Modeling Uplink Coverage and Rate with Aggregation in Machine-to-Machine Communication Networks

Transcription

1 Modelng Uplnk Coverage and Rate wt Aggregaton n Macne-to-Macne Communcaton Networks Derya Malak, Harpreet S. Dllon, and Jeffrey G. Andrews Abstract Macne-to-macne (MM) communcaton s severe power lmtatons callenge te nterconnectvty, access management, and relable communcaton of data. In densely deployed MM networks, coordnatng and aggregatng te generated data s crtcal. We propose an energy effcent data aggregaton sceme for a erarccal MM network wt truncated poweontrol. We optmze te number of erarccal stages and perform a coverage probablty-based uplnk analyss for MM devces. Our analyss exposes te key tradeoffs between te coverage caracterstcs for successve and parallel transmsson scemes tat can be eter alf-duplex or full-duplex. Comparng te rate performances of te transmsson models, we observe tat successve and alf-duplex parallel modes ave betteoverage caracterstcs compared to full-duplex parallel sceme. Index Terms Macne-to-macne communcaton, aggregator, rate, coverage, uplnk, truncated poweontrol, network erarcy, stocastc geometry. I. INTRODUCTION Applcatons nvolvng macne-to-macne (MM) communcaton are rapdly growng, and wll become an ncreasngly mportant source of traffc and revenue n current 4G and future 5G cellular networks. Unlke vdeo applcatons, wc are expected to consume around 70% of all wreless data by te end of te decade [], MM devces wll use a comparatvely smaller fracton. However, MM communcaton as ts own callenges. Te ar nterface desgn for gdata-rate applcatons may not effectvely support vast number of devces expected n MM communcatons, eac usually avng only a small amount of data to transmt. Tus, MM wll requre sopstcated access management and resource allocaton wt QoS constrants to prevent debltatng random access cannel (RACH) congeston []. Unlke most uman generated oonsumed traffc, MM as defned n ts paper s caracterzed by a very large number of small transactons, often from battery powered devces. Te power and energy optmal uplnk desgn for varous access strateges s studed n [3], wle an optmal uncoordnated strategy to maxmze te average trougput for a tme-slotted RACH s developed n [4]. For te small payload szes relevant for MM, a strategy tat transmts bot dentty and data over te RACH s sown to support sgnfcantly D. Malak and J. G. Andrews are wt te Wreless and Networkng Communcatons Group (WNCG), Te Unversty of Texas at Austn, TX, USA. Emal: deryamalak@utexas.edu, jandrews@ece.utexas.edu. H. S. Dllon s wt Wreless@VT, Department of Electrcal and Computer Engneerng, Vrgna Tec, Blacksburg, VA, USA. Emal: dllon@vt.edu. Manuscrpt last revsed: February 9, 06. more devces compared to te conventonal approac, were transmssons are sceduled after an ntal random-access stage. An energy-effcent uplnk desgn for LTE networks n MM and uman-to-uman coexstence scenaros tat satsfes QoS requrements s developed [5]. In MM communcaton, crtcal desgn ssues also nclude energy effcent data aggregaton along wt effcent erarccal organzaton of te devces. Clusterng s a key tecnque to reduce energy consumpton and t ncreases te scalablty and lfetme of te network [6]. Altoug tese ssues ave not been studed n te context of MM communcaton, tere s pror work on dstrbuted networks n te context of wred communcatons. In [7], energy consumpton s optmzed by studyng a dstrbuted protocol for statonary ad oc networks. In [8], a dstrbuton problem wc conssts of subscrbers, dstrbuton and concentraton ponts for a wred network model s studed to mnmze te cost by optmzng te densty of dstrbuton ponts. In [9], a erarccal network ncludng multple snks, aggregaton nodes and sensors s proposed, wc yelds sgnfcant energy savngs. We propose an energy-effcent desgn for MM uplnk were devces ave a maxmum poweonstrant. Snce te MM devces are power lmted and tus range lmted, mult-op routng s a feasble strategy rater tan drect transmssons. However, n desgnng mult-op protocols, te number of ops cannot be ncreased arbtrarly due to te addtonal energy consumpton ncurred by relays; long-op