The evolving explosion in high data rate services and applications will soon require the use of untapped, abundant unregulated spectrum of the visible light for communications to adequately meet the demands of the fifth-generation (5G) mobile technologies. Radio-frequency (RF) networks are proving to be too scarce to cover the escalation in data rate services. Visible light communication (VLC) has emerged as a great potential solution, either in replacement of, or a complement to, existing RF networks, to support the projected traffic demands. Despite the prolific advantages of VLC networks, VLC faces many challenges that must be resolved in the near future to achieve full standardization and to be integrated to future wireless systems. Here, we review the emerging research in the field of VLC networks and lay out the challenges, technological solutions, and future work predictions. Specifically, we first review the VLC channel capacity derivation and discuss the performance metrics and the associated variables. The optimization of VLC networks are also discussed, including resources and power allocation techniques, user-to-access point (AP) association and APs-to-clustered-users-association, APs coordination techniques, non-orthogonal multiple access (NOMA) VLC networks, simultaneous energy harvesting and information transmission using the visible light, and the security issues in VLC networks. Finally, we propose several open research problems to optimize the various VLC networks by maximizing either the sum rate, fairness, energy efficiency, secrecy rate, or harvested energy.
This paper provides, in Section II, an overview of VLC technology, defines and discusses the objectives and constraints that must be taken into account when optimizing VLC networks. Special emphasis is placed on channel capacity derivations, and the unique properties of VLC. We also discuss the variables, parameters, and constraints having an impact on the performance of VLC networks. All optimization techniques are reviewed in Section III, including power and resource allocation, users-to-APs association, cell formation, and AP cooperation used for mitigating the disadvantages of VLC networks to improve performance. This important topic was previously investigated by Li @cite_202 . However, their study was focused on the difference between user-centric and network-centric cell formations, and the interference reduction techniques, whereas in this paper, we place our attention on the techniques, used in RF VLC and in VLC standalone networks, that are aimed at alleviating the limitations in VLC networks. In other words, we show how to formulate optimization problems, what are the techniques used for solving these optimization problems, how the different objectives, limitations, constraints are evaluated, and how added RF APs can remove stand-alone VLC network limitations.
@cite_202
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Universal mobile telecommunication system (UMTS) networks should be deployed according to cost-effective strategies that optimize a cost objective and satisfy target quality-of-service (QoS) requirements. In this paper, we propose novel algorithms for joint uplink downlink UMTS radio planning with the objective of minimizing total power consumption in the network. Specifically, we define two components of the radio planning problem: 1) continuous-based site placement and 2) integer-based site selection. In the site-placement problem, our goal is to find the optimal locations of UMTS base stations (BSs) in a certain geographic area with a given user distribution to minimize the total power expenditure such that a satisfactory level of downlink and uplink signal-to-interference ratio (SIR) is maintained with bounded outage constraints. We model the problem as a constrained optimization problem with SIR-based uplink and downlink power control scheme. An algorithm is proposed and implemented using pattern search techniques for derivative-free optimization with augmented Lagrange multiplier estimates to support general constraints. In the site-selection problem, we aim to select the minimum set of BSs from a fixed set of candidate sites that satisfies quality and outage constraints. We develop an efficient elimination algorithm by proposing a method for classifying BSs that are critical for network coverage and QoS. Finally, the problem is reformulated to take care of location constraints whereby the placement of BSs in a subset of the deployment area is not permitted due to, e.g., private property limitations or electromagnetic radiation constraints. Experimental results and optimal tradeoff curves are presented and analyzed for various scenarios. In order to counteract the explosive escalation of wireless tele-traffic, the communication spectrum has been gradually expanded from the conventional radio frequency (RF) band to the optical wireless (OW) domain. By integrating the classic RF band relying on diverse radio techniques and optical bands, the next-generation heterogeneous networks (HetNets) are expected to offer a potential solution for supporting the ever-increasing wireless tele-traffic. Owing to its abundant unlicensed spectral resources, visible light communications (VLC) combined with advanced illumination constitute a competent candidate for complementing the existing RF networks. Although the advantages of VLC are multi-fold, some challenges arise when incorporating VLC into the classic RF HetNet environments, which may require new system architectures. The user-centric (UC) design principle for VLC environments constitutes a novel and competitive design paradigm for the super dense multi-tier cell combinations of HetNets. The UC concept may be expected to