Hybrid Resource Scheduling for Aggregation in Massive Machine-type Communication Networks

1 Hybrid Resource Scheduling for Aggre gation in Massi v e Machine- type Communication Netw orks Onel L. Alcaraz L ´ opez, Hirle y Alves, Pedro H. J. Nardelli, Matti L atva-aho Abstract —Data aggr egation is a prom ising approach to enable massiv e machine-type com munication (mMTC). Here, we first characterize the aggregation ph ase where a massive number of machine-type devices transmits to their respective aggre gator . By using non-orthogonal multiple access (NOMA), we present a hybrid access sch eme where sev eral machine-typ e devices (MT D s) share th e same orthogonal channel. Then, we assess the relaying phase where the aggr egatted data is forwa rded to the base station. The system perfo rmance is in vestigated in terms of a verage number of MT Ds that are simultaneously serv ed under imperfect successiv e in terference cancell ation (S IC) at the aggrega tor f or two schedu ling schemes, n amely random r esource schedu ling (RRS) and ch annel-depend ent resour ce scheduling (CRS), which is then used to assess the p erf ormance of data forwarding ph ase. Index T erms —data aggr egation, mMTC, resourc e scheduling, NOMA I . I N T R O D U C T I O N Machine-ty pe Commun ication (MTC) is a n inherent part of the fifth generation ( 5G) cellu la r n etworks [1], [ 2], covering automatic data gener ation, exchange, pro cessing and actuation that are the basis of intelligent machine networks. Such MTC networks ar e gr owing at a n im p ressiv e r ate and som e pr edic- tions are pointin g to 2 0 billion machine- type devices (MT Ds) connected to wireless networks in 2023 and beyond [3]. Mas- si ve Mach ine-type Comm u nication (mMTC) are envisaged to cope with that large num ber o f , o ften low-complexity lo w- power , M T Ds that are b ecoming par t of wireless networks [4]. A sur vey on the req uirements, technical challen ges, and existing work on med ium access con trol (MA C) la y er pro- tocols for supportin g th ese new u se cases, is pr e sented in [5], wh ile auth ors describe also the issues related to effi cient, scalable, and fair channel access. In fact, d ifferent strategies have bee n pro posed to provide mo re efficient access, e.g., access class bar ring [ 6], prio ritized ran dom access [7], b ackoff adjustment schem e [ 8], d elay-estimation based ra n dom access [9], distributed queuing [10], d ata agg regation [2], [ 11]. Da ta aggregation co nsists in MTDs that o rganize themselves lo cally to MTC ar ea networks, the n, the ar e a n etworks connect to the Onel L. Alc araz L ´ opez, Hirley Alves and Mat ti Latv a-aho are with the Centre for Wire less Communications (CWC), Uni versit y of Oulu, Finland. { on el.alcara zlopez,hirley . alves,matti.latv a-aho } @oulu.fi Pedro H. J. Nardelli is with Laboratory of Control Engineering and Digital Systems, Lappeenranta Univ ersity of T echnology , Finla nd. pe- dro.nardel li@lut.fi This work is partially supported by Academy of Finland (Aka) (Grants n.303532, n.307492 , n.318927 (6Genesis Flagship)), SRC/ Aka BCDC Energy (n. 292854), as well as the Finnish Foun dation for T echnolog y Promotion, the Finnish Funding Agency for T echnology and Innov ation (T eke s), Bitti um W ireless, K eysight T echnologies Finland, K yynel, MediaT ek Wirel ess, Nokia Solution s and Networks. core n e twork throu gh MTC gatew ays or data aggr egato rs. This alleviates the p roblem of massive signaling overhe a d on the architectura l side and it is a key solutio n strategy to collect, process, and co mmun ica te data in MT C u se cases with static devices, especia lly if the locations of the d evices a r e known, such as smart utility mete r s or vide o surveillance cameras [4], [12], [13]. In [14], auth ors survey data agg r egation strategies in large- scale wireless sensor networks (WSNs), while focu sin g on the proc essing challenges behind the large volume of data. I n [15], an experimental study using state-of- th e-art d rive testing equipmen t is conduc ted in order to capture and an a lyze the im- pact of MTC data aggregation on signaling overhead in cellular networks with focus on static MT Ds such as sma r t m eters and monitorin g senso rs. Autho rs of [16] present a scheme designed to provide data a ggregation fo r hetero geneou s an d concu rrent sets of Constrain ed Application Protoco l [17] (CoAP) data- requests. The problem of energy-op timal routing an d mu ltiple- sink ag g regation is in vestigated in [18], as well as join t ag gre- gation and dissemination of sensor mea su rement data in MT C edge networks. An agg regation scheme is p roposed in [1 1] for capillary networks connected to the L TE ne twork to improve their comm u nication efficiency . Authors ana lyze the tr ade-offs between rand om access interaction, resou rce allocation, and commun ication latency , a nd reveal that acceptin g th e extra latency for accumulatin g p ackets can significantly redu ce the random access req u ests and the required resources fo r th e data transmissions. Notice th at when aggr egating a massi ve num ber of MTDs, the density o f the aggregators, althou g h it is considerab ly smaller comp ared to the density of the MTDs, will still b e large. Hence, the inter ference generated by the d evices sharin g the same resourc e is no t negligible. Th ere is, though, limited literature considering th e interference in mMTC with data aggregation and resour ce sched uling. Authors in [19] partially address th ose issues by considering a multi-cell network scenario, whose key metr ics (namely