}{B}_{u,c}\cdot {\mathit{log}}_{2}\left(1+\frac{{G}_{u,c}\cdot {P}_{u,c}}{{B}_{u,c}^{UL}\cdot {N}_{g}}\right)-{R}_{u}^{min}\right]$$
(19)

where \(P=\left\{{P}_{u,c}\right\}, B=\{{B}_{u,c}\}\).

Now denote \(\overline{P },\overline{B },\overline{\varphi },\overline{\kappa },\overline{\tau },\overline{\zeta },\overline{\epsilon }\) as the optimal value of the differentiable Lagrangian \(\Lambda \left(P,B,\varphi ,\kappa ,\tau ,\zeta ,\epsilon \right)\). Then it should satisfy the following relation from Fritz-John's optimality condition [21].

$$\forall u,\frac{\partial \Lambda }{\partial {\varphi }_{u}}=1-\overline{{\kappa }_{u}}\cdot \overline{{P }_{u}}=0.$$
(20)
$$\forall u,{\kappa }_{u}\cdot \frac{\partial \Lambda }{\partial {\kappa }_{u}}=\overline{{\kappa }_{u}}\cdot \left({w}_{u}\cdot \overline{{R }_{u}}-\overline{{P }_{u}}\cdot \overline{{\varphi }_{u}}\right)=0.$$
(21)
$${\left.\frac{\partial \Lambda }{\partial P}\right|}_{P=\overline{P} }={\left.\frac{\partial \Lambda }{\partial B}\right|}_{B=\overline{B} }=\overline{{\tau }_{u}}\cdot \frac{\partial \Lambda }{\partial {\tau }_{u}}=\overline{{\zeta }_{c}}\cdot \frac{\partial \Lambda }{\partial {\zeta }_{c}}=\overline{{\epsilon }_{u}}\cdot \frac{\partial \Lambda }{\partial {\epsilon }_{u}}=0.$$
(22)

From (20) we get

$$\overline{{\kappa }_{u}}=\frac{1}{\overline{{P }_{u}}}>0, \overline{{P }_{u}}>0,$$
(23)

and from (21) we get

$${w}_{u}\cdot \overline{{R }_{u}}-\overline{{P }_{u}}\cdot \overline{{\varphi }_{u}}=0, \overline{{\varphi }_{u}}= \frac{{w}_{u}\cdot \overline{{R }_{u}}}{\overline{{P }_{u}}}\ne 0.$$
(24)

Equations (23) and (24) show that there is a saddle point, a non-zero solution of the Lagrangian \(\Lambda \left(P,B,\varphi ,\kappa ,\tau ,\zeta ,\epsilon \right)\), and there is some correlation between \(\overline{P },\overline{B },\overline{\varphi },\overline{\kappa }\) at the saddle point.

Now, with the notion that the derivative of the right-hand side of (21)

$${F}_{u}\left(P,B,\kappa ,\varphi \right)={\kappa }_{u}\cdot \left({w}_{u}\cdot {R}_{u}-{P}_{u}\cdot {\varphi }_{u}\right).$$
(25)

The partial derivative \(\frac{\partial {F}_{u}}{\partial {\varphi }_{u}}\) is equal to the right-hand side of (21), it shows that (16), (18) and (12_a–12_c) of the constraint conditions are equivalent to the following system of equation.

$$\underset{P,B}{\mathit{max}}\sum_{u=1}^{U}{F}_{u}\left(P,B,\kappa ,\varphi \right).$$
(26)
$$\forall u, \sum_{c=1}^{C}{P}_{u,c}\le {P}_{u}^{max}.$$
(26_a)
$$\forall u, \sum_{c=1}^{C}{B}_{u,c}{\mathit{log}}_{2}(1+\frac{{G}_{u,c}\cdot {P}_{u,c}}{{B}_{u,c}\cdot {N}_{g}})\ge {R}_{u}^{min}.$$
(26_b)
$$\forall c, \sum_{u=1}^{U}{B}_{u,c}\le {BW}_{c}.$$
(26_c)

It is also considered that (22) is the KKT condition [21] of the (26).

