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Joint congestion control and resource allocation for energyefficient transmission in 5G heterogeneous networks
EURASIP Journal on Wireless Communications and Networking volume 2019, Article number: 227 (2019)
Abstract
The deployment of small cells with carrier aggregation (CA) is a significant feature of fifth generation (5G) mobile communication systems which could be characterized by the multidimensional heterogeneity on their diversified requirements upon different resources. Taking the heterogeneity into account, we consider here a joint optimization problem wherein multiple kinds of resources are concurrently allocated to optimize the system throughput utility while enhancing the network energy efficiency (EE) and maintaining the system stability. Especially, for the highdimensional nondeterministic polynomial (NP)hard allocation problem embedded, we conduct a mathematical programming model involving nonlinear integer constraints to seek the longterm stable utility on throughput and introduce an iterative optimal modulation and coding schemebased (optimal MCSbased) heuristic algorithm as an effective solver. In addition, as data traffic and channel condition will be timevarying in the real world, an admission control based on the Lyapunov technique that requires no prior knowledge on channel information is proposed to reduce the system overhead. Finally, not only the performance bound is derived in theory, but also the numerical experiments are conduced to reveal its characteristics with respect to the system parameter V and the EE requirement.
Introduction
For the next generation of mobile internet connectivity, 5G networks aim to offer increased data rate, shortened latency, improved energy efficiency, reduced cost, and other desired features. To this end, the communication society has proposed many techniques from different aspects such as dense heterogeneous networks, cloudbased radio access networks, energyaware communications, and wireless energy harvesting [1]. Among these, the dense heterogeneous networks (HetNets) based on Long Term EvolutionAdvanced (LTEA) including carrier aggregation (CA) as its key feature would be particularly useful since the aggregation can achieve wider bandwidth and better energy efficiency (EE) [2]. Specially, with the aid of 4G framework, smallcells (SCs) that represent picocells, femtocells, etc., can be more easily deployed to improve the 5G capacity by offloading the traffic from a macro cell (MC) to SCs [3].
Providing these benefits, designing HetNets, however, is a challenging work. One of the hardest challenges is caused by its resource and interference management because both MCs and SCs in a 5G network would tend to utilize the radio resources from the same service provider. To reduce the overhead emerged, the cells would be arranged under the socalled cochannel deployment, i.e., by spatially reusing the available spectrum or, specifically, by using a different set of channels or resource blocks (RBs) for macro base stations (MBSs) and small base stations (SBSs), as noted, e.g., in [4]. However, it is only a step toward the solution to the CAcapable LTE system that allows several component carriers (CCs) to be aggregated. That is, given CA, this system is still complicated by its requirement to modify the radio resource management (RRM) entity, including CC selection, RB allocation, modulation and coding scheme (MCS) assignment, and power allocation. For this complexity, many researches had been done to develop the approaches on RRM that can properly allocate RBs, CCs [5–7], and even MCSs [8] to increase the performance. Now, as the standard evolves, more attentions are paid to the heterogeneous networks wherein the multiple types of resources would be allocated between MCs and SCs that are connected by backhaul links in a multitier sense [9, 10]. In such networks, highcapacity fiber backhaul (e.g., IEEE 802.3av 10GEPON) will play a major role that consistently provides data rates 100 times higher than cellular networks to help in reaching the envisioned 10 Gbps peak data rates required by 5G [10]. Here, we focus on the multicell multitier networks equipped with highcapacity backhaul and introduce a solution based on discrete power control^{Footnote 1}, reflecting the fact that 3GPP LTE cellular networks only support discrete power levels in the downlink via a userspecific datatopilotpower offset parameter [13].
Given that, a joint congestion control and downlink resource allocation problem is particularly considered with the objective to maximize a longterm throughput utility subject to a systemwide EE requirement. The major challenge of this optimization problem is brought by the various constraints that are specific to the LTEA system with CA. For this, a highdimensional allocation problem involved is first formulated as a programming model whose constraints involve integer variables coupled with a nonlinear form, and optimally solving such a model at each transmission time interval (TTI) is impractical. In addition, for the data traffic and channel condition involved would be both timevarying in the system, an admission control is usually required to stabilize the data queue for each user equipment (UE). Thus, to address the combinatorial problem with queueing stability, an iterative optimal MCSbased heuristic algorithm inspired by the iterative linear programmingbased heuristic [14, 15] is proposed to resolve the NPhard allocation problem involved in the low layer. Then, as LTE would be a stochastic system with timevarying traffic and channel as noted, we further address its queueing stability problem at the high layer for the system. This is challenging because unlike deterministic optimization, stochastic optimization is usually hard to solve, and even harder than most wellknown combinatorial optimization problems [16]. Given that, the Lyapunovbased optimization is considered to be a very useful technique to enable constrained optimization of time averages in general stochastic systems [17]. Accordingly, a Lyapunov optimization framework is developed to address the high layer problem focusing on the timevarying data traffic and channel condition without a priori knowledge of arrivals. By combining the solutions from the two layers, we are able to approach the optimal tradeoff with a control parameter V and satisfy the longterm EE requirement simultaneously. More specifically, the characteristics of this work can be summarized as follows:

For the highdimensional resource allocation optimization problem in the 5G LTEA multitier multicell heterogeneous wireless networks that is a NP hard combinatorial problem, we first transform the corresponding nonlinear integer programming model into a linear counterpart that can be solved by conventional techniques.

Then, an iterative optimal MCSbased heuristic algorithm or IOMHA for short inspired by the iterative linear programmingbased heuristic is developed to approach the optima within a time limit. Given that, a twolayer method is proposed for the stochastic programming problem so that the data queue of each UE can be stabilized in the high layer based on the resources efficiently allocated in the low layer.

Using the Lyapunov optimization framework, we realize a formulation to strike a balance between average throughput and average delay while guaranteeing the required EE performance and accommodating both traffic variations in the long term and channel fading in the short term, in the heterogeneous networks.

