Towards Energy Efficiency for IoT: A Cross Layer Design in Cognitive Cellular Networks
With the proliferation of user devices, the user-level energy efficient (EE) design becomes more critical in the era of Internet-of-Things (IoT). Besides, user generated data traffic requires sufficient radio spectrum to support but current spectrum are not fully utilized. In this paper, we propose a novel cognitive radio (CR) mesh network and augment it with the current cellular networks. Under this architecture, we investigate the user-centric, rather than the conventional infrastructure-level, EE design by jointly considering user association, power control, OFDM sub-channel allocation, flow routing and link scheduling. The design is cast into a cross-layer optimization problem which however proves to be difficult to tackle. In light of this, we propose a parametric transformation technique to convert the original problem into a more approachable one while further introducing the -confidence level and integer relaxation approach to address the complexity of the problem. Furthermore, we present a two-layer algorithm, consisting of an inner and outer loop optimization, to solve the transformed problem. A benchmark scheme is used to demonstrate the advantage of our design. We then carry out extensive simulations to exhibit the optimality of our proposed algorithms and the network performance under various settings.
Over last few years, the explosive growth of smart devices is accelerating the advent of IoT. Being a critical element in most IoT applications, smartphone enables people to monitor and control tens of IoT devices and appliances inside home, offices and buildings . Despite the promise, the increased load on smartphone escalates its wireless data traffic, which puts great pressure on the supporting infrastructure (e.g., 4G/LTE). Moreover, the increased wireless traffic causes smartphone’s battery to be drained much quicker than before. This is not only an unpleasant experience for users but also not friendly to the environment. Although one may argue that each smartphone uses a trivial amount of energy, the aggregated annual electricity consumption of solely iPhone 5 in 2012, for instance, is equivalent to the annual electricity usage of 54,000 US households . Thus, prolonging the battery life, or in other words, improving the energy efficiency of smartphone is of vital importance both to the global scale energy saving and to the applicability of IoT applications.
It has been well recognized that the key to increasing network energy efficiency is to optimize the spectrum efficiency per unit power spent . The reason lies in the Shannon’s capacity theorem, which reveals the tradeoff between power and bandwidth. Specifically, link capacity increases only logarithmically with power but linearly with bandwidth, indicating that bandwidth could more effectively bring down the power consumption. However, recent studies show that a large portion of licensed spectrum is not well utilized in certain geographical areas and instead remains idle most of the time . The inefficiency of spectrum usage drives the Federal Communications Commission (FCC) to open under-utilized licensed spectrum for public sharing, which enables unlicensed users to opportunistically access the unoccupied licensed spectrum as long as they do not cause harmful interference to the licensed users. In this norm, the cognitive radio (CR)  has been viewed as the enabling technology to realize dynamic spectrum sharing and increase spectrum efficiency, so the cognitive communication becomes a promising paradigm for achieving energy-efficient communications.
There has been a flux of research studies [6, 7] on how to augment cognitive communications in cellular networks to enhance the network performance in terms of throughput and energy efficiency. However, most of them assume end users are equipped with cognitive radio capabilities to perform spectrum sensing and switching, which in fact have several disadvantages. For instance, the energy consumption due to constant spectrum sensing is so large that the batteries of end devices are drained quickly. Moreover, the harvested spectrum may not be utilized efficiently due to the lack of network-wide coordination. Therefore, a new cognitive radio architecture that can alleviate the burden on end devices while fully utilizing the harvested spectrum is in dire need to improve the performance (e.g., energy efficiency) of future wireless networks.
In this paper, we investigate the user-centric energy efficiency optimization problem in a cognitive mesh cellular network. The cognitive capacity harvesting network (CCHN) was firstly proposed in our previous works  as a novel flexible architecture for cognitive radio networks (CRNs), as shown in Fig.1. The routers, also called CR routers, have CR capabilities and can form a multi-hop mesh network to help the end users who may not have CR capabilities deliver their traffic using harvested spectrum. The network resource management (e.g., harvested spectrum allocation, power control and mobility management) is executed by a centralized secondary service provider (SSP) for better resource utilization. In this paper, we integrate this architecture into a cellular network setting and leverage the benefits of this architecture to improve the network-wide energy efficiency.
Although there have been many research works studying the energy efficiency problems in cellular networks [9, 10, 11, 12, 13], most of them define the energy efficiency as the ratio of the sum of users’ throughput to the sum of infrastructure’s (e.g., base station’s) power consumption [9, 10, 12, 13]. Compared with the grid-powered communication infrastructures, wireless devices are usually battery-powered and more sensitive to power consumption, especially in the era of IoT. It is thus more worthwhile to measure the power consumption w.r.t. the end devices. Moreover, the “sum-to-sum” energy efficiency matric fails to capture the user diversity and is unable to guarantee fairness. To the best of our knowledge, there are barely research works yet to quantify the network-wide EE as the weighted sum of every end user’s EE and while both throughput and power consumption are w.r.t. each individual user. In this paper, we plan to fill in this gap and address the user-centric network-wide EE optimization problem in cognitive mesh cellular networks.
