On the Effects of Frequency Scaling Over Capacity Scaling in Underwater Networks—Part I: Extended Network Model

In this two-part paper, information-theoretic capacity scaling laws are analyzed in an underwater acoustic network with n regularly located nodes on a square, in which both bandwidth and received signal power can be limited significantly. Parts I and II deal with an extended network of unit node density and a dense network of unit area, respectively. In both cases, a narrow-band model is assumed where the carrier frequency is allowed to scale as a function of n, which is shown to be crucial for achieving the order optimality in multi-hop (MH) mechanisms. We first characterize an attenuation parameter that depends on the frequency scaling as well as the transmission distance. Upper and lower bounds on the capacity scaling are then derived. In Part I, we show that the upper bound on capacity for extended networks is inversely proportional to the attenuation parameter, thus resulting in a highly power-limited network. Interestingly, it is shown that the upper bound is intrinsically related to the attenuation parameter but not the spreading factor. Furthermore, we propose an achievable communication scheme based on the nearest-neighbor MH transmission, which is suitable due to the low propagation speed of acoustic channel, and show that it is order-optimal for all operating regimes of extended networks. Finally, these scaling results are extended to the case of random node deployments providing fundamental limits to more complex scenarios of extended underwater networks.     

On the capacity of large Gaussian relay networks

The capacity of a particular large Gaussian relay network is determined in the limit as the number of relays tends to infinity. Upper bounds are derived from cut-set arguments, and lower bounds follow from an argument involving uncoded transmission. It is shown that in cases of interest, upper and lower bounds coincide in the limit as the number of relays tends to infinity. Hence, this paper provides a new example where a simple cut-set upper bound is achievable, and one more example where uncoded transmission achieves optimal performance. The findings are illustrated by geometric interpretations. The techniques developed in this paper are then applied to a sensor network situation. This is a network joint source-channel coding problem, and it is well known that the source-channel separation theorem does not extend to this case. The present paper extends this insight by providing an example where separating source from channel coding does not only lead to suboptimal performance-it leads to an exponential penalty in performance scaling behavior (as a function of the number of nodes). Finally, the techniques developed in this paper are extended to include certain models of ad hoc wireless networks, where a capacity scaling law can be established: When all nodes act purely as relays for a single source-destination pair, capacity grows with the logarithm of the number of nodes.     

Upper bounds to transport capacity of wireless networks

We derive upper bounds on the transport capacity of wireless networks. The bounds obtained are solely dependent on the geographic locations and power constraints of the nodes. As a result of this derivation, we are able to conclude the optimality, in the sense of scaling of transport capacity with the number of nodes, of a multihop communication strategy for a class of network topologies.     

Hierarchical Cooperation Achieves Optimal Capacity Scaling in Ad Hoc Networks

n source and destination pairs randomly located in an area want to communicate with each other. Signals transmitted from one user to another at distance r apart are subject to a power loss of r^-alpha as well as a random phase. We identify the scaling laws of the information-theoretic capacity of the network when nodes can relay information for each other. In the case of dense networks, where the area is fixed and the density of nodes increasing, we show that the total capacity of the network scales linearly with n. This improves on the best known achievability result of n^2/3 of Aeron and Saligrama. In the case of extended networks, where the density of nodes is fixed and the area increasing linearly with n, we show that this capacity scales as n^2-alpha/2 for 2lesalpha<3 and radicn for a alphages3. The best known earlier result of Xie and Kumar identified the scaling law for alpha > 4. Thus, much better scaling than multihop can be achieved in dense networks, as well as in extended networks with low attenuation. The performance gain is achieved by intelligent node cooperation and distributed multiple-input multiple-output (MIMO) communication. The key ingredient is a hierarchical and digital architecture for nodal exchange of information for realizing the cooperation.     

