Bounds on the throughput gain of network coding in unicast and multicast wireless networks

Gupta and Kumar established that the per node throughput of ad hoc networks with multi-pair unicast traffic scales with an increasing number of nodes n as lambda(n) = ominus(1/radic(n log n)), thus indicating that performance does not scale well. However, Gupta and Kumar did not consider network coding and wireless broadcasting, which recent works suggest have the potential to significantly improve throughput. Here, we establish bounds on the improvement provided by such techniques. For random networks of any dimension under either the protocol or physical model that were introduced by Gupta and Kumar, we show that network coding and broadcasting lead to at most a constant factor improvement in per node throughput. For the protocol model, we provide bounds on this factor. We also establish bounds on the throughput benefit of network coding and broadcasting for multiple source multicast in random networks. Finally, for an arbitrary network deployment, we show that the coding benefit ratio is at most O(log n) for both the protocol and physical communication models. These results give guidance on the application space of network coding, and, more generally, indicate the difficulty in improving the scaling behavior of wireless networks without modification of the physical layer.     

Optimal throughput-delay scaling in wireless networks - part I: the fluid model

Gupta and Kumar (2000) introduced a random model to study throughput scaling in a wireless network with static nodes, and showed that the throughput per source-destination pair is /spl Theta/(1//spl radic/(nlogn)). Grossglauser and Tse (2001) showed that when nodes are mobile it is possible to have a constant throughput scaling per source-destination pair. In most applications, delay is also a key metric of network performance. It is expected that high throughput is achieved at the cost of high delay and that one can be improved at the cost of the other. The focus of this paper is on studying this tradeoff for wireless networks in a general framework. Optimal throughput-delay scaling laws for static and mobile wireless networks are established. For static networks, it is shown that the optimal throughput-delay tradeoff is given by D(n)=/spl Theta/(nT(n)), where T(n) and D(n) are the throughput and delay scaling, respectively. For mobile networks, a simple proof of the throughput scaling of /spl Theta/(1) for the Grossglauser-Tse scheme is given and the associated delay scaling is shown to be /spl Theta/(nlogn). The optimal throughput-delay tradeoff for mobile networks is also established. To capture physical movement in the real world, a random-walk (RW) model for node mobility is assumed. It is shown that for throughput of /spl Oscr/(1//spl radic/(nlogn)), which can also be achieved in static networks, the throughput-delay tradeoff is the same as in static networks, i.e., D(n)=/spl Theta/(nT(n)). Surprisingly, for almost any throughput of a higher order, the delay is shown to be /spl Theta/(nlogn), which is the delay for throughput of /spl Theta/(1). Our result, thus, suggests that the use of mobility to increase throughput, even slightly, in real-world networks would necessitate an abrupt and very large increase in delay.     

Capacity of ad hoc wireless networks with infrastructure support

We determine the asymptotic scaling for the per user throughput in a large hybrid ad hoc network, i.e., a network with both ad hoc nodes, which communicate with each other via shared wireless links of capacity W bits/s, and infrastructure nodes which in addition are interconnected with each other via high capacity links. Specifically, we consider a network model where ad hoc nodes are randomly spatially distributed and choose to communicate with a random destination. We identify three scaling regimes, depending on the growth of the number of infrastructure nodes, m relative to the number of ad hoc nodes n, and show the asymptotic scaling for the per user throughput as n becomes large. We show that when m /spl lsim/ /spl radic/n/logn the per user throughput is of order W//spl radic/n log n and could be realized by allowing only ad hoc communications, i.e., not deploying the infrastructure nodes at all. Whenever /spl radic/n/log n /spl lsim/ m /spl lsim/ n/log n, the order for the per user throughput is Wm/n and, thus, the total additional bandwidth provided by m infrastructure nodes is effectively shared among ad hoc nodes. Finally, whenever m /spl gsim/ n/log n, the order of the per user throughput is only W/log n, suggesting that further investments in infrastructure nodes will not lead to improvement in throughput. The results are shown through an upper bound which is independent of the routing strategy, and by constructing scenarios showing that the upper bound is asymptotically tight.     

Towards an information theory of large networks: an achievable rate region

We study communication networks of arbitrary size and topology and communicating over a general vector discrete memoryless channel (DMC). We propose an information-theoretic constructive scheme for obtaining an achievable rate region in such networks. Many well-known capacity-defining achievable rate regions can be derived as special cases of the proposed scheme. A few such examples are the physically degraded and reversely degraded relay channels, the Gaussian multiple-access channel, and the Gaussian broadcast channel. The proposed scheme also leads to inner bounds for the multicast and allcast capacities. Applying the proposed scheme to a specific wireless network of n nodes located in a region of unit area, we show that a transport capacity of /spl Theta/(n) bit-meters per second (bit-meters/s) is feasible in a certain family of networks, as compared to the best possible transport capacity of /spl Theta/(/spl radic/n) bit-meters/s in Gupta et al. (2000), where the receiver capabilities were limited. Even though the improvement is shown for a specific class of networks, a clear implication is that designing and employing more sophisticated multiuser coding schemes can provide sizable gains in at least some large wireless networks.     