More specifically, we consider the uplink transmission where end users can either directly connect to the base station (BS) or associate with the CR routers. For the former case, it is a one-hop transmission and follows the same way as the traditional cellular networks; while in the latter one, an end user is allocated with cellular channels to firstly connect to a CR router and its uplink traffic are then be delivered to the BS via multi-hop transmissions using harvested licensed bands. Throughout this end-to-end (i.e., from end users to the BS) transmission, we investigate the coupling problem of end user association, uplink power control and channel allocations, and CR mesh network multi-hop scheduling and routing. The rationale of studying this problem is that users could use lower transmission power to associate with closer CR routers but may not be able to always receive satisfactory service through the CR mesh network due to the uncertainty and unreliability of the harvested bands; while users could obtain reliable throughput via the BS but may need to apply higher transmission power. Obviously, there is a tradeoff between service quality (e.g., throughput) and power consumption and we will investigate the energy efficient design in this respect, which represents a significant departure from current literature.
Towards this design objective, there are several technical challenges to be addressed. Firstly, the weighted sum-of-ratios form of the objective function is non-convex, which makes the optimization problem difficult to tackle. Secondly, the availability of harvested spectrum is highly unpredictable and the usable bandwidth in this regard should be naturally modeled as a random variable, which results in a stochastic constraint and makes the problem intractable. Thirdly, the association variable is in integer form and tightly coupled with other decision variables so that solving the problem via conventional approaches is highly prohibitive especially when the network size is large. In light of these aforementioned challenges, we propose corresponding solution to each of them, which in turn demonstrates our technical contributions as follows:
We introduce auxiliary variables and transform the original objective function into a parametric subtractive form which bears the desired convexity property. Their equivalence in terms of finding the same solution is further proved.
We reformulate the stochastic constraint of the availability of harvested spectrum as a chance constraint of -confidence level. Then, the feasible region of original optimization problem becomes a convex set.
We address the integer programming part through a two-step procedure: relaxing and then rounding. To decouple the decision variables, we further apply the dual-based approach to make the original problem more tractable.
The rest of the paper is organized as follows. Section II introduces the most recent literature of this topic. Section III describes the system model and then the problem formulations are outlined in Section IV. We propose the solution approaches in Section V and present the performance evaluation in Section VI. Finally, Section VII concludes the paper and points out the future research directions.
Ii Related Works
Over the last years, EE optimization in wireless networks (e.g., cellular networks and CRNs) has become a popular topic under the umbrella of either energy efficiency or green communications [14, 15, 16, 17, 9, 10, 11, 12, 13, 18, 19, 20, 21]. Contrary to the conventional problems of power minimization under rate constraints  and rate maximization under power constraints , the currently well-recognized optimization problem is to maximize EE (defined as the ratio of rate to power consumption) under wireless resource/scheduling constraints . In this section, we focus on such EE maximization problems and particularly restrict our scope to its application in the cellular networks, CRNs and the hybrid cognitive cellular networks.
In the current 4G/LTE orthogonal frequency division multiple access (OFDMA) cellular systems, there are research works studying the EE optimization problem by jointly considering power control and subcarrier allocations for both downlink and uplink transmissions, respectively [9, 10, 11, 12, 13]. Cheung et al.  focused on the multi-relay assisted OFDMA downlink cellular networks and studied the EE maximization problem by formulating EE as the ratio of total network throughput to total power consumption at the BS. The Dinkelbach’s method was then applied to iteratively solve the optimization problem. Xiong et al. also targeted on the same wireless setting but investigated the EE maximization constrained on users’ proportional rate fairness . However, both papers modeled the EE in the “sum-to-sum” form and used the infrastructure’s power consumption as the metric, which lacks emphasis on EE provisioning w.r.t. each individual user. Although Aijaz et al. considered throughput and power consumption both with regard to end users  in the uplink OFDMA cellular systems, they still applied the conventional form of EE metric. Unfortunately, when it comes to modeling network EE as the weighted sum of each user’s EE, the objective is nonconvex and the problem becomes complicated. Recent works [12, 13] are nonetheless exceptions. Zarakovitis et al.  studied the energy efficient design in downlink OFDMA cellular systems by optimizing the EE in weighted sum-of-ratios. They applied the Maclaurin series expansion to transform the objective into a tractable form and then solved it in polynomial time. He et al.  focused on the multi-cell downlink cellular systems with the coordinated beamforming. The authors introduced auxiliary variables to transform the weighted sum-of-ratios objective into a parametric subtractive form, which afterwards became easier to address. However, these works still employed the infrastructure’s rather than end user’s power consumption in the EE characterization.
The energy efficient resource allocation design in CRNs and cognitive cellular networks is still not well investigated. Wang et al. in  considered an OFDM-based CRNs and investigated the energy efficient design by taking channel uncertainty into consideration. Xie et al.  proposed a new cognitive cellular network architecture of both macrocells and femtocells. They utilized game theoretic approaches to investigate the energy efficient research allocation in such a heterogeneous network. Although there are several other works in this context [20, 21], they are more or less the same in the sense that the end users are assumed to have CR capabilities; the design knobs are mainly physical layer power control and subcarrier allocation; and the EE measurement is in the form of the ratio of overall rate to overall power consumption. However, their power consumption model is too simplistic if they assume end users have cognitive radios as the spectrum sensing and channel switching would contribute significant amount of power consumption. On the other hand, they only consider resource allocation in the one-hop communication scheme, which may fall short in the efficient resource utilization compared to the multi-hop communication paradigm.