MTD success pr obabil- ity , av erage nu mber of successful MTDs and pro bability of successful ch a n nel u tilization) are in vestigated for the rand om resource schedulin g (RRS) an d channel-dep endent resource scheduling (CRS) schem es. Another tech nique called non-orthog onal multiple access (NOMA) is seen as a promising tec hnolog y f or th e 5G networks to impr ove th e system spectral efficiency wh ile meeting the dem and of massiv e connectivity d emanded by certain MTC applications (e.g [20]). The key idea behind NOMA is to exploit th e p ower domain f o r multiple access so th a t mu ltiple users can be multiplexed at dif f erent p ower 2 lev els, but at the same time/f requen cy/co de employing SIC to separate superimp osed m e ssages [21]. The perfo rmance of NOMA is e valuated in [22], [ 23] b y using the stoch astic geometry tools. Howev er, the inter-cell inter ference, which is a pervasiv e prob lem in most of the existing wireless n etworks, is not explicitly considered in [22], as many o ther works o n NOMA. I n contrast, auth ors in [2 3] do consider the inter-cell interferen ce when e valuating the perfor mance of NOMA o n coverage prob ability and average ac h iev ab le rate , but on a downlink setup. In [ 24], we p ropose an d an alyze a h ybrid OMA-NOMA scheme fo r mMTC uplink scenario s by exten d- ing the sched uling schemes RRS a n d CRS initially proposed in [19]. Th erein we d eal with th e limited resources and allow u p to two MTDs to share the same o rthogo nal chan nel wh ile we consider im perfect SI C. W e show that ev en wh en th e hy brid scheme would lead to a less reliable system with greater chances of ou tages p er MTD, d u e to the additional intr a- cluster interfer ence, it can sign ificantly improve the nu mber of simultaneou s active MTDs f o r high access d e mand scenarios. Differently from [24], in this work we allow the RRS scheme to contr ol the power coefficients o f the MTDs sharin g the same ch annel, thus bo th, R RS and C RS, hav e the s ame impact in terms of interfe r ence gen erated on the outside network (network outside of th e aggr egation zone o f in terest). Additionally , here we are agnostic of the outside network topolog y , non etheless our pr oposed model captur es the inter- ference coming from outside. W e also include the ev aluation of the relaying phase of the aggr egated data to the base stations, while we fo cus on the average nu mber of simultaneou s active MTDs. That allow u s to highligh t the advantages o f our scheme wh ich aim s to provide massive co nnectivity in scenar- ios with high access d emand, which is not covered by usual OMA setups. Alth ough CRS ac h iev es b etter p erform ance by providing ac c e ss to th e MTDs with best ch annels, RRS could be more practical since the r andom pairing could mode l scenarios wher e so m e MT D s h ave urgency to be served. Results sh ow th at failing to ef ficiently eliminate the intra- cluster interferen ce co u ld reduce significantly the benefits from NOMA while c h allenging its p ractical imp lementation, thus, power con trol plays a main role in these systems. Finally , we attain app roxima te d , yet accu rate, expressions when analyzing the CRS schem e. In contrast to th e time-consuming Mo nte- Carlo s imulation s, o u r analytical deriv ations allow fo r f ast computatio n. Next, Section II introduc e s the system model. Section II I discusses the RRS and CRS scheduling schemes for the aggregation p h ase, while Section IV analyses the r elaying phase and the overall system pe rforman ce. Section V presents the numerical r esults and Section VI concludes th e paper . Notatio n: E [ · ] den otes expectation, Pr( A ) is the probability of ev ent A , Pr( A | B ) is th e P r( A ) con ditioned on B . 1 ( · ) is an indicator function which is equal to 1 if its argument is true and 0 otherwise; while  n k  = n ! k !( n − k )! . Γ( x ) is the gamma f u nction, ψ ( x ) = d [ln(Γ( x ))] dx is the digamma f unction, Q ( a, x ) = 1 Γ( a ) R ∞ x t a − 1 e − t d t is the regularized inco mplete gamma function, and 1 F 1 ( a ; b ; z ) is the Kummer conflu ent hy - pergeometric fun ction. i = √ − 1 and Im { z } is th e imag inary part of z ∈ C . f X ( x ) and F X ( x ) are the Probability Density Function ( PDF) and Cumulative D istribution Function ( CDF) of ran d om variable (R V) X , respectively . X ∼ Exp(1) is a n exponential distributed R V with unit m ean, e . g., f X ( x ) = e − x and F X ( x ) = 1 − e − x ; while Y ∼ Poiss( ¯ m ) is a Poisson distributed R V with mean ¯ m , e.g ., Pr( Y = y ) = 1 y ! ¯ m y e − ¯ m and F Y ( y ) = Q ( y + 1 , ¯ m ) . I I . S Y S T E M M O D E L A N D A S S U M P T I O N S Consider inte r ference- limited 1 uplink of MTC network is divided into two p hases. In the first ph ase (aggregation p hase), the MTD tries to transmit its data with fixed payloa d size b (bits) to its serv ing ag gregator . The MTDs are served thro u gh N ortho gonal channels as in [19]; however , here the same orthog onal ch annel could b e used for more than one MTD. When the access d emand is not so h ig h, the aggregator will be allo cating one MTD per channe l. But, when the access demand exceed s th e av a ilability of ortho g onal channels, some MTDs are allowed to shar e the same orthogon a l channel. This scheme is our propo sed h ybrid OMA-NOMA multiple access scenario [24]. T he number of MTDs requiring service is mo deled as K ∼ Poiss( ¯ m ) . The maximu m n umber of u ser s per ortho gonal ch annel is L , whe re L = 1 r educes to an OMA scen ario, an d for simplicity we focu s on the