Multicast Capacity of Wireless Ad Hoc Networks

Assume that n wireless nodes are uniformly randomly deployed in a square region with side-length a and all nodes have the uniform transmission range r and uniform interference range R > r. We further assume that each wireless node can transmit (or receive) at W bits/second over a common wireless channel. For each node vi , we randomly and independently pick k-1 points pi,j (1 les j les k-1) from the square, and then multicast data to the nearest node for each pi,j. We derive matching asymptotic upper bounds and lower bounds on multicast capacity of random wireless networks. Under protocol interference model, when a ²/r ²=O(n/log(n)), we show that the total multicast capacity is Theta(radic{n/log n}middot(W/radick)) when k=O(n/log n); the total multicast capacity is Theta(W) when k=Omega(n/log n). We also study the capacity of group-multicast for wireless networks where for each source node, we randomly select k-1 groups of nodes as receivers and the nodes in each group are within a constant hops from the group leader. The same asymptotic upper bounds and lower bounds still hold. We also extend our capacity bounds to d -dimensional networks.     

User Capacity Scaling Laws of Multi-user Fading Channels in the Presence of MMSE Channel Estimators

Considering development of delay-sensitive applications in wireless communications and cellular networks along with lack of system resources and environmental factors including fading and noise all together make it impossible for all users to be active and to achieve quality of service requirements simultaneously. To analyze wireless systems performance, user capacity is defined as the maximum number of users that can be activated at the same time. In order to obtain user capacity scaling laws, channel distribution information is required and channel gains can be estimated by different channel estimators such as minimum mean square error estimators. In this paper, estimated channel gains are substituted for true channel gains and effects of imperfect channel estimation errors on the user capacity scaling laws are analyzed. It is shown that the user capacity scales double logarithmically and there is a gap depending on channel estimators accuracy between the lower and upper bounds. Moreover, assuming estimated channels for different users are independent and identically distributed, it is shown that the user capacity scaling laws are asymptotically tight.     

Capacity Scaling of Wireless Networks with Inhomogeneous Node Density: Upper Bounds

We analyze the capacity scaling laws of wireless ad hoc networks comprising significant inhomogeneities in the node spatial distribution over the network area. In particular, we consider nodes placed according to a shot-noise Cox process, which allows to model the clustering behavior usually recognized in large-scale systems. For this class of networks, we introduce novel techniques to compute upper bounds to the available perflow throughput as the number of nodes tends to infinity, which are tight in the case of interference limited systems.     

Capacity of large hybrid erasure networks with random node distribution

The Gupta–Kumar’s nearest-neighbor multihop routing with/without infrastructure support achieves the optimal capacity scaling in a large erasure network in which n wireless nodes and m relay stations are regularly placed. In this paper, a capacity scaling law is completely characterized for an infrastructure-supported erasure network where n wireless nodes are randomly distributed, which is a more feasible scenario. We use two fundamental path-loss attenuation models (i.e., exponential and polynomial power-laws) to suitably model an erasure probability. To show our achievability result, the multihop routing via percolation highway is used and the corresponding lower bounds on the total capacity scaling are derived. Cut-set upper bounds on the capacity scaling are also derived. Our result indicates that, under the random erasure network model with infrastructure support, the achievable scheme based on the percolation highway routing is order-optimal within a polylogarithmic factor of n for all values of m.     

Product Multicommodity Flow in Wireless Networks

We provide a tight approximate characterization of the n-dimensional product multicommodity flow (PMF) region for a wireless network of n nodes. Separate characterizations in terms of the spectral properties of appropriate network graphs are obtained in both an information-theoretic sense and for a combinatorial interference model (e.g., protocol model). These provide an inner approximation to the n ²-dimensional capacity region. Our results hold for general node distributions, traffic models, and channel fading models. We first establish that the random source-destination model assumed in many previous results on capacity scaling laws, is essentially a one-dimensional approximation to the capacity region and a special case of PMF. We then build on the results for a wireline network (graph) that relate PMF to its spectral (or cut) properties. Specifically, for a combinatorial interference model given by a network graph and a conflict graph, we relate the PMF to the spectral properties of the underlying graphs resulting in simple computational upper and lower bounds. These results show that the 1/radicn scaling law obtained by Gupta and Kumar for a geometric random network can be explained in terms of the scaling law of the conductance of a geometric random graph. For the more interesting random fading model with additive white Gaussian noise (AWGN), we show that the scaling laws for PMF can again be tightly characterized by the spectral properties of appropriately defined graphs-such a characterization for general wireless networks has not been available before. As an implication, we obtain computationally efficient upper and lower bounds on the PMF for any wireless network with a guaranteed approximation factor.     