We show that with the EE constraint enforced, the proposed algorithm has its performance advantage especially on EE through our simulation study. In the study, by gradually improving its result, our IOMHA is also shown to resolve the complex allocation problem effectively, trading the optimality of the NPhard optimization problem off against a lower and controllable complexity to approach the optimal solution iteratively, in contrast to the other algorithms shown in, e.g., [5, 8], which would be done only once for obtaining suboptimal solutions to their allocation problems in LTE without a chance for further improvements.
The remainder of this paper is organized as follows. First, the related works are summarized in Section 2. Then, the scheduling constraints and queueing dynamics of the joint optimization problem are formulated in Section 3. The online control method based on Lyapunov driftpluspenalty technique for this problem is proposed in Section 4, and the iterative optimalMCSbased heuristic algorithm involved is introduced in Section 5. Given that, the performance bounds and evaluations of this work are presented in Sections 6 and 7, respectively. Finally, conclusions are drawn in Section 8.
Related works
For 5G networking, there are many networks continuously explored with variant aims on different performance metrics. Among these, energy efficiency (EE) plays a vital role in 5G as the future networks should effectively reduce the overall carbon footprint for the world to be sustainable. With respect to this issue, the authors in [18] had studied energy efficiency of resource allocation in orthogonal frequency division multiple access (OFDMA) downlink networks where the circuit power consumption and the minimum data rate required were both considered. More recently, the authors in [9] investigated energyefficient power allocation and wireless backhaul bandwidth allocation in OFDMA heterogeneous small cell networks. Specifically, they proposed a near optimal iteration resource algorithm to solve the power and bandwidth allocation problem and suggested also a suboptimal lowcomplexity algorithm to this end.
Apart from the above, downlink radio resource allocation methods in LTE system with CA are particularly noted here for their potential on EE even without direct objectives for this aim. As surveyed in [19], a twostep allocation method was considered in [5–7] that first uses a load balancing scheme to assign CCs to UEs, and then schedules RBs of these CCs to reduce the computational complexity for the NPhard RB/CC allocation problem. In addition, different joint allocation approaches had also been done with various efforts to reduce the time complexity. For example, the work in [20] divided the optimization problem into a number of subproblems for each CC to optimize its RB allocation independently. Then, after RB assignment, an iterative resource adjustment algorithm was performed to meet the CA capability requirement for UEs. Despite their differences, these approaches mainly focus on RB/CC allocation and pay no attention to the other constraints specific to LTE/LTEA.
In addition, if categorized by using the number of cells, the authors in [21] have recently proposed for a singlecell scenario a downlink scheduling algorithm aiming to maximize the weighted sum of throughput constrained by the allocation rules of LTE. Similarly, the authors in [8] have addressed a downlink resource scheduling problem that takes also into account the MCS constraint for LTE, with a greedybased algorithm to maximize the system throughput. With the notable performance gains obtained, these algorithms, however, consider no queueing dynamic resulted from the dynamic traffics that should be involved also. Next, as another category, for a multicell scenario, the authors in [22] proposed a resource allocation algorithm that accounts for MCS, RB, and transmit power, with intercell interference coordination, but ignores MCS constraint, CA, and queueing dynamic. In addition, the previous work in [23] considered a dynamic resource allocation algorithm for downlink transmission in a multicell network. However, it considered no discrete power allocation in the downlink and ignored the EE performance that is one of the most important factors impacting the system. Here, for the 5G multitier multicell networks based on LTEA with discrete power levels, we first transform the nonlinear integer scheduling constraints to be involved into their linear counterparts as the previous. Then, an optimalMCSbased heuristic algorithm inspired by the iterative linear programmingbased heuristic is proposed to approach the optima within a time limit. Finally, a driftpluspenalty approach for joint admission control and resource allocation with the requirement on EE and queueing stability is constructed that iteratively resolves the stochastic optimization problem involved for the longterm optimal throughput utility.
Methods
System model and problem formulation
In the sequel, we consider a multitier multicell heterogeneous network as exemplified in Fig. 1, consisting of s base stations (including a MBS and s−1 SBSs) and u UEs located in the service area of these cells. In addition, the network is equipped with a number of c CCs. Each CC has b RBs, and each RB can use one of l MCSs for transmission. Further, there are p discrete power levels (PLs), and MBS/SBSs can choose among P={σ_{1}P_{max},σ_{2}P_{max},....,σ_{p=P}P_{max}} to transmit, where 0<σ_{1}<σ_{2},...,<σ_{p=P}=1 and P_{max} denotes the maximum power as that in [12]. In summary, there are \({\mathcal {U}}, {\mathcal {C}}, {\mathcal {B}}, {\mathcal {L}}, {\mathcal {S}}\), and \({\mathcal {P}}\) to represent the set of UEs, the set of CCs, the set of RBs per CC, the set of MCSs per RB, the set of base stations (BSs), and the set of power levels (PLs) with u, c, b, l, s, and p as their indices, and \({\mathbf {u}} = {\mathcal {U}}, {\mathbf {c}} = {\mathcal {C}}, {\mathbf {b}} = {\mathcal {B}}, {\mathbf {l}} = {\mathcal {L}}, {\mathbf {s}} = {\mathcal {S}}\), and \({\mathbf {p}} = {\mathcal {P}}\) as their cardinalities, respectively. Given that, we focus on downlink transmission in the 5G heterogeneous network based on LTEA and consider a stochastic communication system whose traffic load is changed from time to time, requiring an online admission algorithm for its stability. Further, its channel condition is also timevarying. For this condition, a UE would inspect reference signals currently transmitted from MBS or SBSs to estimate the channel quality of each RB [24]. After that, it will send a feedback report with the channel quality indicator (CQI) whose value would then be mapped to the highestrate MCS adoptable by the UE for receiving the corresponding RB from MBS/SBSs [25]. Then, with the information from UEs and SBSs, MBS is responsible for admission control, resource scheduling, and link adaption. For easy reference, the important symbols for the formulation are summarized in Table 1 in advance.
Multiresource allocation
To show the multiple types of resources involved more concisely, we denote by \(\underline {e}\) a binary variable or an element of the feasible set Ξ representing all possible allocations, where \(\underline {e}_{{u,c,b,l,s,p}} \stackrel {\triangle }{=} (u_{\underline {e}} = u, c_{\underline {e}} = c, b_{\underline {e}} = b, l_{\underline {e}} = l, s_{\underline {e}} = s, p_{\underline {e}} = p\)) with value of 1 exhibits that RB b of CC c on MCS l at PL p of cell s is assigned to UE u, and 0 otherwise. Further, let Ψ_{u,c,b,s,p} be the index of the highestrate MCS allowed among the possible transmissions, \(\underline {\hat {e}}_{{u,c,b,s,p}} = (u,c,b,\hat {l},s,p), \forall \hat {l} \in {\mathcal {L}}\). Given that, the achieved transmission rate with the allocation, \(v(\underline {e})\), is the data rate of an RB on MCS l, r_{l}, for l≤Ψ_{u,c,b,s,p}, and 0 otherwise.
Channel, power, and energy efficiency model
