An Analytical Model for the Intercell Interference Power in the Downlink of Wireless Cellular Networks
Benoit Pijcke, Marie Zwingelstein-Colin, Marc Gazalet, Mohamed Gharbi, Patrick Corlay
11 An Analytical Model for the IntercellInterference Power in the Downlink of WirelessCellular Networks
Benoit Pijcke, Marie Zwingelstein-Colin, Marc Gazalet,Mohamed Gharbi, Patrick CorlayUniversit´e Lille Nord de France, F-59000 LilleUVHC, IEMN/DOAE, F-59313 ValenciennesCNRS, UMR 8520, F-59650 Villeneuve d’Ascq, France [email protected]
Abstract
In this paper, we propose a methodology for estimating the statistics of the intercell interferencepower in the downlink of a multicellular network. We first establish an analytical expression for theprobability law of the interference power when only Rayleigh multipath fading is considered. Next,focusing on a propagation environment where small-scale Rayleigh fading as well as large-scale effects,including attenuation with distance and lognormal shadowing, are taken into consideration, we elaboratea semi-analytical method to build up the histogram of the interference power distribution. From theresults obtained for this combined small- and large-scale fading context, we then develop a statisticalmodel for the interference power distribution. The interest of this model lies in the fact that it can beapplied to a large range of values of the shadowing parameter. The proposed methods can also be easilyextended to other types of networks.
Index Terms
Intercell interference power, statistical modeling, wireless networks, Rayleigh fading, lognormalshadowing.
I. I
NTRODUCTION
In the emerging wireless communication standards LTE-Advanced and Mobile WiMAX, ag-gressive spectrum reuse is mandatory in order to achieve the increased spectral efficiency requiredby IMT-Advanced for the 4th generation of standard telephony. However, since spectrum reusecomes at the expense of increased intercell interference, these standards explicitely requireinterference management as a basic system functionality [1]–[3]. The research area related tothe development and analysis of interference management techniques, mostly in relation with a r X i v : . [ c s . I T ] J u l the more general subject of radio ressource management, is very dynamic, as witnessed by thehigh number of relevant recent contributions in this area [4]–[10]. All these new standards useOFDMA as the modulation and the multiple access scheme. In an OFDMA system, there is nointracell interference as the users remain orthogonal, even through multipath channels. However,when users from different cells are present at the same time on the same subchannel, which isthe case under aggressive frequency reuse, signals superpose, leading to some form of intercellinterference.Providing statistical models of the interference power is essential to allow for an accurateevaluation of networks performances without the need for lenghtly and costly Monte Carlosimulations. The statistical characterization of the interferences has been investigated for a longtime, under lots of different scenarios, and following several approaches. The distribution ofcumulated instantaneous interference power in a Rayleigh fading channel was investigated in[11], where an infinite number of interfering stations was considered. In [12], the interferencepower statistics is obtained analytically for the uplink and downlink of a cellular system, butin the presence of large-scale fading only. Interference modeling when considering only large-scale fading effects has also been investigated in [13]–[15], where the emphasis is on finding agood approximation of the lognormal sum distribution. In [16], an analytical derivation of theprobability density function (pdf) of the adjacent channel interference is derived for the uplink.More recently, in [17] the pdf of the downlink SINR was derived in the context of randomlylocated femtocells via a semi-analytical method. Other contributions have focused directly on theanalysis of a particular performance measure that is influenced by intercell interference, like theprobability of outage and the radio spectrum efficiency [18]–[20]. The analysis of interferencein dense asynchronous networks, such as ad-hoc networks, is also an active research area, forwhich a deep review of the recent developments can be found in [21], [22].In this paper, we derive a semi-analytical methodology to estimate the statistics of the intercellinterference power in a wireless cellular network, when the combined effects of large-scale andsmall-scale multipath fading are taken into consideration. Large-scale effects include attenuationwith distance (path-loss) as well as lognormal shadowing, and the small-scale fading is Rayleighdistributed. We consider a distributed wireless multicellular network, in both cases where powercontrol and no power control is applied. The proposed methodology is semi-analytical, in thatthe statistical estimate of the interference power resulting from N > interferers is obtainedby numerical techniques from an analytically-derived interference model for one interferer. Themethodology is valid in a quite general framework; we have chosen to present it using a hexagonalnetwork layout, although it can handle any other topology. We validate the proposed methodsby comparing the moments of the estimates to the exact moments of the distribution which canbe derived analytically. Using this methodology, we are able to provide a very good estimateof the pdf of the interference power, for different values of the shadowing standard deviation, σ dB . Based on these estimates, we then propose an analytical statistical model of the interference power, based on a modified Burr distribution, which includes 5 parameters. This analytical,parameterized by σ dB , model will hopefully serve as a practical tool for the assesment andsimulation of wireless cellular networks when the effect of shadowing is to be considered.The main contributions of this paper are as follows. • In the special situation where only path-loss and Rayleigh fading are considered (no shad-owing), we derive a very accurate approximated analytical expression for the pdf and thecumulative distribution function (cdf) of the intercell interference power; • We propose a semi-analytical method for the estimation of the pdf of the intercell interferencepower in a multicellular network when the combined propagation effects of path-loss,Rayleigh fading and lognormal shadowing are considered; • Based on this method, we derive an analytical model for the pdf of the intercell interferencepower by slightly modifying a Burr probability distribution. This model is parameterized bythe lognormal standard deviation σ dB and its interest resides in the fact that it is valid onthe whole [0 , -dB range of values.The remainder of this paper is organized as follows. In Section II, we describe the multicelldownlink transmission environment, and we provide the expression of the interference powerfor which we want to find a statistical model. In Section III, the original methodology forestimating the statistics of the interference power is presented. For this purpose, we examine inSection III-A the particular case where path loss and Rayleigh fast-fading are the only fadingphenomena considered. In Section III-B, we include the shadowing effect and we consider in thefirst instance the contribution of one interfering cell. We then generalize to N > interferers.In Section IV, we apply the proposed method to estimate the pdf of the interference power ina typical multicellular network, under two frequency reuse scenarios. Section V is dedicated tothe parametric analytical modeling of the interference power. Section VI concludes the paper bysummarizing the proposed methods and by presenting some perspectives.We will use the following notation for the rest of the paper. Non-bold letters such as x areused to denote scalar variables, and | x | is the magnitude of x . Bold letters like x denote vectors.We use E { X } to denote the expectation of X . The pdf and cdf of the random variable (r.v.) X will be denoted p X ( x ) and F X ( x ) respectively.II. M ULTICELL DOWNLINK TRANSMISSION MODEL
