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# Transmission capacity for dual-hop relaying in wireless ad hoc networks

- Jaeyoung Lee
^{1}, - Hyundong Shin
^{2}, - Joon Tae Kim
^{3}and - Jun Heo
^{1}Email author

**2012**:58

https://doi.org/10.1186/1687-1499-2012-58

© Lee et al; licensee Springer. 2012

**Received:**1 May 2011**Accepted:**22 February 2012**Published:**22 February 2012

## Abstract

To account for *randomly* distributed nodes in a wireless ad hoc network, the transmission capacity is defined as the number of successful transmissions taking place in the network per unit area under an outage constraint. In this paper, we analyze the transmission capacity for dual-hop relaying in a wireless ad hoc network in the presence of *both* cochannel interference and thermal noise, where interferers are spatially distributed following a Poisson distribution. Specifically, we first present the exact outage probability for amplify-and-forward and decode-and-forward protocols in a Poisson field of interferers without neglecting noise at all nodes. We then derive the transmission capacity of such networks, which determines the maximum allowable density of transmitting nodes for each relay strategy at a specified outage probability and data rate. Numerical results demonstrate that the dual-hop relaying is still beneficial in terms of the transmission capacity in wireless ad hoc Poisson networks.

## Keywords

- amplify-and-forward (AF)
- decode-and-forward (DF)
- dual-hop relay
- interference
- outage probability
- Poisson network
- transmission capacity

## 1 Introduction

Cooperative relay communication has recently drawn considerable attention to increase network coverage and reliability [1, 2]. For a decode-and-forward (DF) strategy, a relay node decodes the received signal and transmits it after re-encoding, whereas the relay simply retransmits a scaled version of the received signal in amplify-and-forward (AF) mode [3, 4]. Most of the previous work has focused on noise-limited fading environments, such as Rayleigh fading with additive white Gaussian noise (AWGN). While such work made great strides toward understanding the potential of relay networks, it dealt mainly with ideal configurations with no interference. However, network interference is inevitable in practical wireless networks due to spectral reuse. Motivated by this, relay networks have been recently studied in the presence of cochannel interference [5–7]. In Krikidis et al. [5], the asymptotic performance was analyzed under the interference scenario only at relays. The fixed-gain AF relaying in an interference network was investigated in Zhong et al. [6], where interference at the relay and thermal noise at the destination were ignored for analytical tractability. The exact outage probability for DF relaying was further derived in Si et al. [7] accounting for multiple interferers and noise at both the relay and the destination. In all these work, the locations of network nodes are *deterministic*--without spatial randomness.

To treat the capacity of a decentralized ad hoc wireless network, the transport capacity, defined as the product of the end-to-end sum throughput and distance, has been introduced due to the difficulty in determining the capacity region of a large ad hoc network [8]. More recently, using the stochastic geometry framework, the transmission capacity has been proposed as the maximum density of active transmitting nodes per unit area to satisfy an outage constraint at a given data rate when interferers are randomly scattered and uncoordinated [9]. For a variety of scenarios, this notion of transmission capacity has been used successfully to characterize the physical layer on the ad hoc network [10–14]. A new metric akin to the transmission capacity, called the random access transport capacity, has been also developed for the end-to-end throughput in multi-hop transmission over some distance [15].

In this paper, we consider dual-hop relaying with DF and AF protocols in a wireless ad hoc network in the presence of both interference and noise. Each interfering node in the network independently transmits data and is randomly distributed in a Poisson law over a plane. The motivation behind imposing Poisson interference and noise is threefold: (i) many previous works neglected either interference or noise in analysis of a relay network; (ii) previous work on relay networks with interference considered only a limited number of interferers (iii) randomly distributed nodes(i.e., interferers) allowing a Poisson distribution are suitable for realistic communication model. To the best of our knowledge, relay network to qualify interference using stochastic geometry has not been addressed before. Hence, we analyze exact outage probability of a dual-hop relay with both noise and interference in a wireless ad hoc network. Unlike previous work in a relay network, when the node locations are distributed as a Poisson point process (PPP), we need to analyze relay networks using a metric for decentralized wireless network, termed transmission capacity. Furthermore, as transmission capacity has considered single-hop transmission without noise, we focus on the transmission capacity of dual-hop relay with noise and specific relaying protocols. It is worth of finding maximum successful transmitting nodes per unit area to satisfy outage probability and data rate from transmission capacity of dual-hop relay. Although both noise and interference with Poisson distribution are considered, it is noted that dual-hop relay is still beneficial in terms of transmission capacity.