Accordingly, the allocation (or scheduling) algorithm is conducted to accommodate a slow fading network wherein channel condition would remain unchanged during the resource allocation period (Ch. 6 of [26]), which complies with the highrate network with reduced degree of mobility. In this situation, the signaltonoise ratio (SNR) from BS s to UE u using RB b of CC c at PL p in time t can be represented by
where \(h^{c,b}_{s,u}\) is the channel gain from transmitter (MBS or SBS) s to receiver (UE) u using RB b of CC c, and d_{s, u} is the distance from s to u. The channel is considered to be Rayleigh fading which yields the channel gain following the exponential distribution. In addition, ρ is the pathloss factor and \(N_{s,u}^{c,b}\) is the noise experienced by u when s transmits to u on RB b of CC c. Providing that, an empirical downlink SNR to CQI mapping for LTE such as that in [27, 28] could be used to estimate the CQIs to be returned to BSs. Then, according to the CQIs collected, MBS would decide each MCS index l for the downlink transmission from BS \(s_{\underline {e}} = s\) to UE \(u_{\underline {e}} = u\) using RB \(b_{\underline {e}} = b\) of CC \(c_{\underline {e}} = c\) at PL \(p_{\underline {e}} = p\), in terms of \(\underline {e}\), and transmit the decisions to all SBSs it associates via the backhaul network. Consequently, as 3GPP specifies the transmit data rate of each MCS index l using table representation [24], the data rate \(v(\underline {e})\) would be obtained through a function or table mapping, r_{l}, for each RB on MCS l. Given that and the feasible allocation set Ξ, the total data rate can be given by \(R_{\text {tot}}(t) = \sum _{\underline {e} \in \Xi }\left (\underline {e}(t) \times v\left (\underline {e}(t)\right) \right) \). Similarly, the total power consumption can be obtained by \(P_{\text {tot}}(t) = \sum _{\underline {e} \in \Xi } \left (\underline {e}(t) \left (P^{p}_{s,u}(t) + P^{c}_{s,u}\right) \right)\), where \(P^{p}_{s,u}(t)\) is the transmit power from \(s_{\underline {e}} = s\) to \(u_{\underline {e}} = u\) at power level \(p_{\underline {e}} = p\), and \(P^{c}_{s,u}\) is the constant circuit power for this transmission.
Specifically, in the stochastic system, we are interested in the limits of the timeaverage expectations of the above metrics. That is,
In terms of the longterm metrics, the energy efficiency is considered as the ratio of the longterm aggregated rate to the longterm total energy consumption as
where W is used to accommodate the quantitative difference between the two metrics in the ratio.
Scheduling constraints
For the heterogeneous network with CA, we have the following scheduling constraints. First, as the basic unit for the transmission, each RB can be assigned to a single UE u at most with a certain MCS l. To show this, we let \(\hat {\underline {e}}_{1} \stackrel {\triangle }{=} (\hat {u},c,b,\hat {l},s,p)\) be the binary allocation variables with different \(\hat {u} \in {\mathcal {U}}\) and \(\hat {l} \in {\mathcal {L}}\) while fixing \(c_{\underline {\hat {e}}_{1}} = c, b_{\underline {\hat {e}}_{1}} = b, s_{\underline {\hat {e}}_{1}} = s\), and \(p_{\underline {\hat {e}}_{1}} = p\). Given that, this constraint can be simply shown by
where \(\mathbbm {1}\{x\}\) denotes an indicator function whose value is 1 if x is true, and 0 otherwise. In addition, the notations are given without the time index t for brevity. Further, according to LTEA, it is required that if a UE u is assigned a CC c by a BS s serving it, then all RBs of c allocated to u should use the same MCS l to transmit. More specifically, the MCS constraint based on LTEA is considered as
As noted in Section 3.2, a UE can only use a MCS less than or equal to Ψ_{u,c,b,s,p}. If not, it could lead to an unacceptable bit error rate on transmission, and the transmission should be discarded. Thus, we have the following constraint
Apart from the above without special notion on the number of cells involved, here, we take into account the constraints specific to the multicell environment as well as follows. First, to reduce overheads on backhaul, it is commonly considered that a UE is only served by a single BS s, which implies a monopoly constraint as
Second, even given the spatial reuse principle, it should be still considered that an RB b of CC c already allocated to a BS s can not assigned to its neighboring BSs \(s' \in {\mathcal {N}}_{s}\) to avoid the leading cause of intercell interference. Consequently, it also implies a monopoly constraint as
Moreover, there are two cardinality constraints to be involved. First, each UE u has its own limitation on the number of CC allocated by a BS s, denoted by k_{u}. For example, it can be enforced that a UE of LTE 8/9 can only use 1 CC while a LTEA UE would use 2 CCs. In general, such a constraint can be written by
Similarly, a cardinality constraint for each BS s to equip with at most f_{s} CCs for communication can be represented by
Linear transformation of scheduling constraints
As shown in above, the indicator functions with complex conditions could be nonlinear on the binary integer variables involved. For those especially involving logical operations, we refer to the work in [29] showing that two eitheror constraints f(x_{1},x_{2},...,x_{n})≤0 and g(x_{1},x_{2},...,x_{n})≤0 can be transformed to f(x_{1},x_{2},...,x_{n})≤My and g(x_{1},x_{2},...,x_{n})≤M(1−y) with a large number M and auxiliary binary variable y such that f(x_{1},x_{2},...,x_{n})≤M and g(x_{1},x_{2},...,x_{n})≤M. Here, given a certain l, the condition in the outer indicator function in (6) implies a logic operation to choose among the multiple binary variables \(\underline {\hat {e}}_{8} \stackrel {\triangle }{=} (u,c,\hat {b},l,s,p)\) which satisfy the condition \(l_{\hat {\underline {e}}_{8}} = l\) shown in the inner indicator function can be transformed to \(\sum \underline {\hat {e}}_{8} \leq \mathbf {b} \underline {y}_{1}\), where \(\underline {\hat {y}}_{1} \stackrel {\triangle }{=} (u, c, \hat {l}, s, p)\) is defined to play the role of the auxiliary variable y, and \(\mathbf {b} = {\mathcal {B}}\) defined before plays the role of M. Given that, the constraint that all RBs should be assigned only the same MCS in this context can be done by \(\sum \underline {\hat {y}}_{1} \leq 1\) on the auxiliary variables. Therefore, (6) can be transformed to the linear counterparts as
In addition, the inner indicator functions in (10) and (11) could be also regarded as the logic operations to choose among the binary variables that can satisfy the conditions specified, and can apply a transformation like the above. Specifically, with the aid of the auxiliary variables \(\underline {\hat {y}}_{2}\) in addition to the binary variables \(\underline {\hat {e}}_{9}\), both shown below, (10) can be represented by
Similarly, with the auxiliary variables \(\underline {\hat {y}}_{3}\) and the binary variables \(\underline {\hat {e}}_{10}\) shown below, (11) can be transformed to
Apart from these, the monopoly constraints shown in (8) could be rewritten with linear forms as well. To this end, let \(\sum _{\hat {\underline {e}}_{4} =(u,\hat {c},\hat {b},\hat {l},s,\hat {p})} \hat {\underline {e}}_{4}\) be the first metric for transforming the logical eitheror constraints in [29] (here, \(\hat {\underline {e}}_{4}\) is directly drawn because \(\mathbbm {1}\left \{\hat {\underline {e}}_{4} \right \}= \hat {\underline {e}}_{4}\)) and \(\sum _{\tilde {\underline {e}}_{4} =(u,\tilde {c},\tilde {b},\tilde {l},\tilde {s} \in {\mathcal {S}} \backslash s,\tilde {p})} \tilde {\underline {e}}_{4}\) be the second metric. Then, by introducing also the large number, M=ucblsp, and the auxiliary binary variables \(\underline {\hat {y}}_{4}\), we can transform (8) into its linear counterparts as
Similarly, by introducing the auxiliary binary variables \(\underline {\hat {y}}_{5} \) and M into (9), we have the linear counterparts as
Stochastic system and queue dynamic