We consider the downlink of an OFDMA-based 19-cell cellular network having the 2D hexag-onal layout depicted on Fig. 1. We assume a unit-gain omnidirectional SISO ( single input, singleoutput ) antenna pattern, both for the fixed access points (APs) and the mobile user terminals(UTs) which are supposed to be uniformly distributed over the service area. As OFDM is usedfor intracell communication, we assume an orthogonal transmission scheme within a cell. Weconsider a synchronous discrete-time communication model in which active APs at any given time slot send information symbols to their respective UTs over a shared spectral resource, whichgives rise to an interference-limited environment. In this framework, we will focus on the statisticsof the so-called intercell interference power undergone by a typical UT. In this regard, we willconsider UT in cell (denoted UT , see Fig. 1), for it is surrounded by 18 potential interferers.For UT , the received signal on OFDMA subchannel (cid:96) at time slot m can be modeled as y ( m, (cid:96) ) = h ( m, (cid:96) ) x ( m, (cid:96) ) + N (cid:88) n =1 h n ( m, (cid:96) ) x n ( m, (cid:96) ) + w ( m, (cid:96) ) . Here x ( m, (cid:96) ) represents the information symbol intended to UT and x n ( m, (cid:96) ) , n (cid:54) = 0 , the n th interfering symbol (this symbol is sent from AP n to its respective user). The coefficient h n ( m, (cid:96) ) denotes the instantaneous gain of the (cid:96) th (interfering) subchannel from AP n to UT .Each subchannel (cid:96) is subject to additive white Gaussian noise w ( m, (cid:96) ) . In the following, we willfocus without loss of generality on a single OFDMA subchannel, thereby omitting subchannelindex (cid:96) in all subsequent notations.Two frequency reuse scenarios will be considered (see Fig. 1): • the full frequency reuse pattern, denoted FR1, where all APs in the network transmit at thesame time using the same frequency range ( N = 18 intercell interferers); • a partial frequency reuse pattern, denoted FR3, with reuse factor 3 ( N = 6 interferers).Each channel is assumed to be flat-fading, possibly experiencing small-scale multipath fadingand/or large-scale effects. For the rest of the paper, we concentrate on the instantaneous channelpower gain G n ( r n ) , which is proportional to | h n ( m ) | and can be expressed as a three-factorproduct: G n ( r n ) = G pl ,n ( r n ) G f ,n G s ,n , n = 1 , , . . . , N. (1)In the above equation, r n denotes the distance between UT and AP n (distances r n are func-tions of UT ’s position within its cell). G pl ,n ( r n ) = K (1 /r n ) γ is the (deterministic) path loss(normalized with distance, see Appendix A), where K is a constant and γ represents the pathloss exponent. The Rayleigh fading gain G f ,n is modeled by an exponential distribution withrate parameter equal to , i.e., E { G f ,n } = 1 ; we denote the corresponding pdf by p G f ,n ( x ) . Theshadowing gain G s ,n is modeled by a lognormal distribution whose pdf can be written p G s ,n ( x ) = ξ √ πσ dB x exp (cid:32) − (10 log ( x ) − µ dB ) σ dB (cid:33) , x > , where ξ = 10 / ln(10) [23]. Note that the importance of the shadowing phenomenon is directlyrelated to the standard deviation σ dB . For a given σ dB , the parameter µ dB is determined to ensurea unit mean shadowing gain: E { G s ,n } = 1 , which leads to µ dB = − σ dB / (2 ξ ) . As r.v.’s G f ,n and G s ,n are independent from each other, and as E { G f ,n } = E { G s ,n } = 1 , we have, from (1), As this paper will focus on power gains only, the term power will then be omitted in subsequent paragraphs. E { G n ( r n ) } = G pl ,n ( r n ) , which reflects the fact that the n th interfering channel’s Rayleighfading and shadowing components cause the actual gain G n ( r n ) to fluctuate about its meanvalue G pl ,n ( r n ) .The total interference power undergone by UT can then be written I = (cid:80) Nn =1 P n G n ( r n ) ,where P n = E (cid:8) | x n | (cid:9) is the power emitted by AP n . In what follows, we consider that allAPs transmit at the same power, i.e., P n = P for all n . This corresponds to, e.g., a fast-fadingenvironment where no channel state information feeds back from mobile users to APs, whichresults in a no power control scheme where all APs transmit at the maximum power; althoughcrude, this scheme can be seen as a lower bound on performance for real systems. Consideringthat each AP transmits at the same power P also applies to a more practical scenario where APshave access to channel state information and power control is associated with the opportunisticscheduling policy proposed (and proved to be sum-rate optimal) in [10], when the number ofusers per cell is high (since in this case, it can be expected that the channels between usersscheduled at the same time and their serving APs have about the same power gains). Thus, theinterference simplifies to I = P (cid:80) Nn =1 G n ( r n ) .We now define the interference gain — which will be denoted G — as being the sum of thechannel power gains between the interested user and the N interferers, i.e., G = N (cid:88) n =1 G n ( r n ) = N (cid:88) n =1 G pl ,n ( r n ) G f ,n G s ,n . (2)(Note that G is a function of UT ’s location through the distances r n .) So, as I = P G ,characterizing the interference power I is equivalent to studying the interference gain G . Wewill concentrate on the latter in the subsequent sections.III. M ETHODOLOGY
