OfdmaClosed-Form Rate Outage Probability for OFDMA Multi-Hop Broadband Wireless Networks under Nakagami-m Channels

IJASCSE, VOL 1, ISSUE 4, 2012

Dec. 31

Closed-Form Rate Outage Probability for OFDMA Multi-Hop
Broadband Wireless Networks under Nakagami-m Channels

Mohammad Hayajneh and Najah AbuAli
College of Information Technology
UAE University
Al Ain, United Arab Emirates

the required application data rate. LTE and WiMAX both
Abstract—Rate outage probability is an important performance implement OFDMA as the technique to share the access of
metric to measure the level of quality of service (QoS) in the the media among users.
Generation (4G) broadband access networks. Thus, in this OFDMA is a multiple-access scheme that inherits the
paper, we calculate a closed form expression of the rate outage ability of Orthogonal Frequency Division Multiplexing
probability for a given user in a down-link multi-hop OFDMA-
(OFDM) to combat the inter-symbol interference and
based system encountered as a result of links’ channel
variations. The channel random behavior on different
frequency selective fading channels. In order to reduce the
subcarriers allocated to a given user is assumed to follow effect of selective fading channels in wireless networks,
independent non-identical Nakagami-m distributions. Besides OFDMA divides the total frequency bandwidth into a
the rate outage probability formulas for single hop and multi- number of narrow-band subcarriers in such a way that each
hop networks, we also derive a novel closed form formulas for subcarrier (SC) experiences a flat fading channel. This is
the moment generating function, probability distribution possible by selecting the SC width to be close to the
function (pdf), and the cumulative distribution function (cdf) of coherence bandwidth of the channel. A small end-portion of
a product of independent non-identical Gamma distributed the symbol time, which has duration equal to the maximum
random variables (RVs). These RVs are functions of the delay spread channel, is called a cyclic prefix (CP). The CP
attainable signal-to-noise power ratio (SNR) on the allocated is copied at the beginning of the symbol time duration to
group of subcarriers. For single-hop scenario, inspired by the eliminate the effect of inter-symbol interference (ISI). Thus,
rate outage probability closed formula, we formulate an the inter-symbol interference is dealt with while
optimization problem in which we allocate subcarriers to users orthogonality among the subcarriers is maintained.
such that the total transmission rate is maximized while
catering for fairness for all users. In the proposed formulation, Accordingly, OFDMA is a preeminent access technology
fairness is considered by guaranteeing a minimum rate outage for broadband wireless data networks compared with
probability for each admitted user. (Abstract) traditional access technologies such as TDMA and CDMA.
The main advantages of OFDMA over TDMA/CDMA stem
Keywords-component; OFDMA; Nakagami-m; Outage from the scalability of OFDMA, the uplink orthogonality
probability; Moment generating function and the ability to take advantage of the frequency selectivity
of the channel. Other advantages of OFDMA include
I. INTRODUCTION MIMO-friendliness and ability to support prominent quality
of service (QoS).
The Worldwide Interoperability for Microwave Access
(WiMAX) and Long-Term Evolution (LTE)– also known as The rate outage probability defined as the probability of
Evolved Universal Terrestrial Radio Access (E-UTRA)– are having the attainable link data rate below the required
two promising candidates for providing ubiquitous IP 4G application data rate. Our objective in this work is to derive a
wireless broadband access. The LTE and WiMAX are closed form expression of the rate outage probability for
designed as all IP packet switched networks in order to be Orthogonal Frequency Division Multiple Access (OFDMA)-
capable of providing higher link data bit rate [1], [2]. Thus, based system under the Nakagami-m channel model. Our
all services involving real and non- real time data, are interest in calculating the rate outage probability in an
implemented on top of the IP as a network layer protocol. OFDMA system is motivated by the fact that OFDMA is the
Real time data services transmission such as voice and video common physical layer technology for the 4th generation
require maintaining the wireless link data rate within the (4G) and beyond broadband wireless networks; mainly LTE-
requirement of the rate demands of these applications. Thus, Advanced and IEEE802.16m networks. Calculating a closed-
the outage of the link data rate more than a predefined form rate outage probability expression facilitates designing
threshold is not tolerable. The link data rate is said to be in sound radio resource management schemes for the efficient
an outage if the rate requirements of an application cannot operation of these networks in single and multi-hop contexts.
be satisfied. Hence, we define the rate outage probability as Recently, several attempts are made to compute the
the probability of having the attainable link data rate below outage probability in OFDMA systems. These attempts can
www.ijascse.in Page 22