Now, even though the scheduling constraints can be linearly transformed, the design of 5G heterogenous networks in dynamic is still challenged by stochastic channel condition and timevarying data traffic. Specifically, the random channel gains are considered to be exponentially distributed, and the downlink traffics to UEs in time t are represented by an vector A(t)=△(A_{1}(t),...,A_{u}(t)), according to an independently and identically distributed (i.i.d.) distribution over t whose expectations would be \(\mathbb {E}\{A(t)\} = \lambda \stackrel {\triangle }{=} (\lambda _{1},...,\lambda _{\mathbf {u}})\). In addition, it is assumed that a maximum \(A_{u}^{max}\) exists that any nonnegative traffic arrival A_{u}(t) will not exceed. Given that, however, the statistics of A(t) are still unknown and its capacity region is also hard to estimate for a real system. Thus, without flow control, the data queues can not be stabilized in general. For this issue, an admission control method is proposed here to determine R_{u}(t) out of A_{u}(t), followed by an allocation algorithm introduced next to provide link rates μ_{u}(t) for serving the admitted traffic. To realize this mechanism, the data queueing dynamic for UE \(u \in {\mathcal {U}}\) is formulated first by
Then, the average data queue length on each u would be conducted to be strongly stable as
Note that in (22), the service rate μ_{u} defined for a UE u can be obtained by
Similarly, we ignore the time index t in above for brevity. As a result, \(R_{tot} = \sum _{u \in {\mathcal {U}}} \mu _{u}\). Moreover, we can see that not only the resource scheduling to provide service, but also the throughput \(r_{u}(t) \stackrel {\triangle }{=} \frac {1}{t} \sum _{\tau =0}^{t1} \mathbb {E}\{R_{u}(\tau)\}\) to represent its performance contributes the queueing dynamic (22). Given that, the timeaverage throughput \(\overline {r}_{u}\), which represents the admitted and transmitted data for u in the long term, is considered as the key metric in the timevarying system for optimization:
Problem formulation
Taking all the above into account, we can now formulate the joint congestion control and resource allocation with EEdelay tradeoff problem (JCREEP) for the heterogeneous wireless network by the following stochastic programming model:
In above, (26C1) denotes the strong stability of data queues in the long term. (26C2) and (26C3) exhibit the constraints enforcing that the average and instantaneous throughput to be feasible. (26C4) shows the resource scheduling constraints in linear forms most done in Section 3.5. Note that, even with the linear forms, the constraints (5), (7), and (12)(21) are still involving the specific binary integer variables \(\underline {\hat {e}}, \underline {\hat {y}}\), or both, and deciding these binary variables concurrently for the optimization is a combinatorial problem that is NPhard if no special structures are imposed. Finally, (26C5) ensures that the EE performance will achieve the requirement \(\eta _{EE}^{\text {req}}\) predefined. It is worth noting here that, by using EE in (4) as one of the constraints rather than the objective function, we can maximize the system utility and guarantee EE of the whole system simultaneously, which may not be achieved by simply optimizing the EE metric as the program objective as that in the related works [30, 31].
Optimization for the stochastic system
With the aid of Lyapunov driftpluspenalty technique and the iterative heuristic algorithm to be introduced, we would next develop an online control framework to resolve (26) composed of the resource allocation problem and the traffic admission control problem in the stochastic system.
Equivalent transformation
As shown in (26), JCREEP involves a function \(\phi (\overline {x})\) with a timeaverage parameter, say \(\overline {x}\), rather than a timeaverage function \(\overline {\phi (x)}\) with a pure parameter, say x. To use the Lyapunov driftpluspenalty technique in the optimization as shown in [17], we would reformat JCREEP to involve the latter by first introducing an infinite sequence of random vectors in \(\mathbb {R}\) as γ=(γ_{1}(t),...,γ_{u}(t)). Then, we define a timeaverage metric \(\overline {\gamma }_{u} \stackrel {\triangle }{=} {\lim }_{T \rightarrow \infty } \frac {1}{T} \sum _{t=0}^{T1} \mathbb {E}\{\gamma _{u}(t)\}\) and a timeaverage function \(\overline {\phi (\gamma _{u})} \stackrel {\triangle }{=} {\lim }_{T \rightarrow \infty } \frac {1}{T} \sum _{t=0}^{T1} \mathbb {E}\{\phi (\overline {\gamma }_{u}(t))\}\). With these, JCREEP can be transformed to an equivalent problem, say eJCREEP, as follows:
Virtual queues
In eJCREEP, (27C6) denotes the constraints to ensure the system stability representing the fact that the arrivals would be eventually served. To conform these constraints, we define a virtual queue H_{u} for each \(u \in {\mathcal {U}}\). Specifically, given an initial value H_{u}(0)=0, such a queue will be updated by
In addition, for the EE performance requirement in (27C5), we define a virtual queue Z which evolves as
In terms of queueing dynamic similar to (22), the variables γ_{u}(t) and Wη_{EE}P_{tot}(t) can be regarded as the arrivals of the virtual queues in (28) and (29), while R_{u}(t) and R_{tot}(t) as the service rates of these virtual queues, respectively.
Online control based on Lyapunov driftpluspenalty
Given H_{u}(t),Z(t), and Q_{u}(t) for the online control method, we define \(\Theta (t)\stackrel {\triangle }{=} \left \{Q_{u}(t), H_{u}(t), Z(t): u \in {\mathcal {U}} \right \}\), a vector concatenating all the data and virtual queues involved. Further, for realizing a scalar metric to reflect the queue congestion, we define a quadratic Lyapunov function corresponding to the system as
Here, a small value of L(Θ(t)) implies that the sizes of data queues and virtual queues are all small and that the queues have strong stability. Given that, the queue stability can be ensured by persistently pushing the Lyapunov function toward a lower congestion state. Thus, to stabilize these queues, a oneslot conditional Lyapunov drift can be defined by
Now, apart from satisfying the average constraints and optimizing the system throughput utility, with this drift, our online dynamic control algorithm can observe the data and virtual queues, the current channel conditions, and the traffic states at each slot t so that R_{u}(t) can be determined and the resources be allocated to support γ_{u}(t), by minimizing a bound on the following Lyapunov conditional driftpluspenalty expression:
In above, the system parameter V is a nonnegative weight to represent the emphasis on the utility maximization compared with the queue stability and can be flexibly chosen to make a tradeoff between them. More precisely, with the above queueing dynamics, an upper bound for the driftpluspenaltybased algorithm can be obtained with the following theorem.
Theorem 1
At slot t, for any observed queue state Θ(t), and V≥0, the Lyapunov driftpluspenalty algorithm can satisfy the following inequality:
where \(\Gamma = \frac {1}{2} \left (3 \sum _{u \in {\mathcal {U}}} \left (A_{u}^{\text {max}}\right)^{2} + \sum _{u \in {\mathcal {U}}} \left (\mu _{u}^{\text {max}}\right)^{2} + \left (P_{\text {tot}}^{\text {max}}(t)\right)^{2} + \left (R_{\text {tot}}^{\text {max}}(t)\right)^{2} \right)\), and \(\mu _{u}^{\text {max}}\) denotes the maximum transmission rate that can be obtained on u.
Proof
Please refer to Appendix 1. □
Solving problem by decomposition
By observing the inequality in Theorem 1, we can decide to minimize the bound given in the righthand side (R.H.S.) of (33) at every time slot for the optimization. This is more convenient than directly minimizing the driftpluspenalty function itself because the minimization on R.H.S. could be decoupled to a series of independent subproblems that can be solved independently and simultaneously, as shown as follows.
Auxiliary variables
The first subproblem is to determine the optimal auxiliary variables γ_{u}, conducted to track the stability constraint shown in (27C6). Specifically, the optimal γ_{u} can be resulted from minimizing \( \mathbb {E}\left \{\sum _{u \in {\mathcal {U}}} \left (V\overline {{\phi }(\gamma _{u}(t))}  H_{u}(t) \gamma _{u}(t) \right)  \Theta (t)\right \}\) that is obtained by slightly rearranging the relevant terms in the R.H.S of (33). Clearly, for the minimization, a concave nondecreasing system utility ϕ(·) for each u should be given at first. Here, the wellknown utility function log(1+v_{u}γ_{u}) is considered as an example wherein v_{u} denotes a weight to maintain, e.g., the proportional fairness among UEs. Further, since the variables are independent among UEs, the minimization on γ_{u}(t) can be decoupled from the joint optimization. Finally, by reversing the sign in the objective for minimization, we have an equivalent maximization problem as