干扰消除对无线Ad Hoc网络传输容量的影响

无线Ad Hoc网络是一个干扰受限系统,网络的容量取决于网络中的干扰信号功率的大小。在目前针对无线Ad Hoc网络容量进行研究的文献中没有使用有效的抑制干扰信号功率的方法,使得网络容量受到很大限制。提出通过设置保护区域的方法来降低干扰信号强度,重新对无线Ad Hoc网络进行建模,并推导了传输容量的上下界。通过仿真结果可以知道,设置保护区域能够极大程度降低网络中的干扰信号强度,合理地设置保护区域大小能有效提高传输容量。     

On the path-loss attenuation regime for positive cost and linear scaling of transport capacity in wireless networks

Wireless networks with a minimum inter-node separation distance are studied where the signal attenuation grows in magnitude as 1/ρ/sup δ/ with distance ρ. Two performance measures of wireless networks are analyzed. The transport capacity is the supremum of the total distance-rate products that can be supported by the network. The energy cost of information transport is the infimum of the ratio of the transmission energies used by all the nodes to the number of bit-meters of information thereby transported. If the phases of the attenuations between node pairs are uniformly and independently distributed, it is shown that the expected transport capacity is upper-bounded by a multiple of the total of the transmission powers of all the nodes, whenever δ>2 for two-dimensional networks or δ>5/4 for one-dimensional networks, even if all the nodes have full knowledge of all the phases, i.e., full channel state information. If all nodes have an individual power constraint, the expected transport capacity grows at most linearly in the number of nodes due to the linear growth of the total power. This establishes the best case order of expected transport capacity for these ranges of path-loss exponents since linear scaling is also feasible. If the phases of the attenuations are arbitrary, it is shown that the transport capacity is upper-bounded by a multiple of the total transmission power whenever δ>5/2 for two-dimensional networks or δ>3/2 for one-dimensional networks, even if all the nodes have full channel state information. This shows that there is indeed a positive energy cost which is no less than the reciprocal of the above multiplicative constant. It narrows the transition regime where the behavior is still open, since it is known that when δ<3/2 for two-dimensional networks, or δ<1 for one-dimensional networks, the transport capacity cannot generally be bounded by any multiple of the     

Control of wireless networks with rechargeable batteries [transactions papers]

We consider the problem of cross-layer resource allocation for wireless networks operating with rechargeable batteries under general arrival, channel state and recharge processes. The objective is to maximize total system utility, defined as a function of the long-term rate achieved per link, while satisfying energy and power constraints. A policy with decoupled admission control and power allocation decisions is proposed that achieves asymptotic optimality for sufficiently large battery capacity to maximum transmission power ratio (explicit bounds are provided). We present first a downlink resource allocation scenario; the analysis is then extended to multihop networks. The policy is evaluated via simulations and is seen to perform very well even in the non-asymptotic regime. This policy is particularly suitable for sensor networks, which typically satisfy the asymptotic conditions required by our methodology.     