Dec. 31
formulation for single hop cellular system scenario. In
be categorized into two major categories: the simulation Section 6 we conclude the paper.
modeling and the analytical derivation. An example on II. SYSTEM MODEL
simulation modeling is the work presented in [3]. Simulation
requires collecting sufficient statistics of the network, which We consider an OFDMA system model in two network
is time consuming, then applying a prediction model, which scenarios: namely, single hop cellular radio network, see
risks various prediction error. Hence, we will focus on the Fig. 1 and multi-hop cellular radio network, see Fig. 2. Each
analytical derivation category. The works in [4, 5, 6, 7, 8, cell, the network consists of a Base Station (BS) with
9,and 10] are examples of the analytical derivation attempts omnidirectional antenna located in the center of the cell
for single and multi-hop networks. The authors of [4] and [5] serving multiple Subscriber (SS) stations uniformly
distributed over the cell.
The SSs share the total channel bandwidth, which is
compute uplink and downlink outage probabilities in [4] and divided into N orthogonal subcarriers. SSs and BS
[5] respectively under Rayleigh fading channel model. The
proposal presented in [6] computes downlink outage transmission is on a time frame basis, where each frame
probability of all users in a cell for a particular resource consists of multiple OFDMA symbols. The channel
allocation (subcarrier). In [7], the author computes the condition of the SS is assumed not changing during each
downlink outage probability for a system with a limited time frame. However, it may change from one frame to
channel feedback, and the proposal presented in [8] attempt another. The received low-pass signal at SS k on the nth SC
to describe downlink outage probability related to bandwidth
user demand. The work in [10], and [9] addressed the at OFDMA symbol t is denoted by Rk ,n (t ) and given as:
calculation of outage probability in multi-hop networks. The
authors of [10] provided three lower bounds of end-to-end  jk ,n ( t )
outage capacity over Rayleigh fading channels, while [9] Rk ,n (t ) = hk ,n (t )e S k , n ( t )  zk , n ( t ) (1)
provided an approximation on outage probability through the
numerical inversion of a Laplace transform over Rayleigh where Sk ,n (t ) is the equivalent low-pass transmitted signal
fading channels as well .
 jk ,n ( t )
The aforementioned analytical proposals, although to the k th SS on the n th SC. hk ,n (t )e is the time-
compute outage probability for different objectives, they do
share common procedures. They compute the outage variant transfer function of the equivalent low-pass
probability for a particular resource allocation (OFDMA frequency non-selective channel of the k th SS experienced
subcarrier) and/or employed a simplified channel model  jk ,n ( t )
such as the Rayleigh channel model. Moreover, all proposals on the n th SC. The transfer function hk ,n (t )e is a
are semi-analytical where outage probability computation is
approximated through calculating upper and/or lower complex-valued Gaussian random process in which hk ,n (t )
bounds or through providing numerical solutions.
Nevertheless, none of the above proposals provides a closed is a Nakagami-m distributed random variable and k ,n (t ) is
form expression of the outage probability. In this paper, we
address the void of the current literature by deriving a closed uniformly distributed random variable in (  ,  ) . hk ,n (t )
form formula for the rate outage probability in single (i.e.,
PMP) and multi-hop networks. We compute the outage represents the channel envelope (modulus), while k ,n (t )
probability based on a generalized channel model, namely captures the phase of the equivalent low-pass channel. We
the Nakagami-m model of which Rayleigh is merely a
special case. Finally, we express the outage probability Adopt the Nakagami-m distribution because it fits the
formula as a direct function of the user required data rate and realistic statistical variations of the channel envelope in both
two other parameters; the Nakagami-m fading coefficient line-of-sight (LOS) and non line-of-sight (NLOS)
and the spreading factor. communications. Both, the envelope and the phase of the
Our derivation has several applications in 4G and beyond channel are assumed to be jointly independent. Furthermore,
broadband networks. For example, outage probability can be
considered as a QoS metric to meet a specific connection we assume that hk ,n (t ) and hk ,m (t ) are independent
data rate requirement, or it can be utilized as a performance random variables m  n , which implies that the channel
measure to evaluate the level of meeting the total demands
of users in a cellular network. coherence bandwidth is approximately equal to the
The remainder of this paper is organized as follows. In bandwidth of a SC. zk , n ( t ) is the complex-valued white
Section 2 we present the problem description and system
Gaussian noise process with power spectral density denoted
model, while the computation of the outage probability and
the relevant mathematical formulation is offered in Sections by N o .
3 and 4. In section 5, we introduce an optimization