Let us determine whether (26) is a concave function for the variables P, B, or a convex function.

Theorem 1: The function \({F}_{u}\left(P,B,\kappa ,\varphi \right)\) for P and B in (26) is a concave function with respect to P, B.

The proof of this theorem is given in the Appendix.

Thus, (26) has an optimal solution and can be solved by converting to dual problems using the Lagrange multiplication approach.

$$L\left(\delta ,\vartheta ,\gamma ,{P}_{u,c},{B}_{u,c}\right)=\sum_{u=1}^{U}{\kappa }_{u}\cdot ({w}_{u}\cdot \sum_{c=1}^{C}{B}_{u,c}{\mathit{log}}_{2}\left(1+\frac{{P}_{u,c}\cdot {G}_{u,c}}{{B}_{u,c}\cdot {N}_{g}}\right)-{\varphi }_{u}\cdot \sum_{c=1}^{C}{P}_{u,c})+\sum_{u=1}^{U}{\delta }_{u}\cdot \left\{\left[\sum_{c=1}^{C}{B}_{u,c}\cdot {\mathit{log}}_{2}\left(1+\frac{{G}_{u,c}\cdot {P}_{u,c}}{{B}_{u,c}\cdot {N}_{g}}\right)\right]-{R}_{u}^{min}\right\}-\sum_{c=1}^{C}{\vartheta }_{c}\cdot \left(\sum_{u=1}^{U}{B}_{u,c}-{BW}_{c}\right)-\sum_{u=1}^{U}{\gamma }_{u}\cdot \left(\sum_{c=1}^{C}{P}_{u,c}-{P}_{u}^{max}\right)$$
(27)
$$\underset{\delta ,\vartheta ,\gamma \ge 0}{\mathit{min}}\underset{{P}_{u,c},{B}_{u,c}}{\mathit{max}}L\left(\delta ,\vartheta ,\gamma ,P,B\right)$$
(28)

where the parameters \(\delta ,\vartheta ,\gamma \) are Lagrange multipliers.

Equation (28) is a dual problem. It is solved by decomposing the equation for Lagrange multipliers and the equation for variables \({P}_{u,c},{B}_{u,c}\). For given Lagrange multipliers \(\delta ,\vartheta ,\gamma \), \(\underset{{P}_{u,c},{B}_{u,c}}{\mathit{max}}L\left(\delta ,\vartheta ,\upgamma ,{P}_{u,c},{B}_{u,c}\right)\) is a standard concave problem, and the equation for the Lagrange multipliers \(\delta ,\vartheta ,\gamma \) can be solved by the sub gradient method for a given power and bandwidth assignment. To satisfy (28), the following relation should be established.

$${\left.\frac{\partial L}{\partial {P}_{u,c}}\right|}_{P=\overline{P },B=\overline{B} }=0.$$
(29)

We can obtain \({P}_{u,c}\) for the given value of \(\delta ,\vartheta ,\gamma \) using the (29), then, substituting \({P}_{u,c}\) into (27) and considering (28), the following (30) is obtained.

$$\underset{{B}_{u,c}}{\mathit{max}}{L}^{\mathrm{^{\prime}}}({B}_{u,c}).$$
(30)

The equation is a standard linear programming problem for the variable \({B}_{u,c}\) and can be solved using well-known methods or tools.