In this work, we leverage a multi-hop CRN integrated with the cellular network and herein explore how to maximize users’ EE. This architecture was previously shown in  as an effective approach in many folds to using the CR technology with existing wireless technologies. We anticipate this research will further expand our knowledge horizon by investigating user-level EE design in the era of IoT.
Iii System Model
Iii-a A Cognitive Mesh Cellular Network
In this paper, we consider the cognitive mesh cellular network which consists of a secondary service provider (SSP), one BS, CR routers and end users, as shown in Fig.1. Particularly, the SSP is a wireless service provider, such as an existing cellular operator, which has its own licensed bands, typically called the “basic bands”, for reliable control signalling, handling handovers, accommodating voice traffic and supporting data traffic if available. The SSP could also harvest spectrum bands from other operators via paradigms such as spectrum sensing or spectrum auction. As the centralized coordinator, the SSP observes and collects network information in its coverage area (e.g., users’ traffic demands, channel state information and available radio resources) and then performs network optimization (e.g., power control, channel allocation, link scheduling and routing) to determine the optimal approaches for service provisioning. Note that all these control messages are carried on the basic bands for reliable network management. Furthermore, BSs and CR routers are both deployed by the SSP and play analogous roles as the marco and pico BSs, respectively, as in LTE systems. However, CR routers herein are more intelligent devices with CR capabilities, and form a mesh network capable of using the harvested bands. BSs also have multiple radio interfaces and serve as the gateways to the backbone network for CR routers. In this architecture, the end users may not necessarily have to possess CR capabilities and the CR routers could tune the radio interfaces to what the end users normally use (e.g., cellular bands) and deliver services. Of course, due to the close proximity between CR routers and end users, the frequency reuse of the basic bands and energy efficiency of the network are greatly enhanced, which have the same merits as the heterogenous cellular networsk. For more details of this architecture, interested readers are referred to .
Iii-B Network model
In this paper, we focus on uplink user association aiming to maximize end users’ energy efficiency. Downlink could likewise be considered through a similar approach. For a cell in the cognitive mesh cellular network as shown in Fig.1, suppose a set of users, say , each of which initializes a session whose destination is the BS denoted by . We index the set of sessions as and let and denote the source (i.e., ) and destination (i.e., ) of session , respectively. Also consider the cell consisting of CR routers and together with the BS , we denote as the set of wireless infrastructures. Suppose the network applies single carrier-frequency division multiplexing access (SC-FDMA) for end users’ uplink transmissions, where the network’s basic band is divided into number of orthogonal sub-channels which are shared among end users. Denote the harvested band as and it is allocated to the mesh network to form multihop transmissions.
Our model is applicable to a low mobility and high user density scenario such as in homes of a community or offices of a building. In such an environment, resource allocation process can be conducted during the channel coherence time when channel is regarded as static. In light of this, we can just apply the line-of-sight (LOS) channel model instead of fast fading ones in our problem. Moreover, although signaling and computation overhead is incurred when solving the cross-layer optimization problem, the solution is applicable over a relatively long time due to users’ low mobility and small variation of network parameters, which makes such overhead tolerable in the long run.
Iv Problem Formulation
Given the system model described before, we target at the problem of uplink end-to-end data delivery from end users to the BS, where the user association, uplink power control, channel allocation, routing and link scheduling are jointly considered so as to maximize the network wide end users’ energy efficiency.
Iv-a Uplink SC-FDMA
In practical systems, 3GPP LTE Release 12 specifies the number of sub-channels can vary from 6 to 110, depending on the total available bandwidth . Normally, the set of allowable configurations are and each sub-channel has a bandwidth of 180 KHz . Furthermore, the uplink SC-FDMA implies certain restrictions on power and channel allocations . First, any sub-channel can only be assigned to one user, called exclusiveness. Second, every user’s allocated sub-channels must be continuous, called adjacency. Third, user’s transmit power should be identical on any allocated sub-channel in order to retain a low peak-to-average power ratio (PAPR).
Denote the indicator as the association variable, where implies that user is associated with infrastructural node , and otherwise. Normally, the association is assumed to be performed in a large scale compared to the variation of channel so the fast fading is averaged out over the association time . We also consider a relatively low mobility environment, where the resource allocation is carried out during the channel coherence time so the channel can be regarded as static. Furthermore, in our cognitive mesh cellular network, to simplify the problem, we assume the sub-channels are not reused so as to avoid strong interference.
Iv-A1 User Association
Any end user can only be associated with the BS or a CR router, but not both. This physical constraint can be expressed as follows
Iv-A2 Sub-channel Allocation
Suppose the end user is allocated with number of sub-channels which are contiguous in nature111Note that our work can be naturally extended to incorporate the case of finding the optimal sub-channel allocation pattern (i.e., a specific chunk of sub-channel collections) [26, 11], and we will leave it for the future work.. Due to the property of exclusiveness, the total allocated sub-channels cannot exceed , which is stated in below
where denotes the set of nonnegative integers.