A minimum cost heterogeneous sensor network with a lifetime constraint

We consider a heterogeneous sensor network in which nodes are to be deployed over a unit area for the purpose of surveillance. An aircraft visits the area periodically and gathers data about the activity in the area from the sensor nodes. There are two types of nodes that are distributed over the area using two-dimensional homogeneous Poisson point processes; type 0 nodes with intensity (average number per unit area) /spl lambda//sub 0/ and battery energy E/sub 0/; and type 1 nodes with intensity /spl lambda//sub 1/ and battery energy E/sub 1/. Type 0 nodes do the sensing while type 1 nodes act as the cluster heads besides doing the sensing. Nodes use multihopping to communicate with their closest cluster heads. We determine them optimum node intensities (/spl lambda//sub 0/, /spl lambda//sub 1/) and node energies (E/sub 0/, E/sub 1/) that guarantee a lifetime of at least T units, while ensuring connectivity and coverage of the surveillance area with a high probability. We minimize the overall cost of the network under these constraints. Lifetime is defined as the number of successful data gathering trips (or cycles) that are possible until connectivity and/or coverage are lost. Conditions for a sharp cutoff are also taken into account, i.e., we ensure that almost all the nodes run out of energy at about the same time so that there is very little energy waste due to residual energy. We compare the results for random deployment with those of a grid deployment in which nodes are placed deterministically along grid points. We observe that in both cases /spl lambda//sub 1/ scales approximately as /spl radic/(/spl lambda//sub 0/). Our results can be directly extended to take into account unreliable nodes.     

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.     

Algebraic gossip: a network coding approach to optimal multiple rumor mongering

The problem of simultaneously disseminating k messages in a large network of n nodes, in a decentralized and distributed manner, where nodes only have knowledge about their own contents, is studied. In every discrete time-step, each node selects a communication partner randomly, uniformly among all nodes and only one message can be transmitted. The goal is to disseminate rapidly, with high probability, all messages to all nodes. It is shown that a random linear coding (RLC) based protocol disseminates all messages to all nodes in time ck+/spl Oscr/(/spl radic/kln(k)ln(n)), where c<3.46 using pull-based dissemination and c<5.96 using push-based dissemination. Simulations suggest that c<2 might be a tighter bound. Thus, if k/spl Gt/(ln(n))/sup 3/, the time for simultaneous dissemination RLC is asymptotically at most ck, versus the /spl Omega/(klog/sub 2/(n)) time of sequential dissemination. Furthermore, when k/spl Gt/(ln(n))/sup 3/, the dissemination time is order optimal. When k/spl Lt/(ln(n))/sup 2/, RLC reduces dissemination time by a factor of /spl Omega/(/spl radic/k/lnk) over sequential dissemination. The overhead of the RLC protocol is negligible for messages of reasonable size. A store-and-forward mechanism without coding is also considered. It is shown that this approach performs no better than a sequential approach when k=/spl prop/n. Owing to the distributed nature of the system, the proof requires analysis of an appropriate time-varying Bernoulli process.     

Asymptotic distribution of the number of isolated nodes in wireless ad hoc networks with Bernoulli nodes

Nodes in wireless ad hoc networks may become inactive or unavailable due to, for example, internal breakdown or being in the sleeping state. The inactive nodes cannot take part in routing/relaying, and thus may affect the connectivity. A wireless ad hoc network containing inactive nodes is then said to be connected, if each inactive node is adjacent to at least one active node and all active nodes form a connected network. This paper is the first installment of our probabilistic study of the connectivity of wireless ad hoc networks containing inactive nodes. We assume that the wireless ad hoc network consists of n nodes which are distributed independently and uniformly in a unit-area disk, and are active (or available) independently with probability p for some constant 0

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Capacity of Multichannel Wireless Networks Under the Protocol Model

This paper studies the capacity of a n node static wireless network with c channels and m radio interfaces per node under the protocol model of interference. In their seminal work, Gupta and Kumar have determined the capacity of a single channel network (c=1, m=1). Their results are also applicable to multichannel networks provided each node has one interface per channel (m=c) . However, in practice, it is often infeasible to equip each node with one interface per channel. Motivated by this observation, we establish the capacity of general multichannel networks (m les c). Equipping each node with fewer interfaces than channels in general reduces network capacity. However, we show that one important exception is a random network with up to O(logn) channels, where there is no capacity degradation even if each node has only one interface. Our initial analysis assumes that the interfaces are capable of switching channels instantaneously, but we later extend our analysis to account for interface switching delays seen in practice. Furthermore, some multichannel protocols proposed so far rarely require interfaces to switch, and therefore, we briefly study the capacity with fixed interfaces as well.     