L = 2 setup. Fur thermor e, the scenario with L = 2 seems more practical than L > 2 wh en we take into co n sideration the processing co mplexity fo r SIC r eceiv ers, especially when SIC error prop agation is co nsidered as discussed in [25]. Figs.1 a-c show snap shots of the considered aggr egation phase fo r three different re alizations. The silent MTDs a r e those out of the N · L av ailab le resources being used b y the activ e MTDs. The aggregator imp lements the resource scheduling according to one o f the sch emes p resented in Section III, and the MTDs considered are those with granted acc ess since the rando m access in the network is assumed to b e perf ormed 2 as in [19], [24], [27]. After aggr egating the MTDs’ d ata, the a ggregator acts as an ordinar y cellu lar user and relay s the entire infor mation to its associated BS in the secon d p hase (r elaying p hase) as shown in Fig. 1d. For the aggregation phase with L > 1 , there is both: outside interf erence ρI o, 1 (i.e. interf erence f r om MTDs operating on the same chann els but being served by o ther aggregators), and inside in terference (i. e. interferen ce from MTDs within t he ser v ing area o f the aggregator), which ar e both R Vs. For the r elaying ph ase, let ρI o, 2 be the interf erence at the BS fr om aggregato rs in oth er ce lls operating with the same chann el r esources. Notice that I o,x relies on the outside network top ology , which is ass umed unknown, but with a Laplace transform of I o,x in the fo rm of L I o,x ( s ) = exp  − φ x Γ  1 + 2 α  Γ  1 − 2 α  s 2 α  , x = { 1 , 2 } , (1) 1 The interfere nce from other MTDs is much lar ger than the white noise in the recei vers and, therefore, can be ignored. Howe ver , notice that the impact of the noise can be easily incorpora ted into our analysis. 2 The resource scheduling schemes require that synchroniz ation proce- dures, as well as the random access stage, are performed in adv ance. In fact, the work in [26] proposes a NOMA scheme allowi ng th e combinat ion of random access and data transmissions phases, where our resource scheduling schemes can be easily incorpora ted to improve the ove rall system perfor- mance. The detail s of such implementat ion are out of the scope of this work. 3 BS aggregator silent MTDs active MTDs a) b) c) d) Fig. 1. a), b) and c) Snapshot of the aggre gation phase with ¯ m = 6 , L = 2 and N = 4 . MT Ds with the same shape and color are using the same channel across the entire network. (Differe nt realizati ons: a) K < N , b) N < K < L · N , c) K > L · N ). d) Snapshot of the relaying phase. which is a established result fr o m the stochastic g eometry f or wireless networks generated as PPP with Ray leigh fading , where α is the path-lo ss expon ent, wh ile φ x accounts f or network density , ch a r acteristics of the a g gregation/re laying areas, an d oth ers [19], [28]. Also, (1) ho lds under the assump - tion of using statistical full inv ersion power contr o l [ 29] with parameter ρ , as we assume here to guara ntee a un iform user experience while saving valuable ene rgy . The latter im p lies that devices con trol the ir tr ansmit power such that the average signal power recei ved at the serving aggregator/BS is equal to the pr e defined value ρ . T h us, the instantaneo us received power a t the receiv er si de is ρh , where h ∼ Exp(1) is the channel power gain und er quasi-static Rayleigh fading, and ρ does not impact the p erform ance since we assume the n etwork as in terferen c e-limited. Notice that the process φ x could also be depe n dent of α , as discussed in [30], [31]. I n that case, the Laplace tran sform of the interference for d ifferent po int p rocesses ap pear to be merely h orizontally shifted versions of ea c h other (in dB) as long as their di versity gain is th e same. Thu s, scaling the threshold s by th is SIR gain factor β , 3 we get ( β s ) 2 α , where β 2 α would be included in φ x . By prop e rly selecting φ x , th e outside interferenc e for any given topolo g y cou ld be then characterized . Finally , full channe l state info rmation (CSI) is assumed at re c eiv er side as in [19], [2 2], [2 3]. I I I . A G G R E G AT I O N P H A S E In this section we discu ss th e RRS and CRS sched uling schemes f or our h ybrid access protoco l. A. RRS for th e Hybrid Access Under the RRS scheme, N out of the K instantan eous MTDs req uiring transmissions are indepen d ently an d ran- domly chosen and then matched , one-to- one, with the o rthog- onal ch annels. If K ≤ N , a ll MT Ds get chan n el r esources, and ev en N − K chann els will be unused. Otherwise, if K > N , the channel allocatio n is executed again by allowing the r e maining MTDs to sh are chan n els with the already served MTDs. Th is process is executed rep eatedly until all the MTDs ar e allo cated or the maxim u m numb er of MTDs p er channel, L , is reached for all the cha nnels. The in side interf erence, com ing from 3 β will also depe nd on s , but finding β for a fixed s already gi ves a good approximat ion [31]. the MTDs within the same aggregation zo ne and shar in g the same ch annel, is faced with SIC. T h e SIR, SIR r j,u , of the j th MTD b eing dec oded o n a typical channel, given the number of MTDs u on the same channel an d the RRS scheme , is SIR r 1 , 1 = h I o, 1 , while SIR r 1 , 2 = a 1 max( h 1 , h 2 ) I o, 1 + a 2 min( h 1 , h 2 ) , (2) SIR r 2 , 2 = a 2 min( h 1 , h 2 ) I o, 1 + µa 1 max( h 1 , h 2 ) , (3) where µ ∈ [0 , 1 ] is used to model the impact caused by imperfect SIC [32], while h 1 and h 2 are the channel po wer coefficients of both MTDs