Dec. 31

III. DERIVATION OF SINGLE-HOP
RATE OUTAGE PROBABILITY
We denote to the SS’s k , (k {1,..., K} on subcarrier n ,
n {1,...., N }) data rate by rk ,n (t ) bits per OFDMA In this paper, the measure of the level of QoS
satisfaction is given in terms of the attainable data rate , rk (t )
symbol. rk ,n (t ) depends on the allocated power pk ,n , SS
, against the SS k minimum required data rate, rk ,min .
k on subcarrier n , and the channel gain hk ,n . The channel
Explicitly, a SS is in outage, if the required data rate is not
gain hk ,n is modeled as a Nakagami-m random variable met due to low attainable data rate. The outage of meeting the
with (a probability density function) pdf defined as: required data rate can be considered as a QoS metric or as a
performance measure. For the case of the outage probability
m
mk ,n as a QoS metric, a SS can demand a predefined threshold for
( ) 2
2mk ,k ,n 2 mk ,n 1 k , n the probability of the outage of service. The outage
f h ( ) = n
 e , n  N , (2)
(mk ,n )k ,n
k ,n m probability threshold will be used to control the amount of
n
network resources allocated for the user. Alternatively, the
outage probability can be calculated given the allocated
where k ,n = E{hk ,n } is a coefficient that reflects the
2
resources. In this case the outage probability is considered as
spread of the Nakagami-m distribution and a performance measure. For both cases, our aim is to
N = {1,2, , N } is the indexing set of the data calculate the rate outage probability for a specific SS in
consequence of its variable channel quality and adaptive
transmission.

subcarriers. mk ,n is the fading coefficient that measures the The rate outage probability Pkout of the kth SS can
be expressed as:
severity fading of the channel. mk ,n is defined as:
Pkout = Pr(rk (t )  rk ,min ) (4)
Now, let us define the achievable transmission data rate by
 2 ,n
k subcarrier n of SS k as [11]:
mk ,n = . (3)
E{(hk ,n  k ,n )2 }
2
B
log2(1   k ,n (t )),
rk ,n (t ) = (5)
N
Note that for mk ,n =  , this corresponds to a non- where B is the link bandwidth and N is the total number
fading channel. For mk ,n = 1 , the Nakagami-m distribution of data subcarriers. Total link bandwidth is denoted as B
reduces to the Rayleigh distribution, which is used to model B
and hence, represents the physical bandwidth of each
small scale channel variation in NLOS systems, and to one- N
sided Gaussian distribution for mk ,n = 1 . Further subcarrier.  k ,n (t ) is the Signal to Noise power Ratio (SNR)
2
of SS k at subcarrier n at OFDMA symbol t and is
examples on the generality of the Nakagami-m model is expressed as:
when mk ,n < 1 , in this case Nakagami-m distribution pk ,n H k ,n (t )
 k ,n ( t ) = , (6)
closely approximates the Hoyt distribution, and for mk ,n > 1 No B / N
, it approximates the Rician channel model. Thus, the where pk ,n is the transmit power of the SS k at the
Nakagami-m channel model is capable of capturing a 2
subcarrier n , H k ,n (t ) @ hk ,n (t ) with hk ,n (t ) the channel
realistic wireless channel through the two variables; mk ,n gain between the BS and SS k at subcarrier n at OFDMA
and  k ,n . Hence, Nakagami-m channel is an appropriate symbol t . No is the Additive White Gaussian Noise
model to meet this work objective, which is to facilitate (AWGN) power spectral density at the receiver. As per our
earlier assumption that the channel gains on all subcarriers
practical resource management through trackable closed-
change form one time frame to another time frame, we drop
form expression of the rate outage probability.
the time index t in our next calculations. Note that in (6), the


Dec. 31

interference coming from other users using same subcarriers 
Pkout = Pr Yk | Ck  2 k ,min
r / Bsc
, (15)
as SS k in the same OFDMA symbol has been assumed
negligible. The impact of the interference in OFDMA
signalling can be taken care of in the MAC protocol. Yk is a product of independent not necessarily identically

