On the Effective Capacity of Amplify-and-Forward Multihop Transmission Over Arbitrary and Correlated Fading Channels

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IEEE WIRELESS COMMUNICATIONS LETTERS 1

On the Effective Capacity of Amplify-and-Forward Multihop TransmissionOver Arbitrary and Correlated Fading Channels

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Kostas P. Peppas,Senior Member, IEEE, P. Takis Mathiopoulos,Senior Member, IEEE,

and Jing Yang,Member, IEEE

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AbstractThis letter presents a novel approach for analyzing,5in a unified way, the effective capacity performance of multi-6hop transmission with channel state information (CSI)-assisted7amplify-and-forward (AF) relay systems over arbitrary and corre-8lated fading channels under a maximum delay constraint. Using a9moment generating function (MGF)-based approach, an exact sin-10gle integral expression for the effective capacity is deduced. This11expression can be easily evaluated by means of standard numerical12integration techniques for a great deal of fading distributions, such13as the Nakagami-m, generalized-K, and generealized Gamma.14Several case studies, including uncorrelated and correlated fading15environments, are employed to demonstrate the versatility of the16proposed analytical approach. Extensive numerically evaluated17results accompanied with complimentary Monte-Carlo simula-18

tions are used to substantiate the analytical derivations.19

Index TermsAmplify-and-Forward, delay constraints,20effective capacity, fading channels, multisystems.21

I. INTRODUCTION22

O NE of the major challenges for future generation wireless23 communication systems is to support quality-of-service24(QoS) requirements for different applications. However, many25

important emerging applications, including voice over internet26

protocol (VoIP), interactive gaming, mobile TV and comput-27

ing, as well as interactive and multimedia streaming, are largely28

delay-sensitive. They further impose stringent QoS constraints,29

which typically appear in the form of constraints on queuing30

delays or queue lengths. In the past, the concept of effec-31

tive capacity (EC) has been introduced as a link-layer model32

for supporting QoS requirements of communication systems33

[1]. EC is the dual of effective bandwidth [2], and can be34

interpreted as the maximum constant arrival rate that a wire-35

less channel can support while a given QoS requirement is36

guaranteed.37

Let us now consider the multi-hop relay technology in which38

two users communicate with each other via relay terminals,39

This technology has emerged as an inspiring and powerful40

approach to improve the reliability, coverage and through-41

put of wireless communication systems. In the past, the EC42of amplify-and-forward (AF) relaying systems with channel43

state information (CSI) has been addressed in several research44

Manuscript received September 12, 2015; accepted February 10, 2016. The

associate editor coordinating the review of this paper and approving it for

publication was H. H. Nguyen.

K. P. Peppas is with the Department of Informatics and Telecommunications,

University of Peloponnese, Tripoli 22100, Greece (e-mail: [email protected]).

P. T. Mathiopoulos is with the Department of Informatics and

Telecommunications, National and Kapodistrian University of Athens,

Athens 15784, Greece (e-mail: [email protected]).

J. Yang is with the School of Information Engineering, Yangzhou University,

Yangzhou, China (e-mail: [email protected]).

Digital Object Identifier 10.1109/LWC.2016.2530787

works. In [3], closed form bounds on the EC of dual hop 45

systems operating in Rayleigh fading environments have been 46

deduced. In [4], bounds on the EC of two-way dual-hop systems 47

over Rayleigh fading channels have been presented. It should be 48

noted that both of these papers consider only the dual-hop case 49

and their corresponding analysis cant be easily generalized to 50

multi-hop relaying systems. Furthermore, they consider specific 51

channel models with emphasis given to simple fading channel 52

models such as the multi-path Rayleigh fading model. To the 53

best of the authors knowledge, the more realistic composite 54

multi-path/shadowing channel model has not been yet consid- 55

ered in connection with EC performance analysis studies. In 56

addition, the impact of link correlation on the EC performance 57

of multi-hop systems has not yet been investigated in conjunc- 58

tion with the problem of how to analytically evaluate the EC 59

performance of such communication systems. 60

Motivated by the above, in this contribution, a novel 61

approach for the exact EC performance analysis of CSI-assisted 62

AF multihop systems over arbitrary and correlated fading chan- 63

nels is presented. A single integral expression for the EC which 64

uses a moment generating function (MGF) based approach, 65

is deduced. This expression can be easily evaluated for a 66

variety of fading distributions by means of standard numeri- 67

cal integration techniques. It is noted that, to the best of the 68

authors knowledge, a unified analytical approach for the EC 69analysis of AF relaying systems is not, to date, available in 70