The remainder of the paper is organized as follows. Section 2 presents the system model based on location of nodes, channel models, and distribution of interference. We derive outage probability analysis of DF and AF strategies in Section 3. Based on the outage probability of dual-hop relay, transmission capacity is derived in Section 4. Section 5 compares simulation results with analytical results. Finally, Section 6 concludes the paper.

## 2 System model

We consider a wireless ad hoc network consisting of a source, a relay, and a destination with no direct source-to-destination link. All nodes have a single antenna operating in a common frequency band and are in half-duplex mode.

### 2.1 Node locations

The sets of nodes in a relay network are denoted as $\mathcal{S}:=\left\{S{x}_{n},n\in \mathbb{N}\right\}$ for sources, $\mathcal{R}:=\left\{R{x}_{n},n\in \mathbb{N}\right\}$ for relays, and $\mathcal{D}:=\left\{D{x}_{n},n\in \mathbb{N}\right\}$ for destinations. Now, allow the set of nodes to be divided into two different pairs for each hop transmission: (*Sx*_{
n
}, *Rx*_{
n
}) and (*Rx*_{
n
}, *Dx*_{
n
}), where the distance between two nodes in a pair is *d*_{
SR
}and *d*_{
RD
}, respectively. In a dual-hop relay network, sources and relays are also interferers in each hop. In particular, all nodes independently transmit and distribute their locations following a Poisson distribution. Let the location of *Sx*_{
n
}be *S*_{
n
}, *Rx*_{
n
}be *R*_{
n
}, and *Dx*_{
n
}be *D*_{
n
}. The source set ${\Phi}_{\mathcal{S}}:=\left\{{S}_{n}\right\}$ generating interference at the first time slot is modeled as a homogeneous PPP on the plane of intensity *λ*_{1}. Since *R*_{
n
}is randomly located at a fixed distance *d*_{
SR
}from *S*_{
n
}, the set ${\Upsilon}_{\mathcal{R}}:=\left\{{R}_{n}\right\}$ also follows a homogeneous PPP on a two-dimensional plane with intensity *λ*_{2}. As destination node *D*_{
n
}is placed at *d*_{
RD
}away from the relay node *R*_{
n
}, ${\Psi}_{\mathcal{D}}:=\left\{{D}_{n}\right\}$ is also a homogeneous PPP with intensity *λ*_{3}.

In this paper, we consider reference source *Sx*_{0}, reference relay *Rx*_{0}, and reference destination *Dx*_{0} in a decentralized wireless network that transmits desirable data in the *Φ*_{
S
}, *Υ*_{
R
}, and *Ψ*_{
D
}. According to each hop transmission, reference nodes can be bounded as (*Sx*_{0}, *Rx*_{0}) and (*Rx*_{0}, *Dx*_{0}). For the (*Sx*_{0}, *Rx*_{0}) pair in the first hop, the relay node *Rx*_{0} is located at the origin and source node *Sx*_{0} is placed at *d*_{
SR
}meters away from the relay node *Rx*_{0}. For the (*Rx*_{0}, *Dx*_{0}) pair in the second hop, we place the relay node *Rx*_{0}*d*_{
RD
}meters away from the destination node *Dx*_{0}, where *Dx*_{0} is located at the origin.

