Outage Analysis of Uplink Two-tier Networks
aa r X i v : . [ c s . I T ] A ug Outage Analysis of Uplink Two-tier Networks
Zolfa Zeinalpour-Yazdi and Shirin Jalali
Abstract —Employing multi-tier networks is among the mostpromising approaches to address the rapid growth of the datademand in cellular networks. In this paper, we study a two-tieruplink cellular network consisting of femtocells and a macrocell.Femto base stations, and femto and macro users are assumedto be spatially deployed based on independent Poisson pointprocesses. We consider an open access assignment policy, whereeach macro user based on the ratio between its distances from itsnearest femto access point (FAP) and from the macro base station(MBS) is assigned to either of them. By tuning the threshold,this policy allows controlling the coverage areas of FAPs. Fora fixed threshold, femtocells coverage areas depend on theirdistances from the MBS; Those closest to the fringes will have thelargest coverage areas. Under this open-access policy, ignoringthe additive noise, we derive analytical upper and lower boundson the outage probabilities of femto users and macro users thatare subject to fading and path loss. We also study the effect ofthe distance from the MBS on the outage probability experiencedby the users of a femtocell. In all cases, our simulation resultscomply with our analytical bounds.
Index Terms —Heterogeneous networks, Uplink communica-tion, Outage, Open access policy, Poisson point process
I. I
NTRODUCTION
Wireless cellular networks, originally designed for voicecommunications, are nowadays commonly used for surfing theInternet or communicating image, audio or video files. Thismassive unpredicted overhead load has urged communicationengineers to develop new approaches to design and employ-ment of cellular communication systems. One of such rela-tively new techniques, which has been proved to be successful,is employing multi-tier networks. For instance, in the case oftwo-tier networks, the existing cellular network is overlaid by femtocells , which are employed by users in an ad-hoc mannerat their homes or offices.Analytical performance evaluation of cellular networks hasalways been a complicated task. Modeling various aspectsof cellular networks, such as the physical channel itself,has been a cornerstone of such analysis and therefore thesubject of extensive research for many years. Modeling theusers’ and cells’ locations is another aspect of a cellularnetwork that also plays a major role in analytical evaluations.Traditionally, the idealized grid model has been employed tomodel the locations of the cells and their coverage areas. Thismodel, although simple to describe, is intractable for mostanalytical evaluations and is also arguably not very accurate.This is especially true in heterogenous networks with ad hocemployment of small cells. More comprehensive and recentmodels for spatial distributions of the cells and users are
Z. Zeinalpour-Yazdi is with the Department of Electrical and computerEngineering, Yazd University, Yazd, Iran (e-mail: [email protected]),S. Jalali is with the Department of Electrical Engineering, Princetonuniversity, NJ 08540 (e-mail: [email protected]) models based on stochastic geometric tools such as Poissonpoint process (PPP). Such models are advantageous from twomain perspectives: first, they provide a more realistic modelof cellular networks compared to the traditional grid-basedmodels, second, they make the analysis more tractable.In this paper, we analyze the outage performance of anuplink two-tier network with a MBS located at the center of acircle representing its coverage area; macro users (MU), femtousers (FU) and FAPs are assumed to be spatially distributedwithin the circle randomly and independently according toPPPs with different densities. We consider the open accesspolicy studied in [1], [2] for downlink communication. Thismodel covers closed access policy as a special case and allowsoptimizing the coverage areas of the FAPs when the systemparameters vary. Using this model, we derive tight upper andlower bounds on the outage probabilities of users covered bythe MBS and also FUs and MUs that are covered by FAPs. Toachieve this goal we first derive upper and lower bounds on theLaplace transform of the number of MUs serviced by the basestation. We also derive the Laplace transform of the numberof MUs covered by a FAP located at a specific distance fromthe MBS. Employing the Laplace transforms of the numberof users in each group, plus some geometric analysis, webound our desired outage probabilities. Our simulation resultsconfirm and comply with our bounds.
A. Related work
While employing PPP as a stochastic model for users oraccess points distributions was originally proposed in 1997 in[3]–[5], it was not until recently that this model was used foranalyzing the performance of wireless cellular networks. (Re-fer to [6] for a review of this model.) The stochastic geometric-based models such as PPP was employed by Baccelli et al. in[7] to analyze large mobile ad hoc network (MANET) and byAndrews et al. in [8] to study the downlink performance ofcellular networks. Later, this model was used for analyzingthe downlink performance of multi-tier networks [9]–[11].However, as mentioned in [12], similar analysis for uplinkcommunication has been missing until very recently.Multi-tier networks have been studied from different per-spectives such as power control [13], [14], spectrum allocation[15], [16], and exploiting cognitive radio techniques [17], [18].These are just few examples of some related work and byno means are meant to be a comprehensive review of theliterature. (See [12], [19], [20] and the references therein fora relatively comprehensive review of the literature.)Analytic study of the outage performance of a single-tiernetwork with nodes distributed according to a PPP is done in[21]. Uplink performance of two-tier networks has been stud-ied in the literature under different models and approximations.
While most of the work on this topic has been on traditionalgrid model, recently there has been several results on analyzinguplink performance of two-tier networks under PPP model forusers and access points. Chandrasekhar et al. study outageprobabilities of femto and MUs distributed according to PPPsin a reference macrocell in [22]. The authors consider aCDMA-based model under closed access and approximate theoutage probability. Xia et al. in [23] compare closed accessversus open access policy in an uplink communication. In theiranalysis, they consider a reference macrocell with the basestation located at the center, and a single FAP located at aspecific distance from the base station. The MUs are assumedto be distributed independently at random. They suggest thatwhile for orthogonal multiple access schemes such as TDMAor OFDMA the choice of open versus closed depends on theusers density, in non-orthogonal schemes such as CDMA openaccess is strictly better than closed access.The uplink performance of macrocells overlaid with femto-cells is also studied in [24]. There, while the authors considerPPP spatial distribution for MUs, FUs and FAPs, the usersassignment policy is closed access, and by assuming a TDMAscheme they limit the number of active users in each femtocellper time slot to one. In [25], the authors study the distributionof the signal to interference plus noise ratio (SINR) in bothuplink and downlink, when time division duplex (TDD) isemployed. In their setup, the users of each tier are distributedaccording to a PPP and each user connects to the closest basestation.In an independent work, which the authors became awareof right before submitting this paper, Bao et al. analyzethe interference and outage performance of a two-tier uplinknetwork under closed access policy [26]. The authors of [26]also study the open access policy in a subsequent paper [27].While the ultimate goals in [27] and this paper are the same,there are some major differences betweens the two. First,unlike this paper, in [27], each femtocell is assumed to havea fixed coverage area, and a MU is handed off to the FAPif it falls within that fixed coverage area. Here, we considera different open access policy, where each MU decides toconnect to either its closest FAP or the MBS, based onits distances from them. This policy leads to FAPs havingdifferent coverage areas, depending on their distances to theMBS. This assignment policy introduces new geometricalaspects to our outage analysis. Second, unlike [27], we deriveclosed-form expressions for our upper and lowers bounds onthe outage probabilities of MUs and FUs. For a MU servicedby the MBS, we study and bound its outage performance asa function of its distance of the FAP from the MBS. Finally,here we consider multi-carrier frequency hopping modulation,which provides a decentralized alternative to OFDM. In [27],the authors consider a single shared channel for all users.Finally, one of the reviewers pointed us to the work of ElSawyet al. [28], which has appeared on Arxiv after our initialsubmission. In [28], the authors study the uplink performanceof a multi-tier network under a different access policy whereeach user connects to its closest access point (femto or macro).