routng s a compettve strategy for many networks [0]. Terefore, n ts paper, we analyze a erarccal MM communcaton model wt data aggregaton nvolvng multop transmssons. Despte prevous researc efforts, e.g., [5], [], to te best of our knowledge, tere as been no study focusng on data aggregaton scemes for MM networks togeter wt te rate coverage caracterstcs, especally from an energy optmal desgn perspectve. Provdng suc a study s te man contrbuton of ts paper. We propose a general aggregator model for power lmted MM devces n Sect. II. Usng stocastc geometry, we analyze te SIR and rate coverage caracterstcs of te multstage transmsson process wt truncated poweontrol to determne te optmal number of stages (ops), wc wll be detaled n Sect. III. We consder two possble transmsson tecnques: ) successve sceme, were te erarccal levels are not actve smultaneously, and ) parallel sceme, were eter all te levels are actve smultaneously as n full-duplex transmsson or were te actve levels are nterleaved as n

2 alf-duplex mode, wc wll be detaled n Sect. IV. In Sect. V, we provde a numercal performance comparson n terms of te communcaton rates of te proposed tecnques. II. DATA AGGREGATION AND TRANSMISSION MODEL We consder a cellular-based uplnk model for MM communcaton were te BS and devce locatons are dstrbuted as ndependent Posson Pont Processes (PPPs) wt respectve denstes of BS and wt BS. Devces transmt data to te BS by aggregaton. Te ntal devce process s ndependently tnned by probablty <0.5 to generate te aggregator process a wt densty a and te transmtter process u wt densty u, were = u + a. Eac transmtter s assocated to te closest aggregator,.e., te transmtters n te Vorono cell of te typcal aggregator wll transmt ter payloads to tat aggregator. Te aggregaton process can be extended to multple stages. Eac erarccal level s composed of te transmtter and te aggregator processes, were te aggregator processes of all stages are ntally determned suc tat tey are dsjont from eac oter. At eac stage, after te set of transmtters transmt ter payloads to ter nearest aggregators, and once te transmssons of a erarccal level are completed, te transmtters are turned off and excluded from te process. In te subsequent erarccal level, te aggregators of te prevous stage become te transmtters, and tey transmt ter data to te aggregator process of te new stage. Te aggregaton process s repeated over multple stages to generate te erarccal transmsson model. Te process ends wen all te payload s transmtted to te BS n te last stage of ts mult-op process, wc we call a transmsson cycle. Te transmtter devce densty of te frst stage s found by subtractng te total densty of aggregators P from te ntal K densty of te devce process as u() = k= a(k). By te end of te frst stage, te devces wt a densty of u() wll transmt ter payload to te aggregator process wt a densty of a(). Next, at te second stage, te aggregators of te frst stage form te new transmttng devce process,.e., u() = a (), and tese devces transmt to te devces formng te set of second stage aggregator process wt densty a(). Te aggregaton process can be extended to k>tages n a smlar manner by lettng u (k) = a (k ) for k. Te mportant desgn parameters are tabulated n Table I, were K = P K k= k. In Fg., we llustrate a tree-stage model, were te aggregator process of stage k,.e., a(k), s obtaned by ndependently tnnng wt probablty 0.4 k,.e., a(k) as a densty of 0.4 k. Eac BS as an average coverage area of BS, and eac devce as a fxed payload of M bts to be transmtted to te BS. We assume standard power law pat loss model were te sgnal power decays as d over dstance d, were s te pat loss exponent. We also assume open loop power control wt maxmum transmt poweonstrant under wc te transmt power of a devce located at dstance d from te BS s P T (d) =mn{p Tmax, P T d }, were P Tmax s te maxmum transmt poweonstrant and P T s te receved / power wen d apple P Tmax /P T. Wt ts assumpton, te u(k) ( K) a(k) E [N a(k)] ( K)/ k = apple k apple K k = K k K k BS K / BS t tx(k) ( K)/ k ( K)/ K TABLE I: Desgn parameters. average receved power at te BS from