Iv-A3 Link Capacity
Given the sub-channel allocation, according to Shannon-Hartley theorem, the capacity of the link between an end user and the infrastructral node can be calculated as follows
where is the bandwidth of each sub-channel, is the power spectrum density (of unit ) of the Additive White Gaussian Noise (AWGN), and is the channel fading coefficient. Here, is denoted as the transmit power of the end user and as we know, must be evenly distributed across the allocated sub-channels to retain a low PAPR, which is given by . Therefore, the link capacity is calculated as the sum of all allocated sub-channels’ capacities, as shown in (4).
Iv-A4 Power Control
Due to the hardware constraint, the transmit power of an end user cannot exceed its maximum allowable power level . Since end users have different power capabilities, is a user-dependent variable. Moreover, we do not consider the power control of CR routers so their transmit powers are assumed to be fixed.
Iv-B The Cognitive Mesh Network
When an end user is associated with a CR router, its traffic is delivered to the BS via multi-hop transmissions in the cognitive mesh network. In other words, the SSP allocates the harvested band to the mesh network and performs link scheduling and routing optimization to determine how to assist the end user to complete its session whose destination is the BS . In what follows, we investigate the link scheduling and routing problem in the cognitive mesh network.
Iv-B1 Transmission Range and Interference Range
Following the widely used model , we define the power propagation gain from the CR router () to another infrastructural node (either a CR router or the BS) () as , where is the antenna gain, refers to the Euclidean distance between and , and is the path loss exponent. Let assume that CR routers apply the same constant transmit power and define that the transmission is successful only when the received signal power exceeds a threshold , i.e., . Then, we can obtain the transmission range of CR router as . Accordingly, we define the set of infrastructural nodes being in the transmission range of the CR router () as
On the other hand, to efficiently use harvested bands, the SSP should ensure the transmissions over different links do not conflict with each other. In light of this, we define the interference range in a similar way as before. Suppose the received interference can be ignored only when the received power is less than a threshold , i.e., . Therefore, the interference range of the CR router () can be obtained as . Accordingly, the set of infrastructural nodes being in the interference range of the CR router () is defined as
Iv-B2 Conflict Graph and Independent Sets 
Given the prior definition of the interference range, we can claim that two communication links conflict if the receiver of one link is within the interference range of the transmitter of the other link. A conflict graph is used to characterize the interfering relationship among different infrastructural links. Specifically, each vertex indicates a transmission link (, ) and two links are said to be conflicted if there is an edge connecting the two corresponding vertices.
With this conflict graph being created, we can define an independent set (IS), which consists of a set of vertices and any two of them do not share an edge. In this case, all the transmission links in an IS do not interfere with each other and thus can be carried out successfully at the same time. If adding one more vertex into the IS results in a non-independent one, the set is called the maximal independent set (MIS). We can collect all the MISs of the conflict graph in a set , where represents the total number of MISs, i.e., .
Iv-B3 Link Scheduling
In this paper, we consider different MISs are scheduled with certain time shares (out of unit time) so that the links within each MIS can carry out the transmissions simultaneously. From our previous discussion, we know only one of the MISs can be active at one time instance and we denote the time share allocated to the MIS as . Therefore, we have to satisfy the following constraint
On the other hand, the link capacity of the link can be obtained based on Shannon-Hartley theorem, which is
where is the bandwidth of the harvested band. According to the link scheduling, the actual data rate over the link could be 0 if is not scheduled at a time instance. In other words, we use to represent the achieved data rate over the link when is scheduled, where if and 0 otherwise. Considering the link could exist in all the MISs, the total achieved data rate over the link can be expressed as
Iv-B4 Flow Routing
In this paper, we consider the network level end-to-end (from users to the BS) service provisioning. Suppose the end user initiates a session , the SSP should determine whether to support it by associating directly to the BS through one-hop transmission or connecting to the cognitive mesh network and then arriving at the BS via multi-hop transmissions. Here denote as the supported flow rate for the session over the link at the network level. If node is the source of session , which is the end user (i.e., ), we have the following constraints
The constraint (11) means that the incoming flow rate of any session at the source node is zero since the end user is the initiator of the session. The constraint (12) reflects the first hop from the end user to an infrastructural node , where signifies the achievable data rate of user . Clearly, the association variable is coupled with flow rate in (12), implying that there only exists one wireless link from the end user to one of the infrastructural nodes to support data rate. Besides, the flow rate on a link should be constrained by the link capacity according to (4), which is expressed as
For any infrastructural node , which is the CR router, the flow conservation law (FCL) implies that for any session , the total flow into must be equal to the total flow out of . This can be expressed as
Clearly, if the CR router is directly associated with end user , constraint (14) could be rewritten as according to (12); whereas if the CR router is the intermediate infrastructural node to support , constraint (14) would be equivalent to .
Moreover, all the flows in the cognitive mesh network are completed at the BS, which means that the BS is the common destination, i.e., . Thus, we have another constraints for the destination node described as follows:
Note that if the end user is directly connected to the BS (i.e., ), (16) becomes , indicating that session is not supported through the CR routers. Instead, if , meaning user is associated with the cognitive mesh network, (16) could be rewritten as .