B. Notation
Calligraphic letters such as X and Y represent sets. The sizeof set X is denoted by |X | . Given sample space Ω and event E ⊆ Ω , E is an indicator random variable that is one whenevent E happens. For ≤ i ≤ j ≤ n , x ji , ( x i , x i +1 , . . . , x j ) .Also, for simplicity x i = x i . Uppercase letter characters suchas X and Y are used for matrices and random variables. C. Paper organization
The organization of the paper is as follows. Section IIreviews the network model studied in this paper from variousperspectives: modulation technique, spatial distributions ofusers, channel model and access policy. In Section III, westudy the users density distributions. The results of this sectionis used extensively in the following sections in analyzing theperformance of the system. In Section IV, which containsthe main results of the paper, we analyze the femto andMUs outage probabilities. Section V presents the simulationresults and compares them with our analytical bounds. Finally,Section VI concludes the paper.II. S
YSTEM MODEL
A. MCFH technique
Orthogonal frequency devision multiplexing (OFDM) is awidely popular multiple access method in wireless networks,and has received a lot of attention in recent years. In anOFDM-based multiple access system, the carrier frequenciesare assigned by the central node. (Three different methodsfor assigning frequencies are described in [29].) However,this centralized frequency assignment is not quite desirablefor emerging decentralized wireless cellular networks such asfemtocells, where, due to practical challenges’, it is preferredto minimize the coordination between the central and the femtobase stations.Multicarrier frequency-hopping (MCFH) modulation intro-duced in [30], and later analyzed by various researchers [31]–[34], provides a decentralized alternative to OFDM mod-ulation. In MCFH, similar to OFDM, all sub-carriers areorthogonal to each other. However, unlike OFDM, in a multi-user setup, the carriers are not assigned to the users by a centralnode, and the users are allowed to randomly and independentlyselect their carriers. In addition to being decentralized, anotheradvantage of MCFH to OFDM, as will become clear through-out the paper, is that it makes the model more amenable todirect analysis. The results of such analysis will provide insighton how to select the systems’ parameters in an OFDM-basedsystem as well. In this paper, we assume that all users adoptMCFH modulation. While MCFH is clearly different fromOFDM, most of our results will continue to hold for OFDM-based systems with some mild adjustments.In MCFH, the available bandwidth is divided into n s non-overlapping adjacent subbands. Each subband respectively isdivided into n h equispaced frequencies. Hence, overall, therewill be n s n h available subchannels . (It is usually said that thesystem’s processing gain ( G ) is equal to n s n h .) At each time,each user uniformly at random selects one of the n h carriers in User 2subband 1 subband 2 subband 3User 1
Fig. 1. Depiction of MCFH frequency assignments. each subband. Fig. 1 shows the carrier selections in a simpleMCFH system with n s = 3 and n h = 4 and two users. Asshown in the figure, since unlike OFDM, users select theircarriers independently with no coordination, it is possible thattwo users send data over the same frequency simultaneously. B. Spatial distribution
For spatial distribution of MUs, FUs and FAPs, we followthe model introduced in [1]. We consider a MBS b m located atthe center of a circle of radius R denoted by S m . FAPs A f aredistributed according to a PPP with density λ f . Therefore, thenumber of FAPs ( |A f | ) is distributed as Poiss(¯ n fap ) , where ¯ n fap , πR λ f . Conditioned on |A f | = m , the locations ofthe m FAPs are uniformly distributed over S m . Independently,MUs U m are distributed based on a PPP with density µ m .Note that “MUs” are users that are not inside a home, office,etc. that is equipped with a FAP. However, a MU might beserved by a FAP based on its distance from the MBS and thelocations of surrounding FAPs. Finally, FUs of FAP a f ∈ A f are distributed according to a PPP with density µ f restrictedto a ring of internal radius r f and width ∆ centered at a f .For FAP a f ∈ A f , let U f ( a f ) and U m ( a f ) denote the set ofFUs and MUs serviced by the FAP a f , respectively. Clearly, ∪ a f ∈A f U m ( a f ) ⊆ U m . Various studies indicate that open access policies havesuperior performance both from the perspective of the FUs(in uplink) and MUs (in downlink). Therefore, in this paperwe focus on a two-tier network with open access policy. Thespecific access policy that we consider is described in SectionII-D.