a devce n ts coverage area and located at dstance d from te BS s constant and equal to P R (d) =mn{p Tmax d, P T }. III. SIR COVERAGE PROBABILITY We propose a coverage-based model for data aggregaton, assumng tat te transmsson s successful f te SIR of te devce s above a tresold. We derve te coverage probablty n te uplnk for te proposed data aggregaton model. Ortogonal access s assumed n te uplnk and at any gven resource block, tere s at most one devce transmttng n eac cell. Let u be te process denotng te locaton of devces transmttng on te same resource as te typcal devce. Te uplnk SIR of te typcal devce x u located at dstance kxk from ts aggregator on a gven resource block s SIR = g mn{p Tmax kxk, P T } P g z mn{p Tmax, P T Rz }Dz, () z u\{x} were R z and D z denote te dstance between te transmtter aggregator par and te dstance between te nterferer and te typcal aggregator, respectvely. Te random varable g z g s te small-scale d fadng parameter due to nterferer z. Assumpton. Te actual dstrbuton of R z s very ard to caracterze due to te randomness bot n te area of te Vorono cell of te aggregator and n te number of te devces t serves. Terefore, we approxmate t by te dstance of a randomly cosen pont n R to ts closest aggregator,.e., by a Rayleg dstrbuton []: f Rz (r z )=(r z / )e r z /, r z 0, = p /( a ). () Te uplnk SIR coverage of te proposed system model wt truncated poweontrol s gven by te followng Lemma. Lemma. Te uplnk SIR coverage wt truncated power control: Wt truncated poweontrol and wt mnmum average pat loss assocaton, te uplnk SIR coverage s P(T )=pl Ir (T P T )+ Z L Ir (Tr P )f R (r)dr, (3) were p = exp a rc, R s Rayleg dstrbuted wt parameter = p /( a ), and L Ir (s) e e ( e a (+ ar c ))P T C (sp T ) ( p) ap Tmax E Rz[R z C (sp Tmax R z ) R z>] (4) denotes te Laplace transform of te nterference were C (T )= F,,, T, and F s te Gauss- Hypergeometrc functon. In mnmum average pat loss assocaton, a devce assocates to an aggregator wt mnmum pat loss averaged over te small-scale fadng,.e., te aggregator as mnmum Rz product among all aggregators.

3 λ u () λ u () λ u () λ u (3) () () () (3) () (3) All devces Stage Stage Stage 3 Fg. : A tree-stage aggregaton sceme wt PPP( =) devce process. Te aggregator processes (stage :, stage :, stage 3: ) are obtaned by tnnng te orgnal PPP, and ave denstes a(k) =0.4 k P for stages k=:3. Densty for te ntal transmtter process (stage : ) s u() = 3 k= a(k), and for te transmtter processes for later stages are u(k) = a(k ) for k>, respectvely. Last stage aggregators correspond to te BSs. Proof. See Appendx A. Corollary. For no maxmum poweonstrant, wt open loop poweontrol and wt mnmum average pat loss assocaton, te uplnk SIR coverage s [3] lm P(T ) exp P Tmax! T C (T ) A. Coverage Probablty and Number of Stages. (5) Te number of mult-op stages K s manly determned by te SIR coverage and te dstance coverage, wc we defne as te probablty tat te dstance between a devce and ts nearest aggregator s below a tresold. We provde bounds on K usng tese coverage concepts. Assumpton. Interstage ndependence. Te proposed erarccal aggregaton model ntroduces dependence among te stages of mult-op communcaton snce eac subsequent stage s generated by te tnnng of te prevous stage. For analytcal tractablty, we assume tat te mult-op stages are ndependent of eac oter. Hence, te transmsson s successful f and only f all te ndvdual stages are successful. Te success probablty over K stages s P cov (K) =P(SIR >T,...,SIR K >T)= Y K k= P k(t ), were P k (T ) denotes te coverage probablty at stage k. Wt full cannel nverson and mnmum average pat loss assocaton, te uplnk SIR coverage s ndependent of te nfrastructure densty as gven n (5). For te case P Tmax!, snce P k (T ) n (5) s also ndependent of te devce densty, and only depends on te tresold T and pat loss exponent, s dentcal for all stages, and denoted by P(T ). Lemma. For any maxmum poweonstrant P Tmax, gven a mnmum probablty of coverage requrement P cov (K)> " and an SIR tresold T, te number of stages s upper bounded by log (/( ")) K U =. (6) log(max k P k (T )) Proof. Te upper