2.6 Conclusions
Following [9], we constructed a scheme that achieves the throughput capacity of a large ad hoc wireless network with high probability as the number of nodes increases. The proofs are made simple and more intuitive (we do not resort to the Vapnik-Chervonenkis Theorem for example), and we were able to study the asymptotic behavior of ad hoc wireless networks under a local traffic pattern as well as hybrid wireless networks.
For a local traffic pattern we showed the impact of the mean S-D distance on the throughput. Moreover there is a limit in the throughput improve-ment as the mean path length becomes smaller than the cell side length.
For a local traffic pattern for which Θ
L(n) c(n)
≤ Θ(1), we obtain a per-node throughput larger than log1n, whereas it is larger than 1
L(n)√
logn for ΘL(n)
c(n)
> Θ(1). It seems that the way to increase the throughput ca-pacity is by relaxing the connectivity condition, which can be achieved by decreasingc(n), the cell side length.
In this chapter, we have demonstrated also the benefits of using a hy-brid wireless network in terms of per-node capacity. The base-stations are regularly placed within the network area, and the analysis is based on the subdivision of the network into f(n) clusters, where f(n) is the number of base-stations in the network. Moreover, the infrastructure network is as-sumed to have abundant bandwidth and resources. Inside each cluster, the communications are done in a pure ad hoc mode, whereas if the source and the destination do not belong to the same cluster, packets first reach the base-station in a multi-hop fashion and tunnel through the infrastructure to the base-station nearest to the destination. We obtain a per node through-put larger than q
f(n)
nlogn(e.g. for f(n) = (logn)2, we obtain
qlogn
n ). The gain in performance is mainly due to the reduction in the mean number of hops from the source to the destination.
Finally, an initial step on the study of event-driven traffic patterns and their impact on the throughput expressions is conducted. The results are promising, and further developments and extensions of the model considered are left for future research.
Appendix 2.A Bound on the Interference
Let us bound the interference. Consider a particular cellc∗. If one node from this cell is transmitting, all other simultaneous transmissions may occur in cells belonging to the same set of cells that are at a vertical and horizon-tal distance of exactly some multiple of K. Actually, the interfering cells are placed along the perimeter of concentric squares, whose center isc∗, and each square contains (2lK+1)2, l= 1,2..., S(n) cells and 2lK, l= 1,2..., S(n) interfering cells as depicted in Fig.2.3, whereS(n) is the number of such con-centric squares. For example, take the particular case whereK= 4, the first concentric square contains 8 interfering cells, whereas the second concentric square contains 16 interfering cells. Each node in the intended cellc∗ trans-mits information packets to nodes in the eight neighboring cells. Then, the distance between these nodes (the possible receivers in the eight adjacent cells) and the interfering ones is at least l(K−2)c(n), l = 1,2..., S(n). As we are considering a lower bound, we take the worst-case and neglect edge effects. Then, the number of concentric squares (irrespective of the position of the intended cell, since the worst case is when the intended cell is at one corner of the area) is at most S(n) ≤
&q n
logn
K
'
. We proceed in upper bounding the interference at the receiver:
I = X
2.A. Bound on the Interference 45 wherecis a positive number. The thermal noiseN0 is negligible asn→ ∞, and by combining Eq.(2.7) with Eq.(2.20), the SINR(n) is lower bounded by SINRmin(n) which is a constant and as Cij(n) = Wlog2(1 + SINRmin(n)), we obtain the result Eq.(2.6).
Chapter 3
Connectivity Graph and Conditions for Constant Throughput in Wireless Ad Hoc Networks
3.1 Introduction
In the area of wireless networks that operate in ad hoc mode, a question that has recently attracted significant research interest and activity, is the derivation of scaling laws for the capacity. This activity was sparked by the seminal work of [9], who proposed to model wireless ad hoc networks as ran-dom geometric graphs and examined the scaling of the long-term averaged throughput with the number of nodes n. Their main result is that, given Θ(n) randomly selected destination pairs, the throughput per source-destination pair scales at least as Θ
√ 1 nlogn
and at most as Θ
√1 n
for random networks, where nodes are placed uniformly at random on a given network area. The achievability is proved in the asymptotic sense by design-ing proper routdesign-ing and transmission mechanisms. An insight from [9] is that, to maximize the throughput, we need to minimize the transmission power (range) of each node, while still keeping the network connected. That is, we need to reduce the interference region of each transmission and schedule as many non-interfering concurrent transmissions as possible.
This work was extended by a number of works that established scaling 47
laws assuming different network models. For example, in the model of [9], the geographical area is fixed and the density of nodes is increasing with the number of nodes (dense network). In [35], techniques from percolation theory are used to show that in dense random networks a Θ(1/√
n) through-put is achievable. A different network model assumes that the number of nodes is increasing with the area of the network leading to a fixed density of nodes per area. For this case, in [37] and in [25], information theoretic upper bounds on the rate of communication are derived, as a function of the value of power loss exponents. Other follow-up works include [38, 39, 40, 41].
A result common to the different models is that, for fixed nodes networks, the throughput per source-destination pair vanishes as the number of nodes grows.