### 2.2 Channel model

*x*

_{1}and

*x*

_{2}can be rewritten as

where *F*_{12} captures the small-scale fading which obeys a Rayleigh fading model and *l*(∥*x*_{1} -*x*_{2}∥) = ∥*x*_{1}-*x*_{2}∥^{-α}characterizes the effect of large-scale path loss following the power law with path loss exponent *α*. We assume that channel responses for all nodes are independent and quasi-static.

### 2.3 Distribution of aggregate interference

where h_{
i
}is a instantaneous channel response of interferer *i, Φ* is a homogenous PPP with intensity *λ*, and *P*_{
i
}is interference power.

*MGF*) of total interference power using method in Haenggi and Ganti [16], denoted by ${\varphi}_{I}\left(s\right)\triangleq \bullet \left\{{e}^{-s\text{I}}\right\}$, can be written as

where *λ* is intensity of interferers, *c*_{
d
}is the volume of *d*-dimensional unit ball, *r* is the radius of finite area located interferers, and *δ* ≜ *d*/*α*. Also, *Γ*(*x*) is the Gamma function ${\int}_{0}^{\infty}{t}^{x-1}{e}^{-t}dt$ and *γ*(*s,x*) is the lower incomplete gamma function ${\int}_{0}^{x}{t}^{s-1}{e}^{-t}dt$.

## 3 Outage performance analysis

### 3.1 Decode-and-forward relaying

*y*

_{ R }at a reference relay node

*Rx*

_{0}can be expressed as

_{0}is the transmitted symbol with

*E*[|s

_{0}|

^{2}] =

*P*

_{ S }, h

_{ SR }is the channel between the reference source and the reference relay, and n

_{ R }is an additive white Gaussian noise with an average power of ${\sigma}_{R}^{2}$. In addition, the relay node

*Rx*

_{0}of the reference pair is affected by its interferes s

_{ i }located in a homogenous PPP

*Φ*

_{ S }. Average transmit power of an interferer is

*E*[|s

_{ i }|

^{2}] =

*P*

_{ iR }, and ${\left\{{\text{g}}_{i}\right\}}_{i\in {\Phi}_{S}}$ are the channels from the interferer

*i*to the reference relay

*Rx*

_{0}. Thus, the received signal to interference-plus-noise ratio (

*SINR*) at the relay can be written as

*Dx*

_{0}, a transformed copy of the received signal at the relay, $\mathcal{T}\left({y}_{R}\right)$, is then transmitted to the destination node, which suffers from the other relay's transmission ${\left\{{\text{w}}_{j}\right\}}_{j\in {\Upsilon}_{R}}$ located in a homogeneous PPP

*ϒ*

_{ R }, each average transmitting power of an interferer

*j*is

*E*[|w

_{ j }|

^{2}] =

*P*

_{ jD }. Thus, the received signal at the destination

*Dx*

_{0}is given by

_{ RD }is the channel between the reference relay and the reference destination, n

_{ D }is an additive white Gaussian noise with an average power of ${\sigma}_{D}^{2}$, and {l

_{ j }}

_{j∈Φ}are the channels from the interferer

*j*to the reference destination. In DF, the relay fully decodes the message

*s*

_{0}based on the received signal,

*y*

_{ R }, and then transmits the re-encoded message $\mathcal{T}\left({y}_{R}\right)={\text{s}}_{r}$ with a power of

*P*

_{ R }to the destination terminal. Thus, the received

*SINR*at the destination can be written as

*CDF*) and probability density function (

*P*

_{ S }|h

_{ SR }|

^{2}and $\text{Q}~{\sum}_{{\Phi}_{S}}{\left|{\text{g}}_{i}\right|}^{2}{{P}_{i}}_{R}$. Then, the

*CDF*and

*d*

_{ SR }is distance between reference source and reference relay and

*α*is path loss exponent. As the main physical range of interest for path loss exponent in wireless networks is 2 <

*α*< 5, we consider path loss exponent

*α*= 4 for a relatively lossy environment in an urban area. From (3), the

*MGF*of total interference power $\text{Q}~{\sum}_{{\Phi}_{S}}{\left|{\text{g}}_{i}\right|}^{2}{{P}_{i}}_{R}$ with two-dimensional plane($\delta =\frac{1}{2}$ case and

*c*

_{ d }=

*π*) can be written as

*CDF*and

*M*= 0 can be expressed as

where *λ*_{1} is intensity of source nodes and *r*_{1} denotes the radius of interference area.