On the other hand, the expression updating the parameters \(\delta ,\vartheta ,\gamma \) for \({P}_{u,c}\), \({B}_{u,c}\) is as follows, considering equation set (12_a–12_c) according to the subgradient method.

$$\begin{aligned} & {\delta }_{u}^{(l+1)}={\delta }_{u}^{(l)}+{\nu }_{\delta }\cdot \left\{\left[\sum_{c=1}^{C}{B}_{u,c}^{\left(l\right)}\cdot {\mathit{log}}_{2}\left(1+\frac{{G}_{u,c}\cdot {P}_{u,c}^{\left(l\right)}}{{B}_{u,c}^{\left(l\right)}\cdot {N}_{g}}\right)\right]-{R}_{u}^{min}\right\},\\ & {\delta }_{u}^{\left(l+1\right)}\ge 0.\end{aligned}$$
(31)
$${\vartheta }_{c}^{(l+1)}={\vartheta }_{c}^{(l)}+{\nu }_{\mathrm{\vartheta }}\cdot \left(\sum_{u=1}^{U}{B}_{u,c}^{\left(l\right)}-{BW}_{c}\right),{\vartheta }_{c}^{\left(l+1\right)}\ge 0.$$
(32)
$${\gamma }_{u}^{(l+1)}={\gamma }_{u}^{(l)}+{\nu }_{\gamma }\cdot \left(\sum_{c=1}^{C}{P}_{u,c}^{\left(l\right)}-{P}_{u}^{max}\right),{\gamma }_{u}^{\left(l+1\right)}\ge 0.$$
(33)

Here, denotes the superscript \((l)\) as the lth cycle, and \({\nu }_{x}(x\in \left\{ \delta , \vartheta , \gamma \right\})\) is the positive step interval in the subgradient method. System of (13) or (14-1), (14-2), (13_a)–(13_f) are linear-least squares problems and can be solved as a quadratic programming [22, 23]. The algorithm for solving (26) are illustrated in Table 2. Table 3 shows the algorithm for obtaining global power and bandwidth allocation.

Table 2 The algorithm for solving (26)
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Table 3 The algorithms for obtaining global power and bandwidth allocation
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Figure 3 shows the flowchart of our initial EE scheduling method. When the solution results are obtained, the initial EE values can also be calculated using (7)–(10). The EE, power and bandwidth obtained by OptEE process for the time will be a criterion for resource allocation.

Fig. 3
figure 3

Flowchart of initial energy efficient resource allocation algorithm

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Therefore, the algorithm which determines resource allocation priority and EE update based on this value can achieve high efficiency, allowing UE with high EE to have the high resource allocation priority in uplink.

2.4 Update of EE values considering hysteresis

Using the algorithm proposed in Sect. 2.2, computational times consumed to estimate the EE will be very long. This is unsuitable when considering that the scheduling algorithm should be run in real time at the base station. So, we propose a new algorithm, if there is a new connection, then the algorithm will solve the optimization problem for the initial EE value, otherwise it will calculate the historical mean EE.

In this paper, EE update with optimization solution and hysteresis, providing UE matching based on the real time watch of the new connection, is processed in parallel and the calculated values are exchanged through the buffer. The latest optimum EE stored in the buffer is used in MeanEE equation.

The EE of a UE u denoted \({\varepsilon }_{u}(t)\) at any moment t and the mean EE of the UE u denoted as \(\overline{{\varepsilon }_{u}^{his}(t)}\). The equation for updating the mean EE considering hysteresis is as follows.

$$\overline{{\varepsilon }_{u}(t)}=\left(1-\frac{1}{{T}_{w}}\right)\overline{{\varepsilon }_{u}^{his}\left(t-1\right)}+\frac{1}{{T}_{w}}{\varepsilon }_{u}(t)$$
(34)

where

$$ \overline{{\varepsilon_{u}^{his} \left( {t - 1} \right)}} = {\text{max}}\left( {\overline{{\varepsilon_{u}^{opt} \left( {\left\lfloor {\left( {k - 1} \right) \cdot \frac{{t_{op} }}{TTI}} \right\rfloor } \right)}} ,\;\overline{{\varepsilon_{u} \left( {t - 1} \right)}} } \right) $$
(34.1)

Tw is the size of the window.