sharing the channel whe n u = 2 . W e can weight the p ower of coexistent n odes on the same channel thro ugh a 1 and a 2 coefficients. O f course, so me kind of feedba c k from the agg regators would be requ ir ed after pair in g the MTDs 4 . By letting a 1 + a 2 = δ be a fixed value we imp ose so me kind of total transmission power constraint. This is crucial for NOMA s cenario s, and h ere it is particular important in or d er to con trol the interference generated on close aggregator s in the outside area 5 . Also, lim I o, 1 → 0 SIR r 1 , 2 is unb ounde d , b ut lim I o, 1 → 0 SIR r 2 , 2 ≤ a 2 a 1 µ since min( h 1 , h 2 ) ≤ ma x( h 1 , h 2 ) , thus th e p erform a nce of the second MTDs being decod ed o n the chan nel is strongly limited by the SIC impe rfection p arameter, but by p r operly selecting a 1 , a 2 that situation can b e relaxed. Consider fixed r a te codin g scheme wher e the receiver deco des successfully if the SIR exceeds a th reshold θ > 0 , achieving the in formatio n rate o f log 2 (1 + θ ) [bits/symbol] , we state the following theorem . 4 Since up to 2 MTDs can be scheduled to transmit over the same channe l, acquiri ng and using CSI at the MTD side is not appropriat e. Instead, the aggre gator should acquire the CSI a nd use it f or the scheduli ng and for determin ing the po wer control c oef ficients; while finally fo rwarding suc h informati on back to the MTDs. 5 Notice that po wer constraint s are usually li nked to ea ch devic e in- di viduall y since they are mostly relate d to hardware limitations. In fact a 1 , a 2 ≤ δ , therefore, we are implicitly considerin g also indi vidual power constrai nts. Ho wev er , since one chann el may be occupied by 2 MTDs, by setting a 1 + a 2 = δ we are able of controllin g the interference generated on the giv en channel on close aggre gators in the outside area, and e ven if δ ≈ 1 we are m aking it comparable to the interfe rence that would generate a single MTD if operating alone in that channel. 4 Theorem 1. The RRS succe ss pr ob a bility , p r j,u , of the j th MTD sharing a typical channel co nditione d on u MTDs, is given by p r 1 , 1 = L I o, 1 ( θ ) , (4) p r 1 , 2 = ( 2 a 1 a 1 + θ a 2 L I o, 1 ( θ a 1 ) − a 1 − θ a 2 a 1 + θ a 2 L I o, 1  2 θ a 1 − θ a 2  , if 0 ≤ θ a 2 a 1 < 1 2 a 1 a 1 + θ a 2 L I o, 1 ( θ a 1 ) , if θ a 2 a 1 ≥ 1 , (5) p r 2 , 2 = ( a 2 − θ µa 1 a 2 + θ µa 1 L I o, 1  2 θ a 2 − θ µa 1  , if 0 ≤ θ µa 1 a 2 < 1 0 , if θ µa 1 a 2 ≥ 1 . (6) Pr oof. See App e ndix A. Remark 1. As long as θ 2 µ < 1 , it is a dvisable choosing a 1 , a 2 such that θ < a 1 a 2 < 1 θ µ , a nd bo th MTDs operating on th e same channel get the success pr obability sh own in th e first line of (5) a nd (6) . Notice that by going c lo ser to 1 θ µ we favor the first MTD being decod ed, while if we cho o se a smaller a 1 a 2 , the second MTDs benefi ts. However , find ing the v a lues of a 1 and a 2 for which the MTDs co uld perform with similar reliability for any setup , seems intractable. Theorem 2. The Pr oba bility Mass F unction ( PMF) of the number of active MTDs, K r 1 , is given in (7) at the top of the next page , wher e f 1 ( k r 1 ) = min( k r 1 , 2 N − k ) , f 2 ( k r 1 , r 1 ) = min( k r 1 − r 1 , k − N ) and f 3 ( k r 1 ) = min( k r 1 , N ) . Pr oof. See App e ndix B. B. CRS for th e Hyb rid Acce ss The CRS scheme seeks to ma ke better u se of chann el resources by stro n gly relying on all the C SI a vailable fo r scheduling . Th e MTD with better fading (eq uiv alently , better SIR) will be preferentially assigned with the a vailable chan- nel resources. An aggregato r with K instantaneous MTDs requirin g transm ission has the k nowledge of their fadin g gains. Let { h 1 , ..., h i , ..., h K } denote the d ecreasing ordere d cha n nel gains, where h i − 1 > h i . If K ≤ N all the MTDs will be chosen , but if K > N the ag gregator will p ick the N MTDs with b etter chan nel gains, i.e., h 1 , ..., h N , and then will assign ran d omly the orthogon al ch annels to them [19]. As a continu ation, th e rem aining MTDs can be still allocated sharing those same resou rces, i.e., users N + 1 ,..., K go to the second rou nd for allocation. Th is pro cess is executed repeatedly un til all the MTDs are allocated or the maximu m number of MT Ds p er ch annel, L , is re a ched [2 4]. 6 Under the CRS scheme and using SIC to f ace the inside interferen ce, the SIR, SIR c ( i ) j,u , of th e j th M T D bein g decoded on a typical ch annel, given the first MTD allocated there has the i th larger chann e l coefficient, h i , and there are u MTDs sharing that same chan nel, is given by SIR c ( i ) 1 , 1 = h i I o, 1 and SIR c ( i ) 1 , 2 = a ( i ) 1 h i I o, 1 + a ( i ) 2 h i + N , (8) 6 Notice that imperfect CSI would not only affe ct the informatio n decod- ing procedure under this scheme, but also the resource scheduling stage since the channel coef ficients’ ordering may be af fected. A detai led performance analysi s under imperfect CSI is rega rded as our future work. SIR c ( i ) 2 , 2 = a ( i ) 2 h i + N I o, 1 + µa ( i ) 1 h i . (9) Notice that the bou nd perfo rmance is th e same as previ- ously discu ssed for the RRS schem e. Meanwh ile, the feed- back/signalin g overhead is also the same as for the RRS scheme since th e CSI acquisition would take place at the ag- gregator side, which in turn will o nly fo rward back the channel allocation for each MTD and the power co ntrol coefficients if necessary . W e have assumed that such metadata in formatio n is sufficiently small such that the low-rate feed back is err or-free. Howe ver, p ractical perf ormanc