**Theorem 1**(

*CDF*and

*The CDF and PDF of received SINR at the relay*$\text{X}~{\gamma}_{R}^{\text{DF}}$

*are given by*

*Proof See Appendix A*

**Corollary 1**

*Using the same method, the CDF and PDF of the received SINR at the destination*$\text{Y}~{\gamma}_{D}^{\text{DF}}$

*can be written as*

*where d*_{
RD
}*is the distance between relay and destination and r*_{2}*is radius of finite area with intensity of interferers λ*_{2}.

*SINR*of dual-hop DF relaying

*γ*

_{eq}can be written as

where *R* is end-to-end spectral efficiency.

where *β* is target SINR *β* = 2^{2R}- 1

### 3.2 Amplify-and-forward relaying

_{ RD }is the channel between the reference relay and the reference destination, n

_{ D }is an additive white Gaussian noise with an average power of ${\sigma}_{D}^{2}$ and {l

_{ j }}

_{j∈ϒ}are the channels from the interferer

*j*to the reference destination. The amplification factor

*G*of AF relaying with interference [5] can be expressed as

*y*

_{ D }can be rewritten as

*SINR*at the destination of AF relaying can be given by

*SINR*(21) to make the Macdonald random variable form in Barua et al. [4] as

where $a={\sigma}_{R}^{2}$ and $b={\sigma}_{D}^{2}$.

*SINR*of AF relaying can be written as

The random variable X and Y were defined in DF relaying, and the *PDF* and *CDF* of them are the same as in Theorem 1 and Corollary 1.

**Theorem 2**(Outage probability of AF relaying)

*From the*PDF

*and*CDF

*of*X

*and Y, we can compute outage probability of AF relaying as*

*where*

*Proof See Appendix B*

## 4 Transmission capacity analysis

In this section, we compute transmission capacity *C*(ϵ) based on outage probability of a dual-hop relay. Transmission capacity was defined as the maximum density of the transmitting node to satisfy outage probability and data rate on medium access control (MAC) layer performance. That is, it is efficient to find the maximum available transmitting source and relay nodes to satisfy a given outage probability and data rate (i.e., Quality of Service (QoS)) from the transmission capacity of a dual-hop relay. As transmission capacity in Weber et al. [13] considers single-hop transmission without noise, we focus on the transmission capacity of a dual-hop relay with noise and specific relaying protocols. We assume that the sets of *Φ*_{
S
}, *ϒ*_{
R
}, and *Ψ*_{
D
}are homogeneous PPPs on the two-dimensional plane with the same intensity λ due to the random translation invariance property of PPP [17].

### 4.1 Direct transmission

*C*(ϵ) [13, 14] is defined as

where ϵ is outage probability for target SINR *β* = 2^{
R
}- 1 with spectral efficiency *R, q*^{-1}(ϵ) is the spatial intensity of attempted transmission associated with outage probability ϵ and it is always greater than 0. The transmission capacity *C*(ϵ) is thinned by the probability of success 1 - ϵ.

*SINR*at the destination is represented as

_{ SD }is channel between source and destination, ${\left\{{\text{f}}_{k}\right\}}_{k\in \Phi s}$ are the channels from the interferer

*k*to the reference destination, and

*P*

_{ kD }is each average transmitting power of interferer

*k*to destination. Similar to Theorem 1, the outage probability of direct transmission between source and destination can be given by

where *r* is radius of finite area with the intensity of each interferer λ, and *d*_{
SD
}is the distance between source and destination.