bound s obtaned by combnng (3) wt te condton P cov (K) > " and usng te relaton P cov (K) = Q K k= P k(t ) apple max k P k (T ) K. In addton to te SIR outage, snce te devces are randomly deployed, any devce wll be n outage wen ts nearest aggregator s outsde ts transmsson range. Tus, we am to nvestgate te mnmum number of requred stages gven a dstance outage constrant. A lower bound on te optmal number of mult-op stages s gven by te followng Lemma. Lemma 3. Te number of stages s lower bounded by l PRmn m K L = E[L( a )]E, (7) N a P Tmax were L( a ) denotes te total lengt of te connectons, N a s te random varable tat denotes te number of devces n te Vorono cell of a typcal BS or aggregator, P Tmax s te maxmum transmt power and P Rmn s te mnmum detectable sgnal power at te recever. Proof. Te proof follows from te mean addtve caracterstc assocated wt te typcal cell of te access network model provded n [4, C. 4.5]. Te mean total lengt of connectons n te Vorono cell of an aggregator equals E[L( a )] = kxk e a kxk 3/ dx = u a /. (8) R Gven a maxmum transmtter poweonstrant, P Tmax for eac devce, te maxmum transmsson range s gven by (P Tmax /P Rmn ) /. Dvdng L( a ) by N a and takng ts expectaton wt respect to te dstrbuton of N a, we obtan te mean lengt of connectons, and dvdng ts rato by te maxmum transmsson range, we get te desred result. u Z IV. TRANSMISSION RATE MODELS For an nterference lmted network, te rate of te typcal devce s Rate = W N a log ( + SIR), were W s te total bandwdt of te communcaton cannel, and N a s te load at te typcal aggregator. Te average number of devces served by te typcal aggregator s E[N a ]= u / a. Rate coverage s defned as rate exceedng a gven tresold,.e., P(Rate > )= X P (SIR >T N a = l) P Na (l), (9) l=0 were T = Na W and P Na (l) s te probablty mass functon (PMF) of N a, and s gven by te followng Lemma. Lemma 4. Te PMF of te number of devces served per aggregator of stage k,.e., N a (k), s ( u (k)/ a (k)) l (3.5+l) P Na(k)(l) = (3.5+( u (k)/ a (k))) 3.5+l (3.5) (l + ), (0) For tractablty, we take expectaton over a PPP frst to fnd E[L( a)], and ten multply t by E assumng ndependence. N a

4 were u(k) and a (k) for k are gven n Table I. Proof. Normalzed dstrbuton functon of Vorono cell areas n D can be modeled by [5] as f(y) = (3.5) y 5 e 7 y. Usng te denstes of te transmtters and te aggregators of stage k, te probablty generatng functon (PGF) of te stage k devces n te random area y s [6] G Na(k)(z) =E[exp (( u (k)/ a (k))y(z ))], () wc s of a Posson random varable N a (k) wt mean ( u (k)/ a (k))y. Te PMF of N a (k) can be recovered by takng dervatves of G Na(k)(z) as P Na(k)(l) =G (l) N a(k) (0)/l!. Te key assumpton n our analyss s tat tere s one actve devce per resource block n eac Vorono cell. Usng (0) te probablty of not fndng any devce n te Vorono cell of te typcal aggregator at stage k s P Na(k)(0) = G Na(k)(0) = (3.5+E[N a (k)]) 3.5. (4) Let p t (k) be te probablty tat tere s at least one devce n te Vorono cell of te typcal aggregator n te k t stage. Terefore, te nterference feld of stage k s tnned by p t (k), and te effectve densty of te nterferng devces at stage k s p t (k) u (k). Usng (4), p t (k) = (3.5+ E[N a (k)]) 3.5. From Table I, we can see tat p t (k) s te same for <k<k, and dfferent for k {,K}. We consder two man transmsson protocols, namely ) a successve transmsson protocol were te stages are actvated sequentally,.e., a alf-duplex sequental mode, and ) a parallel transmsson mode, wc s eter a full-duplex protocol were all stages are smultaneously actve, or a alfduplex protocol wt alternatng actve stages. A. Rate Dstrbuton for Successve Stages In ts mode, eac transmsson cycle conssts of te stages operatng n successon. Ts mode may provde lower resource utlzaton, but t as low nterference snce multple stages are not actve smultaneously. Let K = {,...,K} denote te set of stages and R = {Rate,...,Rate K } be te set of transmsson rates at eac stage. Te transmsson rate n successve mode s gven by R S = K mn kk Rate k. Remark. Dependence of erarccal levels. Te erarccal levels are not ndependent from eac oter