保护区域对无线Ad Hoc网络空间进度密度的影响

研究无线Ad Hoc网络容量的文章中,往往不会考虑如何抑制干扰信号功率,使得网络容量的提高受到很大限制。文章使用随机几何对无线Ad Hoc网络进行建模,研究空间进度密度这一容量定义,提出在无线Ad Hoc网络中通过设置保护区域的方式来抑制干扰信号功率,推导了网络的空间进度密度的近似值、上下界和最优保护区域半径等的理论表达式。理论分析与仿真结果说明,中断概率和空间进度密度的近似值和上下界能够很好的逼近仿真值;设置保护区域能够降低无线Ad Hoc网络中的干扰信号功率,提高空间进度密度。     

Transmission capacity of wireless ad hoc networks with outage constraints

In this paper, upper and lower bounds on the transmission capacity of spread-spectrum (SS) wireless ad hoc networks are derived. We define transmission capacity as the product of the maximum density of successful transmissions multiplied by their data rate, given an outage constraint. Assuming that the nodes are randomly distributed in space according to a Poisson point process, we derive upper and lower bounds for frequency hopping (FH-CDMA) and direct sequence (DS-CDMA) SS networks, which incorporate traditional modulation types (no spreading) as a special case. These bounds cleanly summarize how ad hoc network capacity is affected by the outage probability, spreading factor, transmission power, target signal-to-noise ratio (SNR), and other system parameters. Using these bounds, it can be shown that FH-CDMA obtains a higher transmission capacity than DS-CDMA on the order of M/sup 1-2//spl alpha//, where M is the spreading factor and /spl alpha/>2 is the path loss exponent. A tangential contribution is an (apparently) novel technique for obtaining tight bounds on tail probabilities of additive functionals of homogeneous Poisson point processes.     

无线ad hoc网络多性能指标基本性能边界

在无线ad hoc网络中,基本性能边界对路由算法和资源分配协议的分析和评价具有重要的意义。对无线ad hoc网络多性能指标基本性能边界进行了研究,包括理论上最优的性能边界和实际可以得到的性能边界。提出了一种稳定状态(steady state)下的网络基本性能指标分析模型。该模型考虑了无线网络广播特性和无线信道干扰,可同时分析多个性能指标,包括：吞吐量、端到端延迟和能量消耗。基于该模型,针对ad hoc网络中最常见的多流—单/双中继拓扑分析基本性能指标,求解多目标优化问题得到基本性能边界。仿真结果验证了模型的准确性,均方根误差小于10-3量级。     

The transport capacity of wireless networks over fading channels

We consider networks consisting of nodes with radios, and without any wired infrastructure, thus necessitating all communication to take place only over the shared wireless medium. The main focus of this paper is on the effect of fading in such wireless networks. We examine the attenuation regime where either the medium is absorptive, a situation which generally prevails, or the path loss exponent is greater than 3. We study the transport capacity, defined as the supremum over the set of feasible rate vectors of the distance weighted sum of rates. We consider two assumption sets. Under the first assumption set, which essentially requires only a mild time average type of bound on the fading process, we show that the transport capacity can grow no faster than O(n), where n denotes the number of nodes, even when the channel state information (CSI) is available noncausally at both the transmitters and the receivers. This assumption includes common models of stationary ergodic channels; constant, frequency-selective channels; flat, rapidly varying channels; and flat slowly varying channels. In the second assumption set, which essentially features an independence, time average of expectation, and nonzeroness condition on the fading process, we constructively show how to achieve transport capacity of /spl Omega/(n) even when the CSI is unknown to both the transmitters and the receivers, provided that every node has an appropriately nearby node. This assumption set includes common models of independent and identically distributed (i.i.d.) channels; constant, flat channels; and constant, frequency-selective channels. The transport capacity is achieved by nodes communicating only with neighbors, and using only point-to-point coding. The thrust of these results is that the multihop strategy, toward which much protocol development activity is currently targeted, is appropriate for fading environments. The low attenuation regime is open.     

Even One-Dimensional Mobility Increases the Capacity of Wireless Networks

We study the capacity of ad hoc wireless networks with mobile nodes. The mobility model examined is one where the nodes are restricted to move along one-dimensional paths. We examine the scaling laws for the per-user throughput achievable over long time-scales, making this suitable for applications with loose delay constraints. We show that under this regime of restricted mobility, we attain a constant throughput (i.e.,$Theta(1)$) per user, which is significantly higher than the throughput of fixed networks, which decays as$O(1over sqrtn)$with the number of nodes$n$, as shown by Gupta and Kumar.     