the open technical literature. The proposed unified analyti- 71

cal approach is employed to determine the EC performance 72

of CSI-assisted multi-hop AF relaying systems under uncor- 73

related Generalized Gamma, uncorrelated Generalized-K and 74

correlated Nakagami-m fading channels. In fact, as will be 75

explained later on, the analysis is valid for arbitrary and cor- 76

related fading channels as long as the MGF of the reciprocal 77

end-to-end signal-to-noise ratio (SNR) exists. The validity of 78

the proposed analysis is substantiated by extensive numerically 79

evaluated performance results accompanied with equivalent 80

results obtained by means of Monte-Carlo simulations. 81

Notations: E denotes expectation, fX() denotes the 82Probability Density Function (PDF) of the random variable 83

(RV) X, MX() denotes the MGF of the random variable 84X, Ja() is the Bessel function of the first kind and order 85a [5, Eq. (8.402)], Ia() is the modified Bessel function of 86the first kind and order a [5, eq. (8.431)], Ka() is the mod- 87ified Bessel function of the second kind and order a [5, 88

eq. (8.432)], () is the Gamma function [5, Eq. (8.310/1)], 89(,x, b, )=

x r1 exp

r br dr is the extended 90incomplete gamma function [6], p Fq () is the generalized 91hypergeometric function [5, eq. (9.14/1)] and G

m,np,q [] is the 92

Meijers G-function [5, Eq. (9.301)]. 93

2162-2337 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.

See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.


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II . SYSTEMM ODEL94

Let us consider a multi-hop transmission system in which95

a source node is communicating with a destination node via96

N1 AF relay nodes in series. All nodes are assumed to have97slow mobility, are equipped with a single antenna and work98

in half duplex mode. The available bandwidth is allocated to99

the source node for data transmission, which is further divided00into orthogonal sub-channels across time using a time-division01

scheme. It is also assumed that CSI is only known at the receiv-02

ing terminals and that only one terminal transmits in each time03

slot [3].04

As far as its queuing model is concerned, a simple first-input05

first-output (FIFO) buffer with constant arrival rate (source data06

rate) at the source data link layer is considered. By consider-07

ing ideal modulation and coding at the source physical layer,08

the service rate of the buffer will be equal to the instantaneous09

channel capacity which is time varying [3]. No buffering takes10

place at the relays and thus their transmissions are limited to11

the physical layer. Consequently, the arriving signal/packet is12

immediately amplified and sent to the next node without any13

buffering requirements or additional delays.14

Considering the previously described system model and15

assuming that the transmitter sends uncorrelated circularly16

symmetric zero-mean complex Gaussian signals, the instanta-17

neous EC can be deduced as [1], [3, Eq. (9)]:18

R( )= 1TfB

ln E(1+ end)(1)

whereendis the end-to-end instantaneous SNR of the AF CSI-19

assisted multi-hop transmission over arbitrary fading channels,20

=TfB /Nln(2)whereTf is the fading block length, B the21system bandwidth and the exponentially decay rate of the QoS22

violation probability. Note that a large value for corresponds23

to a fast decaying rate thus having more stringent QoS require-24

ments, while a smaller one refers to slower decaying rate and25

thus looser QoS requirements.26

The end-to-end instantaneous SNR,end, is characterized by27

the normalized harmonic mean of the instantaneous SNRs at28

each hop as [7]29

end=

N

n=1

1

n

1(2)

where n denotes the instantaneous SNR of the n-th hop.30

Denoting the reciprocal ofend asend=1/end, (2) can be31expressed asend=

Nn=1n wheren is the reciprocal of the32

instantaneous SNR of the n-th hop.33

In general, the evaluation of the expectation in (1) requires34

the computation of an N-fold integral. The numerical eval-35

uation of such integral is, by all means, computationally36

intractable, even for small values of N (e.g. 3 or 4).37

Furthermore, such integral cannot be expressed as a product of38

single integrals because of the non-linear factor (1+ end) .39Due to this difficulty, existing works dealing with the evalua-40

tion of the EC of AF systems, usually employ bounds on the41EC that become tight at high SNR regions [3], [4]. However, as42

it will be shown next, by using the MGF of the reciprocal end- 143

to-end SNR,end, a generic expression for the exact evaluation 144ofR ( )can be obtained. 145

III . EXACTA NALYTICALE XPRESSIONS 146

Proposition: The EC for AF CSI-assisted multi-hop trans- 147

mission over arbitrary fading channels can be expressed in 148

terms of a single integral as 149

R( )= 1TfB

ln

01 F1(; 1; u)

Mend (u)

udu

.

