Electrical Engineering

We address the fundamental question of whether or not there exist stable operating points in a network in which selfish nodes share a common channel, and if they exist, how the nodes behave at these stable operating points. We begin with a wireless communication network in which nodes (agents), which might have different utility functions, contend for access on a common, wireless communication channel. We characterize this distributed multiple-access problem in terms of a one-shot random-access game, and then analyze the behavior of the nodes using the tools of game theory. We give necessary and sufficient conditions on nodes for the complete characterization of the Nash equilibria of this game for all 2. We show that all centrally controlled optimal solutions are a subset of this game theoretic solution, and almost all (w.r.t. Lebesgue measure) transmission probability assignments chosen by a central authority are supported by the game theoretic solution. We analyze the behavior of the network throughput at Nash equilibria as a function of the costs of the transmitters incurred by failed transmissions. Finally, we conclude the paper with the asymptotic analysis of the system as the number of transmitters goes to infinity. We show that the asymptotic distribution of the packet arrivals converges in distribution to a Poisson random variable, and the channel throughput converges to ((1 +)) ln((1 +)) with 0 being the cost of failed transmissions. We also give the best possible bounds on the rates of convergence of the packet arrival distribution and the channel throughput.

Contention control is an important means to mitigate adverse affects of multiple access interference on transmitted data packets in wireless communication networks. Most practical contention control mechanisms, e.g., IEEE 802.11 and slotted Aloha, rely on the assumption that all users abide with the rules of the protocol. There is also a growing body of work employing game theoretic techniques to gain new design insights for controlling contention when users act selfishly to maximize their perceived performance, as well as to reverse/forward engineer existing contention control protocols. In this paper, it is shown that the asymptotic behaviour of the slotted Aloha protocol when the number of contenders grows large coincides with a Nash equilibrium that is derived from a game theoretic contention model. Furthermore, the relationship between the design parameters in both approaches is established. In particular, it is shown that the back-off exponent in the slotted Aloha based contention control is linearly related to the cost of a collision in the game theoretic approach.

This paper analyzes layer 2 contention resolution strategies for wireless networks with multipacket reception by using noncooperative game theory. Necessary and sufficient conditions are obtained for a strategy profile to be a Nash equilibrium. Applications of the derived equilibrium conditions to predict selfish behavior and the resulting equilibrium performance are illustrated in specific communication scenarios along with various design insights. The collective equilibrium behavior of wireless networks with large user populations is also studied, and a Poisson-Bernoulli type approximation is obtained for the total number of packet arrivals. Finally, random access control with imperfect information structure is considered, the form of equilibrium strategies as well as uniqueness and existence results for general wireless channel models are obtained, and the best-response learning dynamics achieving an equilibrium are illustrated in specific instances.

This paper addresses the following question: how reliable is it to use the unbounded path-loss model G(d) = d −α , where α is the path-loss exponent, to model the decay of transmitted signal power in wireless networks? G(d) is a good approximation for the path-loss in wireless communications for large values of d but is not valid for small values of d due to the singularity at 0. This model is often used along with a random uniform node distribution, even though in a group of uniformly distributed nodes some may be arbitrarily close to one another. The unbounded path-loss model is compared to a more realistic bounded path-loss model, and it is shown that the effect of the singularity on the total network interference level is significant and cannot be disregarded when nodes are uniformly distributed. A phase transition phenomenon occurring in the interference behavior is analyzed in detail. Several performance metrics are also examined by using the computed interference distributions. In particular, the effects of the singularity at 0 on bit error rate, packet success probability and wireless channel capacity are analyzed.