3 QoS and energy-efficiency aware scheduling and resource allocation scheme

The main objective of user prioritizing is to arrange the order of users for resource allocation. We assume that users are classified into two types: users with GBR service and UEs with Non-GBR service, according to the service characteristics.

For GBR UEs large packet delay causes packet loss and low QoS gains. Therefore, in order to enhance the QoS gains for GBR UEs, it is important to ensure that packets are not delayed as much as possible, so this should be taken into account in the order of resource allocation. For Non-GBR UEs, the main cause of packet loss is the lack of buffer capacity. Therefore, in order to enhance the QoS gains for Non-GBR UEs, it is important to allocate resources so that no buffer overflow occurs, and this should be considered in the calculating user priority. We determined the priority factor \({\chi }_{s}\left(t\right)\) considering packet delay and buffer length for GBR and Non-GBR UEs as follows.

$${\chi }_{u}\left(t\right)=\left\{\begin{array}{c}\eta \cdot \frac{{d}_{u,t}}{{D}_{u}}, u\in GBR\\ \frac{{q}_{u,t}}{{Q}_{u}}, u\in non-GBR\end{array}\right.$$
(35)

\({D}_{u}\): delay tolerances of UE,

\({d}_{u,t}\): longest delay time of the packets in the UE buffer,

\({Q}_{u}\): maximum queue size of UE,

\({q}_{u,t}\): current queue length in the buffer of UE,

\(\eta\): This is the parameter for adjusting the weighting values of GBR and Non-GBR UEs, which is predetermined by network operators.

In (35), a threshold (maximum delay threshold for the GBR UEs, maximum buffer length threshold for the Non-GBR UEs) and priority is determined in proportion to the ratio between the current values, thus allowing resource allocation in such a way as to reduce packet loss as possible. Here, a scheduling metric \({\chi }_{u}\left(t\right)\) is proposed aiming to reduce the packet loss rate and increase EE. Scheduling metric is as follows:

$${\psi }_{u}\left(t\right)=\left(\frac{{\left[{\varepsilon }_{u}\left(t\right)\right]}^{\rho }}{{\left[\overline{{\varepsilon }_{u}\left(t\right)}\right]}^{1-\rho }}\right){\chi }_{u}\left(t\right), {u}^{*}=\underset{u}{\mathit{argmax}}{\psi }_{u}(t)$$
(36)

where \({\varepsilon }_{u}\left(t\right)\) is the instantaneous EE achievable in the current time slot, and \(\overline{{\varepsilon }_{u}\left(t\right)}\) is the historical mean EE as a result of resource allocation in the previous time slots, which is obtained in (34). \(\rho \) is the weight parameter introduced to balance the current EE and historical mean EE and take into account fairness, which is \(\rho \in [\mathrm{0,1}]\).

Setting the weight parameter \(\rho \) to 1, which means gives higher user priority to UEs with high EE in the current time slot, resulting in higher priority of UEs with good channel status in scheduling. The reason is that the EE means the throughput per unit energy consumption, since UEs with good channel status can achieve higher signal-to-noise ratios for the same transmission power consumption. However, if the value is set to 0, the mean EE is the determining factor in choosing UEs. In this case, fairness is considered more heavily and all UEs achieve the same EE in the long term.

After that, resource blocks are allocated and the corresponding link adaptation is performed to provide the desired data rate in the order of UEs with higher priority.

4 Simulation results

In the paper, real time simulation is realized in open source simulator LTE-SIM and MATLAB open source Vienna LTE Simulator is used for non-real time simulation.

All simulation model parameters are summarized in Table 4.