We are now interested in finding an estimate of the pdf of the random interference gain (2).Since direct calculation of the pdf does not seem possible, we aim at producing an accuratehistogram for the interference gain G that will then be modeled using a specified statisticaldistribution. Such a histogram is constructed from a set of samples called a typical set, i.e.,a discrete ensemble of values that accurately represents a random phenomenon. Traditionally(and especially in the telecommunications area), this typical set is issued from Monte Carlosimulations, which might, at first sight, produce satisfying results. However, in a propagationenvironment that is subject to intense shadowing (i.e., for large values of the [0 , -dB rangeunder consideration), the classical Monte Carlo method fails at producing a representative setof sampled gains [24], [25]. This can be explained by examining the particular distributioninvolved, for one single as well as for multiple interfering cells. A typical cdf of the interferencegain (single or multiple interferers) for a high value of σ dB belongs to the class of heavy-tailed distributions [26], for which the least-frequently occuring values — also called rare events — are the most important ones, as a proportion of the total population, in terms of moments. A finite-time random drawing process performed on this cdf never produces these rare events because oftheir very low probabilities, which causes the resulting set to be not typical. Hence the need fora new approach.As will be seen in Subsection III-B2, the pdf and the cdf of the interference gain for one singleinterferer may be expressed in its integral form. From this expression, we propose the followingtwo-step approach:1) Produce a typical set of gains for one interferer using the generalized inverse method .This method consists in generating a typical set of samples corresponding to an arbitrarycontinuous cdf F , and is based upon the following property: if U is a uniform [0 , r.v.,then F − ( U ) has cdf F ;2) Produce a typical set for multiple interferers by adequately combining typical sets fromsingle interferers and the Monte Carlo computational technique. A. Special case: No shadowing
We start this section by considering a propagation environment in which the only fadingphenomenon is due to Rayleigh multipath fading. In this particular case, (1) simplifies to G n ( r n ) = G pl ,n ( r n ) G f ,n . (3)We first note that, because of the symmetry of the network geometry, we need only studythe interference power distribution for UT located within one of the twelve triangular sectorsdepicted on Fig. 2; in the following, we will consider the grey-shaded region for illustrationpurposes.We now introduce an original approximation that will help simplify further computations.We can see that in (3), it is UT ’s random position that makes the path loss G pl ,n ( r n ) fluctuate,when the randomness of G f ,n is due to Rayleigh fading. But it is worth noting that, although bothphenomena are random, path loss fluctuations differ from multipath fading in an important way: the path loss takes values in a finite set (related to UT ’s location within its cell) whereas thevariations due to fading have an (theoretically) infinite dynamic range. Since pathloss fluctuations’dynamics are very small compared to fading’s, we propose to approximate (3) by replacing eachgain G pl ,n ( r n ) by its average value, which leads to G n ≈ E r n { G pl ,n ( r n ) } G f ,n = E r ,θ { G pl ,n ( f n ( r , θ )) } G f ,n , (4)using the notation r n = f n ( r , θ ) , n = 1 , , ..., N , where ( r , θ ) are UT ’s polar coordinates, asdepicted in Fig. 2. By examining (4), we see that, under this approximation, G n does not dependon UT ’s varying position anymore. We further note that G n , as expressed in (4), is an exponentially distributed r.v. with rateparameter /λ n [27], λ n — which we call the average path loss — being defined as follows: λ n = E r ,θ { G pl ,n ( f n ( r , θ )) } . (5)Using (5), (4) can also be written G n ≈ λ n G f ,n , (6)and the intercell interference gain (2) can be reduced to a sum of independant (but not identicallydistributed) exponential r.v.’s: G ≈ N (cid:88) n =1 λ n G f ,n . (7) G , as expressed in (7), is a r.v. whose cdf, denoted F G ( g ) , has a closed form expression availablein the literature [28]; it can be expressed as F G ( x ) = 1 − N (cid:88) n =1 A n exp (cid:18) − xλ n (cid:19) , (8)where A n = λ NnN (cid:81) j =1 j (cid:54) = n λ n − λ j , n = 1 ...N. The pdf, denoted p G ( g ) , can be easily calculated by deriving (8): p G ( x ) = N (cid:88) n =1 A n λ n exp (cid:18) − xλ n (cid:19) . (9)In Section IV-A, it is first shown that approximation (4) is valid in the case of one singleinterfering cell. This consequently validates the proposed model (7) in the case of multipleinterfering cells, which we show for both frequency reuse patterns FR1 and FR3. B. General case: Attenuation with distance, shadowing and multipath fading
Let us now focus on characterizing the distribution of the intercell interference gain G in apropagation environment where Rayleigh fading as well as shadowing (due to obstacles betweenthe transmitter and receiver that attenuate signal power) are taken into account. To the best ofour knowledge, no closed form expression for the interference gain G exists in the literature.But, as will be seen in Section III-B2, we determine an analytical formula (under integral form)of the distribution of the interference gain for one interferer. Using this result, we are able toobtain a histogram for G’s distribution in the presence of multiple interferers.For this purpose, we proceed in two steps: first, we compute a typical set for the interference gain produced by one single interferer. As described in Section III-B2, this is done by numericalcomputation (from the integral-form cdf), followed by non-uniform partitioning, and then inver-sion, of the cdf. Then we generate a typical set for N interferers using an appropriate combinationof the (weighted by λ n ) typical sets of each single interferer (Section III-B3). The accuracy ofthe proposed method will be evaluated in both single- and multiple-interferer cases by comparingthe actual moments computed from the typical sets with the exact moments of the interferencegain distribution (which can be formulated analytically, as will be seen in Section III-B1).
1) Preliminaries:
We begin this section by examining two important points.When taking into account multipath fading as well as shadowing as the fading effects in thepropagation environment, a question arises about the validity of the original approximation (6).Fortunately, our approximation is being strengthened by this additional contribution due toshadowing, since this phenomenon is just another source of infinite-dynamics randomness. Takingshadowing into consideration amounts to introducing an additional term in (6) that can now bewritten G n ≈ λ n G f ,n G s ,n . (10)A second point pertains to the moments of both statistical distributions of G n (single interferer)and G (multiple interferers). Using approximation (10), it is shown in Appendix B that the k th-order moment of G n ’s distribution has the following expression: E (cid:110) ( G n ) k (cid:111) = k ! exp (cid:18) k ( k − σ dB (cid:19) . (11)Computation of the k th-order moment of G ’s distribution is done in Appendix C and leads tothe following formula: E (cid:8) G k (cid:9) = k ! (cid:88) a : | a | = k λ a exp (cid:32) σ dB (cid:32) − k + N (cid:88) n =1 α n (cid:33)(cid:33) , (12)where a = ( α , α , . . . , α N ) , α n ∈ N , n = 1 , , . . . , N , is an N -dimensional vector whose sumof components is written | a | = (cid:80) Nn =1 α n , and λ α = λ α λ α . . . λ α N N . So the summation in Eq (12)is taken over all sequences of non-negative integer indices α through α N such that the sum ofall α n is k . Note that the st-order moment, E { G } = N (cid:88) n =1 λ n , (13)is a quantity of particular interest because it is proportional to the average power of the inter-ference signal.As closed form expressions of moments have been determined, they may be used in evaluatingthe accuracy of typical sets for both single- and multiple-interferer statistical laws.