become one of the disruptive techniques to be used in the forthcoming fifth-generation era. This paper provides a comprehensive survey of visible-light-aided OW systems with special emphasis on the design and optimization of VLC networks, where the radically new UC design philosophy is reviewed. Finally, design guidelines are provided for VLC systems. This paper investigates the optimal resource allocation in wireless powered communication network with user cooperation, where two single-antenna users first harvest energy from the signals transmitted by a multi-antenna hybrid access point (H-AP) and then cooperatively send information to the H-AP using their harvested energy. To explore the system information transmission performance limit, an optimization problem is formulated to maximize the weighted sum-rate (WSR) by jointly optimizing energy beamforming vector, time assignment, and power allocation. Besides, another optimization problem is also formulated to minimize the total transmission time for given amount of data required to be transmitted at the two sources. Because both problems are non-convex, we first transform them to be convex by using proper variable substitutions and then apply semi-definite relaxation to solve them. We theoretically prove that our proposed methods guarantee the global optimum of both problems. Simulation results show that system WSR and transmission time can be significantly enhanced by using energy beamforming and user cooperation. It is observed that when the total amount of information of two users is fixed, with the increase of the information amount of the user relatively farther away from the H-AP, the transmission time of the user cooperation scheme decreases while that of the direct transmission increases. Besides, the effects of user position on the system performances are also discussed, which provides some useful insights. Visible light communication (VLC) systems have typically operated at data rates below 20 Gbps and operation at this data rate was shown to be feasible by using laser diodes (LDs), beam steering, imaging receivers and delay adaptation techniques. However, an increase in the computational cost is incurred. In this paper, we introduce fast computer generated holograms (FCGHs) to speed up the adaptation process. The new, fast and efficient fully adaptive VLC system can improve the receiver signal to noise ratio (SNR) and reduce the required time to estimate the position of the VLC receiver. In addition, an imaging receiver and a delay adaptation technique are used to reduce the effect of inter symbol interference (ISI) and multipath dispersion. Significant enhancements in the SNR, with VLC channel bandwidths of more than 36 GHz are obtained resulting in a compact impulse response and a VLC system that is able to achieve higher data rates (25 Gbps) with full mobility in the considered indoor environment. A resource allocation framework is presented for spectrum underlay in cognitive wireless networks. We consider both interference constraints for primary users and quality of service (QoS) constraints for secondary users. Specifically, interference from secondary users to primary users is constrained to be below a tolerable limit. Also, signal to interference plus noise ratio (SINR) of each secondary user is maintained higher than a desired level for QoS insurance. We propose admission control algorithms to be used during high network load conditions which are performed jointly with power control so that QoS requirements of all admitted secondary users are satisfied while keeping the interference to primary users below the tolerable limit. If all secondary users can be supported at minimum rates, we allow them to increase their transmission rates and share the spectrum in a fair manner. We formulate the joint power rate allocation with proportional and max-min fairness criteria as optimization problems. We show how to transform these optimization problems into a convex form so that their globally optimal solutions can be obtained. Numerical results show that the proposed admission control algorithms achieve performance very close to that of the optimal solution. Also, impacts of different system and QoS parameters on the network performance are investigated for the admission control, and rate power allocation algorithms under different fairness criteria. We propose an efficient client-based approach for channel management (channel assignment and load balancing) in 802.11-based WLANs that lead to better usage of the wireless spectrum. This approach is based on a “conflict set coloring” formulation that jointly performs load balancing along with channel assignment. Such a formulation has a number of advantages. First, it explicitly captures interference effects at clients. Next, it intrinsically exposes opportunities for better channel re-use. Finally, algorithms based on this formulation do not depend on specific physical RF models and hence can be applied efficiently to a wide-range of in-building as well as outdoor scenarios. We have performed extensive packet-level simulations and measurements on a deployed wireless testbed of 70 APs to validate the performance of our proposed algorithms. We show that in addition to single network scenarios, the conflict set coloring formulation is well suited for channel assignment where multiple wireless networks share and contend for spectrum in the same physical space. Our results over a wide range of both simulated topologies and in-building testbed experiments indicate that our approach improves application level performance at the clients by upto three times (and atleast 50 ) in comparison to current best-known techniques. The throughput of wireless networks can be