*q*-

^{1}(ϵ) can be a negative number due to thermal noise. Therefore, as density of transmitting nodes cannot be a negative number, transmission capacity with noise can be redefined as follows

*q*

^{-1}(ϵ) >0 can be written as

### 4.2 Dual-hop relaying

Since we are concerned with the outage probability of the relay network after two-hop transmission, the transmission capacity of dual-hop relay is identical to that of single hop except for the target SINR *β.*

*SINR*with Macdonald r.v. for AF strategy (24) can be written as

*SINR*of DF relaying (14). As we focus on the information theoretical capacity, the fact that DF can be the upper bound of AF is established in it. Using this relation, we note that the transmission capacity of DF is the upper bound of AF relaying as follows

*SINR β*= 2

^{2R}-1 with spectral efficiency

*R*. In this paper, we consider a single transmission capacity of dual-hop relay

*C*

_{ R }(ϵ) regardless of its protocols. From outage probability (16), we can obtain transmission capacity of dual-hop relay

*C*

_{ R }(ϵ) as

where ϵ is outage probability of dual-hop relaying for target *SINR β*.

## 5 Numerical and simulation results

In this section, we present some numerical examples of the outage probability and transmission capacity for a dual-hop relaying with both interference and noise in a Poisson network. We consider Rayleigh fading channel and path loss exponent *α* = 4 to illustrate our analytic and simulated results.

### 5.1 Outage probability

*P*

_{ S }=

*P*

_{ R }for DF and AF strategies considering both noise and interference in a Poisson network, respectively. Both figures show that outage probability with end-to-end spectral efficiency

*R*= 1 bps/Hz (target SINR

*β*= 3) and noise variance ${\sigma}_{R}^{2}={\sigma}_{D}^{2}=1$ for various system parameters. Figures 2a and 3a depict the outage probability as a function of transmission power with different intensity of interferers for DF and AF strategies, respectively. Both figures reveal significant decrease of outage performance and have an error floor with increasing intensity of interferes

*λ*. Figure 2b shows that outage probability of DF relaying with different distance pair (

*d*

_{ SR },

*d*

_{ RD }) = (1.5,1.5), (1.5, 2) and (2, 2). It is noted that the outage performance of dual-hop relaying is degraded by increasing their each hop distance

*d*

_{ SR }and

*d*

_{ RD }, respectively. For AF relaying, increasing interference power is greatly influenced by outage probability performance in Figure 3b. In addition, we can see that the analysis agrees exactly with the simulation results from Figures 2 and 3.

### 5.2 Transmission capacity

*d*

_{ SR }is the same as

*d*

_{ RD }in transmission capacity. Figure 4 shows the spatial intensity of attempted transmission associated with outage probability

*q*

^{-1}(ϵ) as a function of outage probability with noise variance ${\sigma}_{R}^{2}={\sigma}_{D}^{2}=1$ and interference power

*P*

_{ iR }=

*P*

_{ S }× 0.05 at the end-to-end spectral efficiency

*R*= 1 bps/Hz. We compare

*q*

^{-1}(ϵ) of a dual-hop relaying with direct transmission for different transmission power

*P*

_{ S }=

*P*

_{ R }= 15 and 20 dB. As we have seen from (32) and (34), this figure shows that

*q*

^{-1}(ϵ) can be zero in relatively low-outage probability regime due to thermal noise. On the contrary, in high-outage probability regime, the curves in Figure 4 have a crossing point at

*P*

_{ S }=

*Pr*= 20 dB due to a loosen outage probability constraint, and it means that the advantage of using relay can be decreasing at communication system which permits low QoS. However, the dual-hop relaying has more transmitting nodes to satisfy outage probability than does direct transmission at most of outage probability regime.