2) Single interferer:
We now turn on to computing a typical set for the interference gainproduced by one interferer. For convenience, the average path loss (5) for this single interfereris normalized to 1, i.e., λ n = 1 , so (10) reduces to G n ≈ G f ,n G s ,n . (14)As G n is the product of two independent r.v.’s, its cdf can be written F G n ( x ) = (cid:90) ∞ p G f ,n ( u ) (cid:34)(cid:90) xu p G s ,n ( y ) d y (cid:35) d u = (cid:90) ∞ p G f ,n ( u ) F G s ,n (cid:16) xu (cid:17) d u, (15)where F G s ,n ( x/u ) denotes the shadowing gain’s cdf. Recalling that G s ,n is modeled as a lognormalr.v., we have, using the same notations as in Section II, F G s ,n ( x/u ) = Q (cid:16)(cid:16) µ dB −
10 log (cid:16) xu (cid:17)(cid:17) /σ dB (cid:17) , where Q ( z ) = 1 / √ π (cid:82) ∞ z exp ( − t /
2) d t is the complementary error function of Gaussianstatistics. Replacing p G f ,n ( u ) and F G s ,n ( x/u ) by their respective expression in (15), we obtainan integral-form expression for the cdf of the intercell interference gain produced by one singleinterfer: F G n ( x ) = (cid:90) ∞ Q (cid:32)
10 log (cid:0) ux (cid:1) σ dB − σ dB ξ (cid:33) exp ( − u ) d u. (16)We are now interested in generating a typical set of the interference gain G n ; we denote thistypical set by S (cid:96)n , where (cid:96) is the number of elements in the set. It was mentioned in Section IIIthat, though widely used in telecommunications, the Monte Carlo computational technique provesinefficient for large values of σ dB . An interesting alternative method is the generalized inversemethod, for which an (cid:96) -element typical set for a given distribution is obtained by an (cid:96) -leveluniform partitioning, followed by inversion, of the cdf. Now we know that, for large valuesof σ dB , the distribution of G n exhibits the heavy-tailed property, which means, as describedbefore, that the least-frequently occuring values (i.e., the highest gains) are the most importantones in terms of moments. Therefore, taking these highest amplitudes into consideration usingthe ’classical’ generalized inverse method would require a finer partitioning of the cdf, whichwould produce a typical set made up of a huge amount of elements.In order to construct a typical set with a reasonable value for (cid:96) , we propose to accomodatethe above-mentioned method by performing a non-uniform partitioning of G n ’s cdf, and, as highamplitudes are important in terms of moments, we proceed with a finer partitioning of the [0 , segment for values close to . The implementation details of the method are described on Fig. 3;they result from a good compromise between accuracy and simplicity. We first divide the interval [0 1] of the cdf into J intervals, numbered j = 1 , . . . , J , of different lengths: the j th intervalhas a length d j = 9 × − j , j = 1 , . . . , J − ; the last interval has a length d J = 10 − J toensure (cid:80) Jj =1 δ j = 1 . We next perform a P -level uniform partitioning on each interval, i.e., eachinterval is now partitioned by P equally-spaced points. Finally, we invert the partitioned cdf toobtain a typical set S (cid:96)n of cardinality (cid:96) = J × P . Also, as the proposed partitioning is non-uniform, S (cid:96)n needs to be associated a probability set: the probability of an element computedfrom the j th interval is δ j = d j /P . It can be shown (see Section IV-B) that using J = 25 intervals containing P = 900 points each — which results in a typical set that contains only (cid:96) = 25 ×
900 = 22 , elements — guarantees that up to third-order moments derived fromthe typical set are within 1% of the exact values for all σ dB .
3) Multiple interferers:
We now focus on finding an L -element typical set — denoted S L —for the interference gain G that must be computed from N typical sets S (cid:96)n , n = 1 , , . . . , N .We first note that interferer n ’s typical set can be directly obtained by weighting each elementof S (cid:96)n by its average path loss λ n ; we will denote interferer n ’s typical set by λ n S (cid:96)n . Let us nowfind a way to produce the ensemble S L from the typical sets λ n S (cid:96)n .Ideally, S L should be constructed by considering all combinations of the elements of thetypical sets λ n S (cid:96)n , but the cardinality of the resulting set, L = (cid:96) N = ( J P ) N , would rapidlybecome prohibitive as the number N of interferers increases.To get rid of this complexity, we point out that the above-mentioned ideal (exhaustive) solutioncan also be viewed as an exhaustive combination of intervals ( J N combinations) associated withan exhaustive combination of elements within each interval combination ( P N combinations). Andwe observe that the most important part of this exhaustive solution pertains to the combinationof intervals , i.e., the combination of elements belonging to interval j of typical set λ n S (cid:96)n withelements belonging to interval k , k (cid:54) = j , of typical set λ m S (cid:96)m , m (cid:54) = n . So a way to constructa (near optimal) typical set for G could be to perform exhaustive combinations of the intervals(as in the exhaustive solution), and to approximate the exhaustive combination of the elementswithin each interval combination by the following procedure: for each of the J N combinationsof N P -point intervals, • Perform a random permutation of the P elements within each of the N P -point intervals ; • Add up these N permuted P -point intervals to obtain one resulting P-element interval.This last P -element interval approximates the P N -element interval that would have resultedfrom an exhaustive combination of elements within the considered interval combination. Now, asthere are J N interval combinations, the resulting typical set would contain J N P elements, whichcan still be prohibitive, so this second solution — which we will refer to as the near-optimal To produce moments of the same accuracy, the traditional uniform partitioning approach would require about (cid:96) = 900 × points. Two interval combinations of the same rank j are supposed to be orthogonal because of the high number of points in eachinterval ( P = 900 ), which guarantees the independance of permutations. solution — can not be applied as such.We eventually propose a novel approach which makes use of this near-optimal solution and isbased on the following two-step algorithm:Step 1 Apply exhaustive combinations of intervals to a subset of M interfering links;Step 2 Perform Monte Carlo simulations for the N − M remaining links.We now detail the principle of the proposed method. In Step 1, we apply the near-optimal solutiondescribed above, but to a subset of M < N interfering links which we will call compelled links.The compelled links are chosen to have the highest average path losses ( λ ≥ · · · λ M ≥ · · · λ N )so as to minimize errors in other (non-compelled) interfering links. The exhaustive combinationof the J intervals for M compelled links obtained from the near-optimal solution thus results in one set of J M P elements. In Step 2, we build up a J M P -element set for each of the N − M remaining, non-compelled, links by performing J M random drawings of intervals according tothe probability set { δ j } , j = 1 , , . . . , J . As in the near-optimal solution, a random permutation ofthe elements is applied at each drawing. The ensemble of amplitudes of the intercell interferencegain G — the so-called typical set S L —is then constructed by adding up these N − M + 1 