significantly improved by multi-channel communications compared with single-channel communications since the use of multiple channels can reduce interference influence. In this paper, we study interference-aware topology control and QoS routing in IEEE 802.11-based multi-channel wireless mesh networks with dynamic traffic. Channel assignment and routing are two basic issues in such networks. Different channel assignments can lead to different network topologies. We present a novel definition of co-channel interference. Based on this concept, we formally define and present an effective heuristic for the minimum INterference Survivable Topology Control (INSTC) problem which seeks a channel assignment for the given network such that the induced network topology is interference-minimum among all K-connected topologies. We then formulate the Bandwidth-Aware Routing (BAR) problem for a given network topology, which seeks routes for QoS connection requests with bandwidth requirements. We present a polynomial time optimal algorithm to solve the BAR problem under the assumption that traffic demands are splittable. For the non-splittable case, we present a maximum bottleneck capacity path routing heuristic. Simulation results show that compared with the simple common channel assignment and shortest path routing approach, our scheme improves the system performance by 57 on average in terms of connection blocking ratio. We propose to add a new dimension to existing wireless multimedia communications systems by enabling competing stations to proactively engage in the resource management game by adapting their cross-layer transmission strategies. For this, we model wireless stations (WSTAs) as rational and selfish players competing for available wireless resources in a dynamic game. We focus on polling-based wireless LAN (WLAN) networks, where developing an efficient solution for managing the available transmission opportunities is of paramount importance. The resource allocation game is coordinated by a network moderator, which deploys a novel resource management based on the Vickrey-Clarke-Groves (VCG) mechanism to determine a) the amount of time to be allocated to the various users and b) the transmission cost associated to the allocated resources. The transmission cost is referred to in the VCG mechanism as "transfer" and depends not on the used resources, but rather on the inconvenience (in terms of utility impact) that it causes to other WSTAs. The transfer is introduced in order to discourage WSTAs from lying about their resource requirements. Importantly, this proposed dynamic resource management approach for wireless multimedia applications changes the passive way stations are currently adapting their cross-layer strategies by enabling them to selfishly influence the wireless systems dynamics by proactively adapting their packet scheduling strategies, error protection strategies, etc. Hence, each wireless station can play the resource management game by adapting its multimedia transmission strategy depending on the experienced channel conditions, derived video quality, attitude towards risk, willingness to pay for resources and available information about the wireless network. Our simulations show that using the VCG mechanism the WSTAs do not have any incentives to lie about their resource requirements as otherwise they will be severely penalized by a high transfer. We also show that deploying advanced cross-layer strategies for playing the resource management game significantly benefit the WSTAs' received video quality. The willingness-to-pay for resources is introduced to provide WSTAs a tool to gather additional resources whenever they need to transmit an important (part of a) video sequence by agreeing to pay for resources an increased cost. A novel risk-aware scheduling scheme is also proposed that provides WSTAs the ability to dynamically avoid network congestion and hence, reduce their incurred transfer In this paper, we study resource allocation for a multicarrier-based cognitive radio (CR) network. More specifically, we investigate the problem of secondary users’ energy-efficiency (EE) maximization problem under secondary total power and primary interference constraints. First, assuming cooperation among the secondary base stations (BSs), a centralized approach is considered to solve the EE optimization problem for the CR network where the primary and secondary users are using either orthogonal frequency-division multiplexing (OFDM) or filter bank based multicarrier (FBMC) modulations. We propose an alternating-based approach to solve the joint power-subcarrier allocation problem. More importantly, in the first place, subcarriers are allocated using a heuristic method for a given feasible power allocation. Then, we conservatively approximate the nonconvex power control problem and propose a joint successive convex approximation-Dinkelbach algorithm (SCADA) to efficiently obtain a solution to the nonconvex power control problem. The proposed algorithm is shown to converge to a solution that coincides with the stationary point of the original nonconvex power control subproblem. Moreover, we propose a dual decomposition-based decentralized version of the SCADA. Second, under the assumption of no cooperation among the secondary BSs, we propose a fully distributed power control algorithm from the perspective of game theory. The proposed algorithm is shown to converge to a Nash-equilibrium (NE) point. Moreover, we propose a sufficient condition that guarantees the uniqueness of the achieved NE. Extensive simulation analyses are further provided to highlight the advantages and demonstrate the efficiency of our proposed schemes. This paper investigates the optimal rate