(3)

Proof: Let us consider the well-known identity [5, 150

Eq. (1.512/4)] 1510

exp [(x+ 1)s] s1ds=()(1+x) . (4)

By substituting (4) into the expectation in (1) and after some 152

simple algebraic manipulations while exploiting the definition 153

of the MGF, i.e., Mend (s) 0 exp(s )fend ( )d, EC 154

can be deduced as 155

R( )= 1TfB

ln

1

()

0

exp(s)s1Mend(s) ds

.

(5)

Integrating [8, Eq. (18)] by parts and employing the identity 156dd

u J0(2

su )=

su1J1(2

su ), the MGF of end can be 157

expressed in terms of the MGF ofendas 158

Mend (s)=

0

J0(2

su )Mend (u)

udu. (6)

By substituting (6) into (5), the integral in (5) can be 159

expressed as 160

0

exp(s)s1

0

J0(2

su )Mend (u)

udu

ds

=

0

Mend (u)

u

0

exp(s)s1J0(2

su )ds

du.

The inner integral, with respect tos , can be evaluated in closed 161

form by employing [5, Eq. (6.643/1)] and [5, Eq. (9.220/2)] 162

yielding (3), which completes the proof. 163 163

It is noted that (3) is valid for arbitrary and correlated fading 164

channels, as long as the MGF ofend exists. For the special 165case of uncorrelatedn , the EC of the considered system can be 166readily evaluated by employing the following corollary. 167

Corollary: When there is no correlation among all hops, the 168

EC of AF CSI-assisted multi-hop system is deduced as 169

R( )= 1TfB

ln

01 F1(; 1; u)

N

n=1

Mn (u)

u

Nk=1k=n

Mk(u)du

. (7)

Proof: When uncorrelated hops are considered, Mend (u) 170

can be expressed as the product of the MGFs ofn , i.e. 171Mend (u)=

Nn=1 Mn (u). Then, (7) is readily obtained from 172

(3) by evaluating the derivative of Mend (u) with respect 173tou . 174 174


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Fig. 3. EC of dual-hop CSI-assisted AF systems over correlated Nakagami-m

fading channels as a function of1, form 1=m 2=2.

Mend (u)= 4

(m)

k=0

um+k

k!(m+k)(12)m+k

1 k

Km+k

2

u

(1 )1

Km+k

2

u

(1 )2

.

(11)

Although the first derivative ofMend (u), which is required in35

(3) for the evaluation of EC, has been obtained in a straight-36

forward way by employing [5, Eq. (8.486/11)], it will not be37

presented here due to lack of space. Note that for the special38

case of independent and identically distributed Nakagami-39

m links, i.e. for 1=2= and = 0, the EC can be40expressed in closed form. In this case, all infinite series terms41 that appear in (11) withk=0,k, vanish and (11) simplifies as42

Mend (u)= 4um

(m)2mK2m

2

u

. (12)

The first derivative ofMend (u) can be deduced by employing43

[5, Eq. (8.486/10)] and [5, Eq. (8.486/11)] as44

Mend (u)

u= 8u

m1/2

(m)2m+1/2Km

2

u

Km1

2

u

.

(13)

In order to deduce a closed-form expression for R( ), the45confluent hypergeometric function in (3) and the product of46

the Bessel functions in (13) are first expressed in terms of47

the Meijers G-function, i.e. 1 F1(; 1; u)=[()]148G1,11,2

u

10,0 [13, Eq. (8.4.45/1)] and Km 2u/ 49Km1

2

u/ = 0.5 G3,01,3 4u/ 01/2,m1/2,m+1/250

[13, Eq. (8.4.24/29)]. Then, by employing [13, Eq. (2.24.1/1)]51

and [13, Eq. (8.2.2/15)], R ( )is deduced in closed form as52

R( )= 1TfB

ln

22m+1

()(m)2G 3,12,3

4

1/2+m,1,m,2m

.