sets;it is of cardinality L = J M P . Associated to S L is a probability set determined as follows: toeach interval is associated a weight which is the product of probabilities δ k of intervals issuedfrom compelled links (for non-compelled links, probabilities are accounted for by means of therandom selection process); these weights are then normalized to obtain probabilities. Finally,the histogram of the interference gain G can be constructed from these resulting amplitude andprobabiliy sets. It is important to note, however, that, as a random drawing process is involved, anumber of iterations might be needed in order for this process to converge (elements of S L andassociated probabilities are averaged at each iteration). We will call this semi-analytical techniquethe Monte Carlo-panel method (MCP, in short) .The MCP method is illustrated on Fig. 4 for N = 4 interfering cells, M = 2 compelled links,and J = 2 intervals per typical set (these intervals — denoted A and B — have probabilities δ = 0 . and δ = 0 . respectively, and each one of them contains P elements). Step 1 of thealogrithm is summarized in the light-grey shaded box: intervals from typical sets S (cid:96) and S (cid:96) (corresponding to compelled interfering links 1 and 2, and weighted by their respective averagepath losses λ and λ ) are combined together, as described in the near-optimal solution, to obtaina set of amplitudes of cardinality P representative of the two compelled links; associated tothis set of amplitudes is a set of weights { . , . , . , . } . The dark-grey shaded boxsummarizes Step 2: for each non-compelled interfering link, a P -element set of amplitudes ismade up by 4 intervals ( A or B ) drawn according to the probability set { . , . } and appliedrandom permutations. The typical set S L (with L = 4 P in our example) is then obtained bysumming up together all these sets. The histogram of the interference gain G is constructed The term ’panel’ refers to survey panels used by polling organizations. from S P and the associated probability set . Note that one random permutation of the interval(permuted intervals have been assigned the prime symbol) is performed at each (compelled orrandom) manipulation of an interval.Implementing the MCP method however requires cautiousness. In non-compelled links, randomdrawings of intervals are performed based on the probability set { δ j } , j = 1 , , . . . , J . In thisprocess, lowest-probability intervals, which contain the highest interference gains, are totallyignored for two reasons. The first reason pertains to the fact that obtaining a significant frequencyof appearance of such rare events would require a prohibitive number of simulation runs. Thesecond reason is due to limitations inherent to software simulation tools which use pseudo-randomnumber generators to generate sequences of ’random’ numbers belonging to a fixed set of values.In order to take into account the ignorance of the contribution of the highest interference gains ofthe N − M non-compelled interfering links in the probability set { δ j } , we suggest the followingworkaround: in these links, we intentionally make exclusive use of the J , ≤ J < J , firstintervals, and we associate them a loaded probability set (cid:8) δ (cid:48) j (cid:9) defined as follows: δ (cid:48) j = αδ j for ≤ j ≤ J for J + 1 ≤ j ≤ J (17)where α = 1 J (cid:80) j =1 δ j (18)is a normalizing constant such that (cid:80) J j =1 δ (cid:48) j = 1 (using the particular non-uniform partitioningdescribed previously, we have: α = 1 / (cid:0) − . J (cid:1) (cid:38) ).Now, as was mentioned before, high amplitudes play an important role in terms of moments.Although the impact of neglecting them in non-compelled links is globally limited because theselinks are weighted by smaller average path losses λ n ( n = M +1 , . . . , N ), it has to be compensatedin order to satisfy the st-order moment constraint (i.e., the sampled mean has to converge tothe exact value ). For this purpose, small (resp. large) amplitudes need to be underweighted(resp. overweighted). Thus, an underweighting multiplicative factor, denoted f − , is applied toamplitudes of the J first intervals of compelled links; similarly, an overweighting multiplicativefactor f + is applied to amplitudes of the last N − J intervals. (Computation details of factors f − and f + are given in Appendix D.)Let us last notice that the choice for values of M and J is a trade-off between differents aspects:cardinality of the resulting typical set (i.e., tractable number of points), number of simulationruns and accuracy of the histogram. We have determined that M = 2 and J = 3 meet all these The probability set is obtained by normalizing the set of weigths. We recall that the mean E { G } is of particular importance because it is proportional to the average interference power. requirements. IV. N UMERICAL RESULTS
In this section, we present numerical results related to the different methods introduced in thepreceding section. In Section IV-A, we first examine the validity of the original approximationintroduced in Section III, stating that the interference gain G n (and, consequently, G ) does notdepend on the user’s position within its cell. For this purpose, we compare the approximation of G given by (6) with the ’exact’ formula (3). Then, in Section IV-B, we obtain the histogram of theinterference gain G n (one single interferer) by applying the non-uniform partitioning generalizedinverse method described in III-B2. Finally, the MCP method (see III-B3) is used to build upthe histogram of the interference gain for multiple interferers in Section IV-C.We use the following simulation parameters. We consider a system functioning at 1 GHz.We fix the cell radius to R = 700 m, d = 10 m, and the pathloss exponent to . , whichcorresponds to a typical urban environment, as described in the COST-231 reference model [29].The reference distance is chosen to be equal to R . Average path losses λ n , n = 1 , , . . . , N , aredetermined numerically using (5) and are summarized in Table I. A. No shadowing
In this section, we evaluate the proposed approximation (6) against Monte Carlo simulationsperformed on (3). We first consider the contribution of one interfering cell and, in this regard, weexamine two opposite scenarios: one for which the investigated interferer (i.e., AP 1) producesthe largest dynamic range for the intercell interference power undergone by a user in the grey-shaded triangular area of Fig. 2; the other one for which the investigated cell (i.e., AP 13) has thesmallest dynamics. Obviously, both dynamics differently impact the accuracy of our model. Notethat, in both cases, the sum of interference gains (7) reduces to one exponential r.v. Modeled andsimulated pdf’s for above-mentioned cases (a) and (b) are plotted in Fig. 5 and Fig. 6 respectively,and the good match of the curves shows that the proposed method is a good approximation.We then consider the whole set of interfering cells ( N interferers) under frequency reusepatterns FR1 and then FR3, for which results are shown in Fig. 7 and Fig. 8 respectively. Wesee that simulated and modeled probability laws (2) and (7) respectively closely match for bothfrequency reuse patterns. We also note that simulated and approximated curves are closer to oneanother for FR3 than they are for FR1. As explained before for the single-interferer scenario,fluctuations of actual pathlosses G pl ,n ( r n ) , n = 7 , ..., , can be assumed to have about the samedynamic range, but these dynamics are smaller than those of gains G pl ,n ( r n ) , n = 1 , ..., . B. Shadowing, one interferer