*P*

_{ iR }=

*P*

_{ S }× 0.05 at the end-to-end spectral efficiency

*R*= 1 bps/Hz is plotted in Figures 5 and 6. Figure 5 also compares direct transmission in terms of transmission capacity for different distance

*d*

_{ SR }=

*d*

_{ RD }= 0.5, 1.5, and 2. Likewise Figure 4, the transmission capacity of dual-hop relay performs better than that of direct transmission, especially in the relatively low outage probability regime and long distance. That is, using a dual-hop relay has more successful transmission nodes to satisfy outage probability and data rate. Conversely, the transmission capacity of dual-hop relaying is similar or lower than direct transmission in high-outage probability regime and short distance. But, since we utilize a relay to transmit over long distance and concern low-outage probability regime, the dual-hop relaying can be still efficient from transmission capacity point of view. Figures 4 and 5 reveal significant gain of dual-hop relaying compared to direct transmission in terms of density of attempted transmission and transmission capacity.

*P*

_{ S }=

*Pr*= 20 and 25 dB and distance

*d*

_{ SR }=

*d*

_{ RD }= 1.5 and 2. Figure 6 also shows that the existing region (

*q*

^{-1}(ϵ) > 0) of transmission capacity of a dual-hop relay is growing with increasing transmitting power

*P*

_{ S }=

*Pr*and decreasing distance

*d*

_{ SR }=

*d*

_{ RD }. This is due to the transmission capacity with thermal noise can be zero for small transmission power and large distance. In common, the maximum transmission capacity is achieved at a specific outage probability in Figures 5 and 6. It is because that the transmission capacity consists of spatial density

*q*

^{-1}(ϵ) and success probability (1 -ϵ) from the definition

*C*(ϵ) =

*Rq*

^{-1}(ϵ)(1-ϵ). In other words, as we have seen from Figure 4,

*q*

^{-1}(ϵ) increases consistently due to less outage constraint, but the success probability (1 - ϵ) goes to zero as the outage probability increases. Therefore, the transmission capacity decreases due to the success probability in high-outage probability regime and has maximum value at a specific outage probability.

## 6 Conclusion

This paper considers a dual-hop relaying in the presence of both noise and interference simultaneously, allowing a Poisson interference model. The outage probability of DF and AF strategies have been derived, especially we verified that the end-to-end *SINR* of AF relaying has Macdonald r.v's. The analytic and simulated results showed that outage probability of dual-hop relay had an error floor in high density of transmitting nodes and performance was greatly influenced by distance and interference power. Furthermore, we consider a metric of decentralized wireless network, called transmission capacity, for dual-hop relay with AF and DF strategies. We redefine the transmission capacity, because spatial intensity of attempted transmission associated with outage probability *q*^{-1}(ϵ) can be negative number due to thermal noise. Hence, we note that the existing region (*q*^{-1}(ϵ) > 0) of transmission capacity of dual-hop relay is growing with increasing transmission power. Our results reveal that transmission capacity of a dual-hop relay considering both noise and interference has better performance than direct transmission in a wireless ad hoc network.

## Appendix

### A Proof of Theorem 1

*CDF*of end-to-end

*SINR*at the relay $\text{X}~{\gamma}_{R}^{\text{DF}}$ is given by

*A*can be changed to closed-form using following equation in Gradshteyn and Ryzhik [18]

*n*= 1, $p=\frac{{d}_{SR}^{\alpha}x}{{P}_{S}},q=\frac{{\left[\text{erf}\left(\sqrt{{R}_{1}}\right)\right]}^{2}{\pi}^{4}{\lambda}_{1}^{2}{P}_{iR}}{16}$. Then, the

*CDF*of X can be written as

Finally, we can obtain *CDF* and *PDF* of *X* in Theorem 1.

### B Proof of Theorem 2

*CDF*of X and Y, we can compute outage probability of AF relaying as

Using *y* - *z* = *u*, we can obtain outage probability of AF relaying in Theorem 2.

## Declarations

### Acknowledgements

This research was supported by the KCC (Korea Communications Commission), Korea, under the "Development of highly efficient transmission technology for the next generation terrestrial 3D HDTV" support program supervised by the KCA (Korea Communications Agency) (KCA-2011-10912-02002).