This paper studies optimum power control and sumrate scaling laws for the distributed cognitive uplink. It is first shown that the optimum distributed power control policy is in the form of a threshold based water-filling power control. Each secondary user executes the derived power control policy in a distributed fashion by using local knowledge of its direct and interference channel gains such that the resulting aggregate (average) interference does not disrupt primary's communication. Then, the tight sum-rate scaling laws are derived as a function of the number of secondary users N under the optimum distributed power control policy. The fading models considered to derive sum-rate scaling laws are general enough to include Rayleigh, Rician and Nakagami fading models as special cases. When transmissions of secondary users are limited by both transmission and interference power constraints, it is shown that the secondary network sum-rate scales according to 1 en h log log (N), where n h is a parameter obtained from the distribution of direct channel power gains. For the case of transmissions limited only by interference constraints, on the other hand, the secondary network sum-rate scales according to 1 eγg log (N), where γg is a parameter obtained from the distribution of interference channel power gains. These results indicate that the distributed cognitive uplink is able to achieve throughput scaling behavior similar to that of the centralized cognitive uplink up to a pre-log multiplier 1 e , whilst primary's quality-of-service requirements are met. The factor 1 e can be interpreted as the cost of distributed implementation of the cognitive uplink. Index Terms-Cognitive radio, multiple access channels, power control, throughput scaling, distributed algorithms. 1 The terms cognitive uplink and cognitive multiple access network are used interchangeably throughout the paper.

This paper introduces an analytical framework for two small-world network models, and studies the delay of targeted social search by considering messages traveling between source and target individuals in these networks. In particular, by considering graphs constructed on different network domains, such as rectangular, circular and spherical network domains, analytical solutions for the average social search delay and the delay distribution are obtained as a function of source-target separation, distribution of the number of long-range connections and geometrical properties of network domains. Derived analytical formulas are first verified by agent-based simulations, and then compared with empirical observations in small-world experiments. Our formulas indicate that individuals tend to communicate with one another only through their short-range contacts, and the average social search delay rises linearly, when the separation between the source and target is small. As this separation increases, long-range connections are more commonly used, and the average social search delay rapidly saturates to a constant value and stays almost the same for all large values of the separation. These results are qualitatively consistent with experimental observations made by Travers and Milgram in 1969 as well as by others. Moreover, analytical distributions predicted by our models for the delay of social search are also compared with corresponding empirical distributions, and good statistical matches between them are observed. Other somewhat surprising conclusions of the paper are that hubs have limited effect in reducing the delay of social search and variation in node degree distribution adversely affects this delay.

This paper studies the structure of downlink sum-rate maximizing selective decentralized feedback policies for opportunistic beamforming under finite feedback constraints on the average number of mobile users feeding back. Firstly, it is shown that any sum-rate maximizing selective decentralized feedback policy must be a threshold feedback policy. This result holds for all fading channel models with continuous distribution functions. Secondly, the resulting optimum threshold selection problem is analyzed in detail. This is a non-convex optimization problem over finite dimensional Euclidean spaces. By utilizing the theory of majorization, an underlying Schur-concave structure in the sum-rate function is identified, and the sufficient conditions for the optimality of homogenous threshold feedback policies are obtained. Applications of these results are illustrated for well known fading channel models such as Rayleigh, Nakagami and Rician fading channels, along with various engineering and design insights. Rather surprisingly, it is shown that using the same threshold value at all mobile users is not always a ratewise optimal feedback strategy, even for a network with identical mobile users experiencing statistically the same channel conditions. For the Rayleigh fading channel model, on the other hand, homogenous threshold feedback policies are proven to be rate-wise optimal if multiple orthonormal data carrying beams are used to communicate with multiple mobile users simultaneously.

This paper investigates the problem of Gaussian approximation for the wireless multi-access interference distribution in large spatial wireless networks. First, a principled methodology is presented to establish rates of convergence of the multi-access interference distribution to a Gaussian distribution for general bounded and power-law decaying path-loss functions. The model is general enough to also include various random wireless channel dynamics such as fading and shadowing arising from multipath propagation and obstacles existing in the communication environment. It is shown that the wireless multi-access interference distribution converges to the Gaussian distribution with the same mean and variance at a rate 1 √ λ , where λ > 0 is a parameter controlling the intensity of the planar (possibly non-stationary) Poisson point process generating node locations. An explicit expression for the scaling coefficient is obtained as a function of fading statistics and the path-loss function. Second, an extensive numerical and simulation study is performed to illustrate the accuracy of the derived Gaussian approximation bounds. A good statistical fit between the interference distribution and its Gaussian approximation is observed for moderate to high values of λ. Finally, applications of these approximation results to upper and lower bound the outage capacity and ergodic sum capacity for spatial wireless networks are illustrated. The derived performance bounds on these capacity metrics track the network performance within one nats per second per hertz.