Besides, for the link from one CR router to another infrastructural node , the total flow rate on that link (by aggregating all the rates of supported sessions) should not exceed the link capacity. From our previous discussion, we know that the link capacity is dependent on the scheduled time share on that link, which is described as (10). Thus, we have the following constraint
Iv-C User-centric Network-wide Energy Efficiency Optimization
Our work aims to maximize the user-centric network-wide (i.e., the weighted sum of end users’) energy efficiency by considering user diversity in power capability (e.g., residual energy, maximal allowable power). In the current literature, the mathematical formulation for energy efficiency (EE) is generally via two approaches: difference-based  and ratio-based . The former definition represents EE as the difference between the data rate and the term of a pricing constant (with dimension of bit/s/W) multiplying the transmit power. However, the EE in this method is measured in bit/s, which lacks physical rationale. In light of it, we apply the ratio-based approach, where the EE is measured in bit/s/Joule and defined as the ratio of data rate to power consumption which in this work are both with respect to the end users. Thus, we coin it as the user-centric EE metric, in contrast to the previous works that applied BS’s power consumption in the EE definition [12, 13]. Furthermore, contrary to the conventional EE definition as the ratio of the system sum rate to the sum power consumption [9, 29], our user-centric network-wide EE measures the weighted sum of end users’ EE, such that the heterogenous EE requirements from different users of various power capabilities can be investigated.
Before presenting the optimization problem, we first define the power consumption model of end user as , where is the efficiency of the transmit power amplifier when it operates in the linear region, whereas is the circuitry power dissipated in all other circuit blocks (e.g., mixer, oscillator, DAC and etc.) which is independent of the transmit power and normally a constant value. Then, the EE for the end user can be obtained as , where denotes its achieved data rate according to (12). Finally, we introduce a weighting factor associated with user EE, which provides a means for service differentiation as well as fairness. Particularly, the weights could be determined inversely proportional to users’ residual energy so that less power capable users are allocated with higher EE priorities.
By considering the user association, power control, channel allocation, routing and link scheduling constraints introduced previously, we can thus formulate the following optimization problem to achieve the maximal User-centric Network-wide EE (UNEE-Max)
where , , , , and are optimization decision variables. Clearly, UNEE-Max is a cross-layer optimization problem involving coupled variables from the physical layer to the network layer. In the next section, we elaborate several difficulties in addressing UNEE-Max problem and introduce techniques to solve it accordingly.
V Overview of The UNEE-Max Problem
V-a Complexity of The UNEE-Max
We first highlight several key difficulties in solving the UNEE-Max problem.
V-A1 NP-completeness for searching all MISs
Under constraint (7), we need to search all the MISs for link scheduling. However, finding all the MISs in a conflict graph is NP-complete, which is the common obstacle encountered in multi-hop wireless networks . Although we can apply brute-force search when the size of is small, it is highly prohibitive when becomes large. Therefore, it requires a cost-effective approach to find MISs so as to make the problem tractable.
V-A2 Uncertainty of the harvested band
In CRNs, SUs are allowed to access PUs’ spectrum bands only when these bands are not occupied by PUs. SUs must immediately evacuate when PUs reclaim the spectrum. In practice, the availability of these harvested bands is highly unpredictable due to the uncertainty of PUs’ activity and SSP’s statistical inference model (i.e., false alarm / miss detection probabilities) . Therefore, the bandwidth of the harvested band (defined in (9)) is a random variable, whose probability distribution could be derived from statistical characteristics of these PUs’ bands from some observations and experiments [32, 4]. However, the randomness of makes (17) a stochastic constraint, which causes the feasible region of UNEE-Max to be both random and nonconvex.
V-A3 Combinatorial nature of user association
The indicator variable in constraints (1) and (2) enforces unique association, which makes the problem combinatorial. Although the classical branch-and-bound approach can be applied to solve general integer programming problems, due to the tight coupling between the association and the resource allocation (i.e., power control, link scheduling and routing) in UNEE-Max, it is difficult to solve using traditional approaches.
V-A4 Nonconcavity of objective function
The objective function in the UNEE-Max problem is in the form of weighted sum of linear fractional functions (WSoLFF), which is generally nonconcave [33, 34]. An immediate consequence is that the powerful tools from the convex optimization theory do not apply to the UNEE-Max, and the KKT conditions are only necessary conditions for optimality . Therefore, we need to transform the UNEE-Max to a certain form from which approximate solution to the UNEE-Max can be found.
V-B The UNEE-Max Relaxation Algorithm
After outlining the difficulties in solving UNEE-Max problem, we introduce the relaxation or transformation techniques to make the UNEE-Max tractable.
V-B1 Critical MIS set
Although there exists exponentially many MISs in a conflict graph, Li et. al.  proved that only a small portion of MISs, termed as critical MIS set, can be scheduled in the optimal resource allocation. Instead of searching all MISs, we thus apply the SIO-based approach proposed in [30, 36] to return the critical MIS set in polynomial time, where . Therefore, we can replace with in constraint (1), and (17) to make the UNEE-Max problem more tractable. Note that SIO-based approach may only give a fraction of in a limited searching time leading to a loss in solution optimality. In light of this, we could deliberately allow a longer searching time as the SIO-based approach can be run offline.