allocation problem with end-to-end delay constraints in multi-hop wireless networks. We introduce Virtual Link Capacity Margin (VLCM), which is the gap between the schedulable link capacity and the maximum allowable flow rate over a link, for link delay control. We formulate the problem as a utility maximization framework with two sets of constraints: 1) capacity and schedulability constraints and 2) end-to-end delay constraints. By dual decomposition of the original optimization problem, we present a control algorithm that jointly tunes the flow rates and VLCMs, through a double-price scheme derived with regard to the constraints: the link congestion price reflecting the traffic load of a link, and the flow delay price reflecting the margin between the average packet delay and the delay requirement of a flow. We prove that the algorithm converges to a global optimum where the aggregate network utility defined over the flow rate set is maximized, while the delay constraints are satisfied. A key feature of our algorithm is it does not rely on a specific traffic model. The algorithm is implemented distributedly via joint wireless link scheduling and congestion control. Simulation results show that our algorithm outperforms heuristic rate allocation algorithms while satisfying the end-to-end delay constraints. We investigate a wireless system of multiple cells, each having a downlink shared channel in support of high-speed packet data services. In practice, such a system consists of hierarchically organized entities including a central server, Base Stations (BSs), and Mobile Stations (MSs). Our goal is to improve global resource utilization and reduce regional congestion given asymmetric arrivals and departures of mobile users, a goal requiring load balancing among multiple cells. For this purpose, we propose a scalable cross-layer framework to coordinate packet-level scheduling, call-level cell-site selection and handoff, and system-level cell coverage based on load, throughput, and channel measurements. In this framework, an opportunistic scheduling algorithm--the weighted Alpha-Rule--exploits the gain of multiuser diversity in each cell independently, trading aggregate (mean) down-link throughput for fairness and minimum rate guarantees among MSs. Each MS adapts to its channel dynamics and the load fluctuations in neighboring cells, in accordance with MSs' mobility or their arrival and departure, by initiating load-aware handoff and cell-site selection. The central server adjusts schedulers of all cells to coordinate their coverage by prompting cell breathing or distributed MS handoffs. Across the whole system, BSs and MSs constantly monitor their load, throughput, or channel quality in order to facilitate the overall system coordination. Our specific contributions in such a framework are highlighted by the minimum-rate guaranteed weighted Alpha-Rule scheduling, the load-aware MS handoff cell-site selection, and the Media Access Control (MAC)-layer cell breathing. Our evaluations show that the proposed framework can improve global resource utilization and load balancing, resulting in a smaller blocking rate of MS arrivals without extra resources while the aggregate throughput remains roughly the same or improved at the hot-spots. Our simulation tests also show that the coordinated system is robust to dynamic load fluctuations and is scalable to both the system dimension and the size of MS population. This paper considers jointly optimal design of crosslayer congestion control, routing and scheduling for ad hoc wireless networks. We first formulate the rate constraint and scheduling constraint using multicommodity flow variables, and formulate resource allocation in networks with fixed wireless channels (or single-rate wireless devices that can mask channel variations) as a utility maximization problem with these constraints. By dual decomposition, the resource allocation problem naturally decomposes into three subproblems: congestion control, routing and scheduling that interact through congestion price. The global convergence property of this algorithm is proved. We next extend the dual algorithm to handle networks with timevarying channels and adaptive multi-rate devices. The stability of the resulting system is established, and its performance is characterized with respect to an ideal reference system which has the best feasible rate region at link layer. We then generalize the aforementioned results to a general model of queueing network served by a set of interdependent parallel servers with time-varying service capabilities, which models many design problems in communication networks. We show that for a general convex optimization problem where a subset of variables lie in a polytope and the rest in a convex set, the dual-based algorithm remains stable and optimal when the constraint set is modulated by an irreducible finite-state Markov chain. This paper thus presents a step toward a systematic way to carry out cross-layer design in the framework of “layering as optimization decomposition” for time-varying channel models. In this paper, we study resource allocation for multiuser multiple-input–single-output secondary communication systems with multiple system design objectives. We consider cognitive radio (CR) networks where the secondary receivers are able to harvest energy from the radio frequency when they are idle. The secondary system provides simultaneous wireless power and secure information transfer to the secondary receivers. We propose a multiobjective optimization framework for the design of a Pareto-optimal resource allocation algorithm based on the weighted Tchebycheff approach. In particular, the algorithm design incorporates three important system