Remark 1:
In our analysis we consider a single MBSlocated at the center of a circle of radius R . In reality ofcourse there are more MBSs. The placement of the macrocellscan be modeled either as a deterministic process or randombased on an independent PPP with density λ m . In both cases,it is reasonable to assume that each MU connects to itsclosest MBS, and hence divide the plane based on the Voronoipartition determined by the locations of MBSs. Assuming thatthe MBSs employ one of the known frequency reuse methods,are hence orthogonalize the users of neighboring macrocells,then, without loss of generality, in the analysis one can focuson the case where there is only one MBS. In a random setting,where MSBs are employed according to a PPP of density λ m ,it is proved in [35] that the expected number of FAPs in a“typical” macrocell becomes equal to λ f /λ m . Based on thisresult, choosing λ m = 1 / ( πR ) , the expected number of FAPsin a “typical” macrocell is equal to λ f /λ m = πR λ f , which isconsistent by our model in this paper. By controlling radius R ,we can study the effect MBSs’ density λ m on the performance. C. Channel Model
We consider both small scale fading and path loss. Let h iu,a f and h iu,b m denote the fading coefficients corresponding to thechannel in subband i ∈ [1 : n s ] from user u to FAP a f andto MBS b m , respectively. We consider a slow-fading channelmodel, and assume that the fading coefficients remain constantduring the whole coding block. Furthermore, we assume thatthe coefficients corresponding to different subbands and alsodifferent channels are all independent. The channel coefficientsare assumed to have a Rayleigh distribution. That is, thepower attenuation coefficient | h iu,a | , where a ∈ { a f , b m } ,is exponentially distributed as P (cid:0) | h iu,a | > x (cid:1) = e − x/σ ,for x ≥ . The path loss affecting the signal transmittedby user u to base station a , a ∈ { a f , b m } , is modeled as PL u,a = L d αu,a , where L is path loss at unit distance, and α > denotes the attenuation factor [2]. D. Access policy
Consider macro user u ∈ U m . For user u and FAP or MBS a , let d ( u, a ) denote their Euclidian distance. Further, let d ( f ) u denote the distance between user u and its nearest FAP, i.e., d ( f ) u , min { d ( u, a f ) : a f ∈ A f } . As mentioned earlier, wefocus on open access policy, where MUs can also be servicedby FAPs. We consider the following open access policy, whichwas considered in [1]. Let κ < be a parameter of the system.Then, according to this assignment policy,1) if d ( f ) u m < κd ( u m , b m ) , then MU u m is assigned to itsclosest FAP,2) if d ( f ) u m ≥ κd ( u m , b m ) , then MU u m is assigned to theMBS b m .Letting κ = 0 , requires all MUs to be serviced by thebase station, which is equivalent to having a closed accessassignment policy. κ controls the coverage areas of FAPs andincreasing it enlarges the coverage areas.As defined earlier, U m ( a f ) ⊂ U m denotes the set ofMUs that are serviced by FAP a f ∈ A f . Let U ( − f ) m de-note all MUs that are not serviced by FAPs, i.e., U ( − f ) m = U m \ ( ∪ a f ∈A f U m ( a f )) .III. U SERS DENSITY DISTRIBUTION
Before analyzing the signal to interference ratios (SIR)experienced by different users in different groups, in thissection we study the distributions of N a f f , |U f ( a f ) | , N a f m , |U m ( a f ) | and N b m m , |U ( − f ) m | . By our assumption, the FUsare distributed in a ring of width ∆ and internal radius r f .Hence, N a f f ∼ Poiss(¯ n fu ) , where ¯ n fu , π (( r f + ∆) − r f ) µ f denotes the expected number of FUs in each FAP, and itsLaplace transform Φ N aff ( s ) is equal to Φ N aff ( s ) , E[e − sN aff ] = e ¯ n fu (e − s − . (1)Also, let ¯ n mu , πR µ m denote the expected number of allMUs.Consider MBS b m and FAP a f shown in Fig. 2 that arelocated at distance r from each other. MU u m is served by FAP a f instead of b m , if d ( u m , a f ) < κd ( u m , b m ) , where κ < .Translating this condition into cartesian coordinate dimensions PSfrag replacements b m a f r r c Fig. 2. MUs served by FAP a f located at distance r from MBS b m . with the origin located at b m and the x -axis along the lineconnection b m to a f , we obtain ( x − r ) + y ≤ κ ( x + y ) ,or (1 − κ ) x + (1 − κ ) y − rx + r ≤ . In other words, ( x − r − κ ) + y ≤ r (cid:16) − κ ) − − κ (cid:17) , which is equivalent to a circle of radius r c = κr − κ centeredat ( r/ (1 − κ ) , . Therefore, the coverage area of each FAPdepends on its distance from the base station. As the distanceincreases, the coverage area, and the expected number ofcovered MUs increase as well. In summary, given FAP a f located at distance d f = d ( b m , a f ) , the coverage area of a f for MUs, i.e., the area in which MUs are serviced by FAP a f is a circle of radius √ γd f , where γ , κ (1 − κ ) . (2)Therefore, N a f m ∼ Poiss( πγd f µ m ) , where ¯ n f mu , πγd f µ m ,and Φ N afm ( s | d f ) , E[e − sN afm | d ( a f , d f ) = d f ] = e ¯ n f mu (e − s − . (3)In some cases, in our analysis we are interested in thedistribution of N a f m − , conditioned on N a f m ≥ . For thatwe define, Φ + N afm ( s | d f ) , E (cid:2) e − s ( N afm − (cid:12)(cid:12) d ( a f , d f ) = d f , N a f m ≥ (cid:3) = ( e ¯ n f mu (e − s − − e − ¯ n f mu − e − ¯ n f mu )e s . (4)Finally, we study N b m m . Conditioned on A f (locations ofFAPs), N b m m is a Poisson random variable of mean µ m S − f ,where S − f denotes the area exclusively covered only by b m and not FAPs. Therefore, Φ N bmm ( s ) = E[e µ m S − f (e − s − ] . Theorem 1:
For s ≥ , the Laplace transform of N b m m , Φ N bmm ( s ) , satisfies Φ N bmm ( s ) ≥ e ¯ n mu (e − s − and Φ N bmm ( s ) ≤ e ¯ n mu (e − s − n fap ( τ ( s ) − +e γ − − ¯ n fap + γ − log( γ ¯ n fap ) , where τ ( s ) , e (1 − e − s ) γ ¯ n mu − − e − s ) γ ¯ n mu , (5)and γ is defined in (2). Proof:
Since N b m m ≤ |U m | always holds, for s ≥ , e − sN bmm ≥ e − s |U m | . Therefore, Φ N bmm ( s ) ≥ E[e − s |U m | ] =e ¯ n mu (e − s − . If the coverage areas of the FAPs do not overlap,then S − f = π ( R − γ P a f ∈A f d ( a f , b m )) . In general,the regions might overlap, and therefore S − f ≥ π ( R − γ P a f ∈A f d ( a f , b m )) . This lower bound clearly is a function of the locations of the FAPs, and can be negative. Let E denote the event that |A f | ≤ γ − . However, if E holds,then γ P a f ∈A f d ( a f , b m ) ≤ γR |A f | ≤ R , and π ( R − γ P a f ∈A f d ( a f , b m )) ≥ . We employ this observation toderive an upper bound on Φ N bmm ( s ) . By the law of totalexpectation Φ N bmm ( s ) = E[e µ m S − f (e − s − |E ]P( E )+E[e µ m S − f (e − s − |E c ]P( E c ) ≤ E[e µ m S − f (e − s − |E ] P( E ) + P( E c ) . (6)On the other hand, E[e µ m S − f (e − s − |E ] ≤ E[e µ m π ( R − γ P af ∈A f d ( a f ,b m ))(e − s − |E ]= e ¯ n mu (e − s − P ( E ) ⌊ γ − ⌋ X n =0 E[e − µ m πγ (e − s − n P i =1 d i ] P( |A f | = n ) (a) = e ¯ n mu (e − s − P ( E ) ⌊ γ − ⌋ X n =0 τ n ( s ) P( |A f | = n ) ≤ e ¯ n mu (e − s − P ( E ) ∞ X n =0 τ n ( s ) P( |A f | = n )= e ¯ n mu (e − s − P ( E ) e ¯ n fap ( τ ( s ) − , (7)where ( a ) holds because E[e (1 − e − s ) µ m πγd ( a f ,b m ) ] = Z R e (1 − e − s ) µ m πγr rR dr = e (1 − e − s ) γ ¯ n mu − − e − s ) γ ¯ n mu = τ ( s ) . By the Chernoff bound, for x > , P( E c ) = P( |A f | > γ − ) ≤ E[e x |A f | ]e x/γ = e ¯ n fap(e x − e x/γ . Optimizing the bound by choosing x as the solution of ¯ n fap e x − γ − = 0 , we obtain P( E c ) ≤ e γ − − ¯ n fap + γ − log( γ ¯ n fap ) . (8)Combining (6), (7) and (8) yields the desired result.In our outage analysis presented in the proceeding sections,in some cases we study the case where we know that thereexists one FAP a f at distance d f from b m . In those cases, itwill be useful to define the conditional Laplace transform of N b m m as Φ N bmm ( s | d f ) . Theorem 2:
For s ≥ and d f ∈ (0 , R ) , we have Φ N bmm ( s | d f ) ≥ e ¯ n mu (e − s − , and Φ N bmm ( s | d f ) ≤ e (¯ n mu − ¯ n f mu )(e − s − (1 − e − ¯ n fap ) τ ( s ) e ¯ n fap ( τ ( s ) − + e γ − − ¯ n fap + γ − log( γ ¯ n fap ) − e − ¯ n fap , where τ ( s ) and γ are defined in (5), and (2), respectively,and as before ¯ n f mu = πγd f µ m , ¯ n mu = πR µ m , and ¯ n fap = πR λ f . Proof:
The proof closely follows the steps of the proofof Theorem 1. The only difference is that here we know that |A f | ≥ and that one FAP is located at distance d f from b m .Therefore, S − f = π ( R − γd f − γ P a ′ f ∈A f \ a f d ( a ′ f , b m )) . Define event E as before. Then, again by the law of totalexpectation, Φ N bmm ( s | d f ) ≤ E[e µ m S − f (e − s − |E , |A f | >
0] P(
E||A f | > E c ||A f | > . Note that
E[e µ m S − f (e − s − |E , d f ] ≤ E[e µ m π ( R − γd f − P a ′ f ∈A f \ af d ( a ′ f ,b m ))(e − s − |E , d f ]= e (¯ n mu − ¯ n f mu )(e − s − P ( E||A f | > ⌊ γ − ⌋ X n =1 E[e − µ m πγ (e − s − n − P i =1 d i ] . P( |A f | = n ||A f | > (¯ n mu − ¯ n f mu )(e − s − P ( E||A f | > − e − ¯ n fap ) ⌊ γ − ⌋ X n =1 τ n − ( s ) P( |A f | = n ) ≤ e (¯ n mu − ¯ n f mu )(e − s − P ( E||A f | > − e − ¯ n fap ) ∞ X n =0 τ n − ( s ) P( |A f | = n ) ≤ e (¯ n mu − ¯ n f mu )(e − s − P ( E||A f | > − e − ¯ n fap ) τ ( s ) e ¯ n fap ( τ ( s ) − . (9)Furthermore, P( E c ||A f | ≥
1) = P( E c )P( |A f |≥ ≤ e γ − − ¯ n fap+ γ − γ ¯ n fap) − e − ¯ n fap , where the last step follows from (8).IV. O UTAGE ANALYSIS
In this section we analyze the outage performance of MUsand FUs in the uplink network described in Section II. Weassume that every user equipment employs power control tocompensate for the effect of path loss. By power control,MUs serviced by the MBS intend to achieve received powerlevels of P m . Similarly, FUs and MUs serviced by FAPs adjusttheir transmitted power to achieve received power of P f . Wefurther assume that the performance of the users is primarilylimited by the interference caused by other users of both tiers.Therefore we ignore the effect of additive Gaussian noise inour analysis.To bound the outage probability, in each case, we firstcompute the signal to interference ratio (SIR) experiencedby user equipments. The derived SIRs are probabilistic anddepend on channel coefficients, and users locations. Then, webound the outage probabilities by employing results we provedin the previous section. A. MU served by a FAP
Consider FAPs a f and ˆ a f ∈ A f \ a f . The distance betweenuser u that is covered by ˆ a f is usually much smaller thanthe distance between u and a f . That is, d ( u, ˆ a f ) ≪ d ( u, a f ) ,or d ( u, ˆ a f ) d ( u,a f ) ≪ . Therefore, in evaluating the performance ofusers covered by a f , unless the density of FAPs ( λ f ) is verylarge, the term corresponding to the interference caused byusers (macro or femto) covered by other FAPs is negligiblecompared to the other terms. Making this approximation, theupload SIR experienced by user u m ∈ U m ( a f ) in subband i ∈ { , , . . . , n s } is equal to SIR m,f = P f | h ium,af | n s I m,f , (10)where I m,f = X u f ∈U f ( a f ) P f | h iu f ,a f | G + X ˆ u m ∈U m ( a f ) \ u m P f | h i ˆ u m ,a f | G + X ˆ u m ∈U ( − f ) m (cid:16) d (ˆ u m , b m ) d (ˆ u m , a f ) (cid:17) α P m | h i ˆ u m ,a f | G . (11)In (11), the interference terms are caused by the FUs of FAP a f , the other MUs of FAP a f , and the MUs serviced bythe MBS, respectively. In our model, from the perspectiveof the outage performance of MUs served by a FAP, thereis no difference between MUs and FUs covered by thatFAP. Therefore, statistically, (10) and (11) also describe theperformance experienced by FUs of a f . Remark 2:
While it might seem that we have assumedthe same attenuation factor α for all different links in thenetwork, in fact, the results do not change in general wherethe path-loss exponent of outdoor and cross-wall (outdoor-indoor) transmissions are assumed to be equal ( α ) and largerthan the path-loss exponent of indoor transmissions ( β ). Toobserve this, note that in (10), exponent β only affects usersin U f ( a f ) . However, due to our power control assumption,terms like d β ( u f , a f ) do not appear in the interference I m,f .The same is true for our analysis presented in the next sectioncorresponding to MUs served by the MBS.Consider FAP a f ∈ A f positioned at distance d f = d ( a f , b m ) from MBS b m . In the rest of this section, we deriveupper and lower bounds on the outage probability of MU u m ∈ U m ( a f ) as a function of d f . As just mentioned, thesame bounds hold for FUs covered by a f , as well.For MU ˆ u m ∈ U ( − f ) m , since ˆ u m is directly serviced by MBS b m , instead of one of the FAPs such as a f , we must have κd (ˆ u m , b m ) ≤ d ( f )ˆ u m ≤ d (ˆ u m , a f ) . Therefore, d (ˆ u m ,b m ) d (ˆ u m ,a f ) ≤ κ . Let δ ˆ u m , d (ˆ u m ,b m ) d (ˆ u m ,a f ) , where, as a reminder, b m and a f denote the base station and the FAP at distance d f from b m ,respectively. As we just argued, δ ˆ u m ≤ κ − , for all ˆ u m ∈U ( − f ) m . Given the complicated distribution of δ ˆ u m , in order tocharacterize the outage probability, we quantize δ ˆ u m .Consider the setup shown in Fig. 3, where FAP a f islocated at distance d f from b m . The coverage area of b m isshown by the black circle of radius R centered at b m . Thegreen circle on the right represents points with δ ˆ u m = κ − .Similarly, the points on the green circle on the left have δ ˆ u m = κ . The other pairs of circles correspond to some othervalues of κ ′ > κ . Points on the black line have δ ˆ u m = 1 .Note that, by our assumption, all MUs are located inside theblack circle. Therefore, the parts of colored circles that areoutside of the black circle have zero probability. Consider −1000 −500 0 500 1000 1500−1000−800−600−400−20002004006008001000 R d f Fig. 3. Partitioning the coverage area κ , κ < κ < . . . < κ t − < κ t , . Let ˆ δ u ˆ u m = κ − i , if κ − i +1 < δ ˆ u m ≤ κ − i ,κ i +1 if κ i < δ ˆ u m ≤ κ i +1 ,κ if δ ˆ u m ≤ κ, (12)and ˆ δ l ˆ u m = κ − i +1 , if κ − i +1 < δ ˆ u m ≤ κ − i ,κ i if κ i < δ ˆ u m ≤ κ i +1 , δ ˆ u m ≤ κ. (13)Then by construction, ˆ δ l ˆ u m ≤ δ ˆ u m ≤ ˆ δ u ˆ u m , for all ˆ u m ∈ U ( − f ) m ,and, unlike δ ˆ u m , ˆ δ u ˆ u m and ˆ δ l ˆ u m are finite-alphabet randomvariables. Let S i and s i , i = 1 , . . . , t , denote the regioncorresponding to (ˆ δ u ˆ u m , ˆ δ l ˆ u m ) = ( κ − i − , κ − i ) , and its area,respectively. Similarly, define S i and s i , i = − t, . . . , − tocorrespond to the region with (ˆ δ u ˆ u m , ˆ δ l ˆ u m ) = ( κ − i , κ − i − ) .Finally, S and s correspond to (ˆ δ u ˆ u m , ˆ δ l ˆ u m ) = ( κ, . InAppendix A, given t ∈ N + , we present analytic expressionsfor computing s − t , s − t +1 , . . . , s t − , s t . Lemma 1:
For i = 1 , . . . , t , p i , P(ˆ δ u ˆ u m = κ − i − ) =P(ˆ δ l ˆ u m = κ − i ) = s i s , for i = − t, . . . , − , p i , P(ˆ δ u ˆ u m = κ − i ) = P(ˆ δ l ˆ u m = κ − i − ) = s i s , and p , P(ˆ δ u ˆ u m = κ ) =P(ˆ δ l ˆ u m = 0) = s s , where s , s + P ti =1 ( s i + s − i ) . Proof:
Let S denotes the whole circuit of radius R minusthe coverage area of a f . Hence the area of S is equal to s = s + P ti =1 ( s i + s − i ) . For i = − t, . . . , − , we have P(ˆ δ u ˆ u m = κ − i )= E[ ˆ u m ∈S i ]= E[E[ ˆ u m ∈S i |A f ]]=E (cid:20) s i − S i,m s − S m (cid:21) , where S m and S m denote the region in S that is covered bythe MBS b m , and its area, respectively. (This is of course thearea that is not covered by FAPs.) Also, S i,m denotes the areaof S i ∩ S m . Note that S i,m and S m are both random variablesthat depend on the locations of the FAPs. To derive the desiredresult, we employ the tower property one more time: P(ˆ δ ˆ u m = κ i ) = E (cid:20) E h s i − S i,m s − S m | S m i(cid:21) = E (cid:20) s i − ( s i /s ) S m s − S m (cid:21) = s i s . The proof of the rest of the theorem follows from the sameargument.Let P m,f out ( d f ) denote the outage probability experienced bya MU covered by a FAP located at distance d f of b m . Weemploy Lemma 1 and our upper-bounding and lower-boundingquantizations of δ ˆ u m to derive the following theorem thatpresents both an upper bound and a lower bound on P m,f out ( d f ) . Theorem 3:
Let T h , Tn h . For t ∈ N + , define q l ( s ) , p sκ α /η + t X i =1 (cid:16) p i s/ ( ηκ αi − ) + p − i sκ αi /η (cid:17) , (14) q u ( s ) , p + t X i =1 (cid:16) p i s/ ( ηκ αi ) + p − i sκ αi − /η (cid:17) , (15)where ( p − t , . . . , p t ) are defined and characterized in Lemma1, η , P f /P m , and τ o = τ ( − log q l ( T h σ )) , with τ ( · ) definedin (5). Then, P m,f out ( d f ) ≤ − (1 + T h )(e ¯ n f mu / (1+ T h ) − ¯ n f mu − · e − ¯ n fu T h / (1+ T h ) − ¯ n mu (1 − q u ( T h /σ )) and P m,f out ( d f ) ≥ − (1 + T h )(e ¯ n f mu / (1+ T h ) − ¯ n f mu − − ¯ n fu T h / (1+ T h ) · (cid:16) e (¯ n mu − ¯ n f mu )( q l ( Thσ ) − n fap ( τ o − (1 − e − ¯ n fap ) τ o + e γ − − ¯ n fap + γ − log( γ ¯ n fap ) − e − ¯ n fap (cid:17) . Proof:
The details of the proof is presented in Ap-pendix B, but the outline of the proof is as follows. First, wederive upper and lower bounds on the upload SIR experiencedby macro user u m ∈ U m ( a f ) , namely SIR m,f . To achieve thisgoal, we employ the quantizations of δ ˆ u m defined in (12) and(13). Then, we connect the outage probability with the Laplacetransform of the bounds on SIR. Finally, we use the fact thatthe support sets of the locations of MUs served by MBS andFAPs do not overlap to prove that independence of the numberof interfering users in different groups. Remark 3:
In Theorem 3, t corresponds to the number ofpartition levels of the MBS’s coverage area, and is a parameterthat can be selected arbitrarily. In other words, the bounds holdfor any t ∈ N + , but choosing higher value of t leads to tighterupper and lower bounds. Remark 4:
In Theorem 3, the terms in the bounds thatdepend on distance d f are ¯ n f mu , q u ( T h /σ ) and q l ( T h /σ ) .As we will see in the numerical results presented in SectionV, both the upper and lower bounds are not monotonic in d f .This follows from the non-monotonic behavior of q u ( T h /σ ) and q l ( T h /σ ) . The term (1+ T h )(e ¯ n f mu / (1+ T h ) − / (e ¯ n f mu − ,which appears in both upper and lower bound, is usually veryclose to one, and in monotonically decreasing in d f . Therefore,the other terms in each bound are the dominant terms. Remark 5:
To gain more insight on the effect of differentparameters on the bounds in Theorem 3, we can considertheir approximate values for typical set of parameters, when the number of carriers is large. As mentioned in Remark4, (1 + T h )(e ¯ n f mu / (1+ T h ) − / (e ¯ n f mu − is close to one,especially if the number of carriers is large. We can alsoapproximate τ o as − e − log( ql ( Thσ γ ¯ n mu log( q l ( Thσ )) γ ¯ n mu by approximating of e − s in τ ( s ) as − s . Employing these approximations, andignoring the other non-dominant terms, the upper and lowerbound can be simplified as − e − ¯ n fu T h − ¯ n mu (1 − q u ( T h /σ )) and − e − ¯ n fu T h − (1 − q l ( T h /σ ))(¯ n mu − . γ ¯ n fap ¯ n mu − ¯ n f mu ) , respectively.When the FAP gets close to the MBS, i.e., d f ≪ R , δ ˆ u m ≈ , and hence, q u ( T h /σ ) , q l ( T h /σ ) ≈ T h / ( σ η ) ,or − q u ( T h /σ ) , − q l ( T h /σ ) ≈ T h / ( σ η ) , which furthersimplifies the upper and lower bounds to − e − T h (¯ n fu + ¯ n mu ησ ) ,and − e − T h (¯ n fu + ¯ n mu ησ − ¯ nf mu ησ − γ ¯ n fap ¯ n mu2 ησ ) , respectively. B. MU served by the MBS
The upload SIR experienced by user u m ∈ U ( − f ) m at theMBS b m in subband i is equal to SIR m,m = P m | h ium,bm | n s I m,m , (16)where, for i ∈ { , . . . , n s } , I m,m = X a f ∈A f X ˆ u m ∈U m ( a f ) (cid:16) d (ˆ u m , a f ) d (ˆ u m , b m ) (cid:17) α | h i ˆ u m ,b m | E[ c ˆ um [ i ]= c um [ i ] ] P f n s + X ˆ u m ∈U ( − f ) m \ u m P m n s | h i ˆ u m ,b m | E[ c ˆ um [ i ]= c um [ i ] ]= X a f ∈A f X u m ∈U m ( a f ) (cid:16) d ( u m , a f ) d ( u m , b m ) (cid:17) α | h iu m ,b m | P f G + X ˆ u m ∈U ( − f ) m \ u m P m G | h i ˆ u m ,b m | . (17)In deriving (17), we have ignored the interference caused bythe FUs. The reason is that in most cases the distance betweenFU u f and its FAP is much smaller than the distance between u f and b m . Theorem 4:
Let P m,m out , P(SIR m,m < T ) . Then, P m,m out ≥ − (1 + T h )(e − ¯ n mu T h / (1+ T h )+¯ n fap ( τ ′ o − +e γ − − ¯ n fap + γ − log( γ ¯ n fap ) ) , and P m,m out ≤ − (1 + T h )(e ¯ n mu / (1+ T h ) − ¯ n mu − where τ ′ o = e γ ¯ n mu Th/ (1+ Th ) − γ ¯ n mu T h / (1+ T h ) . Proof:
The proof is relegated to Appendix C. Similar tothe proof of Theorem 3, here too, we derive upper and lowerbound on the experienced SIR,
SIR m,m . In this case, when amacro user is served by the MBS, d ( u m , a f ) ≤ κd ( u m , b m ) .Employing this bound, yields a lower bound on SIR m,m . Toderive the upper bound, we only consider the interferencecaused by the other MUs served by the MBS.
Remark 6:
Typically, the second term in the upper boundis negligible compared to the first term and can be ignored.Approximating by the first-order Taylor expansion, τ ′ o can beapproximated as τ ′ o ≈ . γ ¯ n mu T h , which holds for large number of carriers per subband. Using these approximations,the lower bound can be simplified to − e − ¯ n mu T h (1 − γ ¯ n fap ) .V. N UMERICAL RESULTS
In this section, to investigate the uplink network perfor-mance and to verify our upper and lower bounds, we presentsome simulation results. Monte-Carlo computer simulationswith realizations are carried out to validate our analyticalbounds and illustrate the accuracy of our approximations. Theconsidered scenario is a two-tier network in a circle of radius R = 1 Km with the MBS located at the center. In the ensuingplots, we use the default values in Table I, unless otherwisestated. Fig. 4 shows a realization of the network with thespecified parameters. Note that the size of the coverage areaof a femtocell depends on its distance from the MBS. In ourmodel, if a MU falls in the coverage areas of more than oneuser, it is serviced by the closest one. −1000 −500 0 500 1000−1000−50005001000 Fig. 4. Sample realization of the network with the parameters specifiedin Table I. (Blue x: MU, green x: FU, circles (except for the largest one):coverage area of a FAP.)
Figs. 5 and 6 show the conditional outage probabilities ofMUs serviced by a FAP located at d f = 700 m from b m , andthe average outage probabilities of MUs served by the MBS,respectively. Different curves in these figures correspond to TABLE IS
IMULATIONS PARAMETERS
Sym. Description Default Values λ f density of FAPs × − m − µ m density of macrocell users × − m − µ f density of FUs .
01 m − ∆ ring width of FUs placement R f ring internal radius of FUs placement
10 m α path loss exponent 4 d f distance between considered FAP and MBS
700 m T SIR threshold level n s number of subbands n h number of subchannels in each subbands η power ratio between FAPs and MBS κ handover parameter . O u t age p r obab ili t y o f M U s e r v ed b y F AP Threshold Level (T)
Analytical (upper bound)Analytical (lower bound)Simulation µ m =25*10 −6 µ m =15*10 −6 µ m =5*10 −6 Fig. 5. Conditional outage probability of MUs served by FAPs located atdistance d f = 700m from the MBS versus threshold level ( T ) for differentMU densities. O u t age p r obab ili t y o f M U s e r v ed b y M BS Threshold Level (T)
Analytical (upper bound)Analytical (lower bound)Simulation µ m =25*10 −6 µ m =15*10 −6 µ m =5*10 −6 Fig. 6. Average outage probability of MUs served by MBS versus thresholdlevel ( T ) for different MUs densities. different MUs densities. Figs. 5 and 6 reflect that our analyticalupper and lower bounds (solid and dot curves) are reasonableapproximations for all considered SIR thresholds.As expected, increasing the threshold level increases theprobability of outage. Clearly this does not imply that theperformance can be improved by lowering T , as its reductiondecreases the achieved rate as well. In general, there is a trade-off between expected capacity [35], [36] and threshold T .The problem of maximizing the expected rate by optimizing T is studied in [37] for a single-tier MCFH system . Weleave extending those results to multi-tier networks for futureresearch.Fig. 7 demonstrates how the average outage probabilities ofMUs vary with handover parameter κ . Here too the boundsare consistent with the simulation results, but the gap increasesslightly as κ increases. In contrast to the downlink scenario[1], where the outage probability is not monotonic in κ , here,increasing κ improves the performance for both MUs and FUs.This result is consistent with [23], where the authors arguethat in non-orthogonal setups, open access is strictly better O u t age p r obab ili t y o f M U s κ Analytical (upper bound)Analytical (lower bound)Simulation
Served by MBSServed by FAP
Fig. 7. Average outage probability of MUs versus κ .