V-B2 -confidence level
where indicates the confidence level for stochastic constraint (17) to be satisfied and . Suppose represent the cumulative distribution function (CDF) of random variable (r.v.) . We could then obtain the as the CDF for the r.v. , which is the link capacity of . Thus, by integrating the critical MISs, the above inequality could be reformulated as
V-B3 Integer relaxation and rounding
In the first phase, we assume that end users can be associated with the BS and CR routers at the same time. In other words, we relax the integer association variable to the continuous domain of . Under this assumption, we also introduce the sub-channel auxiliary variable , where and represents set of all nonnegative numbers, the power auxiliary variable and the flow auxiliary variable , so that , , . Therefore, we can eliminate association constraint (1) and (2) and rewrite the UNEE-Max problem as a Relaxed-UNEE-Max problem which is described as follows
In the second phase, we develop a rounding scheme, as what will be discussed in Section VII, to convert the output of the Relaxed-UNEE-Max problem into a feasible value that satisfies the constraints of original problem UNEE-Max in (18).
V-B4 Parametric subtractive transformation
It is clear that the constraints in the Relaxed-UNEE-Max problem (20) form a convex feasible set w.r.t. the optimization variable set . 222For the sake of brevity, we define the vectors of optimization variables as , , , and . However, it is still challenging due to the sum-of-ratio form in the objective . To overcome this difficulty, we firstly transform the objective function in (20) into an intermediate form by introducing an auxiliary variable and reformulate the Relaxed-UNEE-Max problem as
Although the objective is an affine function w.r.t. , problem (21) is not a convex optimization yet due to the fractional constraint. Thus, we further convert (21) into a parametric subtractive form and show in the following theorem its equivalence to the weighted sum maximization problem (21) with fractional constraint.
Suppose is the solution to problem (21), there exist such that for the parametric variables and , satisfies the KKT conditions of the following problem
Also, the following system equations hold for the parametric variables () and the tuple :
See Appendix A. ∎
Based on Theorem V.1, we can address the problem (21) by solving (22) while guaranteeing (23), such that the solution of the Relaxed-UNEE-Max could be obtained. Furthermore, it is worth noting that if the solution is unique, it is also the global solution . Toward solving (22), we apply a dual-based approach, which has been widely adopted in various network settings for its simplicity of implementation , and augment it with the parametric programming to form inner loop and outer loop iterative update processes. The detailed steps are described in the following section.
Vi Algorithm for The Relaxed-UNEE-Max
Based on our prior discussion, the Relaxed-UNEE-Max problem is equivalent to problem (21), whose solution is identical to (22) when satisfying (23). Hence, we focus on solving problem (22), and for the presentation clarity, we first outline the general idea of the solution algorithm.
The whole algorithm is split into an inner loop and an outer loop optimization problem. The algorithm starts with initializing the parametric variables and . For the given and , (22) becomes a convex optimization problem with an affine objective and a convex feasible set. The inner loop applies dual decomposition approach to solve this convex optimization problem, and each iteration of the dual-based method is termed as the inner loop iterations. Multiple inner loop iterations are performed till the optimal dual and primal solutions are reached. The output of inner loop, which is , are then fed back to the outer loop to update the parametric variables and . The overall algorithm terminates if the convergence condition for and (we will elaborate it later) are met. Otherwise, the algorithm continues by solving the inner loop optimization problem again using the updated and .
Vi-a Algorithm for The Inner Loop Optimization Problem
Suppose the parametric variables are and at the outer loop iteration, the inner loop procedure starts with introducing a partial Lagrange multiplier w.r.t. the third constraint (nonlinear capacity constraint) in problem (20) attempting to decouple the decision variables. We denote the partial Lagrangian by and express it as
The dual function can be then obtained as
Since problem (22) is convex and Slater’s condition for constraint qualification is assumed to hold, it follows that there is no duality gap  and thus the primal problem can be solved via its dual
Vi-A1 Dual problem
We solve the dual variables via the projected subgradient method. First, let us denote the primal variables obtained at inner loop iteration as . Then, the dual variables at inner loop iteration are updated as follows
where is the step size and denotes the projection into the set of non-negative real numbers.
In what follows, we focus on solving the primal variables given the dual variables at each inner loop iteration.
Vi-A2 Primal problem
We now argue that the primal problem
can be reorganized into a routing subproblem and a physical layer resource allocation subproblem. Thus, solving the primal problem is equivalent to solving two independent subproblems, each of which is fairly straightforward. Toward this end, we rewrite original partial Lagrangian as follows
and we can represent it as . Thus, we can separate the primal optimization problem into the following subproblems
which is the routing subproblem, while
which is the physical layer resource allocation (i.e., power control and channel allocation) subproblem.
It is clear that with the given dual variables and the parametric variables and , the routing subproblem (26) is a linear optimization problem w.r.t. the decision variables, which can be easily solved by many softwares, such as CPLEX. On the other hand, the resource allocation subproblem belongs to the general convex optimization problem with a concave objective and a convex feasible region. Thus, it also can be easily solved via the interior point method, for instance.