design objectives: total transmit power minimization, energy harvesting efficiency maximization, and interference-power-leakage-to-transmit-power ratio minimization. The proposed framework takes into account a quality-of-service (QoS) requirement regarding communication secrecy in the secondary system and the imperfection of the channel state information (CSI) of potential eavesdroppers (idle secondary receivers and primary receivers) at the secondary transmitter. The proposed framework includes total harvested power maximization and interference power leakage minimization as special cases. The adopted multiobjective optimization problem is nonconvex and is recast as a convex optimization problem via semidefinite programming (SDP) relaxation. It is shown that the global optimal solution of the original problem can be constructed by exploiting both the primal and the dual optimal solutions of the SDP-relaxed problem. Moreover, two suboptimal resource allocation schemes for the case when the solution of the dual problem is unavailable for constructing the optimal solution are proposed. Numerical results not only demonstrate the close-to-optimal performance of the proposed suboptimal schemes but unveil an interesting tradeoff among the considered conflicting system design objectives as well. This paper first provides a brief survey on existing traffic offloading techniques in wireless networks. Particularly as a case study, we put forward an online reinforcement learning framework for the problem of traffic offloading in a stochastic heterogeneous cellular network (HCN), where the time-varying traffic in the network can be offloaded to nearby small cells. Our aim is to minimize the total discounted energy consumption of the HCN while maintaining the quality-of-service (QoS) experienced by mobile users. For each cell (i.e., a macro cell or a small cell), the energy consumption is determined by its system load, which is coupled with system loads in other cells due to the sharing over a common frequency band. We model the energy-aware traffic offloading problem in such HCNs as a discrete-time Markov decision process (DTMDP). Based on the traffic observations and the traffic offloading operations, the network controller gradually optimizes the traffic offloading strategy with no prior knowledge of the DTMDP statistics. Such a model-free learning framework is important, particularly when the state space is huge. In order to solve the curse of dimensionality, we design a centralized @math -learning with compact state representation algorithm, which is named @math -learning. Moreover, a decentralized version of the @math -learning is developed based on the fact the macro base stations (BSs) can independently manage the operations of local small-cell BSs through making use of the global network state information obtained from the network controller. Simulations are conducted to show the effectiveness of the derived centralized and decentralized @math -learning algorithms in balancing the tradeoff between energy saving and QoS satisfaction. This paper studies the newly emerging wireless powered communication network in which one hybrid access point (H-AP) with constant power supply coordinates the wireless energy information transmissions to from a set of distributed users that do not have other energy sources. A "harvest-then-transmit" protocol is proposed where all users first harvest the wireless energy broadcast by the H-AP in the downlink (DL) and then send their independent information to the H-AP in the uplink (UL) by time-division-multiple-access (TDMA). First, we study the sum-throughput maximization of all users by jointly optimizing the time allocation for the DL wireless power transfer versus the users' UL information transmissions given a total time constraint based on the users' DL and UL channels as well as their average harvested energy values. By applying convex optimization techniques, we obtain the closed-form expressions for the optimal time allocations to maximize the sum-throughput. Our solution reveals an interesting "doubly near-far" phenomenon due to both the DL and UL distance-dependent signal attenuation, where a far user from the H-AP, which receives less wireless energy than a nearer user in the DL, has to transmit with more power in the UL for reliable information transmission. As a result, the maximum sum-throughput is shown to be achieved by allocating substantially more time to the near users than the far users, thus resulting in unfair rate allocation among different users. To overcome this problem, we furthermore propose a new performance metric so-called common-throughput with the additional constraint that all users should be allocated with an equal rate regardless of their distances to the H-AP. We present an efficient algorithm to solve the common-throughput maximization problem. Simulation results demonstrate the effectiveness of the common-throughput approach for solving the new doubly near-far problem in wireless powered communication networks. With the increasing development of IEEE 802.11 based wireless local area network (WLAN) devices, large-scale multi-cell WLANs with a high density of users and access points (APs) have emerged widely in various hotspots. Providing resilient data transmission has been a primary challenge for scaling the WLANs because the high density of users and APs results in too many collisions. In this paper, we analyze and point out the defect of the single association mechanism defined in IEEE 802.11 on transmission reliability from a network perspective. Then, we propose a "multi-AP" architecture with which a MAC layer device called an AP controller (AC) is employed to enable each user to associate and cooperate