Variable-Range Transmission Power Control in Wireless Ad Hoc Networks

In this paper, we investigate the impact of variable-range transmission power control on the physical and network connectivity, on network capacity, and on power savings in wireless multihop networks. First, using previous work by Steele (1988), we show that, for a path attenuation factor a = 2, the average range of links in a planar random network of A m² having n nodes is ~aradicA/n1. We show that this average range is approximately half the range obtained when common-range transmission control is used. Combining this result and previous work by Gupta and Kumar (2000), we derive an expression for the average traffic carrying capacity of variable-range-based multihop networks. For a = 2, we show that this capacity remains constant even when more nodes are added to the network. Second, we derive a model that approximates the signaling overhead of a routing protocol as a function of the transmission range and node mobility for both route discovery and route maintenance. We show that there is an optimum setting for the transmission range, not necessarily the minimum, which maximizes the capacity available to nodes in the presence of node mobility. The results presented in this paper highlight the need to design future MAC and routing protocols for wireless ad hoc and sensor networks based, not on common-range which is prevalent today, but on variable-range power control     

On the capacity of network coding for random networks

We study the maximum flow possible between a single-source and multiple terminals in a weighted random graph (modeling a wired network) and a weighted random geometric graph (modeling an ad-hoc wireless network) using network coding. For the weighted random graph model, we show that the network coding capacity concentrates around the expected number of nearest neighbors of the source and the terminals. Specifically, for a network with a single source, l terminals, and n relay nodes such that the link capacities between any two nodes is independent and identically distributed (i.i.d.) /spl sim/X, the maximum flow between the source and the terminals is approximately nE[X] with high probability. For the weighted random geometric graph model where two nodes are connected if they are within a certain distance of each other we show that with high probability the network coding capacity is greater than or equal to the expected number of nearest neighbors of the node with the least coverage area.     

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.     

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.     

Capacity and delay of hybrid wireless broadband access networks

An optical network is too costly to act as a broadband access network. On the other hand, a pure wireless ad hoc network with n nodes and total bandwidth of W bits per second cannot provide satisfactory broadband services since the pernode throughput diminishes as the number of users goes large. In this paper, we propose a hybrid wireless network, which is an integrated wireless and optical network, as the broadband access network. Specifically, we assume a hybrid wireless network consisting of n randomly distributed normal nodes, and m regularly placed base stations connected via an optical network. A source node transmits to its destination only with the help of normal nodes, i.e., in the ad hoc mode, if the destination can be reached within L (L /spl geq/ 1) hops from the source. Otherwise, the transmission will be carried out in the infrastructure mode, i.e., with the help of base stations. Two transmission modes share the same bandwidth of W bits/sec. We first study the throughput capacity of such a hybrid wireless network, and observe that the throughput capacity greatly depends on the maximum hop count L and the number of base stations m. We show that the throughput capacity of a hybrid wireless network can scale linearly with n only if m = Ω(n), and when we assign all the bandwidth to the infrastructure mode traffics. We then investigate the delay in hybrid wireless networks. We find that the average packet delay can be maintained as low as Θ(1) even when the per-node throughput capacity is Θ(W).     

Optimal Unicast Capacity of Random Geometric Graphs: Impact of Multipacket Transmission and Reception

We establish a tight max-flow min-cut theorem for multi-commodity routing in random geometric graphs. We show that, as the number of nodes in the network n tends to infinity, the maximum concurrent flow (MCF) and the minimum cut-sparsity scale as θ(n²r³(n)/k), for a random choice of k = ω(n) source-destination pairs, where n and r(n) are the number of nodes and the communication range in the network respectively. The MCF equals the interference-free capacity of an ad-hoc network. We exploit this fact to develop novel graph theoretic techniques that can be used to deduce tight order bounds on the capacity of ad-hoc networks. We generalize all existing capacity results reported to date by showing that the per-commodity capacity of the network scales as θ(1/r(n)k) for the single-packet reception model suggested by Gupta and Kumar, and as θ(nr(n)/k) for the multiple-packet reception model suggested by others. More importantly, we show that, if the nodes in the network are capable of (perfect) multiple-packet transmission (MPT) and reception (MPR), then it is feasible to achieve the optimal scaling of θ(n²r³(n)/k), despite the presence of interference. In comparison to the Gupta-Kumar model, the realization of MPT and MPR may require the deployment of a large number of antennas at each node or bandwidth expansion. Nevertheless, in stark contrast to the existing literature, our analysis presents the possibility of actually increasing the capacity of ad-hoc networks with n even while the communication range tends to zero!     