With the primal variables obtained at each iteration, they are fed back to the dual variable update process according to (25), and we keep iterating the inner loop iterations till a predefined stopping criterion is met.
Vi-A3 Stopping criterion and step size
First, we define the stopping criterion for the inner loop algorithm as , where denotes a predefined threshold. On the other hand, the choice of step size affects the convergence rate of the solution. Normally, we could apply diminishing step size  or constant but sufficiently small step size , which are both guaranteed to converge to the optimal solutions. We will examine the impact of step size on the convergence rate in the performance evaluation section.
Vi-B Algorithm for The Outer Loop Optimization Problem
The outer loop optimization problem is in a parametric subtractive form as the objective in problem (22). The goal is to iteratively obtain the parametric variables and , where the iteration here is termed as the outer loop iteration. Parameter may be intuitively viewed as the “price” of power consumption while parameter is introduced as the Lagrange multiplier for the fractional constraint in (21). Here, we apply the gradient method  to update the parametric variables in a following way:
where and are the converged values of decision variables after inner loop iterations. Similar to the inner loop optimization, another small threshold value is selected and the stopping criterion is set to and . The convergence of the outer loop optimization can be guaranteed by the gradient method  and the step size should be selected to be sufficiently small. Later, we will give the convergence analysis in the performance evalualtion section.
Vii User Association and Integer Rounding
To this end, the problem (22) is solved via the prior algorithm whose solution is identical to the one in the Relaxed-UNEE-Max problem (20). However, due to the physical constraint that every user can only be associated with one infrastructural node, the previously obtained solution should be converted to a feasible one for the original problem. Besides, the integer property of the number of allocated OFDM sub-channels also requires a further rounding procedure to the obtained solution. Inevitably, this step could introduce performance degradation, but in the performance evaluation section, we will show that its impact on the performance is quite limited.
First, we present the association rule as
The above operation indicates that we associate the user with the infrastructural node which provides the largest value of EE. In other words, if end user obtains the highest EE from node , we set the association variable while for . In so doing, we can fix the association variables and the original problem UNEE-Max in (18) could be simplified significantly. Here, we coin this simplified problem by fixing the association variables as Asso-UNEE-Max and it can be similarly addressed by the prior algorithm in Fig.2. In later section, the comparison between the network performance of Asso-UNEE-Max and the one obtained by solving Relaxed-UNEE-Max will be demonstrated.
Next, we introduce the integer rounding function as
where the operator rounds the input to the greatest integer that is less than or equal to the input. Besides, the reason we apply function is to guarantee that every end user can at least be assigned with one sub-channel for fairness. The rounding operation is applied to the solution obtained from solving the Asso-UNEE-Max problem, so that the OFDM channel allocation can be determined accordingly. However, the flow variables obtained from Asso-UNEE-Max may not be feasible anymore when doing integer rounding. Therefore, we need to re-solve the UNEE-Max problem (18) and get the calibrated flow values which are the feasible ones. Noticing that for the fixed channel allocation, sub-problem (27) can be easily addressed by classical iterative water-filling algorithm , which is just a one-dimensional (i.e., power) optimization problem. Here, we denote this solution as the one from a so-called Rnd-UNEE-Max problem. Its performance will be compared with the ones obtained from Asso-UNEE-Max and Relaxed-UNEE-Max, respectively, in the evaluation section.
Viii Performance Evaluation
Viii-a Simulation Setup
We consider a area served by one BS and 12 CR routers, where the BS is put in the center while CR routers represented in squares are placed around it, as shown in Fig.3. We also randomly scatter 35 end users in this area whose locations are shown by dots. The end users’ devices are assumed to have an identical circuitry power consumption and power amplifier efficiency . We assume the users’ allowable transmit power may vary and its impact on system performance will be examined later. To provide fairness for end users, all the weighting factors are set to 1. On the infrastructure side, the CR routers are assumed to employ fixed power for transmission and their antenna gain is set to . The power interference threshold is set to while the receiving power threshold is set to . The transmission environment between infrastructural nodes are assumed to have path loss exponent . Given these system parameters, the interference/communication range can be calculated numerically and we could obtain the conflict graph in this regard. We utilize the OFDM channel model for wireless link between end user and infrastructural node as dBm where is in kilometers . Following the standard, we set the bandwidth of each sub-channel as . The noise power spectral density is set to . In addition, for the harvested band, we consider that the availability of it follows a uniform distribution.
As for the algorithmic parameter settings, we set the stopping threshold and as 0.01 and 0.8, respectively; while the step size .
Viii-B Benchmark Setting
To demonstrate the advantage of our proposed cognitive mesh cellular network in improving end users’ energy efficiency, we leverage the basic cellular network (e.g., 4G/LTE) as the benchmark to compare with. In other words, we consider the same end users within this geographical area served by the small cell BS as shown in Fig.3 (excluding CR routers). Similarly, the benchmark UNEE maximization problem, coined as Ben-UNEE-Max, can be proposed as follows:
In this benchmark setting, end users have to be served by the BS so the design of user association, link scheduling and flow routing is eliminated. Rather, we only consider the power control and channel allocation in this one-hop transmission scenario. To effectively solve (30), the same transformation approach can be applied to firstly convert it into a tractable one, which is then solved via the water-filling algorithm .