Obviously, it is a convex optimization problem. To find its optimum, we can first differentiate the objective function \(V \overline {{\phi }(\gamma _{u}(t))}  H_{u}(t) \gamma _{u}(t)\) with respect to γ_{u}(t) and then make the result equal to zero. For the log utility function just exemplified, we can solve the equation resulted to obtain its solution as
Admission control
Recall that for the system stability, our algorithm can admit only R_{u}(t) out of A_{u}(t) arrivals to transmit. For the traffic admission control subproblem in hand, we can observe the second and third expectations in R.H.S. of (33) to minimize \(\mathbb {E}\{ \sum _{u \in {\mathcal {U}}} R_{u}(t) \left (Q_{u}(t)  H_{u}(t) \right)  \Theta (t)\}\), which leads to the optimal traffic admission control at each TTI, as follows:
This is clearly a linear problem, and a simple thresholdbased admission control strategy for this problem can be derived as
As the threshold would imply, only when the virtual queue H_{u}(t) is accumulated larger than the data queue Q_{u}(t), the new arrival A_{u}(t) can then be admitted; otherwise, they will be denied to ensure the data traffic stability. That is, with the simple threshold, the admission control will be conducted to reduce H_{u}(t) to push γ_{u}(t) toward R_{u}(t) and increase the throughput R_{u}(t) to improve the system utility simultaneously.
Resource allocation for energy efficient transmission
As the kernel issue of eJCREEP, how to concurrently determine the multiple kinds of resources at each TTI for EE transmission is a NPhard combinatorial problem, in general, without special structures imposed. Here, with the aid of the driftpluspenalty technique developed, such a highdimensional allocation subproblem can be decomposed as minimizing \( \mathbb {E}\left \{\sum _{u\in {\mathcal {U}}} Q_{u}(t) \mu _{u}(t) + Z(t) \left (R_{\text {tot}}(t)  W \eta _{_{EE}}^{\text {req}} P_{tot}(t) \right)  \Theta (t) \right \}\) without knowing the channel states in advance. Similarly, by negating the objective, we have an equivalent maximization problem as
where \(\alpha _{u}(t) = Q_{u}(t) + Z(t), \beta (t) = W \eta _{_{EE}}^{\text {req}} Z(t)\), and \(P_{u}(t) = \sum _{s \in {\mathcal {S}}} P_{s,u}^{p} + P^{c}_{s,u}\). As shown in Sections 3.4 and 3.5, the scheduling constraints are composed by the binary integer variables involved, and the combinatorial problem would be NPhard, despite the optimization tools. Thus, instead of directly using an integer programming tool to solve this problem which would be still timeconsuming when the inputs are not small enough, we design in the sequel a more computationally efficient algorithm based on the iterative linear programmingbased heuristic (ILPH) to obtain a suboptimal solution that can be done within a time limit required.
Iterative optimal MCSbased heuristic algorithm
As shown in [15], iterative linear programmingbased heuristic (ILPH) is a useful approach to resolve 01 integer programs, which is done by solving a series of small subproblems obtained from linear programming relaxations. Specifically, at each iteration, ILPH will conduct an LPrelaxation of the current problem P to generate one constraint. Then, a reduced problem induced from an optimal solution of the LPrelaxation is solved to obtain a feasible solution for the initial problem. After that, if the stopping criterion is satisfied, then the solutions found are returned. Otherwise, a pseudocut is added to P and the process is repeated.
In our work, the binary variable \(\underline {e}\) for resource allocation is highly dimensional so that even solving a corresponding LPrelaxation problem could be timeconsuming unless the input size is trivially small. Thus, a MCSbased reallocation approach is conducted here to reduce the overhead. For doing so, we define \(J^{0}(\underline {e}) = \{j \in (u,c,b,l,s,p): \underline {e}_{j} =0 \}, J^{1}(\underline {e}) = \{j \in (u,c,b,l,s,p): \underline {e}_{j} =1 \}\), and \(J(\underline {e}) = J^{0}(\underline {e}) \cup J^{1}(\underline {e})\) similar to that in [15]. Then, an iterative optimal MCSbased heuristic algorithm (IOMHA) is introduced to restrict the search process to visiting the optimal solutions already generated from the timelimited optimization on P by adding a pseudocut at each iteration. As tabulated in Algorithm ?? with details, IOMHA first solves the maximization problem instance P in (38) to find a feasible solution \(\underline {e}^{*}\) with utility ν^{∗}. If the solution is not optimal, it might be improved by boosting the MCS of remaining RBs to find the largest MCS usable by all considered RBs [8]. However, instead of using the primitive method, IOMHA further attempts to make the utility contributed by the UE larger by releasing more RBs of the considered CC to render its remaining RBs able to employ an even higherrate MCS. To this end, consider the utility \(h(\underline {e}) = ((Q_{u} + Z) v(\underline {e})  W\eta _{EE}^{\mathrm { req}}Z P_{u})\) without the time index t for brevity. Given that, if a UE u served by a BS s has some RB(s) of CC c^{∗} at PL p reallocated to UE u^{∗}, we search the MCS l^{′} that makes the largest the total UE utility contributed by all remaining RBs of c^{∗} assigned to u among all maximum MCSs employable by these RBs (lines 5–7). Then, we reassign MCS l^{′} to UE u on the transmission of CC c^{∗} from BS s and release the allocations without any utility contribution, producing \(J^{1}(\underline {e}^{*})\) and \(J^{0}(\underline {e}^{*})\) (lines 8–9). The reassignment further forms a new set of constraints \(\{\hat {f} \underline {e} = C \}\), where \(\hat {f}_{j} = 1, \forall j \in J\) while C_{j}=1 if j∈J^{1} and 0 if j∈J^{0} (line 10), and we solve the corresponding problem \(Q = (P\{\hat {f} x = C \})\) with the time limit T_{l} to obtain a feasible (or an optimal) solution \(\hat {\underline {e}}\) giving utility \(\hat {\nu }\) (line 11). If the improvement \(I = \frac {\hat {\nu }} {\nu ^{o}}\) does not exceed a given low bound I_{B}, the process would stop. Otherwise, based on Propositions 1 and 2 in [15], a pseudocut \(\{ f \underline {e} \leq J^{1}(\underline {e}^{*}) 1 \}\), where \(f_{j} = 2 \underline {e}^{*}_{j}  1\) if \(\underline {e}^{*}_{j} \in J(\underline {e}^{*})\) and 0 otherwise, will be added when the remaining time t=t−2T_{l} allows, and the problem will be updated as \(P = (P\{f \underline {e} \leq J^{1}(\underline {e}^{*}) 1 \})\) that would be solved to seek further improvements (lines 12–14). Finally, the allocation result \(\underline {e}\) corresponding to the best utility found during the searching process will be returned (line 15).
Performance bounds
As shown in above, IOMHA is an approximation algorithm to resolve the highdimensional allocation subproblem involved. However, if the optimal solutions can be given, the overall algorithm for eJCREEP can operate under the performance bounds on, e.g., data queue lengths, as shown in the following theorem.
Theorem 2
Given arbitrary traffic arrival rates and an energy efficiency requirement, the algorithm solving eJCREEP with a fixed control parameter V≥0 can guarantee the bounds on data queue lengths as
Proof
Please refer to Appendix 2. □
Apart from the above, the other performance bounds for the Lyapunov driftpluspenalty framework can be also found in a similar way. For example, a driftpluspenalty approach had been shown, e.g., in [32], to achieve O(ε) approximation with a convergence time of O(1/ε^{2}) with ε=1/V.
Results and discussion
Environment setting