es would be upper bo unded by our results. Now , assuming that the rece iver can deco de successfu lly (SIR exceeds a threshold θ ), we state the following theo rem. Theorem 3. Given the fi rst MTD allocate d ha s the i th larges t channel coefficient, h i , and th at u MT Ds sh ar e th at same channel, the CRS success pr obability , p c ( i ) j,u , of the j th MTD being d ecoded on a typica l chann el is a ppr o x imated as p c ( i ) j, u ≈ 1 2 − 1 π ∞ Z 0 e − χ cos( π α ) ϕ 2 α sin  χ sin( π α ) ϕ 2 α − ϕB ( i,K ) j,u  ϕ d ϕ, (10) wher e χ = φ 1 Γ  1 + 2 α  Γ  1 − 2 α  , and B ( i,K ) 1 , 1 = ψ ( K + 1) − ψ ( i ) θ , (11) B ( i,K ) 1 , 2 =  a ( i ) 1 θ − a ( i ) 2  ψ ( K + 1) + a ( i ) 2 ψ ( i + N ) − a ( i ) 1 θ ψ ( i ) , (12) B ( i,K ) 2 , 2 =  a ( i ) 2 θ − µa ( i ) 1  ψ ( K + 1) + µa ( i ) 1 ψ ( i ) − a ( i ) 2 θ ψ ( i + N ) . (13) Pr oof. Theo r em 3 in [24] states that p c ( i ) j,u ≈ 1 2 − 1 π ∞ Z 0 1 ϕ Im  L I o, 1 ( − i ϕ ) e − i ϕB i,K j,u  d ϕ. (14) Substituting ( 1) into (14) along with some algebraic tran sfor- mations, e.g ., χ = φ 1 Γ  1 + 2 α  Γ  1 − 2 α  , ( − i ) 2 α = cos( π α ) − i sin( π α ) and Im { pe − q i } = − p sin( q ) , renders (10). Theorem 4. The PMF of the number of active MTDs, K c 1 , is appr oximated b y (15) at the top o f the next page a n d below (7) , where f z ( · ) for z = 1 , 2 , 3 ar e given in Theor em 2 and ¯ p c 1 , 1 ( k ) = 1 2 N − k N X i = k − N +1 p c ( i ) 1 , 1 , (16) ¯ p c 1 j, 2 ( k ) = 1 k − N k − N X i =1 p c ( i ) j, 2 , (17) ¯ p c 2 j, 2 ( k ) = 1 N N X i =1 p c ( i ) j, 2 . (18) Pr oof. The fact that the su ccess probabilities for the CRS scheme, p c ( i ) j,u , dep end on i an d k complicates h eavily the problem . Finding their average with regard the ind ex i allows us to use the same procedu re wh e n der iving Theorem 2 while attaining an accu rate appr oximation . Now whe n N < K < 2 N in (34), we sub stitute each p robability value, p r 1 , 1 and p r j, 2 , respectively by (16) and (17) since the succ e ss probab ility 5 Pr( K r 1 = k r 1 ) = e − ¯ mp r 1 , 1 ( ¯ mp r 1 , 1 ) k r 1 " 1 k r 1 ! − ( N − k r 1 + 1)  N +1 k r 1   ( N − k r 1 )! − Γ  N − k r 1 + 1 , ¯ m (1 − p r 1 , 1 )   ( N + 1)! # + 2 N − 1 X k = N + 1 f 1 ( k r 1 ) X r 1 =0 f 2 ( k r 1 ,r 1 ) X r 2 =0 " 1  k r 1 ≤ k − N + r 1 + r 2   2 N − k r 1   k − N r 2   k − N k r 1 − r 1 − r 2  ( p r 1 , 1 ) r 1 ( p r 1 , 2 ) r 2 ( p r 2 , 2 ) k r 1 − r 1 − r 2 (1 − p r 1 , 1 ) 2 N − k − r 1 (1 − p r 1 , 2 ) k − N − r 2 · · (1 − p r 2 , 2 ) k − N − k r 1 + r 1 + r 2 e − ¯ m ¯ m k k ! # +  1 − Q (2 N , ¯ m )  f 3 ( k r 1 ) X r 1 =0 1  k r 1 ≤ N + r 1   N r 1  N k r 1 − r 1  ( p r 1 , 2 ) r 1 ( p r 2 , 2 ) k r 1 − r 1 (1 − p r 1 , 2 ) N − r 1 (1 − p r 2 , 2 ) N − k r 1 + r 1 . (7) Pr( K c 1 = k c 1 ) ≈ e − ¯ mp r 1 , 1 ( ¯ mp r 1 , 1 ) k r 1 " 1 k r 1 ! − ( N − k r 1 + 1)  N +1 k r 1   ( N − k r 1 )! − Γ  N − k r 1 + 1 , ¯ m (1 − p r 1 , 1 )   ( N + 1)! # + 2 N − 1 X k = N + 1 f 1 ( k r 1 ) X r 1 =0 f 2 ( k r 1 ,r 1 ) X r 2 =0 " 1  k c 1 ≤ k − N + r 1 + r 2  · ·  2 N − k r 1   k − N r 2   k − N k c 1 − r 1 − r 2   ¯ p c 1 , 1 ( k )  r 1  ¯ p c 1 1 , 2 ( k )  r 2  ¯ p c 1 2 , 2 ( k )  k c 1 − r 1 − r 2  1 − ¯ p c 1 , 1 ( k )  2 N − k − r 1  1 − ¯ p c 1 1 , 2 ( k )  k − N − r 2  1 − ¯ p c 1 2 , 2 ( k )  k − N − k c 1 + r 1 + r 2 · · e − ¯ m ¯ m k k ! # + ∞ X k =2 N f 3 ( k c 1 ) X r 1 =0 1  f 5 ( k c 1 ) ≤ N   N r 1  N f 5 ( k c 1 )   ¯ p c 2 1 , 2 ( k )  r 1  ¯ p c 2 2 , 2 ( k )  f 5 ( k c 1 )  1 − ¯ p c 2 1 , 2 ( k )  N − r 1  1 − ¯ p c 2 2 , 2 ( k )  N − f 5 ( k c 1 ) e − ¯ m ¯ m k k ! . (15) depend s on the numb e r of MTDs requir in g transmission. When K ≥ 2 N we do similar b y rep lacing p r j, 2 by (1 8) in (34). C. Optimum Scheduling: Is it F easible? Notice that previous schedulin g schemes do not guaran tee an optimum p erform ance. This is o bvious fo r th e case of RRS since such sch eme relies entire ly o n rando m p airing, while CRS, e ven when it exploits CSI for making the p airing decisions, is also sub-o ptimal. As an example, notice th at fo r K > 2 N a better scheduling when µ = 0 a n d I o, 1 ≈ 0 will pro bably be the one p airing th e N M T Ds with better fading co nditions with the ones having the worst fading. This is because such pairing benefits always the MTD to be de c o ded first, while the MT D to be decod ed secon d is not goin g to be affected by sig n ificant outside in terference n either b y residu al interferen ce fr om SIC. Conseque n tly , it is expected that as µ and/or I o, 1 take mean ingful values, the CRS’ s pe r forman ce approa c h es (but not necessary reaches) the optim um. The o ptimum sched uling requir es an exhaustiv e search over all the feasib le schedulin g outcomes in ord e r to ad opt th e one offering maximu m perfo rmance. Notice that the dimension of the search sp ace, D K,N , dep ends on K and N since • If K ≤ N the r e is only one feasible allocation, which is granting individual ch a nnel resources to all MTDs; • if N < K ≤ 2 N , th ere are 2 N − K MTDs that will be scheduled alone in their chan nels. T hus, there is a total of  K 2 N − K  for making such selection, while the remain ing 2( K − N ) MTDs n eed to b e paired between each other s to share K − N ch a n nels; which can b e p e r formed in  2( K − N ) − 1  ( K − N ) different ways; • if K > 2 N , it is necessary selecting the 2 N MTDs that will get the channe l resources for which there are  K 2 N  possibilities; and also making the pairing b y testing all the N (2 N − 1) different alternatives. Therefo re, 10 20 30 40 50 60 70 10 0 10 10 10 20 10 30 10 40 10 50 10 60 Fig. 2. A verage dimension