In this section, we make use of the non-uniform partitioning generalized inversion methodintroduced in Section III-B2 to obtain a typical set for the interference gain of one interferer. Table II presents the three first moments computed from typical set S (cid:96)n , as compared with theexact moments of the distribution of the interference gain G n . We see that moments issuedfrom the typical set are far beyond the 1% accuracy requirement. The proposed method alsooutperforms the Monte Carlo simulation technique, which cannot be guaranteed to converge forsuch a small number of points.Histograms of the interference gain G n computed from typical set S (cid:96)n is illustrated on Fig. 9for different values of σ dB . C. Shadowing, multiple interferers
We now evaluate the MCP method developed in Section III-B3. We have determined that , iterations of the base MCP algorithm guarantee that the st-order moment computed fromany typical set (whatever σ dB value is considered) converges to its exact value (13). Table IIpresents the values of the st-order moment of G , both exact (analytical) and approximated(computed from the typical set). We can see that the proposed method performs very well forthe whole range of σ dB values.Histograms of the interference gain G computed from typical sets obtained by the MCP methodare illustrated on Fig. 10 (FR1 scenario) and Fig. 11 (FR3 scenario) for different values of σ dB .V. S TATISTICAL MODEL
In Section III, we developed analytical and numerical methods to build up a good approx-imation of the histogram of the interference gain G . In this section, we aim at using thisresult to elaborate a statistical model for G , i.e., a closed form expression of the probabilitylaw, characterized by the shadowing parameter σ dB . This task is challenging in that one singleparametric law is required, that is valid for propagation environments which considerably varydepending upon the shadowing phenomenon (parameter σ dB ), and that is applicable to variousfrequency reuse scenarios (FR1 and FR3).We initialize the modeling process by extracting usefull information from a carefull analysisof the histograms of the interference gain G (see Fig.’s 10 and 11). We first note that G is apositive continuous r.v. We then observe that all curves are asymmetric, and this property is evenmore pronounced for large values of σ dB . In this case, G’s pdf’s also have a sharper peak anda longer, fatter tail, the last of which being a characteristic of heavy-tailed distributions (a.k.a. power distributions ), as already mentioned.Due to the strongly skewed nature of the interference gain distribution for large σ dB ’s, apower-type statistical model turns out to be suitable here. In this regard, a Pareto-like distributionseems to be a good candidate, so we focus, in first approximation, on a 3-parameter Burr-type XII distribution [27]. The Burr distribution has a flexible shape and controllable location andscale, which makes it appealing to fit any given set of unimodal data that exhibits a heavy-tail behavior (e.g., it is an appropriate model for characterizing insurance claim sizes). However, as3 parameters seem to not be sufficient to correctly characterize the interference gain distributionunder those particularly tight constraints, another law is required, which offers greater flexibilityto match the whole range of σ dB values. Such a flexibility is provided by introducing an additionalshape parameter into the Burr distribution, based on the following property [30]: if F ( x ) is acdf, so is ( F ( x )) η , ∀ η > . Thus, we have established a new Burr-based probability law, whosecdf — denoted F G ( x ) — is given by F G ( x ) = − (cid:18) (cid:18) xβ (cid:19) α (cid:19) k η , x > η > α, k, β > (19)where η , α and k are the shape parameters, and β is the scale parameter of the distribution. G’spdf — denoted p G ( x ) — can be easily obtained by deriving (19): p G ( x ) = ηαkβ (cid:18) xβ (cid:19) α − (cid:32)(cid:18) (cid:18) xβ (cid:19) α (cid:19) k − (cid:33) η − (cid:18) (cid:18) xβ (cid:19) α (cid:19) kη +1 . (20)We next establish a parametric family of functions (parameterized by σ dB ) for the interferencegain G by determining empirical formulas for parameters η , α , k , and β . For this purpose, wepropose that all parameters (whatever frequency scenario is considered) be modeled by the same6-parameter function f that has the following expression: f ( σ dB ) = a + a · − σ dB a (cid:18) (cid:18) σ dB a (cid:19) a (cid:19) a ·
11 + (cid:18) σ dB a (cid:19) a , (21)where coefficients a i , i = 1 , , . . . , have been determined empirically and are summarizedin Table III. Corresponding empirical laws f , as functions of σ dB , are plotted on Fig. 12 (FR1scenario) and Fig. 13 (FR3 scenario). The pdf’s of the proposed statistical model are superimposedon histograms obtained by the MCP method for different values of the shadowing parameter σ dB on Fig. 14 (resp. Fig. 15) for the FR1 (resp. FR3) scenario. We now come to the last step ofour modeling process. As seen earlier, MCP-obtained histograms and the proposed Burr-baseddistributions closely match for the whole range of σ dB . However, care must be taken in definingthe range of gains for which our model is valid. And indeed, the Burr-based statistical law needsto be truncated at a maximum value — denoted x t — defined in such a way that the st-order moment constraint holds, which we can write (cid:90) x t xp G ( x ) d x = E { G } , where E { G } is the exact mean (13). As a consequence of this truncation process, a normalizingfactor, A = 11 − P ( x > x t ) , (22)has to be incorporated in both the cdf and pdf of the elaborated model, which are then written AF G ( x ) and Ap G ( x ) respectively. Regarding the empirical law x t as a function of σ dB , we alsopropose the same 5-parameter function for both FR1 and FR3 scenarios: x t ( σ dB ) = a · exp (cid:18)(cid:18) σ dB a (cid:19) a (cid:19) · exp (cid:32) exp (cid:32) − (cid:18) σ dB − a a (cid:19) (cid:33)(cid:33) , (23)where coefficients a i , i = 1 , , . . . , have been determined empirically and are summarized inTable IV. Empirical laws x t , as functions of σ dB , are plotted on Fig. 16 (FR1 scenario) andFig. 17 (FR3 scenario). The normalizing factor may be easily computed by replacing x t by itsactual value in (22). VI. C ONCLUSION AND F UTURE WORK
In this paper, we have proposed a methodology to estimate the statistics of the intercellinterference power in the downlink of a multicellular network. In a propagation environmentsubject only to path loss and multipath Rayleigh fading, we have established an accurate ap-proximated analytical expression for the interference power distribution. Then, considering thecombined effects of path loss, lognormal shadowing and Rayleigh fading, we have proposed asemi-analytical method for the estimation of the pdf of the interference power. Finally, we havedeveloped a statistical model parameterized by the shadowing parameter σ dB and valid on a largerange of values ( [0 , dB). It is our hope that the methods described in this paper are sufficientlydetailed to enable the reader to apply them to other types of environments.A future work will pertain to improving the statistical interference power model by more closelylinking the proposed model developed for a combined Rayleigh fading–lognormal shadowingenvironment to the ’exact’ analytical formula obtained in the case where only Rayleigh fadingwas considered. Another perspective is to apply the proposed methods to other wireless networktopologies (e.g., ad hoc networks,...). A PPENDIX