The overall achievable transmission data rate, rk in (i.n.i.d) Gamma distributed random variables, X k ,n . Thus, to
its OFDMA symbol, of SS k on all its allocated subcarriers calculate the rate outage probability, it is required to find the
in the subset C : n = 1,....M k ; M k  N
k is: probability density function (pdf) of Yk for a given allocated

rk = r k ,n , (7) subcarriers’ subset Ck . To derive the pdf of Yk , we first
nCk
derive the moment generating function (MGF) Y ( s ) of the
k
B
=
N
 log (1  
nCk
2 k ,n ), (8) random variable Yk as illustrated by the following theorem:
Theorem 1: The MGF of Yk defined as a product of i.n.i.d
B  
= log 2  (1   k ,n )  ,
Gamma distributed random variables is given by
(9)
N  nC  1
 k  Y ( s ) = (16)
where M k =| C | , i.e. the cardinality of the subset Ck .
k
k
(mk ,n )
nCk

Using (9) and (4) the rate outage probability Pkout can be  1 mk ,1 ,...,1 mk , M 
GM   s  k ,n
1, M k
expressed as:
k ,1 0
k

 /B 
 nCk
 

Pkout = Pr  (1   k ,n ) | Ck  2 k ,min sc  , (10)
r

 nC 
1, M k
where GM [.] is a Meijer G-function which is included as
 k  k ,1
a standard built-in function in most of the mathematical
B
where Bsc @ is the bandwidth of each subcarrier. software packages such as MATHEMATICA and MAPLE.
N
Define sYk
Proof: The MGF Y ( s ) is the expectation of e , i.e.
X k ,n @1   k ,n , (11) k

Y ( s) = E{e
sYk
} (17)
k
   s x1 x2 ... xM
Given that  k ,n is based on a Nakagami-m channel model, it =  ... e k
0 0 0

follows that the random variable X k ,n has the following  f X ( x1 ) f X ( x2 )... f X ( xM )dx1...dxM
k ,1 k ,2 k ,M k k k
Gamma distribution:
mk ,n 1 Define J i @ xi xi 1... xM and
( x  1) x 1 k
f X ( x) = exp(  ), (12)  m 1  x / 
 k ,n 1 @  x
mk ,n s x1 J 2
 (mk ,n )
k ,1 1 k ,1
k ,n
k ,n 1 e e dx1 (18)
0
where which is the innermost integral after dropping the constant
1 1 p  k m
1 / (mk ,1 )  k ,1 ,1 , for simplicity. The integral in (18) can be
 k ,n = E{ X k ,n } = (1  k ,n k ,n ).
mk ,n mk ,n N o Bsc rewritten in terms of the Meijer-G function [12] as:
 m 1 x1
Moreover, define the following new random variable, Yk as: 1 =  x1 k ,1 G0,1 [
1,0  1,0 
|0 ]G0,1 [ sx1 J 2 |0 ]dx1 (19)
Yk @  X k ,n , (13)
0  k ,1
nCk
Using the result provided in [13] for calculating
Thus, the achievable rate of the k th SS rk (t ) is given by: integrals of hypergeometric type functions, one can find that
B the integral in (19) can be solved as:
rk = log2 Yk  (14)
1 =  k ,1k ,1 G1,1   s  k ,1 J 2 
m
1,1 1mk ,1
N (20)
and equation (10) is reduced into
  0 




Dec. 31
The integration on x2 defined as 2 is given as: Proof: The pdf of Yk can be found as the inverse Laplace
 m 1  x / 
transform of Y (  s ) , that is
2 @  x 2
k ,2
e2 k ,2
1  dx2 (21) k

 
0
 m 1  x /  fY ( y ) = L1 Y (  s), y

(30)
= k ,1 x2
k ,2 2 k ,2
e (22) k |Ck k
0
Using the formula for the inverse Laplace transform of the
G   s  k ,1 J 2 dx
1,1 1mk ,1
Meijer G-function from [14], we obtain the result in (29).