Table 4 Simulation Parameters
Full size table

Simulation for single cell is conducted by varying its size with the number of UE ranging from small cell to micro cell. In the case of small cell, the number varied in the range of 5 and 20, from 20 to 40 for Pico cell and from 40 to 50 for Micro cell. As shown in the Table 4, the system has some traffic load due to the low bandwidth for each CC, though the number of subscribers is not too high, and it will leads the delay violation to some degree, thus resulting in packet loss. VoIP communication is used as a GBR service and FTP service is chosen as non-GBR service. Simulation time is 10000 TTI (10 s). the movement direction of UE is based on uniformly random distribution. For small cell, UE moves 3 km/h and it is increased as its size gets bigger in the simulation.

The accuracy in solutions of simulation equation is set to 10−4.

All UEs are normally distributed in small area and they have selected randomly VoIP and FTP service. The buffer size is 25 Kbytes for mobile with FTP service and 40 Kbytes for VoIP, respectively. The maximum delay threshold is 40 ms.

In the paper, the comparison between the proposed method and the existing methods like EPF, EE, H-PSSS in EE, Packet loss and throughput with the number of UEs.

4.1 The evaluation of efficiency of the OptEE process based on non-real time simulation

In this section of paper, we have evaluated the availability of OptEE process. Non-real time simulation is conducted with MATLAB in this paper.

The time required for the optimization process OptEE is dependent on the performance of the processor (rate, clock, RAM), algorithm, coding technology and the number of UEs and therefore there is no significance of considering the time of optimization solving calculation.

In the paper, the performance of the proposed method is evaluated for various top. we have examined the performance of the proposed method according to top compared idealized OptEE, top of which is less than 1 ms, in MATLAB simulation.

In addition to comparison between idealized OptEE, we have also compared with the method using only mean EE with hysteresis. Parameters are exactly same in all simulation cases. In other words, the radius is 20 m, the number of UEs are 10 and mobility is 3 km/h in the small cell case, but radius 500 m, 60 UEs and mobility 72 km/h in the macro cell case.

Figure 4 shows the simulation results. Figure 4 illustrates the results of Eq. (37) below in small cell case (Fig. 4a) and macro cell case (Fig. 4b), respectively.

Fig. 4
figure 4

EE trend of the proposed method along optimization solution time

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$$Relative deviation(\mathrm{\%})= \left\{\begin{array}{c}\frac{\left|OptEE-Proposed\right|}{Proposed}\times 100\\ \frac{\left|MeanEE-Proposed\right|}{Proposed}\times 100\end{array}\right.$$
(37)

As can be seen in the figure, the consumed time for OptEE the longer, the difference between the proposed and the idealized OptEE method the larger. This mean that EE of proposed methods decreases less and less as increasing top, considering EE of idealized OptEE is the best.

On the other hand, we can see that the proposed scheme will approach to MeanEE with top. The reason is that the longer the estimation time top, the older our proposed method used optimum EE, the larger its difference with corresponding ideal optimum EE.

As can be seen in Fig. 4b, As the movement speed becomes faster, the difference gets bigger. In the Fig. 4, the left one and the right one clearly shows the distinction. That’s because the movement speed of UE is slow and therefore the channel state changes very slowly along the time in small cell. On the other hand, the movement speed is very fast in Macro Cell and the channel state also changes very quickly. If the speed of UE is slow in Macro cell, the difference between the OptEE and the proposed method might be significantly small since the channel state change is very slow. However, the number of UEs will increase in Macro cell and it causes much longer calculation time and it will affect to the performance of the proposed method even if the movement of UE is slow.

From the result of the comparison, we can conclude that the channel state changes very slow and the more effective, the smaller the number of UE is.

In the paper, we have analysed the ultimate effectiveness of the proposed method by conducting real-time simulation based on LTE-SIM from the evaluation of non-real time simulation result.

4.2 Comparison in EE

In the paper, LTE-SIM based real-time simulation is realized. The radius of Macro cell is about 500 m and the number of UE varies from 10 to 50 to compare with the previous methods about different criterion. The UE’s speed is from 3 to 36 km/h from the conclusion and other parameters are set as above.

