Throughput Scaling Laws for Wireless Networks with Fading Channels

📝 Abstract

A network of n communication links, operating over a shared wireless channel, is considered. Fading is assumed to be the dominant factor affecting the strength of the channels between transmitter and receiver terminals. It is assumed that each link can be active and transmit with a constant power P or remain silent. The objective is to maximize the throughput over the selection of active links. By deriving an upper bound and a lower bound, it is shown that in the case of Rayleigh fading (i) the maximum throughput scales like $\log n$ (ii) the maximum throughput is achievable in a distributed fashion. The upper bound is obtained using probabilistic methods, where the key point is to upper bound the throughput of any random set of active links by a chi-squared random variable. To obtain the lower bound, a decentralized link activation strategy is proposed and analyzed.

💡 Analysis

A network of n communication links, operating over a shared wireless channel, is considered. Fading is assumed to be the dominant factor affecting the strength of the channels between transmitter and receiver terminals. It is assumed that each link can be active and transmit with a constant power P or remain silent. The objective is to maximize the throughput over the selection of active links. By deriving an upper bound and a lower bound, it is shown that in the case of Rayleigh fading (i) the maximum throughput scales like $\log n$ (ii) the maximum throughput is achievable in a distributed fashion. The upper bound is obtained using probabilistic methods, where the key point is to upper bound the throughput of any random set of active links by a chi-squared random variable. To obtain the lower bound, a decentralized link activation strategy is proposed and analyzed.

📄 Content

their simplest forms, have been among the most difficult problems facing the network information theory community for many years.

Followed by the pioneering work of Gupta and Kumar [1], considerable attention has been paid to investigate how the throughput of wireless networks scales with n, the number of nodes, when n is large. This has been done assuming different network topologies, traffic patterns, protocol schemes, and channel models [1]- [10]. Most of these works consider a channel model in which the signal power decays according to a distance-based attenuation law [1]- [7]. However, in a wireless environment the presence of obstacles and scatterers adds some randomness to the received signal. This random behavior of the channel, known as fading, can drastically change the scaling laws of a network in both multihop [8]- [10] and single-hop scenarios [11,Chapter 8], [12], [13].

In this paper, we follow the model of [10], [11], where fading is assumed to be the dominant factor affecting the strength of the channels between nodes. Despite the randomness of the channel, we are only interested in events that occur asymptotically almost surely, i.e., with probability tending to one as n → ∞. Such a deterministic approach to random wireless networks has been also adopted in [5], [7], where the nodes’ locations are random.

We consider a single-hop scenario, i.e., a network structure in which the transmitters send data to their corresponding receivers directly and without utilizing other nodes as routers. It is assumed that each link can be active and transmit with a constant power P or remain silent. The objective is to maximize the throughput over all sets of active links. We propose a threshold-based link activation strategy in which each link is active if and only if its channel gain is above some predetermined threshold. The decision on being active can be made at the receivers, where their own channel gains are estimated and a single-bit command data is fed back to the transmitters.

Hence, there is no need for the exchange of information between links. Consequently, this method can be implemented in a decentralized fashion. We analyze this method for a general fading model and show how to obtain the value of the activation threshold to maximize the throughput.

As an example, we derive a closed form expression for the achievable throughput in the Rayleigh fading environment.

Using probabilistic methods, we derive an upper bound on the achievable throughput when the channel is Rayleigh fading. Interestingly, this upper bound scales the same as the lower bound achieved by the proposed strategy. This proves the asymptotic optimality of the proposed technique among all link activation strategies.

In addition to the channel modeling, [10] is a relevant work in the sense that transmissions occur with the same power and the objective is to maximize the throughput. However, they allow multihop communication in their scheme. Their proposed scheme requires a central unit which is aware of all channel conditions and decides on active source-destination pairs and the paths between them. Despite this complexity, the achievable throughput of their method in the popular model of Rayleigh fading is by a factor of 4 less than the value obtained in this work for a more restricted configuration, i.e., single-hop networks with decentralized management 1 .

The rest of the paper is organized as follows: In Section II, the network model and problem formulation are presented. By proposing a decentralized link activation strategy, a lower bound on the network throughput is derived in Section III. In Section IV, we prove the optimality of the proposed decentralized method in a Rayleigh fading environment. Finally, we conclude the paper in Section V.

Notation: N n represents the set of natural numbers less than or equal to n; log(•) is the natural logarithm function; P(A) denotes the probability of event A; E(x) represents the expected value of the random variable x; ≈ means approximate equality; for any functions f (n) and

We consider a wireless communication network with n pairs of transmitters and receivers.

These n communication links are indexed by the elements of N n . Each transmitter aims to send data to its corresponding receiver in a single-hop fashion. The transmit power of link i is denoted by p i . It is assumed that the links follow an on-off paradigm, i.e., p i ∈ {0, P }, where P is a constant. Hence, any power allocation scheme translates to a link activation strategy (LAS). Any LAS yields a set of active links A, which describes the transmission powers as

The channel between transmitter j and receiver i is characterized by the coefficient g ji . This means the received power from transmitter j at the receiver i equals g ji p j . We assume that the channel coefficients are independent identically distributed (i.i.d.) random variables drawn from a pdf f (x) with mean µ and variance σ 2 . The channel betw

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