Beamforming techniques utilizing only partial channel state information (CSI) has gained popularity over other communication strategies requiring perfect CSI thanks to their lower feedback requirements. The amount of feedback in beamforming based communication systems can be further reduced through selective feedback techniques in which only the users with channels good enough are allowed to feed back by means of a decentralized feedback policy. In this paper, we prove that thresholding at the receiver is the rate-wise optimal decentralized feedback policy for feedback limited systems with prescribed feedback constraints. This result is highly adaptable due to its distribution independent nature, provides an analytical justification for the use of threshold feedback policies in practical systems, and reinforces previous work analyzing threshold feedback policies as a selective feedback technique without proving its optimality. It is robust to selfish unilateral deviations. Finally, it reduces the search for rate-wise optimal feedback policies subject to feedback constraints from function spaces to a finite dimensional Euclidean space.

In cognitive multiple access networks, primarysecondary feedback links are needed to convey secondary transmitter primary base station (STPB) channel gains from the primary base station (PBS) to the secondary base station (SBS). To reduce the amount of feedback exchange between PBS and SBS, this paper proposes a feedback control protocol calledsmallest channel gains (-SCG) feedback protocol in which the PBS feeds back the smallest STPB channel gains, out of of them, to the SBS. We study the performance of-SCG feedback protocol for total power and interference limited (TPIL) networks when transmit powers of secondary users (SUs) are optimally allocated. In TPIL networks, transmit powers of SUs are limited by an average total power constraint as well as a constraint on the average total interference power that they cause to the PBS. It is shown that for = with ∈ (0, 1),-SCG feedback protocol is asymptotically optimal, i.e., secondary network throughput under-SCG and full feedback protocols scales according to 1 ℎ log log () where ℎ is a parameter obtained from the distribution of secondary transmitter secondary base station (STSB) channel power gains, and is the number of SUs. It is also shown that for = (), the interference power at the PBS converges to zero almost surely and in mean as becomes large. This result implies that for large enough, the secondary network just requires the indices of SUs corresponding to the smallest STPB channel gains for performing jointly optimal user scheduling and power allocation rather than the actual realizations of STPB channel gains.

This paper studies optimal distributed power allocation and scheduling policies (DPASPs) for distributed total power and interference limited (DTPIL) cognitive multiple access networks in which secondary users (SU) independently perform power allocation and scheduling tasks using their local knowledge of secondary transmitter secondary base-station (STSB) and secondary transmitter primary base-station (STPB) channel gains. In such networks, transmission powers of SUs are limited by an average total transmission power constraint and by a constraint on the average interference power that SUs cause to the primary base-station. We first establish the joint optimality of water-filling power allocation and threshold-based scheduling policies for DTPIL networks. We then show that the secondary network throughput under the optimal DPASP scales according to 1 en h log log (N), where n h is a parameter obtained from the distribution of STSB channel power gains and N is the total number of SUs. From a practical point of view, our results signify the fact that distributed cognitive multiple access networks are capable of harvesting multiuser diversity gains without employing centralized schedulers and feedback links as well as without disrupting primary's quality-of-service (QoS).

This paper studies the outage capacity of a network consisting of a multitude of heterogenous mobile users, and operating according to the classical opportunistic beamforming framework. The base station is located at the center of the cell, which is modeled as a disk of finite radius. The random user locations are modeled using a homogenous spatial Poisson point process. The received signals are impaired by both fading and location dependent path loss. For this system, we first derive an expression for the beam outage probability. This expression holds for all path loss models that satisfy some mild conditions. Then, we focus on two specific path loss models (i.e., an unbounded model and a more realistic bounded one) to illustrate the applications of our results. In the large system limit where the cell radius tends to infinity, the beam outage capacity and its scaling behavior are derived for the selected specific path loss models. It is shown that the beam outage capacity scales logarithmically for the unbounded model. On the other hand, this scaling behavior becomes double logarithmic for the bounded model. Intuitive explanations are provided as to why we observe different scaling behavior for different path loss models. Numerical evaluations are performed to give further insights, and to illustrate the applicability of the outage capacity results even to a cell having a small finite radius.