10 20 30 40 50 60 70 80 900.20.30.40.50.60.70.80.91 A v e r age ou t age p r obab ili t y o f M U s η Analytical (upper bound)Analytical (lower bound)Simulation
Served by FAP Served by MBS
Fig. 8. Average outage performance of MUs versus the power ratio betweenFAPs and MBS ( η = P f /P m ). than closed access policy. The difference between uplink anddownlink arises from the fact that in the downlink scenarioas the MUs get farther away from the MBS, their receivedpowers and hence SIRs decrease. On the other hand, in theuplink scenario, as they become farther away from the MBS,due to power control, their transmit powers increase as wellto compensate for the path loss. Naturally, increasing thehandover parameter leads to more MUs being covered by FAPsand hence to lower co-tier interference.Note that for plotting the average probability experiencedby MUs served by FAPs, we have taken the expected valueof the upper and lower bounds mentioned in Theorem 3 byconsidering the randomness in d f .Fig. 8 shows the average outage performance of MUs asa function of η = P f /P m , the power ratio between FAPsand MBS. In these plots we have fixed the transmit powerof MBS and because of this, the outage curves of the MUsserved by the MBS are almost constant. Obviously the outageof MUs served by FAPs improves by increasing FAPs transmitpowers. Note that although increasing FAPs powers increasesthe interferences level, but its effect is not significant for MUs, O u t age p r obab ili t y o f M U s s e r v ed b y F AP Normalized Distance from MBS
Analytical (upper bound)Analytical (lower bound)Simulation
Fig. 9. Conditional outage probability of a MU served by a FAPs as afunction of the normalized distance of FAP from the MBS. −5 O u t age p r obab ili t y o f M U s e r v ed b y F AP µ m Analytical (upper bound)Analytical (lower bound)Simulation n h =256 n h =512n h =1024 Fig. 10. Conditional outage probability of MUs served by a FAP located atdistance d f = 700m from the MBS versus their density. whose performance is mainly limited by other MUs and notFUs.Fig. 9 illustrates the conditional outage probability of MUsserved by a FAP, as a function of the FAP’s normalizeddistance from the MBS. As it can be observed from the figure,at first, the outage probability increases as the MU gets fartherform the MBS. In fact because of the assumption of constantreceived power by the MBS in the uplink scenario, as the MUgets farther from the MBS, it will transmit at a higher power,which leads to the degradation in the performance of FUs andalso MUs served by FAPs. However, as the femtocells getclose to the fringes of the cell, their users outage probabilitiesstart to improve as well. The reason is that femtocells thatare far away from the MBS have larger coverage areas andtherefore, in those regions most MUs are serviced by nearbyFAPs, which results in lower interference caused by them.Figs. 10 and 11 show the conditional outage probability ofMUs served by a FAP located at d f = 700 m , and the averageoutage probability of MUs served by the MBS, respectively,as a function of MUs density µ m . Obviously, increasing themacrocell users density will increase their outage probabilities −5 O u t age p r obab ili t y o f M U s e r v ed b y M BS µ m Analytical (upper bound)Analytical (lower bound)Simulation n h =1024n h =256 n h =512 Fig. 11. Average outage probability of MUs served by MBS versus theirdensity. as well, because of more co-tier interferences. However theirperformance can be greatly improved by boosting the numberof available sub-channels as can be seen in the figure, too.VI. C
ONCLUSIONS
In this paper we investigated the uplink performance of two-tier networks consisting of a macrocell overlaid by femtocells.We considered a stochastic spatial distribution for MUs, FUsand FAPs, and assumed that they are generated by independentPPPs. For cell association, we considered an open accesspolicy, where each MU is assigned to its nearest FAP iftheir distance is less than its distance from the MBS timessome factor κ < . Under this model we studied the outageperformance of the system and derived analytical upper andlower bounds on the outage probabilities of both FUs andMUs. The bounds were shown to be tight by our simulations.Throughout the paper we considered a fixed threshold κ forall MUs. A more general model is when κ is not fixed anddepends on the FAP. In other words, since κ determines thecoverage area of FAPs, it is conceivable to consider a scenariowhere FAPs are heterogeneous and can choose their coverageareas. For instance, a FAP can move toward a closed accesspolicy by lowering its corresponding κ , i.e., by only acceptingMUs that are very close.FAPs are connected to a central gateway via wired connec-tions. The capacity constraints imposed by this backhaul wirednetwork can potentially affect the cell selection procedure andmay impede some femtocells to service all MUs that fell intheir coverage area. Characterizing the effect of this constrainton the system’s performance is another interesting questionthat is left for future research.A CKNOWLEDGMENTS