Throughput-Optimal Configuration of Fixed Wireless Networks

In this paper, we address the following two questions concerning the capacity and configuration of fixed wireless networks: (i) given a set of wireless nodes with arbitrary but fixed locations, and a set of data flows, what is the max–min achievable throughput? and (ii) how should the network be configured to achieve the optimum? We consider these questions from a networking standpoint assuming point-to-point links, and employ a rigorous physical layer model to model conflict relationships between them. Since we seek capacity results, we assume that the network is operated using an appropriate schedule of conflict-free link activations. We develop and investigate a novel optimization framework to determine the optimal throughput and configuration, i.e., flow routes, link activation schedules and physical layer parameters. Determining the optimal throughput is a computationally hard problem, in general. However, using a smart enumerative technique we obtain numerical results for several different scenarios of interest. We obtain several important insights into the structure of the optimal routes, schedules and physical layer parameters. Besides determining the achievable throughput, we believe that our optimization-based framework can also be used as a tool, for configuring scheduled wireless networks, such as those based on IEEE 802.16.      

Overcoming untuned radios in wireless networks with network coding

The drive toward the implementation and massive deployment of wireless sensor networks calls for ultralow-cost and low-power nodes. While the digital subsystems of the nodes are still following Moore's Law, there is no such trend regarding the performance of analog components. This work proposes a fully integrated architecture of both digital and analog components (including local oscillator) that offers significant reduction in cost, size, and overall power consumption of the node. Even though such a radical architecture cannot offer the reliable tuning of standard designs, it is shown that by using random network coding, a dense network of such nodes can achieve throughput linear in the number of channels available for communication. Moreover, the ratio of the achievable throughput of the untuned network to the throughput of a tuned network with perfect coordination is shown to be close to 1/e. This work uses network coding to leverage the fact that throughput equal to the max-flow in a graph is achievable even if the topology is not know a priori. However, the challenge here is finding the max-flow of the random graph corresponding to the network.     

Network information flow with correlated sources

Consider the following network communication setup, originating in a sensor networking application we refer to as the "sensor reachback" problem. We have a directed graph G=(V,E), where V={v/sub 0/v/sub 1/...v/sub n/} and E/spl sube/V/spl times/V. If (v/sub i/,v/sub j/)/spl isin/E, then node i can send messages to node j over a discrete memoryless channel (DMC) (X/sub ij/,p/sub ij/(y|x),Y/sub ij/), of capacity C/sub ij/. The channels are independent. Each node v/sub i/ gets to observe a source of information U/sub i/(i=0...M), with joint distribution p(U/sub 0/U/sub 1/...U/sub M/). Our goal is to solve an incast problem in G: nodes exchange messages with their neighbors, and after a finite number of communication rounds, one of the M+1 nodes (v/sub 0/ by convention) must have received enough information to reproduce the entire field of observations (U/sub 0/U/sub 1/...U/sub M/), with arbitrarily small probability of error. In this paper, we prove that such perfect reconstruction is possible if and only if H(U/sub s/ | U/sub S(c)/) < /spl Sigma//sub i/spl isin/S,j/spl isin/S(c)/ for all S/spl sube/{0...M},S/spl ne/O,0/spl isin/S(c). Our main finding is that in this setup, a general source/channel separation theorem holds, and that Shannon information behaves as a classical network flow, identical in nature to the flow of water in pipes. At first glance, it might seem surprising that separation holds in a fairly general network situation like the one we study. A closer look, however, reveals that the reason for this is that our model allows only for independent point-to-point channels between pairs of nodes, and not multiple-access and/or broadcast channels, for which separation is well known not to hold. This "information as flow" view provides an algorithmic interpretation for our results, among which perhaps the most important one is the optimality of implementing codes using a layered protocol stack.     

Capacity and delay tradeoffs for ad hoc mobile networks

We consider the throughput/delay tradeoffs for scheduling data transmissions in a mobile ad hoc network. To reduce delays in the network, each user sends redundant packets along multiple paths to the destination. Assuming the network has a cell partitioned structure and users move according to a simplified independent and identically distributed (i.i.d.) mobility model, we compute the exact network capacity and the exact end-to-end queueing delay when no redundancy is used. The capacity-achieving algorithm is a modified version of the Grossglauser-Tse two-hop relay algorithm and provides O(N) delay (where N is the number of users). We then show that redundancy cannot increase capacity, but can significantly improve delay. The following necessary tradeoff is established: delay/rate/spl ges/O(N). Two protocols that use redundancy and operate near the boundary of this curve are developed, with delays of O(/spl radic/N) and O(log(N)), respectively. Networks with non-i.i.d. mobility are also considered and shown through simulation to closely match the performance of i.i.d. systems in the O(/spl radic/N) delay regime.