of the search space ¯ D N as a function of N for ¯ m ∈ { 10 , 20 , 60 , 120 } . D K,N =    1 , if K ≤ N  K 2 N − K  2( K − N ) − 1  ( K − N ) , if N < K ≤ 2 N  K 2 N  N (2 N − 1) , if K > 2 N , (19) while on average the dimension of the search space is ¯ D N = P ∞ k =0 D k,N Pr( K = k ) , which can be stated a s in (20) at the to p of the next p age. Notice that ( a ) came from using (19), while ( b ) followed from using the C DF of K , ev aluating the sums an d using the d efinition of the Kummer confluent hyp ergeometric function , a n d performing some alg ebraic tran sformatio n s and simplifications. Fig. 2 shows ¯ D N as a fun ction o f N fo r different values of ¯ m . Notice that unless ¯ m ≪ N , the dimension of the search space beco m es extremely large on average, wh ic h makes the exhaustiv e search unfeasible. Since the schedu ling problem appears exactly when the contrary occur, e.g., wh en ¯ m is compara b le or greater than N since otherwise 1 MTD per channel is frequen tly viab le, we can conc lude that in deed the optimum scheduling th rough br ute f orce is un feasible. In the following section we discuss the overall system perfor mance after analy zing the relaying pha se, in which all 6 ¯ D N ( a ) = Pr( K ≤ N ) + 2 N X k = N + 1  k 2 N − k   2( k − N ) − 1  ( k − N ) Pr( K = k ) + N (2 N − 1) ∞ X k =2 N +1  k 2 N  Pr( K = k ) ( b ) = Q ( N + 1 , ¯ m ) + e − ¯ m ¯ m 1+ N 6( N − 1)!  3 1 F 1  1 − N , 3 2 , − ¯ m 4  + ¯ m ( N − 1) 1 F 1  2 − N , 5 2 , − ¯ m 4   +  1 − e − ¯ m  ¯ m 2 N N (2 N − 1) (2 N )! . (20) collected data is fo rwarded to the BS. I V . R E L A Y I N G P H A S E & O V E R A L L P E R F O R M A N C E In the relaying phase, the aggr egator transmits its aggr egated data to the BS 7 , 8 . The aggregated data can be successfu lly decoded by the BS if SI R meets th e following conditio n log 2 (1 + SIR rel ) ≥ τ K 1 , where τ = b T W in bits per channel use per MTD (bpcu /MTD) with T being the relay ing transmission time and W is th e available bandwid th for that transmission. SIR rel is the SIR of th e signal received at the BS. Then, we write the relaying success pr obability con ditioned on K 1 activ e ag gregated M TDs as p rel ( K 1 ) = Pr  SIR rel ≥ 2 τ K 1 − 1  ( a ) = E I o, 2 h Pr  h ≥ (2 τ K 1 − 1) I o, 2     I o, 2 i ( b ) = E I o, 2 h exp  − (2 τ K 1 − 1) I o, 2  i ( c ) = L I o, 2 (2 τ K 1 − 1) ( d ) = exp  − φ 2 Γ  1 + 2 α  Γ  1 − 2 α  2 τ K 1 − 1  2 α  , (21) where ( a ) comes f rom using SIR rel = g /I o, 2 assuming th at h denotes the ch annel p ower gain of the link b etween the ag- gregator and th e BS, ( b ) com e s fr om u sing F H ( h ) = 1 − e − h , ( c ) follows after u sing th e definitio n of the Laplace transform , and finally ( d ) com e s from using (1). W e now are able to ev aluate the average nu mber of successful MTDs, which is an overall perfo rmance metric embracing bo th , the aggregation and relayin g phases. That metric e valuates th e average nu mber o f MTDs being served by the ag gregator, whose data can be successfully rec e i ved b y the BS. W e can fo rmally write this metric as ¯ K a&r = E h K 1 1  SIR rel > 2 τ K 1 − 1  i = 2 N X k 1 =1 k 1 Pr( K 1 = k 1 ) p rel ( k 1 ) , (22) where Pr( K 1 = k 1 ) is g iv en in (7) and (15) fo r the RRS an d CRS schemes, resp ectiv ely . V . N U M E R I C A L R E S U LT S Both, simu lation an d an alytical resu lts, are p resented in this section to inv estigate the perform ance of ou r hybrid scheme as 7 For simplicity , we assume that each aggregat or has no buf fer and transmits all its aggrega ted data in one go as in [19], [24]. 8 Notice that this transmission occurs over only one BS serving channel, therefo re, the aggregat ion topology is reducing the number of BS channel alloc ations to the MTC de vices in a clu ster , from N in the c ase of no aggre gation, down to 1 . T he importance of such approach is highlight ed in [2], [11]. 10 20 30 40 50 60 70 0 5 10 15 20 25 30 35 40 45 50 simulation results are with markers Fig. 3. A verage number of successful MTDs as a function of the number of channe ls for τ ∈ { 0 . 1 , 0 . 3 } bpcu/MTD. a fu nction of th e system parameters while comparing it with an OM A setup. Un less stated o therwise, results a re o btained by setting ¯ m = 6 0 , α = 3 . 6 , µ = 10 % , θ = 1 , τ = 0 . 2 bpcu/MTD and a 1 = a ( i ) 1 = a 2 = a ( i ) 2 = δ / 2 . W e set φ 1 = − 10 dB, which matches the scenario where all the outside aggregators, serving areas of radius 40 m, are o perating with on e MTD per chan nel, while forming a PPP with den sity 10 − 4 . 4 / m 2 . Also, δ = 1 su c h that the average consumed po wer per ortho gonal ch annel keeps the same for either th e L = 1 or L = 2 setup, while the interferen ce g enerated over MT Ds sharing th e same channel but o utside the serving zo ne keeps similar as in th e OMA setup. For th e r elaying phase we set φ 2 = − 26 dB, wh ich m atches the scenario wher e BSs are serving circular ar e a s of appro x imately 50 0 m, while formin g a PPP with d ensity 1 π 500 / m 2 . Simulation results are genera ted using 20000 Mon te Carlo run s 9 . Fig. 3 shows that the hybrid scheme f or the aggr egation phase ca n impr ove the spectral efficiency by provid ing service to a g reater number of MTDs when the access deman d increases, e.g., ¯ m & 2 N . This claim co mes from [24], wh e re only the a g gregation p hase was an alyzed. W e now extend it by considering the relaying phase, where spite the fact that multiplexing a greater number of MTDs with the same rate degrades the system reliability , th e advantage of the hybrid scheme over the p urely OMA setup hold s. Notice that spectral efficiency for b o th setups, e.g ., L = 1 and L = 2 , d egrades by decreasing φ 1 , φ 2 as c an b e ob served in Fig. 4, and/or by increasing τ . In fact, when τ increases the degradation of the relaying ph ase performan ce d ue to mo re data that it is bein g transmitted could b e faster than the incre a se in the num ber of acti ve MTDs in th e aggr egation p hase wh en N increa ses, thus, the overall per forman ce may w orsen as it is sho wn in Fig. 3 for the case of τ = 0 . 