A. Normalized channel power gain
In this paper, we concentrate on the channel power gain H n ( r n ) = | h n ( m ) | , where h n ( m ) is the instantaneous gain of the channel between AP n and UT . H n ( r n ) can be expressed as a three-factor product: H n ( r n ) = H pl ,n ( r n ) G f ,n G s ,n , (24)where r n represents the distance between UT and AP n (distances r n are functions of UT ’sposition within its cell), and H pl ,n ( r n ) , G f ,n and G s ,n represent the path loss, multipath Rayleighfading and shadowing components respectively. We now further describe these last three com-ponents.The (deterministic) path loss H pl ,n ( r n ) diminishes as the distance r n between UT and AP n increases, based on the common power law [23] H pl ,n ( r n ) = K (cid:18) d r n (cid:19) γ , (25)where K = ( c/ (4 πf d )) is a dimensionless constant, with c being the speed of light, f , theoperating frequency, and d , a reference distance for the antenna far-field; and γ represents thepath loss exponent. In order to make our study independent from the antenna characteristics andthe cell size, we rewrite (25) under the following form: H pl ,n ( r n ) = K (cid:18) d d ref (cid:19) γ (cid:18) d ref r n (cid:19) γ , (26)where d ref is a reference distance, and we introduce the normalized path loss G pl ,n ( r n ) , definedas follows: G pl ,n ( r n ) = (cid:18) d ref r n (cid:19) γ . (27)From (26) and (27), we establish the following relationship: G pl ,n ( r n ) = 1 K (cid:16) d d ref (cid:17) γ H pl ,n ( r n ) . (28)In a similar manner, we define the normalized instantaneous power gain G n ( r n ) as follows: G n ( r n ) = 1 K (cid:16) d d ref (cid:17) γ H n ( r n )= G pl ,n ( r n ) G f ,n G s ,n , (29)where (29) derives from (24) and (28). B. Computation of moments for one interferer
We find the closed form expression of the k th-order moment E (cid:110) ( G n ) k (cid:111) of the statisticaldistribution of the interference gain G n (one interfering cell). We have: E (cid:110) ( G n ) k (cid:111) = E (cid:110) ( G f ,n G s ,n ) k (cid:111) = E (cid:110) ( G f ,n ) k (cid:111) E (cid:110) ( G s ,n ) k (cid:111) , (30)where (30) follows from the independance property of the r.v.’s G f ,n and G s ,n . As G f ,n isexponentially distributed with unit mean, its k th-order moment is given by: E (cid:110) ( G f ,n ) k (cid:111) = k ! (31)As for G s ,n , it has a lognormal distribution with parameters − σ dB / and σ dB ; its raw momentcan be written: E (cid:110) ( G s ,n ) k (cid:111) = exp (cid:18) k ( k − σ dB (cid:19) . (32)Replacing (31) and (32) in (30) leads to (11). C. Computation of moments for multiple interferers
We establish the analytical formula of the k th-order moment E (cid:8) G k (cid:9) of the statistical dis-tribution of the interference gain G (multiple interferers). Using approximation (10), we canwrite: E (cid:8) G k (cid:9) = E (cid:32) N (cid:88) n =1 λ n G f ,n G s ,n (cid:33) k = E (cid:88) a : | a | = k k ! a ! Z a , (33)where the following notation is used: • a = ( α , α , . . . , α N ) , α n ∈ N , n = 1 , , . . . , N , is an N -dimensional vector whose sum ofcomponents is | a | = N (cid:88) n =1 α n ; • the multifactorial a ! is such that a ! = N (cid:89) n =1 ( α n !) ; • the variable Z a is defined as follows: Z a = ( λ G f , G s , ) α ( λ G f , G s , ) α · · · ( λ N G f ,N G s ,N ) α N . Using (30), we can further develop (33), which gives (12).
D. Computation of correction factors
We determine the correction factors used in the MCP method described in Section III-B3.Recall that the technique consists, for non-compelled links, in randomly selecting intervals froma subset containing only the J highest-probability (i.e., smallest-amplitude) intervals. But, ashigh-amplitude intervals never appear in this random process, small amplitudes get overweightedin non-compelled links, which must be compensated in compelled links, where small (resp. large)amplitudes need to be underweighted (resp. overweighted), in such a way that the st-ordersampled moment converges to its exact value. Thus, in order to satisfy the mean constraint, anunderweighting multiplicative factor, denoted f − , is applied to amplitudes of the J first intervalsof compelled links; similarly, an overweighting multiplicative factor f + is applied to amplitudesof the last N − J intervals. We now compute these two correction factors.Let us first see how each interfering link contributes to the st-order moment of the intercellinterference gain G . For each compelled link n , n = 1 , . . . , M , we can write : E { G n } = J (cid:88) j =1 δ j g j = J (cid:88) j =1 δ j g j (cid:124) (cid:123)(cid:122) (cid:125) A + J (cid:88) j = J +1 δ j g j (cid:124) (cid:123)(cid:122) (cid:125) B = 1 , where G n = G f ,n G s ,n (approximation (14), with λ n = 1 ), and, by construction of the typicalset S (cid:96)n , A + B = 1 , ∀ σ dB . For each non-compelled link n , n = M + 1 , . . . , N , G n ’s mean is E { G n } = J (cid:88) j =1 δ (cid:48) j g j < , where the probability set (cid:8) δ (cid:48) j (cid:9) is given by (17). So, if no correction factors are introduced, thecontribution of all (compelled and non-compelled) links to the intercell interference gain G gives Note that, for the sake of simplification, each P -element interval is reduced to its center of mass — denoted g j . the following mean: E { G } = M (cid:88) n =1 λ n E { G n } (cid:124) (cid:123)(cid:122) (cid:125) =1 + N (cid:88) n = M +1 λ n E { G n } (cid:124) (cid:123)(cid:122) (cid:125) < < N (cid:88) n =1 λ n , where N (cid:88) n =1 λ n = A N (cid:88) n =1 λ n + B N (cid:88) n =1 λ n (34)is the exact mean (13).Let us now introduce the correction factors f − and f + into compelled links, as describedpreviously. G ’s st-order moment — denoted E cor { G } — then becomes: E cor { G } = M (cid:88) n =1 λ n (cid:32) J (cid:88) j =1 δ j f − g j + J (cid:88) j = J +1 δ j f + g j (cid:33) + N (cid:88) n = M +1 λ n J (cid:88) j =1 αδ j g j = M (cid:88) n =1 λ n (cid:0) Af − + Bf + (cid:1) + N (cid:88) n = M +1 λ n αA = A (cid:32) f − M (cid:88) n =1 λ n + α N (cid:88) n = M +1 λ n (cid:33) + Bf + M (cid:88) n =1 λ n . (35)In order for both exact and actual means to be equivalent (i.e., (34) ≡ (35)), we need to solve thefollowing system: f − M (cid:88) n =1 λ n + α N (cid:88) n = M +1 λ n = N (cid:88) n =1 λ n f + M (cid:88) n =1 λ n = N (cid:88) n =1 λ n which leads to f − = 1 − ( α − N (cid:80) n = M +1 λ nM (cid:80) n =1 λ n (36) f + = 1 + N (cid:80) n = M +1 λ nM (cid:80) n =1 λ n . (37)Note that we have f + > and, as α (cid:38) , f − (cid:46) . R EFERENCES [1] N. Himayat, S. Talwar, A. Rao, and R. Soni, “Interference management for 4G cellular standards [WiMAX/LTE UPDATE],”
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VERAGE PATH LOSSES λ n , n = 1 , , . . . , N , DEFINED BY (6),
IN DECREASING ORDER OF IMPORTANCE . E
ACH INDEX m OFCOLUMN AP m CORRESPONDS TO INDEX OF AVERAGE PATH LOSS λ n ( n (cid:54) = m , IN GENERAL ). FR ( N = 18 ) FR ( N = 6 ) n λ n AP m n λ n AP m .
467 1 1 0 .
568 82 3 .
588 2 2 0 .
426 183 1 .
708 6 3 0 .
307 104 1 .
069 3 4 0 .
219 165 0 .
767 5 5 0 .
178 126 0 .
663 4 6 0 .
158 147 0 .
568 88 0 .
426 189 0 .
316 710 0 .
307 1011 0 .
260 912 0 .
219 1613 0 .
188 1714 0 .
178 1215 0 .