As shown in Fig. 5, the proposed algorithm provides higher EE compared to EPF, H-PSSS and MeanEE algorithms. Because the algorithm gets the higher gain in term of EE, compared to the approaches considering only the QoS in user priority. Especially, the proposed algorithm provides a similar high energy efficiency as the EE algorithm, because of allocating resource to approximate optimum EE. On the other hand, when the network traffic increases versus the number of UEs, the slightly lower EE gain compared to the EE algorithm aimed at optimizing the EE is related to the fact that, due to the influence of the priority factor considering the QoS, allocating resource to UEs with poor channel conditions resulted in some loss in aspect of EE. Another important reason is that the difference between the time of the MeanEE process using optimum solving value in the proposed method and the time of the OptEE process starting to estimate will be more and more large, because of taking a long time of calculating in the OptEE process with a large of number of UE. In other words, the effectiveness of optimization will be less and less fall according to top.

Fig. 5
figure 5

System EE versus the number of UEs

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4.3 Comparison in packet loss rate

Figure 6 shows the packet loss rate of the system versus the total number of users while setting the number of users with VoIP sessions and FTP services equal to each other. The results showed that the packet loss rate of the EE algorithm increases rapidly with increasing number of UEs and increasing system load, but the packet loss rate of the proposed algorithm increases with similar characteristics as the EPF and H-PSSS, MeanEE algorithms. This is also because of reducing the risk of packet loss due to packet delay and buffer overflow, calculating the user priority which is considered the packet loss.

Fig. 6
figure 6

Packet loss rate versus the number of users

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4.4 Comparison in throughput

Figure 7 shows how the system throughput changes versus the number of UE increases. As shown in the figure, throughput of the proposed algorithm is larger than the previous EPF, EE and MeanEE approaches, although it is almost similar to H-PSSS. It is because the EE of the proposed method is relatively higher and the resource blocks are allocated to minimize packet loss rate simultaneously. Increasing packet loss results in retransmission and retransmission causes a reduction of traffic.

Fig. 7
figure 7

System throughput versus the number of users

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4.5 Comparison in transmission power

Figure 8 shows the UE base power versus the number of UEs. It can be seen from the figure that although the energy consumption of the proposed method is slightly higher than that of the previous EE approach, it is much smaller than that of the EPF, H-PSSS and MeanEE schemes. It is because the EE of the proposed scheme is higher.

Fig. 8
figure 8

UE base power versus the number of users

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Finally, simulation results show that the proposed algorithm is a compromised resource allocation method considering both EE and packet loss rate. Especially, the proposed method has effectiveness when the number of UEs is small.

5 Conclusion

In this paper, firstly, we have formulated optimization problem of EE in LTE-A uplink and proposed the new algorithm for solving it. It is one of the main contributions of this paper to propose the framework for mixing of mean EE scheme (MeanEE process in the paper) and optimization scheme of EE (OptEE process in the paper). Of course, the algorithm to solve the NP-Hard optimization of EE is also the other contribution we have achieved. In order to increase EE of uplink in LTE-A, the scheduler has prioritized UEs for resource allocation based on EE, considering QoS. In case of small cells, the simulation results ensure that the proposed scheme can achieve significantly improvement in criterion of EE.

We have proposed a scheduling metric considering both packet loss rate and EE, based on calculating the EE of UEs with limited transmission power in LTE-A uplink. The proposed metric is the trade-off of the EE and QoS satisfaction. Combination of proposed metric and EE optimization scheme is able to improve EE, as well as, reduce packet loss rate. The simulation results showed that the proposed algorithm have better performance than that of previous approaches. The algorithm has the better characteristics in terms of power consumption, throughput and packet loss rate compared with previous approaches. The proposed method has a significant effect on reducing the energy consumption of the system for different services and achieving higher throughput while satisfying the QoS provision.

We should research on a method to solve quickly the optimization problem, while will continuously study on approaches to improve EE, considering UEs with high speed mobility and Ultra Dense Network in 5G, in the future.