 1,1 0  2

 m 1  x2 / k ,2 Corollary 2: The cumulative distribution function (cdf),
=  k ,1  x2 k ,2 e (23)
0 FY ( y ) , of Yk , is given by:
k |Ck

G1,1   s  k ,1 J 3 x2 0 dx
1,1 1mk ,1

  2  
   ( y  1) 1 
M k ,1
 m 1 x2  G   mk ,1 ,mk ,2 ,...,mk ,M k ,0 
  k ,n
=  x2 k ,2 G0,1 [1,0 1, M k 1
| ] (24)
0  k ,2 0 
FY ( y) =  nCk  (31)
G   s  k ,1 J 3 x2

1,1 1mk ,1
dx
 2
(25)
k |Ck
 ( mk ,n )
 1,1 0
 nCk

Similarly, using the result from [13], the integral in (21) is
given by:

2 =  k ,1k ,1  k ,2 ,2 G1,2   s  k ,1  k ,2 J 3 
m k m 2,1 1mk ,1 ,1mk ,2 Proof: By definition, the cdf FY ( y ) is the probability
(26) k |C
  0 

k

Following the same argument used for calculating the second that the random variable Yk is in the interval [1, y ] , that is
integral, the M-fold integral in (17),  M , is given by FY ( y ) =  fY
y
( z )dz (32)
k
k |Ck 1 k |Ck
induction as: Therefore,
 mk 
 M =   k ,n ,n  (27)  
k  nC 
 k  M k ,0  z  
G   mk ,1 ,mk ,2 ,...,mk ,M k 
  k ,n
 1 mk ,1 ,...,1 mk ,M 
0, M k

G1,M   s k ,n 0 
M k ,1 k
 y 1
 nCk  dz
k
 nCk
 
 FY ( y )=
k |Ck 0 z (mk ,n )
Given  M , the closed-form formula of the MGF Y ( s ) nCk
k k
is: 1 y 1
M ( )   z  1dzd
Y ( s ) = k
(28)  k ,n
0

 (mk ,n ) nCk
k mk ,n
k ,n
1 1
(m   )
nCk nCk

(mk ,n ) 2 j U nCk k ,n
=
Substituting (27) in (28) we get the result in (16).
The probability density function (pdf) of the random variable nCk
Yk is given in the following corollary.
Corollary 1: The pdf of Yk is given by:
1 ( y  1) 
  ( )  d
M k ,0  ( y  1) 
( y  1) 1 G0,M 
  k ,n 
mk ,1 ,mk ,2 ,...,mk , M 
  k ,n
nCk
k  k

 nCk . (  ) ( y  1) 
fY |C ( y ) =  ( ) d (33)
k k
 (mk ,n ) (1   )  k ,n
nCk nCk

(29) where in the last step in (33) we used the equality



Dec. 31
(1   ) = ( ). outage probability in an OFDMA system.

U is an appropriate integration contour, and j = 1 . IV. DERIVATION OF MULTI-HOP RATE OUTAGE
PROBABILITY
Solving the integral in (??) will yield the solution in (31).
In this Section, using the results of section 3, we
Using the CDF, FY ( y ) = Pr (Yk  y ) , in (31) derive the rate outage probability of OFDMA multi-hop
k
and (4), the closed form of the rate outage probability for a communication system under Nakagami-m channel models.
given Ck is given by:
For multi-hop scenario, a minimum requested
out rk ,min / Bsc
P = FY (2 ) (34) transmission rate requirement for a given SS is satisfied when
k k |Ck
the minimum supported transmission rate on an intermediate
 r /B  link (bottle-neck) in the routing path between the BS and this
M k ,1  (2 k ,min sc  1) 1  particular SS is greater than the requested rate. Otherwise, a
G  mk ,1 ,mk ,2 ,...,mk , M ,0 
  k ,n
1, M k 1
 k