158 1416 0 .
145 1117 0 .
118 1518 0 .
107 13 [25] S. Asmussen, K. Binswanger, and B. Hojgaard, “Rare event simulation for heavy-tailed distributions,”
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Journal ofProbability and Statistical Science , vol. 7, no. 2, pp. 153–171, August 2009. TABLE IIE
XACT AND APPROXIMATED MOMENTS FOR ONE SINGLE INTERFERER AND FOR MULTIPLE INTERFERERS . no shadowing( σ dB = 0 dB) intense shadowing( σ dB = 12 dB)exact approximated exact approximated E { ( G n ) } . E (cid:8) ( G n ) (cid:9) . · . · E (cid:8) ( G n ) (cid:9) . · . · E { ( G ) } (FR1) .
25 17 .
10 17 .
25 17 . E { ( G ) } (FR3) .
857 1 .
857 1 .
857 1 . TABLE IIIC
OEFFICIENTS a i , i = 1 , , . . . , , OF THE EMPIRICAL LAWS OF PARAMETERS η , α , k , AND β (FR1 AND
FR3
SCENARIOS ). FR FR a a a a a a a a a a a a η α .
93 0 .
87 65 1 7 . . .
38 0 .
94 39 .
90 2 .
00 8 .
30 3 . k .
65 2 .
18 3 . .
39 4 .
75 2 .
06 0 12 .
70 2 .
35 2 .
07 11 .
00 6 . β .
04 16 .
44 13 .
45 9 6 .
35 2 .
56 1 .
81 24 .
35 3 .
60 2 .
77 1 .
77 1 . TABLE IVC
OEFFICIENTS a i , i = 1 , , . . . , , OF THE EMPIRICAL LAWS OF PARAMETER x t (FR1 AND
FR3
SCENARIOS ). a a a a a FR .
56 6 .
06 1 .
84 5 .
27 2 . FR .
71 5 .
10 1 .
89 6 .
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Fig. 1. Hexagonal model for a 19-cell cellular network. The largest distance from a user to its serving AP is denoted R . Westudy the interference power undergone by the mobile receiver UT in the central cell (numbered ).UT r n r θ AP nR Fig. 2. Because of the particular symmetry of the network geometry, we need only study the interference gain distribution fora user located within one the twelve dashed triangular areas. For illustration purposes, we will consider the grey-shaded sector. δ = 10 − δ = 9 · − ∆ i = 10 − δ = 9 · − ∆ i = 10 − (a) (b) (1 ≤ i ≤ P )( P + 1 ≤ i ≤ P ) . . . . . . . . . . . . . . . . . . . Fig. 3. Illustration of the general inverse method with non-uniform partitioning ( J = 3 , P = 9 ): (a) non-uniform partitioningof the [0 , segment; (b) uniform partitioning of interval I . random drawings and permutations δ δ λ B (cid:48) λ A (cid:48) interferer 1 (compelled) δ δ λ A (cid:48) λ B (cid:48) interferer 2 (compelled) δ δ A B typical set S (cid:96)n amplitudesprobabilities δ δ δ δ δ δ δ δ δ δ δ δ . λ A (cid:48) + λ B (cid:48) λ B (cid:48) + λ B (cid:48) λ B (cid:48) + λ A (cid:48) λ A (cid:48) + λ A (cid:48) λ A (cid:48) λ A (cid:48) λ B (cid:48) λ A (cid:48) λ A (cid:48) + λ A (cid:48) λ A (cid:48) + λ B (cid:48) λ B (cid:48) + λ A (cid:48) λ B (cid:48) + λ B (cid:48) + λ A (cid:48) + λ A (cid:48) + λ B (cid:48) + λ A (cid:48) + λ A (cid:48) + λ A (cid:48) + λ A (cid:48) + λ A (cid:48) interferers 1 and 2 (compelled) λ A (cid:48) λ A (cid:48) λ A (cid:48) λ A (cid:48) interferer 3interferer 4 S TE P S TE P typical set S L Fig. 4. Illustration of the MCP method for N = 4 interfering cells, M = 2 compelled links, and J = 2 intervals per link(denoted A and B , with respective probabilities δ and δ ). Each A (cid:48) (resp. B (cid:48) ) represents one random permutation of A (resp. B ). simulationmodel
400 10 20 30 50 g . . . p G AP ( g ) Fig. 5. Simulated vs. modeled pdf of the intercell interference power with no shadowing when AP is the only interferer.Since AP produces the largest dynamics for the interference power undergone by a user in the grey-shaded sector of Fig. 2with only one interfering cell, these curves correspond to the worst-case scenario for validating our approximation. simulationmodel g . . . . . . p G AP ( g ) Fig. 6. Simulated vs. modeled pdf of the intercell interference power with no shadowing when AP is the only interferer.AP produces the smallest dynamics for the interference power undergone by a user in the grey-shaded sector of Fig. 2 withonly one interfering cell (best match for our model). simulationmodel g
10 6050403020 p G FR ( g )0 . . . . . . . Fig. 7. Simulated vs. modeled pdf of the intercell interference power G for frequency reuse pattern FR1. simulationmodel g p G FR ( g )0 . . . . . Fig. 8. Simulated vs. modeled pdf of the intercell interference power G for frequency reuse pattern FR3. σ dB = 0 σ dB (cid:37) σ dB = 4 σ dB = 12 ··· gp G n ( g ) E { G n } = 1 Fig. 9. Histograms of the interference gain G n (one interferer) for different values of σ dB . σ dB = 12 σ dB (cid:37) gσ dB = 0 p G ( g ) E { G } = 17 . Fig. 10. Histograms of the interference gain G obtained by the MCP method (FR1 scenario). σ dB (cid:37) σ dB = 12 σ dB = 0 gp G ( g ) E { G } = 1 . Fig. 11. Histograms of the interference gain G obtained by the MCP method (FR3 scenario). σ dB σ dB σ dB σ dB αβ k . . η . . . . . . . . . Fig. 12. Empirical laws η , α , k , and β as functions of σ dB (FR1 scenario). σ dB σ dB σ dB σ dB αβ k . η . . . Fig. 13. Empirical laws η , α , k , and β as functions of σ dB (FR3 scenario). σ dB = 12 σ dB (cid:37) gσ dB = 0 simulation E { G } = 17 . model p G ( g ) Fig. 14. Comparison of MCP histograms and modeled cdf of the interference gain G for σ dB = 0 , , , , (FR1 scenario). σ dB (cid:37) σ dB = 12 σ dB = 0 simulation gp G ( g ) model E { G } = 1 . Fig. 15. Comparison of MCP histograms and modeled cdf of the interference gain G for σ dB = 0 , , , , (FR3 scenario). σ dB , , x t Fig. 16. Truncation gain x t as a function of σ dB (FR1 scenario). σ dB x t Fig. 17. Truncation gain x t as a function of σ dBdB