In this section, we numerically evaluate our optimization algorithm through a simulation topology as shown in Fig. 2, wherein 1 MBS and 3 SBSs are deployed, and each of them initially serves 3 UEs located within its transmission range for their downlink transmissions before the resource allocation. In addition to s=4 and u=3 just indicated, there are also c=5, b=10, l=29, and p=3, other resources contributing to the overall complexity that is significant enough to evaluate the highdimensional allocation problem involved. Further, each UE in cell is conducted to dynamically change its position according to the random waypoint (RWP) model [33], and the channel condition is assumed to be varied time to time on each RB as that in [34]. Given the timevarying environment, MBS is conducted to perform the proposed algorithm with T_{l}=1000,W=1,v_{u}=1, along with the other key parameters summarized in Table 2. Based on the above setting, the performance results are resulted and summarized in the sequel.
Result analysis
To be specific, the performance metrics include timeaverage utility, throughput, data queue length, and energy efficiency (EE) denoted by \(\overline {\phi }, \overline {\gamma }, \overline {Q}\), and \(\overline {\eta _{EE}}\), respectively; each of them is represented by its mean value obtained from all UEs carrying out 100 times of the algorithm per experiment. Given these metrics, our algorithm is then conducted by varying V and \(\eta _{EE}^{\text {req}}\) to focus on the performance tradeoffs among throughput, data queue length (or delay), and energy efficiency (EE) in the experiments exemplifying the performance trends. To this end, \(A_{u}(t), \forall u \in {\mathcal {U}}\) at each slot t is randomly generated by the Poisson distribution with the mean value obtained by the maximum TBS=680 multiplying with a given constant C_{1}=14 which represents a possible varying traffic arrival at time t under the maximum allowable rate \(A_{u}^{\text {max}} = TBS * C_{2}\), where C_{2}=20. Following that, the timevarying Rayleigh channel conditions are simulated by using the random channel gains obtained by the exponential distribution with the mean value of 1. Consequently, a wide range of V sampled at [10^{1},10^{3},10^{6},10^{7},10^{8},10^{9}10^{11}10^{15}], and that of \(\eta _{EE}^{\text {req}}\) at [1,2,4,8] are combinatorially examined to know their impacts on the algorithm in general.
The experiment results are summarized in Fig. 3. Specifically, from Fig. 3a and b, we can see that as V increases, the utility and throughput improve significantly and converge to their maximum levels for larger V. This is expected because the achieved utility would increase to optimum at the speed of O(1/V) when V increases, which implies a control emphasizing more on the throughput. However, as shown by the curves remaining nearly the same for large V, we can see also that the improvement will diminish with an excessive increment of V which may then aggravate the congestion as the data queue length would rise as V increases. In addition, it can be noted that as V increases, the system would more emphasize on the throughput utility as noted before, which could increase γ_{u} (with (35)) and then H_{u} (with (28)), leading to more arrivals to be admitted (with (37)) and eventually an increased data queue length (with (22)). Specifically, Fig. 3c exhibits that the increasing data queue length due to the increment of V would increase the average delay, and thus, the tradeoff between throughput and delay emerges, which well confirms Theorem 2.
On the other hand, the performance differences on EE obtained by different EE requirements versus different V are exhibited in Fig. 3d. To show its implication, we note that in the simulations, the EE value obtained from different V without any EE requirement is 3.58 on average, denoted here by EE threshold. Clearly, when \(\eta _{EE}^{\text {req}} = 1\) and 2 that are smaller than the threshold, the EE values actually obtained shown in this figure as well as the throughputdelay tradeoffs shown in the above are very similar, despite V. On the other hand, when \(\eta _{EE}^{\text {req}}\) increases to 4 and 10 that are larger than the threshold, the average throughput would increase especially when V is smaller (see Fig. 3b). This phenomenon can be explained by the aid of Fig. 4 that is obtained with V=10^{1}. As shown therein (Fig. 4), to guarantee \(\eta _{EE}^{\text {req}}\), the network would decrease the transmit power level, and thus encourage the transmissions of small cells by allocating more RBs to SBSs that achieve a higher spectrum reuse gain, followed by the increment of the EE obtained and the average throughput. When V is smaller (such as V=10^{1} as exemplified), the EE performance gain obtained by a higher EE requirement \(\left (\eta _{EE}^{\text {req}}\right)\) is more significant. On the other hand, as V increases, the system would more emphasize on the throughput utility and pay less attention to EE, and hence, the EE gain would decrease and become less significant (see Fig. 3d). These results confirm that our algorithm actually represents a controllable method that can approach the optimal throughput while satisfying the EE requirement by simply manipulating the parameter V to achieve the performance tradeoff required by the system.
Performance comparison
As our IOMHA is conducted to concurrently allocate multiple types of resources in the multitier multicell networks, which can hardly correspond to an existing method in the related works that did not consider the resources: UEs, RBs, CCs, MCSs, cells, and PLs, and the EE constraint at the same time. However, to explore its performance benefits in eJCREEP, we extend the greedy algorithm in [8] (called Greedy), and the LL+RS algorithm introduced therein for comparison, to involve the multiple cells and discrete power levels in this question, resulting in more comparable methods for our work, when compared with the other algorithms possessing certain properties such as continuous power allocation that is hard to changed for the sake of comparison. As introduced in [8], in the first step, the LL+SS algorithm based on [5] will perform CC assignment with the concept of Least Load (LL) by which each UE is assigned the CCs with the least number of UEs. In the second step, it assigns RBs of each CC to UE by its packet scheduling function while resolving the MCS constraint in the scheduling. Given that, LL+SS as well as Greedy still did not consider allocating CCs to multiple cells and utilizing discrete PLs. To address this issue, we first allocate CCs to different cells with the objective of maximizing the sum of SNR values of CCs perceived by cells while complying with our multicell constraint that a UE can be only served by a single BS s and that each BS s can equip with at most f_{s} CCs, as the first level of the extension. After allocating CCs to each cell, Greedy and LL+SS can then be run to play the role of IOMHA in eJCREEP with discrete PLs, respectively, to solve the allocation problem in Section 4.4.3, as the second level of the extension.
In addition, for a more general condition to be encountered, we do not restrict ourselves to consider only the SNR values based on the distances and channel models in the previous set of experiments. Instead, we assume that SNR of each RB perceived by UE would be a random variable uniformally distributed in the range between − 5 and 22.38 according to the SNRCQI index mapping in [27], exemplifying an allocation that can involve all possible mapping values and their results in the simulation. In this case, we solve the allocation problem (38) with an optimization tool for the optimum without limiting its solving time while approaching the optimal result by using IOMHA with a reasonable time constraint represented by T_{B}=1000 and T_{l}=500 and obtaining the suboptimal solutions based on Greedy and LL+SS, respectively, to see their performance differences varied with different V. Specifically, in view of the results revealed in the previous experiment set, we use V={10^{1},10^{6},10^{15}} to exemplify a possible low/midle/high system parameter causing the performance tradeoff in the same spectrum of V from 10^{1} to 10^{15} considered in Section 7.2, while fixing \(\eta ^{\text {req}}_{EE}=10\) and remaining the other parameters.