Viii-C Results and Analysis
First, we examine the convergence behaviors for both inner loop and outer loop optimizations. Since the inner loop is a dual-based (i.e., Lagrangian) algorithm, we also compare its convergence rate under different selection of step sizes. The results are shown in Fig.4. For demonstrative purposes, we only randomly select 10 users for this simulation and use their average EE as the metric to show the convergence performance. Here, the user’s maximum allowable power is set as 1.5W while CR routers’ transmit power is set as 1W. The bandwidth of harvested band is 100KHz and we set the confidence level , while the number of OFDM sub-channels is selected to . Moreover, the data in Fig.3(a) is collected at the last iteration of the outer loop optimization.
As we can see from Fig.3(a), the average EE monotonically increases till the algorithm converges and the EE remains relatively constant (i.e., the difference not exceeding the threshold) afterwards. It can be observed that the algorithm can be guaranteed to converge to the same value under three different step-size settings, but gives the fastest convergence rate (around 48 iterations). This is because as long as the step size is sufficiently small to guarantee convergence, an even smaller step size is not necessary as it slows down the rate to the optimal value. On the other hand, Fig.3(b) illustrates the convergence performance for the outer loop algorithm. For the notational convenience, we take the reciprocal of so that its unit now becomes , while the unit of is naturally being according to (23). It can be seen that the converged optimal value of is exactly the same as the one in Fig.3(a), which proves the overall convergence of Algorithm 1. On the other hand, we observe that the average transmit power for end users is around 0.28W at convergence, which is a small value compared to .
Given the feasibility of Algorithm 1, we now conduct the performance comparison from solving Relaxed-UNEE-Max, Asso-UNEE-Max and Rnd-UNEE-Max utilizing Algorithm 1 and Algorithm 2, respectively. Besides, by solving (30), we obtain the network-wide energy efficiency in the 4G/LTE cellular network, which is used as the benchmark. The evaluation is conducted under different network sizes in terms of the number of end users. We also employ the same values of , , and confidence level as the previous simulation. The results are shown in Fig.5. It can be seen that these curves demonstrate the same relationship between the network size and the network-wide energy efficiency: as the the number of users increase linearly, the network-wide energy efficiency first grows exponentially and then increases slowly. The reason is that the network resources in terms of OFDM sub-channels and harvested band are sufficient when the network size is small and introducing more users will increase the resource utilization efficiency, thus increasing the total network EE. As the number of users keeps increasing, the network becomes congested in the sense that scheduling and routing in the cognitive mesh network becomes the major bottleneck to further boost the network performance. In the later evaluation, we will demonstrate this phenomenon.
On the other hand, we can see that the solution to the Relaxed-UNEE-Max problem yields the highest network-wide EE since every user can be associated with several infrastructural nodes to take full advantage of network diversity. However, by fixing the association variables and solving the Asso-UNEE-Max problem does not sacrifice too much network performance, as shown in Fig.5. Based on the solution of the Asso-UNEE-Max problem, further applying rounding procedure and solving the Rnd-UNEE-Max problem, gives an even lower network-wide EE. Nevertheless, the optimality gaps between the solutions of Relaxed-UNEE-Max and Rnd-UNEE-Max reduces from 19.35% to 7.14% as the number of users increases from 10 to 35, which means our proposed approximation algorithm for association and rounding works well when the network size scales up.
Furthermore, the network-wide energy efficiency in our proposed cognitive mesh cellular network is much higher (e.g., 143% more in the scenario of 25 end users) than that in the traditional cellular network. Such a significant gain in the energy efficiency on one hand attributes to the additional harvested spectrum while on the other hand is due to the close proximity between end users and CR routers allowing lower transmit power for users.
Next, we analyze how the number of OFDM sub-channels and uncertainty of harvested band could affect the user association decision and network performance. For the user association evaluation, we randomly select 13 users just for demonstrative purposes and set and , while keeping other parameters the same as before. The result is shown in Fig.5(a), which illustrates the number of users connected to the BS and to the cognitive mesh network. It can be seen that when the confidence level increases, more users are switched from the cognitive mesh network to the BS. The reason is that higher confidence level means more strict requirement on constraint (17), which in other words means that the usable harvested band becomes more limited. Therefore, some users are re-associated with the BS so that their throughput would be higher although they may use higher transmit power.
The harvested band affects the backbone capability, while the number of OFDM sub-channels impacts the capacity of the first hop from end users and infrastructural nodes. As shown in Fig.5(b), we examine how different OFDM sub-channel patterns (e.g., ) influences the network-wide EE. It can be observed that for the fixed uncertainty of the harvested band (i.e., available bandwidth), network-wide EE increases in a decreasing rate as the number of OFDM sub-channels increases. The reason is that as the number of OFDM sub-channels becomes sufficiently large, the available harvested band allocated to the cognitive mesh network becomes the bottleneck to support the traffic on the first hop links. This also explains the observation that the network-wide EE increases as the confidence level decreases for the fixed number of OFDM sub-channels.