The authors would like to thank the anonymous reviewersfor their helpful comments and suggestions, especially one ofthe reviewers who pointed us to reference [38]. A PPENDIX AD ERIVATION OF s i As defined in Section IV, s i , i = 1 , . . . , t , denotes the areaof the region corresponding to ˆ δ u ˆ u m = κ − i − and ˆ δ l ˆ u m = κ − i or equivalently κ − i ≤ δ ˆ u m ≤ κ − i − . We showed earlier thatthe points that satisfy δ ˆ u m = κ − , are located on a circle ofradius κd f − κ centered at ( d f − κ , . Hence, s i is the macrocellcoverage area surrounded by two such circles with κ = κ i and κ = κ i − . Therefore, for i = 1 , , ..., t − , s i = πd f κ i (1 − κ i ) − κ i > − dfR f ( κ i d f − κ i , R, d f − κ i ) − i − X j =0 s j , and s t = R ( θ − sin(2 θ )) − P t − j =0 s j , where θ = arccos( d f R ) and f ( a, b, c ) , a sec − ( 2 acb − a − c ) − b sec − ( 2 bcb + c − a )+ 12 p ( a + b + c )( b + c − a )( c + a − b )( a + b − c ) [39]. Similarly, s i , for i = − t, . . . , − denotes the area ofthe region where κ − i − ≤ δ ˆ u m ≤ κ − i , and ˆ δ l ˆ u m = κ − i − .Here, the geometrical location of the points satisfying δ ˆ u m = κ is a circle with the origin at ( − κ d f − κ , and radius κd f − κ .Therefore the area of the macrocell zone enclosed by twosuch the circles with κ = κ − i and κ = κ − i − is given by s i = πd f κ − i (1 − κ − i ) − κ − i > RR + df f ( κ − i d f − κ − i , R, d f κ − i − κ − i ) − P j = i +1 s j , for i = − t + 1 , ..., − , and s − t = R ( π − θ + sin(2 θ )) − P j = − t +1 s j , where again θ = arccos( d f R ) .A PPENDIX BP ROOF OF T HEOREM E = { d ( a f , b m ) = d f , N a f m ≥ } . Then, bydefinition, P m,f out ( d f ) = P(SIR m,f < T |E ) Combining thequantizations defined in (12) and (13) with (10) and (11), wehave n h | h iu m ,a f | K u ≤ SIR m,f ≤ n h | h iu m ,a f | K l (B.1)where K u , X u f ∈U f ( a f ) | h iu f ,a f | + X ˆ u m ∈U m ( a f ) \ u m | h i ˆ u m ,a f | + 1 η X ˆ u m ∈U ( − f ) m | h i ˆ u m ,a f | (ˆ δ u ˆ u m ) α , (B.2)and K l , X u f ∈U f ( a f ) | h iu f ,a f | + X ˆ u m ∈U m ( a f ) \ u m | h i ˆ u m ,a f | + 1 η X ˆ u m ∈U ( − f ) m | h i ˆ u m ,a f | (ˆ δ l ˆ u m ) α , (B.3) From (B.1), we have P m,f out ( d f ) ≤ P (cid:16) n h | h iu m ,a f | K u < T |E (cid:17) = E (cid:20) E h | h ium,af | < KuTnh |E , K u i(cid:21) (a) = 1 − E h e − TKunhσ (cid:12)(cid:12)(cid:12) E i = 1 − Φ K u ( Tn h σ | d f ) , (B.4)where ( a ) follows from our assumption that channel coeffi-cient | h iu m ,a f | has a Rayleigh distribution. Here Φ K u ( s | d f ) , E[e − sK u |E ] denotes the conditional Laplace transform of K u defined in (B.2). Since conditioned on the number of usersin each category, the channel coefficients and { ˆ δ ˆ u m } ˆ u m ∈U ( − f ) m are all independent of each other, it follows that Φ K u ( s | d f )=E h(cid:18)
11 + sσ (cid:19) N aff + N afm − · q u ( s ) N bmm (cid:12)(cid:12)(cid:12) E i . (B.5)where q u ( s ) is defined in (14). To derive the lower bound,again from (B.1), and following a similar steps as in (B.4),we derive P m,f out ( d f ) ≥ P (cid:16) n h | h iu m ,a f | K l < T |E (cid:17) = 1 − E h e − TKnhσ (cid:12)(cid:12)(cid:12) E i = 1 − Φ K l ( Tn h σ | d f ) , (B.6)where Φ K l ( s | d f ) , E[e − sK l |E ] . Also, as argued before inderiving (B.5), we have Φ K l ( s | d f )=E h(cid:18)
11 + sσ (cid:19) N aff + N afm − · q l ( s ) N bmm (cid:12)(cid:12)(cid:12) E i , (B.7)where q l ( s ) is defined in (15).Conditioned on the location of a f , N a f f , N a f m and N b m m are independent random variables. The independence of N a f f and ( N a f m , N b m m ) follows from our initial assumption that theprocess of drawing MUs and FUs are independent. To see theindependence of N a f m and N b m m , note that U m ( a f ) denotes theusers that are located in a circle of radius d f p (1 − κ ) − − .(Refer to Fig. 2.) On the other hand, conditioned on thelocation of a f , U ( − f ) m denotes users that are not located inany of the circles corresponding to different FAPs, one ofwhich is the mentioned circle corresponding to a f . Therefore,the support sets of the locations of MUs in U ( − f ) m and theMUs in U m ( a f ) do not have any overlap. Therefore, since themacro users are generated by a PPP process, N a f m and N b m m are independent random variables. As a result, Φ K u ( s | d f ) =Φ N aff (log(1 + sσ ))Φ + N afm (log(1 + sσ ) | d f ) · Φ N bmm ( − log q u ( s ) | d f ) . (B.8)Similarly, Φ K l ( s | d f ) =Φ N aff (log(1 + sσ )))Φ + N afm (log(1 + sσ ) | d f ) · Φ N bmm ( − log q l ( s ) | d f ) . (B.9)Combining the bounds derived for Φ N bmm ( s | d f ) in Theorem 2 with (1), (3), (B.4), (B.6), (B.8) and (B.9) completes the proofof Theorem 3. A PPENDIX CP ROOF OF T HEOREM u m ∈ U m ( a f ) , by our assignment policy, weshould have d ( u m , a f ) ≤ κd ( u m , b m ) . Therefore, I m,m canbe upper-bounded as I m,m ≤ X a f ∈A f X ˆ u m ∈U m ( a f ) κ α | h i ˆ u m ,b m | P f G + X ˆ u m ∈U ( − f ) m \ u m P m G | h i ˆ u m ,b m | ≤ X ˆ u m ∈U m \ u m P m G | h i ˆ u m ,b m | , (C.10)where the last line follows since κ < . Also, clearly, I m,m ≥ P ˆ u m ∈U ( − f ) m \ u m P m G | h i ˆ u m ,b m | . Hence, n h | h iu m ,b m | ¯ K u ≤ SIR m,m < n h | h iu m ,b m | ¯ K l , where ¯ K l = P ˆ u m ∈U ( − f ) m \ u m | h i ˆ u m ,b m | and ¯ K u = P ˆ u m ∈U m \ u m | h i ˆ u m ,b m | , and − Φ ¯ K l ( Tn h σ ) ≤ P m,m out ≤ − Φ ¯ K u ( Tn h σ ) . To derive the upper bound, note that Φ ¯ K u ( s ) = E[e − s ¯ K u ] =E[(E[e − s | h i | ]) |U m |− ||U m | ≥
1] = e a ¯ n mu − a (e ¯ n mu − , where a =1 / (1+ sσ ) . For the lower bound, we employ the upper boundon Φ U ( − f ) m presented in Theorem 1. The only difference herecompared to Theorem 1 is that here we need to condition on N b m m ≥ . However, E h e − sN bmm (cid:12)(cid:12)(cid:12) N b m m ≥ i ≤ E[e − sN bmm ] . The reason is that for any positive integer-valued randomvariable X with p i = P ( X = i ) , i = 0 , , . . . , we have E[e − X ] = p + P ∞ i =1 p i e − i and E[e − X | X ≥
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