## Authors’ Affiliations

## References

- Sendonaris A, Erkip E, Aazhang B: User cooperation diversity--part I: system description.
*IEEE Trans Commun*2003, 51(11):1927-1938. 10.1109/TCOMM.2003.818096View ArticleGoogle Scholar - Laneman JN, Tse DNC, Wornell GW: Cooperative diversity in wireless networks: efficient protocols and outage behavior.
*IEEE Trans Inform Theory*2004, 50(12):3062-3080. 10.1109/TIT.2004.838089MathSciNetView ArticleGoogle Scholar - Bletsas A, Shin H, Win MZ: Cooperative communications with outage-optimal opportunistic relaying.
*IEEE Trans Wirel Commun*2007, 6(9):3450-3460.View ArticleGoogle Scholar - Barua B, Ngo HQ, Shin H: On the sep of cooperative diversity with opportu-nitic relaying.
*Electon Lett*2008, 12(10):727-729.Google Scholar - Krikidis I, John SM, Thompson S, Geortz N: Max-min relay selection for legacy amplify-and-forward systems with interference.
*IEEE Trans Wirel Commun*2009, 8(6):3016-3027.View ArticleGoogle Scholar - Zhong C, Jin S, Wong KK: Dual-hop system with noisy relay and interference-limited destination.
*IEEE Trans Commun*2010, 58(3):764-768.MathSciNetView ArticleGoogle Scholar - Si J, Li Z, Liu Z: Outage probability of opportunistic relaying in rayleigh fading channels with multiple interferers.
*IEEE Signal Process Lett*2010, 17(5):445-448.View ArticleGoogle Scholar - Gupta P, Kumar PR: Transport capacity.
*IEEE Trans Inform Theory*2000, 46(2):388-404. 10.1109/18.825799MathSciNetView ArticleGoogle Scholar - Weber SP, Yang X, Adrews JG, de Veciana G: Transmission capacity of wireless ad hoc networks with outage constraints.
*IEEE Trans Inform Theory*2005, 51(12):4091-4102. 10.1109/TIT.2005.858939MathSciNetView ArticleGoogle Scholar - Weber S, Adrews JG, Jindal N: The effect of fading, channel inversion, and threshold scheduling on ad hoc networks.
*IEEE Trans Inform Theory*2007, 53(11):4127-4149.MathSciNetView ArticleGoogle Scholar - Hunter AM, Adrews JG, Weber S: Capacity scaling of ad hoc networks with spatial diversity.
*IEEE Trans Wirel Commun*2008, 7(12):5058-5071.View ArticleGoogle Scholar - Huang K, Adrews JG, Robert J, Heath W, Guo D, Berry RA: Spatial interference cancelation for mobile ad hoc networks.
*IEEE Trans Inform Theory*2008. [http://arxiv.org/abs/0807.1773]Google Scholar - Weber S, Adrews JG, Jindal N: A tutorial on transmission capacity.
*IEEE Trans Inform Theory*2009. [http://arxiv.org/abs/0809.0016]Google Scholar - Vaze R, Troung KT, Weber S, Robert J, Heath W: Two-way transmission capacity of wireless ad-hoc networks.
*IEEE Trans Inform Theory*2010. [http://arxiv.org/abs/1009.1460]Google Scholar - Adrews JG, Weber S, Kountouris M, Haenggi M: Random access transport capacity.
*IEEE Trans Wirel Commun*2010, 9(6):2101-2111.View ArticleGoogle Scholar - Haenggi M, Ganti RK:
*Interference in Large Wireless Networks*. now - The essence of knowledge, Massachusetts; 2010.Google Scholar - Daley D, Vere-Jones D:
*An Introduction to the Theory of Point Processes*. Springer, Berlin; 2003.Google Scholar - Gradshteyn IS, Ryzhik IM:
*Table of Integrals Series and Products*. 6th edition. Academic, San Diego; 2000.Google Scholar

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