The comparison results are now summarized in Fig. 5. As shown in Fig. 5a, while complying with the performance trend shown in Section 7.2, IOMHA exhibits its throughput to approach the optimal value which is significantly higher than that to be achieved by Greedy and LL+SS despite V. This confirms the benefit of the joint optimization that can concurrently decide the CC allocation to cells and the allocation of the RBs involved to UEs while complying with the MCS constraint and the other constraints. In contrast to the joint approach, the related works [5, 8] usually schedule RBs with or without the MCS constraint, based on the assumption of preallocated CCs. Here, without the joint optimization gain, Greedy is worse than IOMHA as a result, but it still outperforms LL+SS which is consistent with the observation shown in [8].
In Fig. 5b, the data queue length is shown in log10 magnitude to focus on the performance differences brought by the different methods in this metric. If applying a normal scale, the larger queue lengthes resulted from a high V (10^{15}) would be the focus of the figure, making the results from a lower V (10^{1} or 10^{6}) insignificant even though the relative differences among them are all large enough despite V. In this representation, it is clearly shown that IOMHA yields a lower queue length than Greedy and LL+SS throughout the three V parameters, which also denotes a lower delay to be obtained by our method.
Finally, in Fig. 5c, the decreasing trend for the EE performance is exhibited to be the same as that observed from Fig. 3d. While all the methods under comparison have the same trend as expected, the EE performance resulted from IOMHA in eJCREEP is only slightly lower than the optimum, and Greedy has the result lower than ours but still outperforms LL+SS significantly. Taking all the tree metrics (throughput, queue length or delay, and EE) into account, it could be noted that using IOMHA with a proper time constraint to resolve the resource allocation problem and gradually improve the result would be a good method to trade the optimality for the eJCREEP that is NP optimization problem off against a lower and controllable complexity. That is, using IOMHA in eJCREEP would be better than simply adopting onthefly methods such as Greedy and LL+SS in this problem that can be done only once for a suboptimal solution to the complex allocation problem involved without a chance for further improvements.
Conclusions
In this work, we have addressed an optimization problem on the throughput utility while satisfying the EE requirement with timevarying channel condition and data traffic realized by the carrier aggregation technique in 5G heterogeneous wireless networks. For obtaining a practical solution, the highdimensional NPhard allocation problem involved was first formulated with a programming model involving nonlinear integer constraints, and then reformatted to be an equivalent problem involving only linear integer constraints. However, finding an optimal solution for the mixed integer programming model without special structures imposed would be still NPhard and timeconsuming, even with the linear form. For this challenge, an iterative optimal MCSbased heuristic algorithm (IOMHA) was proposed to approach the optimum within a limited period of time demanded by the user, in the low layer. Given that, a Lyapunov optimization framework was developed to resolve the problem in the high layer that can admit timevarying traffics without a priori knowledge of arrivals. Then, with the solutions from the two layers, we completed an approach that can make an optimal tradeoff with a system control parameter V and satisfy the longterm EE requirement simultaneously. Finally, the proposed framework was verified to reveal the performance tradeoffs among throughput, delay, and energy efficiency, showing that it can serve as an efficient way to address such a complex optimization problem, exhibiting the performance trends on the tradeoffs for the future works. In particular, as a resource allocation problem for nowadays stochastic networks becomes more challenging to meet fast convergence and tolerable delay requirement, a machine learning approach involving batch training could be developed as our future work while preserving the stochastic network optimization context guarantees queue stability with our Lyapunov driftpluspenalty framework that can take the advantage of the iterative optimal MCSbased heuristic algorithm proposed to flexibly adjust its convergence time required by the system.
Appendix 1
Proof of Theorem 1:
By leveraging the fact that A≥0,b≥0,Q≥0,(max{Q−b,0}+A)^{2}≤Q^{2}+A^{2}+b^{2}+2Q(A−b), we can square both sides of (22), (28), and (29), and sum the squares for (22) and (28) over all u, leading to
Let \(A_{u}^{\text {max}}\) and \(\mu _{u}^{\text {max}}\) be the upper bounds of A_{u}(t) and μ_{u}(t),∀t, respectively. Further let \(R_{tot}^{\text {max}}(t)\) be \(\sum _{\forall \underline {e} \in \Xi } v(\underline {e}(t))\), and \(P_{\text {tot}}^{\text {max}}(t)\) be \(W \eta _{_{EE}}^{\text {req}} \left (\sum _{\forall \underline {e} \in \Xi } \mathbf {I}(\underline {e}) \left (P^{p}_{s, u}(t) + P^{c}_{s,u} \right)\right)\), where I(x)=1,∀x. In addition, consider \(R_{u}(t) \leq A_{u}^{\text {max}}\) and \(\gamma _{u}(t) \leq A_{u}^{\text {max}}\). After taking these definitions and considerations into (40), (41), and (42), we can then combine the resulted equations and take the expectation with respect to Θ(t) on both sides of the combination, which eventually leads to the oneslot conditional Lyapunov drift as follows:
where \(\Gamma \!\,=\, \frac {1}{2} \left (\!3 \sum _{u \in {\mathcal {U}}} \left (A_{u}^{\text {max}}\right)^{2} \!\,+\, \sum _{u \in {\mathcal {U}}} \left (\mu _{u}^{\text {max}}\right)^{2} \!\,+\, \left (P_{\text {tot}}^{\text {max}}(t)\right)^{2} + (R_{\text {tot}}^{\text {max}}(t))^{2} \right)\). Finally, (33) is obtained by adding \( V \mathbb {E} \left \{\sum _{u \in {\mathcal {U}}} \overline {\phi (\gamma _{u}(t))}  \Theta (t) \right \}\) on both sides of (43).
Appendix 2
Proof of Theorem 2
For the performance bound, we would first show that \(H_{u}^{\text {max}}\stackrel {\triangle }{=} v_{u} V + A_{u}^{\text {max}}\), will be the upper bound of H_{u}(t). It is done by induction, showing that if this bound holds at time slot t, it will be true at time t+1 also. More specifically, because γ_{u}(t) can not exceed \(A_{u}^{\text {max}}\), the algorithm can increase H_{u}(t) with an amount of at most \(A_{u}^{\text {max}}\) at slot t based on (37), and thus, if H_{u}(t)≤v_{u}V, H_{u}(t+1) will not exceed \(v_{u} V + A_{u}^{\text {max}}\). Otherwise, if H_{u}(t)>v_{u}V,γ_{u}(t) will be 0 according to (35). In this case, H_{u}(t) will not increase in t+1, and hence, H_{u}(t+1)≤H_{u}(t) which is bounded above by \(H_{u}^{\text {max}}\).
Next, we proceed to prove Q_{u}(t) to be bounded with respect to \(H_{u}^{\text {max}}\) shown above, which can be also done by induction. First, this bound is assumed to be true at t. Given the induction hypothesis and the relationship \(R_{u}(t) \leq A_{u} (t) \leq A_{u}^{\text {max}}, Q_{u}\) will increase by at most \(A_{u}^{\text {max}}\) in one slot. Recall that \(H_{u}^{\text {max}} \stackrel {\triangle }{=} v_{u} V + A_{u}^{\text {max}}\) is the upper bound of H_{u}(t). Then, if \(Q_{u}(t) \leq H_{u}^{\text {max}}\), then Q_{u}(t+1) will not exceed \(H_{u}^{\text {max}} + A_{u}^{\text {max}} = \left (v_{u} V + A_{u}^{\text {max}}\right) + A_{u}^{\text {max}} = v_{u} V + 2 A_{u}^{\text {max}}\), according to the data queueing dynamic (22) which increases Q_{u}(t) by at most R_{u}(t) while R_{u}(t) can increase by at most \(A_{u}^{\text {max}}\) based on (37). Otherwise, if \(Q_{u}(t) > H_{u}^{\text {max}}\), then R_{u}(t) will be 0 according to (37) as well. Finally, both cases confirm \(Q_{u}^{\text {max}} \stackrel {\triangle }{=} H_{u}^{\text {max}} + A_{u}^{\text {max}} = v_{u} V + 2 A_{u}^{\text {max}}\) to be the bound shown in (39), and the proof is done.
Notes
 1.
Abbreviations
 3GPP:

3rd Generation Partnership Project
 5G:

Fifth generation
 BS:

Base station
 CA:

Carrier aggregation
 CC:

Component carrier
 CQI:

Channel quality indicator
 EE:

Energy efficiency
 eJCREEP:

Equivalent joint congestion control and resource allocation with EEdelay tradeoff problem
 HetNet:

Heterogeneous network
 IOMHA:

Iterative optimalMCSbased heuristic algorithm
 RWP:

Random waypoint
 JCREEP:

Joint congestion control and resource allocation with EEdelay tradeoff problem
 LTE:

LongTerm Evolution
 LTEA:

LongTerm EvolutionAdvanced
 MBS:

Macro base station
 MC:

Macro cell
 MCS:

Modulation and coding scheme
 NP:

Nondeterministic polynomial time
 OFDMA:

Orthogonal frequency division multiple access
 PL:

Power level
 RB:

Resource block
 RRM:

Radio resource management
 SBS:

Small base station
 SC:

Smallcell
 SNR:

Signaltonoise ratio
 TTI:

Transmission time interval
 UE:

User equipment
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Funding
This work was supported by the Ministry of Science and Technology, Republic of China, under grant MOST 1052221E126001MY3.
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Liu, J., Lin, C. & Huang, H. Joint congestion control and resource allocation for energyefficient transmission in 5G heterogeneous networks. J Wireless Com Network 2019, 227 (2019). https://doi.org/10.1186/s136380191532z
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Keywords
 Heterogeneous wireless networks
 Joint optimization
 Energy efficiency
 Carrier aggregation