3 b pcu/MTD when N & 45 . On th e other hand all the curves ten d to overlap w h en N 9 Note that simulatio ns, proposed analyt ical expressions and approxima- tions fit well in all the cases depicted Fig. 3-7, which valid ates our findings. 7 -34 -32 -30 -28 -26 -24 -22 -20 -18 -16 -14 0 5 10 15 20 25 30 35 40 45 50 simulation results are with markers Fig. 4. Relaying phase: av erage number of successful MT Ds as a function of φ 2 for φ 1 ∈ {− 15 , − 5 } dB and N = 30 . increases since the pro bability of ha v ing two MT Ds sharin g the same o rthogo nal ch annel decre a ses so that perform ance is similar to OMA setup. The fluctu ations o bserved in th e CRS scheme for d ifferent v alues of N is co nsequen ce of high er depend ence/sensitivity on power control co efficients compared to RRS scheme. Thus, a caref ul selection of th ose p arameters is required for each system setup. For the CRS scheme we were ab le to reach closed-f orm expressions in [24, Eqs. ( 33), (34)] for the power contr o l coefficients while attainin g similar reliability for both MTDs sharing the same channel. Notice that when u sing the RRS scheme, all MTDs have chance to transmit in d epend e n tly of the channel co nditions, wh ich cou ld model more realistic scen arios wh ere so me MT Ds r equire be served urgently . Also, RRS has a slight d ecreased p erform ance compare d to CRS scheme in the ag gregation phase [24]. Fig. 5 inv estigates the req uired a 1 for RRS to attain either similar r eliability for both MTDs shar ing the same cha nnel or a maximum a verage number of simu ltaneously acti ve MTDs in the aggregatio n p hase ¯ K [2 4, Eq. (17) ], as a fu nction of φ 1 . As o utside inter f erence increases, the req uired power contro l coefficient for th e first M T D, a 1 , d ecreases when similar reliability is the g oal, since the perform ance of the second MTD deterio rates faster and the p ower con trol coefficient a 2 should increase. Otherwise, when the goal is to max imize the number of simultan e ously active MTDs, th e performance of the second MTD need s to be sacrificed, even mor e so when the outside inter ference incr eases until the hybr id scheme perfor ms as the OMA setup, e.g., a 1 = 1 , a 2 = 0 . Notice that almost all the time, a g reater SIC imperfection leads to a reduction in the required a 1 , decreasing its impact on the perfor mance of the seco n d decod e d MTD. Only when reachin g ¯ K max is the goal an d the outside in terference is sufficiently large, a greater SIC imper fection acc e le r ates the tran sition to OMA b y increa sin g a 1 . Also, increasing φ 1 deteriorates ¯ K (see numbere d lab els in Fig. 5). The reach able maximum a verage of simu ltaneously served MTDs as a function total power constraint coefficient, δ , is shown in Fig. 6. Ther ein, we find the coefficients a 1 and a 2 that m aximize ¯ K a & r such that a 1 + a 2 = δ . Notice that increasing δ has a po siti ve imp act on the system perfor mance as lon g as the appro priate values of a 1 and a 2 are s elected, which can be ded u ced from nu mbered labe ls in the figure. For the ca se of τ = 0 . 3 bpcu/MT D this effect is no t evident -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 55 54 50 46 51 49 45 43 42 40 39 35 28 25 25 22 14 7 14 7 simulation results are with markers Fig. 5. a 1 ( a 2 = δ − a 1 ) as a function of φ 1 for the RRS scheme to attai n either similar reliabil ity for both MT Ds sharing the same channel or a maximum aver age number of simultaneously acti ve MTDs in the aggrega tion phase. µ = { 0 , 10 } % and N = 30 . 0.5 1 1.5 2 0 10 20 30 40 50 60 Fig. 6. Maximum av erage of simultaneously served MT Ds as a function of δ , for N = 30 and τ ∈ { 0 . 10 . 3 } bpcu/MTD. 0 5 10 15 20 25 30 35 40 45 50 20 25 30 35 40 45 50 Fig. 7. A verage number of succe ssful MTDs as a funct ion of the SIC imperfect ion parameter for N = 30 . since the relaying phase is limitin g the system p erform a nce much mor e than the agg r egation phase. On the other ha nd, increasing δ is not always feasible, e.g., d ue to transmit hardware limitation s, or even adv isable, e.g. , du e to the extra interferen ce that might be g enerated over o ther OMA network s or b ecause o f a low e n ergy efficiency p erform ance. Therefo re, the ap propr iate selectio n of δ is of param o unt impor ta n ce. Fig. 7 shows the average num ber of simultaneo u sly served MTDs as a function of the SIC imperfectio n coefficient. W e set N = 30 such that eac h chan nel is o p erating with two MTDs almost all th e time, which are more sensitive to the interferen ce and imperfection of the SI C. Since SIC is only 8 related with the L = 2 setup, the OMA setup curves are shown with straight lines. Of co urse, when µ inc reases, the perfor mance of the L = 2 setu p d eteriorates, specifically if the power co efficients are not tune d ac c o rding ly . This is because those coefficients work well for certain system parameter s but others will be require d if they chang e, e.g. different µ in this case. It is expe c ted th at a smaller a 1 , hence larger a 2 , work