rate outage will occur. Therefore, the multi-hop rate outage

=  nCk . probability is the probability that the supported transmission
 ( mk ,n )
nCk
rate on the bottle-neck link is less than the minimum
requested transmission rate by a particular SS. In order to
As an indicative example for the calculated outage mathematically express this probability, let us define the
probability, the outage probability is plotted in Figure 3 as a followings. For the intermediate link (hop) i,
function of the minimum required data rate per user for i{1,2, , N h ,k } with N h ,k is the number of hops
different values of the channel physical bandwidth. The
observed results evidently show that the outage probability (distance) between the BS and kth SS, let
decreases with the increase of the channel physical
bandwidth for the same required data rate. Moreover, For Yki @  X k ,n , (35)
larger channel bandwidths, for example when B is 10 MHz
nCi
k
and 5 MHz, the difference in the outage probability values
become tighter, as shown in Figure 3, with outage probability
4
values of 2  10 and 2  10 for a bandwidth of 5MHz
5 where Cki is the subset of data subcarriers allocated to carry
and 10MHz respectively. the data of the kth SS on the ith hop. Henceforth, the
We performed simulation experiments based on the
system model described in Section 2 to compare the achievable rate per OFDMA symbol of the k th SS rki on ith
calculated rate outage probability in (34) against the rate hop is:
outage probability evaluated from the simulation. The
network consists of one BS and multiple SS. The BS
log2 Yki 
allocates the available SCs per user based on the users’ data
B
rki = (36)
rate requirement and their channel quality. This process is N
repeated each frame duration. For our simulation, we used a
frame size of 10 ms. The matrix of SCs is generated every The minimum supported transmission rate on the
frame based on the Nakagami-m channel model and the SCs route from the BS to the SS k is defined as
are allocated per user to meet the users minimum data rate
requirements. The attainable data rate is calculated given the N h ,k
allocated SCs per user. The simulated outage probability is rkbn =min{rk1, rk2 , , rk } (37)
calculated as the difference between the minimum required
data rate and the attainable data rate. The calculated outage From the above, we can figure out that the multi-hop rate
probability is obtained from (34) by substituting the
bandwidth of the same SCs used in simulation per user and outage probability of the kth SS, Pkout mh , is given by
the Nakagami-m pdf parameters. Results are shown in Figure
Pkout mh = Pr  rkbn | C < rk ,min 
4 for the calculated outage probability against the simulated
outage probability for different values of the minimum k (38)
required data rates. The results in Figure 4 clearly show that
the rate calculated outage probability matches that obtained or
from the simulation for different values of the user required
data rate. Hence, the calculated rate outage probability is
verified to provide a closed form expression of the rate



Dec. 31


Pkout mh = Pr Ykbn | Ck < 2 k ,min
r / Bsc
 (39)
same subcarrier has been assigned again in the network after
some hops (to eliminate interference).

Here, In line with the single-hop simulation results
provided in 4, we performed simulation experiment to
N h ,k compare the calculated multi-hop rate outage probability in
Ck = Cki (41). We used the same environment and simulation
i =1 parameters as that used for the single hop. Our simulation is
based on Type-II LTE-relay and the transparent 802.16j relay
N
Assuming that {Yki }i =1,k are independent random variables,
h mode. We chose these modes to allow centralized SCs
allocation at the BS. Type-I LTE-relay and non-transparent
then
802.16j relay mode do not entail much difference from the

 ,
r / Bsc
point of view of verifying the correctness of the outage
Pkout mh = 1  Pr Ykbn | Ck  2 k ,min (40) probability derivation. Note that we derived the closed form
rate outage probability for a given user assuming the
allocated sub-carriers’ set is pre - allocated. The algorithm for

 
N h ,k

= 1  Pr Yki | Cki  2 k ,min
r / Bsc allocating subcarriers to network users is out of the scope of
i =1
this work. Algorithms for allocating subcarriers could be
performed by the relay node in the non-transparent mode

 
(distributed SCs allocation) or by the BS (centralized SCs

N h ,k

= 1   1  Pr Yki | Cki < 2 k ,min
r / Bsc
allocation). For the sake of simplicity, we simulate here the
i =1 transparent relay mode. Transparent relay mode restricts the
number of hops in the network into 2 hops. The BS allocates
Note that 
Pr Yki | Cki < 2 k ,min
r / Bsc
 is the rate outage
SCs for the relay node and the SS given the Nakagami-m
channel condition at each subcarrier at the relay and the
access link. The attainable rate of the allocated SCs at each
probability at the ith hop, i.e. Pkout ,i , thus (40) can be written
link is compared to the minimum required data rate to
as identify the bottleneck link. The simulated outage probability
is calculated as the difference between the required data rate
N h ,k