with multiple APs. In this way, the users can benefit from the diversity effect of multipaths with independent collisions and transmission errors. This paper concentrates on the theoretical analysis of performance comparison between the proposed ldquoMulti-APrdquo architecture and that in IEEE 802.11. Extensive simulation results show that the proposed ldquomulti-APrdquo architecture can obtain much better performance in terms of the throughput per user and the total throughput, and the performance gain is position dependent. Moreover, the unfairness issue in traditional WLANs due to capture effect can be alleviated properly in the ldquomulti-APrdquo framework. In Long Term Evolution (LTE), the provision of Quality of Service (QoS) strongly depends on the efficiency with which control resources are allocated. Unfortunately, the design of the control channel, the Physical Downlink Control Channel (PDCCH), is not flexible enough to support Intercell Interference Coordination (ICIC) and hence, its degradation at cell edges (a well known issue in LTE) severely jeopardizes system performance. This problem acquires especial relevance in realistic large scale deployments where Signal to Interference plus Noise Ratio (SINR) distributions are much worse than the ones predicted by means of hexagonal layouts. Thus, this paper investigates the challenge of improving SINR levels at cell edges in order to enhance the performance of the PDCCH. The proposed scheme adjusts the power allocated to the PDCCH at each cell in order to improve SINR distributions (minimizing the usage of control resources) and, due to its multiobjective nature, it considers several perspectives of the problem. The results show that gains in the order of 25 can be obtained in severely interfered cells while overall energy savings are around 50 . Joint allocation of spectrum and user association is considered for a large cellular network. The objective is to optimize a network utility function such as average delay given traffic statistics collected over a slow timescale. A key challenge is scalability: given n access points (APs), there are O(2n) ways in which the APs can share the spectrum. The number of variables is reduced from O(2n) to O(nk), where k is the number of users, by optimizing over local overlapping neighborhoods, defined by interference conditions, and by exploiting the existence of sparse solutions in which the spectrum is divided into k+1 segments. We reformulate the problem by optimizing the assignment of subsets of active APs to those segments. An l o constraint enforces a one-to-one mapping of subsets to spectrum segments, and an iterative (reweighted l 1 ) algorithm is used to find an approximate solution. Numerical results for a network with 100 APs serving several hundred users show the proposed method achieves a substantial increase in total throughput relative to benchmark schemes. Next-generation cellular networks will provide higher cell capacity by adopting advanced physical layer techniques and broader bandwidth. Even in such networks, boundary users would suffer from low throughput due to severe intercell interference and unbalanced user distributions among cells, unless additional schemes to mitigate this problem are employed. In this paper, we tackle this problem by jointly optimizing partial frequency reuse and load-balancing schemes in a multicell network. We formulate this problem as a network-wide utility maximization problem and propose optimal offline and practical online algorithms to solve this. Our online algorithm turns out to be a simple mixture of inter- and intra-cell handover mechanisms for existing users and user association control and cell-site selection mechanisms for newly arriving users. A remarkable feature of the proposed algorithm is that it uses a notion of expected throughput as the decision making metric, as opposed to signal strength in conventional systems. Extensive simulations demonstrate that our online algorithm can not only closely approximate network-wide proportional fairness but also provide two types of gain, interference avoidance gain and load balancing gain, which yield 20 100 throughput improvement of boundary users (depending on traffic load distribution), while not penalizing total system throughput.We also demonstrate that this improvement cannot be achieved by conventional systems using universal frequency reuse and signal strength as the decision making metric. The goal of this paper is to determine the data routes, subchannel schedules, and power allocations that maximize a weighted-sum rate of the data communicated over a generic OFDMA wireless network in which the nodes are capable of simultaneously transmitting, receiving and relaying data. Two instances are considered. In the first instance, subchannels are allowed to be time-shared by multiple links, whereas in the second instance, each subchannel is exclusively used by one of the links. Using a change of variables, the first problem is transformed into a convex form. In contrast, the second problem is not amenable to such a transformation and results in a complex mixed integer optimization problem. To develop insight into this problem, we utilize the first instance to obtain efficiently computable lower and upper bounds on the weighted-sum rate that can be achieved in the absence of time-sharing. Another lower bound is obtained by enforcing the scheduling constraints through additional power constraints and a monomial approximation technique to formulate the design problem as a geometric program. Numerical investigations show that the obtained rates are higher when time-sharing is allowed, and that the lower bounds on rates in the absence of time-sharing are relatively tight.