better as µ increases as shown previously in Fig. 5. It is clear that failing to efficiently eliminate the inside interfer ence could reduce s ignifican tly th e b enefits f r om NOMA, and can be a challengin g issue f or implem enting NOM A in practice . V I . C O N C L U S I O N W e analyzed th e d ata aggr egation and relayin g in interferen ce-limited mMTC n etwork. W e ev a luate a hybrid access scheme, OMA-NOM A, while inv estigating its perfor- mance in terms of av erage numb er o f simu ltaneously served MTDs. Power control coefficients are inc orpor a ted to th e practical-inter est RRS sch eme, while we inves tigate them numerically . The n umerical results also sh ow th at our hyb rid access scheme aims at providing massive con nectivity in scenarios with hig h access dem and. Howe ver, inter-cluster in- terference co uld red uce significantly the b enefits fro m NOMA, and challenging its implem entation in practice. A P P E N D I X A P R O O F O F T H E O R E M 1 As in [2 4, Th. 1] , let us write th e success p robabilities as p r 1 , 1 = E I o, 1 [Pr( h > θ I o, 1 | I o, 1 )] = E I o, 1 h e − θ I o, 1    I o, 1 i , (2 3) p r 1 , 2 = E I o, 1 h Pr  max( h 1 , h 2 ) − θ a 2 a 1 min( h 1 , h 2 ) > θ a 1 I o, 1 | I o, 1  i = E I o, 1 h Pr  v 1 > θ a 1 I o, 1 | I o, 1  i = 1 − E I o, 1 h F V 1  θ a 1 I o, 1 i ( a ) =    E h 2 a 1 a 1 + θ a 2 e − θ a 1 I o, 1 − a 1 − θ a 2 a 1 + θ a 2 e − 2 θ a 1 − θa 2 I o, 1    I o, 1 i , if 0 ≤ θ a 2 a 1 < 1 E h 2 a 1 a 1 + θ a 2 e − θ a 1 I o, 1 | I o, 1 i , otherwise (24) p r 2 , 2 = E I o, 1 h Pr  min( h 1 , h 2 ) − θ µa 1 a 2 max( h 1 , h 2 ) > θ a 2 I o, 1 | I o, 1  i = E I o, 1 h Pr  v 2 > θ a 2 I o, 1 | I o, 1  i = 1 − E I o, 1 h F V 2  θ a 2 I o, 1 i ( b ) = ( E h a 2 − θ µa 1 a 2 + θ µa 1 e − 2 θ a 2 − θµa 1 I o, 1    I o, 1 i , if 0 ≤ θ µa 1 a 2 < 1 0 , otherwise , (25) where p r j,u = Pr  SIR r j , u > θ  , v 1 = max( h 1 , h 2 ) − θ a 2 a 1 min( h 1 , h 2 ) and v 2 = min ( h 1 , h 2 ) − θ µa 1 a 2 max( h 1 , h 2 ) , Fig. 8. Region of intersec tion. a) 0 ≤ θa 2 a 1 < 1 (top), b) θa 2 a 1 ≥ 1 (bottom). while ( a ) and ( b ) co me from using th e ir CDF expression s, which are ob tained next. F V 1 ( v 1 ) = P r(max( h 1 , h 2 ) − θ a 2 a 1 min( h 1 , h 2 ) ≤ v 1 ) = Pr  max( h 1 , h 2 ) ≤ v 1 + θ a 2 a 1 min( h 1 , h 2 )  = Pr  h 1 ≤ v 1 + θ a 2 a 1 min( h 1 , h 2 ) \ h 2 ≤ v 1 + θ a 2 a 1 min( h 1 , h 2 )  = Pr  min( h 1 , h 2 ) ≥ ( h 1 − v 1 ) a 1 θa 2 \ min( h 1 , h 2 ) ≥ ( h 2 − v 1 ) a 1 θa 2  = Pr  h 1 ≥ ( h 1 − v 1 ) a 1 θa 2 \ h 2 ≥ ( h 1 − v 1 ) a 1 θa 2 \ h 1 ≥ ( h 2 − v 1 ) a 1 θa 2 \ h 2 ≥ ( h 2 − v 1 ) a 1 θa 2  = Pr  h 1 (1 − θ a 2 a 1 ) ≤ v 1 \ h 2 ≥ h 1 a 1 θa 2 − v 1 a 1 θa 2 \ h 2 ≤ h 1 θ a 2 a 1 + v 1 \ h 2 (1 − θ a 2 a 1 ) ≤ v 1  . (26) Lets consider the following cases: • 0 ≤ θ a 2 a 1 < 1 , then we can continue f rom equation (26) 9 as f ollows. F V 1 ( v 1 ) = Pr  h 1 ≤ v 1 1 − θ a 2 a 1 \ h 2 ≥ h 1 a 1 θa 2 − v 1 a 1 θa 2 \ h 2 ≤ h 1 θ a 2 a 1 + v 1 \ h 2 ≤ v 1 1 − θ a 2 a 1  (27) and th e intersectio n region is shown in Fig . 8a an d notice that h 1 , h 2 ≥ 0 are also re stric tio ns. Therefore we ca n transform (27) to attain the result in (28) at the top of the n ext page. • θ a 2 a 1 ≥ 1 , th en we can continu e from equa tion (26) as follows. F V 1 ( v 1 ) = P r  h 1 ≥ v 1 1 − θ a 2 a 1 \ h 2 ≥ h 1 a 1 θa 2 − v 1 a 1 θa 2 \ h 2 ≤ h 1 θ a 2 a 1 + v 1 \ h 2 ≥ v 1 1 − θ a 2 a 1  , (29) Notice that re gions h 1 ≥ v 1 1 − θ a 2 a 1 < 0 and h 2 ≥ v 1 1 − θ a 2 a 1 do not say nothing ne w because we already know that h 1 , h 2 ≥ 0 . Theref o re we can write (29) as F V 1 ( v 1 ) = P r  h 2 ≥ h 1 a 1 θa 2 − v 1 a 1 θa 2 \ h 2 ≤ h 1 θa 2 a 1 + v 1  = Pr  h 1 a 1 θa 2 − v 1 a 1 θa 2 ≤ h 2 ≤ h 1 θa 2 a 1 + v 1  , (30) and the intersection region is shown in Fig. 8b. Thu s, b y working on (30) we reach (31) a t the top o f next p a g e (below (28)). By comb in ing (28) and (31) we attain the g e neral expression for F V 1 ( v 1 ) , which is F V 1 ( v 1 ) =      1 − 2 a 1 a 1 + a 2 θ e − v 1 + a 1 − a 2 θ a 1 + a 2 θ e − 2 a 1 v 1 a 1 − a 2 θ , if 0 ≤ a 2 θ a 1 < 1 1 − 2 a 1 a 1 + a 2 θ e − v 1 , if a 2 θ a 1 ≥ 1 , (32) while F V 2 ( v 2 ) =      1 − a 2 − θ µa 1 a 2 + θ µa 1 e − 2 v 2 a 2 a 2 + θ µa 1 , if 0 ≤ θ µa 1 a 2 < 1 1 , if θ µa 1 a 2 ≥ 1 (33) can b e attained b y f ollowing a similar pro cedure. Finally , b y tak ing the expectation in (2 3), (24) an d (2 5) we attain (4), (5) and (6), respectively . A P P E N D I X B P R O O F O F T H E O R E M 2 For the system mo d el in Section II th e outside interfe rence on each channel is in depend ent. Let K be the numbe r of MTDs requirin g data tr a n smission, th e c ondition al distribution of K r 1 is giv en in (34) ( on th e top of the next p age) wh ere f 1 ( k r 1 ) = min( k r 1 , 2 N − K ) , f 2 ( k r 1 , r 1 ) = min( k r 1 − r 1 , K − N ) and f 3 ( k r 1 ) = min( k r 1 , N ) . Notice that when K ≤ N all the MTDs are oper ating alone in the ir channels, while wh en N < K < 2 N , 2 N − K MTDs will be op erating alone and K − N will be s haring their chan nels. Add itionally , when K ≥ 2 N , the N M T Ds with channel resources are sharing their chan nels. Thus, (34) comes from combin atorial and pro bability theories. The indicato r f unction guar antees operatio n in the app ropriate regions, e.g., N < K < 2 N and K ≥ N , according to the particular case. After using Pr( K r 1 = k r 1 ) = P ∞ k =0 Pr( K r 1 = k r 1 | k ) Pr( K = k ) along with som e algeb raic man ipulations we attain th e resu lt in (7). R E F E R E N C E S [1] L. Atzori, A. Iera, and G. 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