= 1   1  Pkout ,i  ,
out mh and the bottleneck attainable data rate. The calculated outage
P k (41)
probability is calculated from (41) given the allocated SCs
i =1
used in simulation and the Nakagami-m pdf parameters.
with Results are shown in Figure 5. Figure 5 shows similar results
to those obtained for single-hop outage probability. The
calculated results almost match the simulated results for low
  and high required data rates. Hence, we can safely deduce
 (2rk ,min / Bsc  1) 1
M k ,i ,1  that equation (41) provides a closed form expression of the
G1, M 1  ,0  rate outage probability in multi-hop OFDMA systems.
  k , n
mk ,1 , mk ,2 ,..., mk , M
k ,i k ,i 

 nCk

i


Pkout ,i  V. OUTAGE PROBABILITY A PPLICATIONS
 (mk ,n )
i The closed form of the rate outage probability can be
nCk
utilized in several aspects. For example, it can be integrated
where M k ,i = Cki , i.e. the number of data subcarriers into a subcarrier allocation scheme, where the BS allocates the
allocated to carry the data of SS k on the ith hop. It should subcarriers and subcarriers’ power level to the active SSs in the
be noted that the assumption of the independence of system to achieve a requested outage probability bound. Also,
N it can be used as a performance measure to evaluate the
{Yki }i =1,k can result from assigning each subcarrier once in
h
efficiency of different resource allocation algorithms, or to
the whole network, or from the multiuser diversity even if the improve the performance of a certain resource allocation
algorithm through adapting the resource allocation based on the


Dec. 31
feedback information of the channel status and the outage  
probability of users. As a final example, The SS rate outage 1 M k ,0  y  
( y) G   mk ,1 ,mk ,2 ,...,mk ,M k 
  k ,n
probability can be used to calculate the network rate outage 0, M k
probability which can be used latter in efficient dynamic 
admission control algorithms. f ( y )=  nCk 
k ,eff ( j )|Ck

nCk
(mk ,n )
In this Section, as an example, we consider a
centralized subcarrier allocation scheme that is aware of the y  0, (48)
outage probability of each user, called SCA-OUT, for single
and accordingly, the CDF is given by:
hop cellular network. In SCA-OUT, the BS allocates the
subcarriers to the K active SSs in the system and then loads  
each SC with an optimal power level such that the overall DL M k ,1  y 1 
G   mk ,1 ,mk ,2 ,...,mk ,M k ,0 
  k ,n
1, M k 1
achievable transmission rate is maximized. The constraint can
be the exponential average transmission rate. This is assured to 
F ( y) =  nCk  , (49)

be bounded from above and from below by a given probability.
Mathematically speaking, the problem is formulated as
k ,eff ( j ) |Ck (mk ,n )
follows: nCk


 K 

To be able to solve the optimization problem in SCA-
SCA-OUT: p ,C  Bsc   log 2(1   k ,n (t ))  (42)
max OUT, we need to provide a closed form expression of the
 k =1 nCk  probability given in (42). Therefore, we need to find the pdf of
k ,n k
 
s.t. : Rk (t ) , which is a weighted sum of independent random

Pr  rk ,min  Rk (t )  k rk ,min | C    (43) {rk (n)}n=t .
n
variables =0 Note that the independence of the
k

pk ,n  0, C   random variables {rk (n)}n=t is a result of the assumption that
n
=0
k
the channel gains over all subcarriers between the k th SS and
where 1  k  rk ,max / rk ,min and BS only change on a time frame basis. Next, we introduce
1 1 some preliminary results that lead us to an approximate closed
Rk (t ) = (1  ) Rk (t  1)  rk (t ) , (44) form formula of the constraint in (43).
T T
is the exponential average received rate over a window size of With the deployment of the AMC module at both the BS and
T which is an integer multiple of the time frame T f . Using SS to keep the bit error rate (BER) at a prescribed value, the
achievable rate is a discrete random variable, see Table 1.
the fact that Rk ( 1) = 0 one can rewrite (44) as follows:
Henceforth, the probability mass function (pmf) of rk (n) at
t
1 1
Rk (t ) = (1  )t n rk (n) (45) the n th time frame denoted to  i , and is given by:
T n=0 T
i @ Pr  rk (n) =  i  (50)
From (9), the achievable rate at the j th frame of the k th SS, = Pr ( SNRi   k ,eff (n) < SNRi 1 ) (51)
rk ( j ) can be expressed as follows: SNRi 1
= f ( z )dz
SNRi k ,eff ( j ) |Ck
B
rk ( j ) = log2(Yk ( j )) (46)
= F ( SNRi 1 )  F ( SNRi )
N k ,eff ( j ) |Ck k , eff ( j ) |Ck
B
= log2(1   k ,eff ( j )) (47) where {SNRi }i7=1 are the thresholds of the 7 non-overlapping
N
regions shown in Table 1 with 8 =  , 0 = 0, and
where  k ,eff ( j ) = Yk ( j )  1 is the effective SNR over all k
 i {0.5,1,1.5,2,3,4,4.5} is the number of information
th SS’s allocated subcarriers. Henceforth,  k ,eff ( j ) is a
bits on the M -ary QAM symbol. Providing a closed formula
random variable with the following probability density for the probability in (42) requires the probability density
function:
function (pdf) of Rk (t ) . Assuming that first and second order
moments of Rk (t ) are finite, for a large t , and according to