Substituting the derivative ofMend (u) in (3), the EC per- 253

formance has been obtained and is depicted in Fig. 3, for 254

the cases where m=2 and {0, 0.8}. The infinite series in 255(11) are truncated to 40 terms for numerical accuracy. Again 256

here numerically evaluated results and computer simulations 257

are in excellent agreement. It is also noted that the impact of 258

correlation on the EC performance is rather small. 259

V. CONCLUSIONS 260

This letter, presented a comprehensive and unified analytical 261

approach for the computation of the EC of CSI-assisted multi- 262

hop communication systems over arbitrary and correlated fad- 263

ing channels employing the AF protocol. Using the MGF of the 264

reciprocal end-to-end SNR, a novel generic integral expression 265

for the EC, was deduced. Analytical performance evaluation 266

results obtained by numerical techniques have been in excellent 267

agreement with those obtained via Monte-Carlo simulations, 268

thus verifying the accuracy of the theoretical analysis. 269

REFERENCES 270

[1] D. Wu and R. Negi, Effective capacity: A wireless link model for sup- 271port of quality of service,IEEE Trans. Wireless Commun., vol. 2, no. 4, 272pp. 630643, Jul. 2003. 273

[2] C.-S. Chang, Stability, queue length, and delay of deterministic and 274stochastic queueing networks, IEEE Trans. Autom. Control, vol. 39, 275no. 5, pp. 913931, May 1994. 276

[3] S. Efazati and P. Azmi, Effective capacity maximization in multi- 277relay networks with a novel cross layer transmission framework and 278power allocation scheme, IEEE Trans. Veh. Technol., vol. 63, no. 4, 279pp. 16911702, Nov. 2013. 280

[4] G. G. Ozcan and M. Gursoy, Effective capacity analysis of fixed- 281gain and variable-gain AF two-way relaying, in Proc. IEEE 78th Veh. 282

Technol. Conf. (VTC Fall), Las Vegas, NV, USA, 2013, pp. 15. 283[5] I. Gradshteyn and I. M. Ryzhik,Tables of Integrals, Series, and Products, 284

6th ed. New York, NY, USA: Academic, 2000. 285[6] M. A. Chaudhry and S. M. Zubair, On a Class of Incomplete Gamma 286

Functions With Applications, 1st ed. London, U.K.: Chapman & 287Hall/CRC, 2002. 288

[7] M. O. Hasna and M.-S. Alouini, Outage probability of multihop trans- 289mission over Nakagami fading channels,IEEE Commun. Lett., vol. 7, 290no. 5, pp. 216218, May 2003. 291

[8] M. D. Renzo, F. Graziosi, and F. Santucci, A unified framework for 292performance analysis of CSI-assisted cooperative communications over 293fading channels,IEEE Trans. Commun., vol. 57, no. 9, pp. 25512557, 294Sep. 2009. 295

[9] F. Yilmaz, O. Kucur, and M.-S. Alouini, Exact capacity analysis of mul- 296tihop transmission over amplify-and-forward relay fading channels, in 297Proc. IEEE Int. Symp. Pers. Indoor Mobile Radio Commun. (PIMRC) , 2982010, pp. 22932298. 299

[10] K. Peppas, F. Lazarakis, A. Alexandridis, and K. Dangakis, Moments- 300based analysis of dual-hop amplify-and-forward relaying communica- 301tions systems over generalised fading channels,IET Commun., vol. 6, 302no. 13, pp. 20402047, Sep. 2012. 303

[11] K. Peppas, Accurate closed-form approximations to generalised-k sum 304distributions and applications in the performance analysis of equal-gain 305combining receivers, IET Commun., vol. 5, no. 7, pp. 982989, May 3062011. 307

[12] M. Nakagami, The m-distributionA general formula of intensity 308distribution of rapid fading, in Statistical Methods in Radio Wave 309Propagation. New York, NY, USA: Pergamon, 1960, pp. 336. 310

[13] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev,Integrals and Series 311Volume 3: More Special Functions, 1st ed. New York, NY, USA: Gordon 312and Breach, 1986. 313


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On the Effective Capacity of Amplify-and-Forward Multihop Transmission Over Arbitrary and Correlated Fading Channels