This paper analyzes the behavior of selfish transmitters under imperfect location information. The scenario considered is that of a wireless network consisting of selfish nodes that are randomly distributed over the network domain according to a known probability distribution, and that are interested in communicating with a common sink node using common radio resources. In this scenario, the wireless nodes do not know the exact locations of their competitors but rather have belief distributions about these locations. Firstly, properties of the packet success probability curve as a function of the node-sink separation are obtained for such networks. Secondly, a monotonicity property for the bestresponse strategies of selfish nodes is identified. That is, for any given strategies of competitors of a node, there exists a critical node-sink separation for this node such that its bestresponse is to transmit when its distance to the sink node is smaller than this critical threshold, and to back off otherwise. Finally, necessary and sufficient conditions for a given strategy profile to be a Nash equilibrium are provided.

This paper studies a class of random access games for wireless channels with multipacket reception. First, necessary and sufficient equilibrium conditions for a contention strategy profile to be a Nash equilibrium for general wireless channels are established. Then, applications of these equilibrium conditions for well-known channel models are illustrated. Various engineering insights and design ideas are provided. Finally, the results are extended to an incomplete information game setting, and best-response learning dynamics leading to Nash equilibria are investigated.

This paper studies optimum power control and sum-rate scaling laws for the distributed cognitive uplink. It is first shown that the optimum distributed power control policy is in the form of a threshold based water-filling power control. Each secondary user executes the derived power control policy in a distributed fashion by using local knowledge of its direct and interference channel gains such that the resulting aggregate (average) interference does not disrupt primary's communication. Then, the tight sum-rate scaling laws are derived as a function of the number of secondary users N under the optimum distributed power control policy. The fading models considered to derive sum-rate scaling laws are general enough to include Rayleigh, Rician and Nakagami fading models as special cases. When transmissions of secondary users are limited by both transmission and interference power constraints, it is shown that the secondary network sum-rate scales according to 1 en h log log (N), where n h is a parameter obtained from the distribution of direct channel power gains. For the case of transmissions limited only by interference constraints, on the other hand, the secondary network sum-rate scales according to 1 eγg log (N), where γ g is a parameter obtained from the distribution of interference channel power gains. These results indicate that the distributed cognitive uplink is able to achieve throughput scaling behavior similar to that of the centralized cognitive uplink up to a pre-log multiplier 1 e , whilst primary's quality-of-service requirements are met. The factor 1 e can be interpreted as the cost of distributed implementation of the cognitive uplink.

In cognitive multiple access networks, feedback is an important mechanism to convey secondary transmitter primary base station (STPB) channel gains from the primary base station (PBS) to the secondary base station (SBS). This paper investigates the optimal sum-rate capacity scaling laws for cognitive multiple access networks in feedback limited communication scenarios. First, an efficient feedback protocol called K-smallest channel gains (K-SCGs) feedback protocol is proposed in which the PBS feeds back the K N smallest out of N STPB channel gains to the SBS. Second, the sum-rate performance of the K-SCG feedback protocol is studied for three network types when transmission powers of secondary users (SUs) are optimally allocated. The network types considered are total-power-and-interference-limited (TPIL), interference-limited (IL) and individual-power-and-interference-limited (IPIL) networks. For each network type studied, we provide a sufficient condition on K N such that the K-SCG feedback protocol is asymptotically optimal in the sense that the secondary network sum-rate scaling behavior under the K-SCG feedback protocol is the same with that under the full-feedback protocol. We allow distributions of secondary-transmitter-secondary-base-station (STSB), and STPB channel power gains to belong to a fairly general class of distributions called class C-distributions that includes commonly used fading models. It is shown that for K N = N δ with δ ∈ (0, 1), the K-SCG feedback protocol is asymptotically optimal in TPIL networks, and the secondary network sum-rate scales according to 1 n h log log (N), where n h is a parameter obtained from the distribution of STSB channel power gains. In this case, it is also shown that the average interference power at the PBS can be made arbitrarily small without losing anything from this optimal sum-rate scaling behavior. For IL networks, the K-SCG feedback protocol is asymptotically optimal if K N = O (1). In this case, the secondary network sum-rate scales optimally according to 1 γg log (N), where γ g is a parameter obtained from the distribution of STPB channel gains. Finally, for IPIL networks, it is proven that the K-SCG feedback protocol is asymptotically optimal if K N = O (1), and the secondary network sum-rate scales according to min 1, 1 γg log (N). An extensive simulation and numerical study is also performed to illustrate the established sum-rate capacity scaling laws for finite networks.