Capacity scaling and spectral efficiency in wide-band correlated MIMO channels

The dramatic linear increase in ergodic capacity with the number of antennas promised by multiple-input multiple-output (MIMO) wireless communication systems is based on idealized channel models representing a rich scattering environment. Is such scaling sustainable in realistic scattering scenarios? Existing physical models, although realistic, are intractable for addressing this problem analytically due to their complicated nonlinear dependence on propagation path parameters, such as the angles of arrival and delays. In this paper, we leverage a recently introduced virtual representation of physical models that is essentially a Fourier series representation of wide-band MIMO channels in terms of fixed virtual angles and delays. Motivated by physical considerations, we propose a D-connected model for correlated channels defined by a virtual spatial channel matrix consisting of D nonvanishing diagonals with independent and identically distributed (i.i.d.) Gaussian entries. The parameter D provides a meaningful and tractable measure of the richness of scattering. We derive general bounds for the coherent ergodic capacity and investigate capacity scaling with the number of antennas and bandwidth. In the large antenna regime, we show that linear capacity scaling is possible if D scales linearly with the number of antennas. This, in turn, is possible if the number of resolvable paths grows quadratically with the number of antennas. The capacity saturates for linear growth in the number of paths (fixed D). The ergodic capacity does not depend on frequency selectivity of the channel in the wide-band case. Increasing bandwidth tightens the bounds and hastens the convergence of scaling behavior. For large bandwidth, the capacity scales linearly with the signal-to-noise ratio (SNR) as well. We also provide an explicit characterization of the wide-band slope recently proposed by Verdu. Numerical results are presented to illustrate the key theoretical results.     

On the impact of IEEE 802.11 MAC on traffic characteristics

IEEE 802.11 medium access control (MAC) is gaining widespread popularity as a layer-2 protocol for wireless local-area networks. While efforts have been made previously to evaluate the performance of various protocols in wireless networks and to evaluate the capacity of wireless networks, very little is understood or known about the traffic characteristics of wireless networks. In this paper, we address this issue and first develop an analytic model to characterize the interarrival time distribution of traffic in wireless networks with fixed base stations or ad hoc networks using the 802.11 MAC. Our analytic model and supporting simulation results show that the 802.11 MAC can induce pacing in the traffic and the resulting interarrival times are best characterized by a multimodal distribution. This is a sharp departure from behavior in wired networks and can significantly alter the second order characteristics of the traffic, which forms the second part of our study. Through simulations, we show that while the traffic patterns at the individual sources are more consistent with long-range dependence and self-similarity, in contrast to wired networks, the aggregate traffic is not self-similar. The aggregate traffic is better classified as a multifractal process and we conjecture that the various peaks of the multimodal interarrival time distribution have a direct contribution to the differing scaling exponents at various timescales.     

On capacity of random wireless networks with physical-layer network coding

Throughput capacity of a random wireless network has been studied extensively in the literature. Most existing studies were based on the assumption that each transmission involves only one transmitter in order to avoid interference. However, recent studies on physical-layer network coding (PLNC) have shown that such an assumption can be relaxed to improve throughput performance of a wireless network. In PLNC, signals from different senders can be transmitted to the same receiver in the same channel simultaneously. In this paper, we investigate the impact of PLNC on throughput capacity of a random wireless network. Our study reveals that, although PLNC scheme does not change the scaling law, it can improve throughput capacity by a fixed factor. Specifically, for a one-dimensional network, we observe that PLNC can eliminate the effect of interference in some scenarios. A tighter capacity bound is derived for a two-dimensional network. In addition, we also show achievable lower bounds for random wireless networks with network coding and PLNC.