On the contrary, if nodes are allowed to move, a constant Θ(1) through-put can be achieved per source-destination pair even if the number of nodes grows to infinity [26]. This result assumes a 2-dimensional mobility pattern, where the trajectory of each node is an independent, stationary and ergodic random process with uniform distribution on the unit disk. That is, the mobility pattern is homogeneous with respect to each node, and the sample path of each node covers all the space over time. In the analysis, trans-mission is restricted to the closest nodes, and at each given instance, each link between any two nodes is activated with probability Θ(1/n). Then, a two-phase scheduling policy is employed. In the first phase, source nodes transmit the packets to the closest receiver (which can be a relay or a desti-nation node) and in the second phase transmitters forward the packets that have as destination their closest receiver. Thus, for any source destination pair, (n−2) relay nodes receive and transmit packets at rate Θ(1/n) while source nodes also transmit directly to the destination at a rate Θ(1/n). Note that the flow between each source-destination pair sums up to a fixed rate Θ(1). It is clear that network capacity can be drastically improved when mobility is effectively exploited.
A natural question to ask is whether this good performance is specific to the particular generous mobility pattern, or whether it can be achieved un-der more restricted mobility conditions. The work in [42] made progress in answering this question, by demonstrating that the same order of through-put can still be achieved under restricted 1-dimensional mobility. In their mobility pattern, each node is restricted to move on a randomly and
indepen-3.1. Introduction 49 dently chosen great circle on the unit sphere. However, the general question still remains, which is, under what conditions on the mobility patterns of the users a throughput of Θ(1) is achievable.
Recently, another model of restricted mobility has been examined in [51], where nodes are confined to overlapping neighborhoods, and the throughput scaling as a function of the neighborhood size is analyzed. In [54], the impact of the mobility pattern on the relay throughput (i.e. the maximum rate at which a node can relay data from the source to the destination) is studied. It is shown that the relay throughput depends on the node mobility pattern only via its stationary node position distribution and that a node mobility pattern that results in a uniform steady-state distribution for all nodes achieves the lowest relay throughput.
Independently, in computer science, the problem of multi-commodity flow has received significant attention. In this problem, we are given a graph with edges of a fixed capacity, a set of source-destination pairs, each with its own demand, and we are asked to maximize the amount of information flow that can be simultaneously routed for all source-destination pairs. The emphasis of the work in the literature is in deriving min-cut bounds on the achievable rates and characterizing under what conditions these bounds are tight, see for example [44, 45, 46]. Recently, the area of network coding has emerged, where it is demonstrated that by allowing flows to mix, we may achieve significant throughput benefits for the multi-commodity problem over directed graphs [47]. At this time, the prevailing conjecture is that network coding does not offer benefits for the case of the undirected multi-commodity problem [48]. For our results we will not use network coding techniques, i.e., we will assume that nodes can only forward and not combine their incoming information flows.
In this chapter, our interest is in mobile ad hoc and wireless sensor net-works. Our main contribution is a method that allows to check whether, for a given mobility pattern, a constant Θ(1) throughput per source-destination pair is possible. Intuitively, the reason we get a decreased throughput in fixed nodes networks, is that the average number of hops that a packet needs to traverse from a source to a destination scales with n (the number of nodes in the network). On the contrary, in [26, 42], mobility enables a routing strategy where the number of hops is limited to at most two. Our work is motivated by the observation that in a complete graph between each
source-destination pair there exists a set of max-flow paths whose length is upper bounded by two. This set consists ofn−2 paths of length two and one path of length one. A graph is complete if any two nodes are connected with an edge, which intuitively seems to correspond to the fact that in the uniform mobility pattern any two nodes can become neighbors and success-fully transmit to each other. In other words, the routing approach in [26, 42]
seems to exactly correspond to the routing approach we would use to solve the multi-commodity flow problem on a complete graph of equal capacity edges.
To make this loose connection precise, we first decompose the communi-cation problem into a “transmission policy” and a “scheduling policy”. As we argue in Section 3.3, this decomposition does not affect whether constant throughput is achievable asymptotically. We then introduce the connectiv-ity graph, that does not represent the actual physical network, but rather the available communication resources, for a given transmission policy. The connectivity graph offers an abstraction of the communication capabilities of the ad hoc network: we can study the long-term averaged throughput be-tween source-destination pairs in the actual ad hoc network, by examining information flows in the connectivity graph. Thus, by mapping the ad hoc network problem into a graph problem, we establish a bridge between the multi-commodity flow and the ad hoc network literature, that can be used in both directions.
The focus of this chapter is in using the properties of the connectivity graph to develop a set of necessary and sufficient conditions under which constant Θ(1) throughput is possible. We illustrate how these conditions apply to a number of different topologies, including the topologies in [9, 26, 42]. We then try to understand what structural properties these conditions imply for the connectivity graph and how they translate into properties for the underlying mobility pattern. Interestingly, we provide an example where constant throughput may be possible to achieve by mobile nodes that have a restricted number of neighbors. That is, each node may successfully communicate with at mostn1/t other nodes, for a finitet.
Although we focus on scaling results when the number of nodes of the network increases, the same approach can be used to analyze throughput and design routing over finite networks. In fact, independent of our work [49], the authors in [50] proposed the use of a structure similar to our “connectivity
3.2. Problem Statement 51