Dec. 31
the Central Limit Theorem (CLT), the pdf of Rk (t ) can be

 K 

SCA-OUT: p ,C  Bsc   log 2(1   k ,n (t ))  (57)
approximated as a Normal distributed random variable as max
follows:
 k =1 nCk 
k ,n k
Rk (t )  R  
k (t )
 N (0,1), (52)
R 1  r
 erf  k ,min
 R ( t ) 
k (t ) s. t. : k 
2  2 R ( t ) 
where N (0,1) denotes a normal distribution with zero mean
  k 
and variance equal to 1 , with Rk ( t ) being the mean of
 rk ,min  R ( t )  
Rk (t ) and  2
is the variance of Rk (t ) . Both R ( t ) and  erf  k   
Rk ( t ) k  2 R ( t )  
 
 Rk ( t )
2 k

can be found as functions of r and  r2 ( n ) which pk ,n  0, C   (58)
k (n) k k

are the mean and the variance of rk (n) , respectively: This problem can be either solved analytically or by a heuristic
algorithm, which is beyond the scope of the paper.
1 t 1
Rk ( t ) = (1  T )t n rk ( n )
T n=0
(53) VI. CONCLUSIONS
We introduced an analytical model to calculate a
1 t 1 7
= (1  )t n  i i
tractable closed form rate outage probability expression of
 a SS in a single and multi-hop cellular network. The
T n=0 T i =1
closed form expression of the rate outage probability is
and obtained by analyzing the desired data rate of each user
2
1 t
1 against the link attainable transmission rate. The problem
R
2
k (t )
= 
T 
(1  T )
n =0
2( t n )
 r2 ( n )
k
is formulated as a function of SNR per OFDMA subcarrier
and the Nakagami-m channel model. Simulation results
where, verified the match of the calculated outage probability
 r2 ( n ) = E{rk2 (n)}  r2 ( n ) (54)
with the simulated one in an OFDMA system. Moreover,
we formulated an optimization problem for single-hop
k k
2 scenarios in which subcarriers are assigned to admitted
7
 7 
=  i    i i 
 i
2
 users in the network such that the total transmission rate is
i =1  i =1  maximized while catering for fairness for all users.
Fairness was achieved by guaranteeing a minimum
Therefore, the approximate pdf of the random variable average rate outage probability over a time window of
Rk (t ) is given as: length equal to integer multiple of the time frame for each
admitted user.
 z
  
2

exp   
1 Rk ( t )
ACKNOWLEDGMENT
f R (t ) ( z) = (55)
2 R ( t )  2 R ( t )
2 
 
k
This work was made possible by NPRP grant #
k
 k
 NPRP4-553-2-210 from the Qatar National Research Fund
Next we rewrite an approximate closed form expression for (a member of The Qatar Foundation). The statements
equation (42) as follows: made herein are solely the responsibility of the authors
Pr  rk ,min  Rk (t )  k rk ,min | C 
k
REFERENCES
k rk ,min
= fR ( z )dz (56)
rk ,min k (t )
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Figure 1: Single hop cellular radio network



Dec. 31

Figure 2: Multi-hop cellular radio network abstraction

Figure 5: Multi-hop Outage Probability against
the Required Data Rate, calculated and simulated

Figure 3: Single hop Outage Probability against the Required
Data Rate for different channel bandwidth

Figure 4: Single hop Outage Probability against
the Required Data Rate, calculated and simulated


OfdmaClosed-Form Rate Outage Probability for OFDMA Multi-Hop Broadband Wireless Networks under Nakagami-m Channels

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