This paper studies the outage capacity of a network consisting of a multitude of heterogenous mobile users, and operating according to the classical opportunistic beamforming framework. The base station is located at the center of the cell, which is modeled as a disk of finite radius. The random user locations are modeled using a homogenous spatial Poisson point process. The received signals are impaired by both fading and location dependent path loss. For this system, we first derive an expression for the beam outage probability. This expression holds for all path loss models that satisfy some mild conditions. Then, we focus on two specific path loss models (i.e., an unbounded model and a more realistic bounded one) to illustrate the applications of our results. In the large system limit where the cell radius tends to infinity, the beam outage capacity and its scaling behavior are derived for the selected specific path loss models. It is shown that the beam outage capacity scales logarithmically for the unbounded model. On the other hand, this scaling behavior becomes double logarithmic for the bounded model. Intuitive explanations are provided as to why we observe different scaling behavior for different path loss models. Numerical evaluations are performed to give further insights, and to illustrate the applicability of the outage capacity results even to a cell having a small finite radius.

This paper derives tight performance upper and lower bounds on the downlink outage efficiency of K-tier heterogeneous cellular networks (HCNs) for general signal propagation models with Poisson distributed base stations in each tier. In particular, the proposed approach to analyze the outage metrics in a K-tier HCN allows for the use of general bounded path-loss functions and random fading processes of general distributions. Considering two specific base station (BS) association policies, it is shown that the derived performance bounds track the actual outage metrics reasonably well for a wide range of BS densities, with the gap among them becoming negligibly small for denser HCN deployments. A simulation study is also performed for 2-tier and 3-tier HCN scenarios to illustrate the closeness of the derived bounds to the actual outage performance with various selections of the HCN parameters.

This paper studies the achievable throughput performance of the cognitive uplink under a limited primary cooperation scenario wherein the primary base station cannot feed back all interference channel gains to the secondary base station. To cope with the limited primary cooperation, we propose a feedback protocol called K-out-of-N feedback protocol, in which the primary base station feeds back only the K N smallest interference channel gains, out of N of them, to the secondary base station. We characterize the throughput performance under the K-out-of-N feedback protocol by analyzing the achievable multiuser diversity gains (MDGs) in cognitive uplinks for three different network types. Our results show that the proposed feedback mechanism is asymptotically optimum for interference-limited (IL) and individual-power-and-interferencelimited (IPIL) networks for a fixed positive K N. It is further shown that the secondary network throughput in the IL and IPIL networks (under both the full and limited cooperation scenarios) logarithmically scales with the number of users in the network. In total-power-and-interference-limited (TPIL) networks, on the other hand, the K-out-of-N feedback protocol is asymptotically optimum for K N = N δ , where δ ∈ (0, 1). We also show that, in TPIL networks, the secondary network throughput under both the limited and full cooperation scales logarithmically double with the number of users in the network. These results indicate that the cognitive uplink can achieve the optimum MDG even with limited cooperation from the primary network. They also establish the dependence of pre-log throughput scaling factors on the distribution of fading channel gains for different network types.