OfdmaClosed-Form Rate Outage Probability for OFDMA Multi-Hop Broadband Wireless Networks under Nakagami-m Channels
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IJASCSE, VOL 1, ISSUE 4, 2012Dec. 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
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IJASCSE, VOL 1, ISSUE 4, 2012Dec. 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 jk ,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 . jk ,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 jk ,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 www.ijascse.in Page 23
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IJASCSE, VOL 1, ISSUE 4, 2012Dec. 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 www.ijascse.in Page 24
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IJASCSE, VOL 1, ISSUE 4, 2012Dec. 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 nCk derive the moment generating function (MGF) Y ( s ) of the k B = N log (1 nCk 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 nC 1 k Y ( s ) = (16) where M k =| C | , i.e. the cardinality of the subset Ck . k k (mk ,n ) nCk 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 nCk Pkout = Pr (1 k ,n ) | Ck 2 k ,min sc , (10) r nC 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 nCk 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 1mk ,1 N (20) and equation (10) is reduced into 0 www.ijascse.in Page 25
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IJASCSE, VOL 1, ISSUE 4, 2012Dec. 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 ) = L1 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 1mk ,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 1mk ,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) = nCk (31) G s k ,1 J 3 x2 1,1 1mk ,1 dx 2 (25) k |Ck ( mk ,n ) 1,1 0 nCk 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 1mk ,1 ,1mk ,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 nC 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 nCk dz k nCk FY ( y )= k |Ck 0 z (mk ,n ) Given M , the closed-form formula of the MGF Y ( s ) nCk k k is: 1 y 1 M ( ) z 1dzd Y ( s ) = k (28) k ,n 0 (mk ,n ) nCk k mk ,n k ,n 1 1 (m ) nCk nCk (mk ,n ) 2 j U nCk k ,n = Substituting (27) in (28) we get the result in (16). The probability density function (pdf) of the random variable nCk 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 nCk k k nCk . ( ) ( y 1) fY |C ( y ) = ( ) d (33) k k (mk ,n ) (1 ) k ,n nCk nCk (29) where in the last step in (33) we used the equality www.ijascse.in Page 26
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IJASCSE, VOL 1, ISSUE 4, 2012Dec. 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 = nCk . probability is the probability that the supported transmission ( mk ,n ) nCk 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 nCi 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 www.ijascse.in Page 27
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IJASCSE, VOL 1, ISSUE 4, 2012Dec. 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 nCk i Pkout ,i V. OUTAGE PROBABILITY A PPLICATIONS (mk ,n ) i The closed form of the rate outage probability can be nCk 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 www.ijascse.in Page 28
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IJASCSE, VOL 1, ISSUE 4, 2012Dec. 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 )= nCk k ,eff ( j )|Ck nCk (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) = nCk , (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: nCk 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 nCk 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 www.ijascse.in Page 29
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IJASCSE, VOL 1, ISSUE 4, 2012Dec. 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 nCk 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 ) [1] IEEE Standard 802.16 Working Group. In IEEE 802.16-2005e Standard for Local and Metropolitan Area Networks: Air interface 1 r R ( t ) rk ,min R ( t ) for fixed broadband wireless access systems - amendment for = erf k ,min k erf k physical and medium access control layers for combined fixed and mobile operation in licensed bands, Dec. 2005. 2 2 R ( t ) 2 R ( t ) k k [2] 3GPP TS 36.300 Version 9. In Technical Specification 3rd Generation Partnership Project; Technical Specification Group Radio 2 where erf () = exp( x )dx is the error 2 Access Network; Evolved Universal Terrestrial Radio Access (E- 0 UTRA) and Evolved Universal Terrestrial Radio Access Network (E- UTRAN), Overall description; Stage 2 (Release 9), Jun. 2009. function. Then, the problem SCA-OUT can be rewritten as follows: [3] J.J. Yang and H. Haas, “ Outage probability for uplink of cellular www.ijascse.in Page 30
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IJASCSE, VOL 1, ISSUE 4, 2012Dec. 31 ofdm-fdma system,” IEEE 59th Vehicular Technology Conference, symposium on Symbolic and algebraic computation, pages 212–224, 2004. VTC 2004-Spring. 2004, 4, May 2004. 1990. [4] I. Viering, A. Klein, M. Ivrlac, M. Castaneda, and J.A. Nossek, “ On [14] A. Prudnikov, Y. Brychkov, and O. Marichev, “ Integral and Series: uplink intercell interference in a cellular system,” IEEE International Inverse Laplace Transforms,” volume 5. Gordon and Breach, 1 Conference on Communications, 2006. ICC ’06., 5, June 2006 edition, 1992. [5] M. Castaneda, M.T. Ivrlac, J.A. Nossek, I. Viering, and A. Klein, “ On downlink intercell interference in a cellular system,” IEEE 18th Table 1: Modulation and Coding Schemes for IEEE 802.16. International Symposium on Personal, Indoor and Mobile Radio Communications, 2007. PIMRC 2007., Sept. 2007. [6] J. Brehmer, C. Guthy, and W. Utschick, “ An efficient approximation Modulation Info Required SNR of the ofdma outage probability region,” IEEE 7th Workshop on (coding) bits/symbol Signal Processing Advances in Wireless Communications, 2006. SPAWC ’06., July 2006. BPSK(1/2) 0.5 6.4 [7] J. Leinonen, J. Hamalainen, and M. Juntti, “ Outage capacity analysis of downlink odfma resource allocation with limited feedback,” IEEE Information Theory Workshop, 2008. ITW ’08., May 2008. QPSK(1/2) 1 9.4 [8] P.P. Hasselbach and A. Klein, “An analytic model for outage probability and bandwidth demand of the downlink in packet QPSK(3/4) 1.5 11.2 switched cellular mobile radio networks,” IEEE International Conference on Communications, 2008. ICC ’08., May 2008. QAM(1/2) 2 16.4 [9] I. Lee and D. Kim, “ Outage probability of multi-hop MIMO relaying with transmit antenna selection and ideal relay gain over Rayleigh fading channels,” IEEE Transactions on Communications, QAM(3/4) 3 18.2 57(2):357 – 360, Feb. 2009. QAM(2/3) 4 22.7 [10] K. Moshksar, A. Bayesteh, and A. Khandani, “Randomized vs. orthogonal spectrum allocation in decentralized networks: Outage analysis,” pages 1–41, Nov. 2009. QAM(3/4) 4.5 24.4 [11] C. E. Shannon. A mathematical theory of communication. Tech. Rep., Bell Sys. Tech. Journal, 27, 1948. . [12] I. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series and products. Academic Press, fifth edition, 1994. . [13] Adamchik V. S. and Marichev O. I, “ The algorithm for calculating integrals of hypergeometric type functions and its realization in reduce system,” In ISSAC ’90: Proceedings of the international Figure 1: Single hop cellular radio network www.ijascse.in Page 31
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IJASCSE, VOL 1, ISSUE 4, 2012Dec. 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 www.ijascse.in Page 32
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