and ence, t s not tractable to analyze te jont rate dstrbuton for successve stages. Instead, we defne te rate outage as n (5) were transmsson rates are assumed ndependent. Wtout trackng te pat of te bts (payload) transmtted, we only consder f te erarccal transmsson process s successful. Transmsson from a devce s successful f ts payload s delvered to te BS at te end of K stages. Wt te ndependence assumpton n Remark, te rate coverage for successve stages s P(R S > ) Y X P k ( K l W )P Na(k)(l). (5) kk l=0 If we let P Tmax lm P k( K l W P Tmax!!, P k ( K l W ) exp ) s smplfed to K l W C ( K l W ). B. Rate Dstrbuton for Full-Duplex Parallel Stages In full-duplex mode, transmssons are not nterrupted durng a transmsson cycle unlke te sequental mode. All stages operate n parallel, te st stage devces only transmt, and te rest of te devces bot transmt and aggregate smultaneously, durng all stages of te mult-op transmsson. Due to te smultaneous transmssons at all levels of te erarccal model, te nterference at eac stage s due to ) te nterferers of tat stage,.e., te ntra-stage nterference, and ) te remanng transmttng devces of te oter stages,.e., te nter-stage nterference. Te erarccal levels are determned n te same manner smlar to te successve mode, and also dependent n ts mode, and ence, te nter-stage nterference s correlated. Altoug full-duplex parallel mode offers g resource utlzaton compared to successve mode, t as ger nterference snce all te stages are actve. Te ntra-stage nterference n te parallel mode can be obtaned n te smlar manner as n te successve mode. Lemma 5 (See next page.) provdes te analytcal expressons for te Laplace transforms of ntra-stage and nter-stage nterference. Corollary. Te Laplace transform of te total nter-stage nterference for P Tmax!s P (K T ). L Ik c (s) exp B (sp T ) C (sp T ) (6) Lemma 6. Te uplnk SIR coverage probablty for te fullduplex parallel mode s gven by P(SIR P >T) Y p k L Ik (T P T )L Ik c (T P T ) kk Z + L Ik (Tr P )L Ik c (Tr P )f Rk (r)dr, (7) were f Rk (r) =(r/ k )e r / k for k = p /( a (k)). Proof. Te proof s smlar to tat of Teorem I n [3]. However, nstead of one servng stage, te servng stages cange sequentally. Result follows from te evaluaton of () usng (3). Snce te total nter-stage nterference n (3) s ndependent from te ntra-stage nterference, ts Laplace transform s ncorporated as a multplcatve term. Lemma 7. Lettng T l = l W, te rate coverage probablty for te full-duplex parallel transmsson mode s gven by P(R P > )= Y X P(SIR k >T l N a (k) =l)p Na(k)(l). (8) kk l=0 Remark. Te models descrbed above,.e., te sequental mode wc s alf-duplex by desgn, and te full-duplex parallel mode, are te two prncple desgn scemes tat manly dffer n terms of ter total rate coverages. Altoug te full-duplex strategy s probably not feasble for MM communcaton, t s elpful to ave a comparson of te rate dstrbutons of bot models, and provded foompleteness. Next, we ntroduce te alf-duplex parallel transmsson. C. Rate Dstrbuton for Half-Duplex Parallel Stages Te full-duplex parallel mode can be transformed nto a alf-duplex communcaton sceme. In ts mode, at a

5 Lemma 5. Te Laplace transforms of te ntra-stage nterference and te nter-stage nterference are gven as follows: (a) Te Laplace transform of te ntra-stage nterference at stage k s L Ik (s) exp ( e eff a (k) ( + e a (k)rc))p T C (sp T ) +( p k ) e a (k)p Tmax E Rzk Rz k were p k = exp( a (k)rc) and e a (k) =p t (k) a (k). (b) Te Laplace transform of te total nter-stage nterference from stages {l l K,l 6= k} s L Ik c (s) Y exp ( e eff a (l) ( + e a (l)rc))(b (sp T )+ P T lk c were B (s) = F,, +, s ( p l ) e a (l)e Rzl R z l C (sp T )) + P sptmax ( )Rz C l Rz l sptmax B Rz l s obtaned usng te Gauss-Hypergeometrc functon. Proof. Proof s skpped due to space constrants. It s provded n te extended verson of ts paper [7]. sptmax C Rz R zk >, () k R zl >, (3) partcular tme slot, only te even or odd stages are actve. Analyss of ts model s qute smlar to te parallel mode analyss, usng only te actve stages wen calculatng te nter-stage nterference. Te SIR coverage of te alf-duplex mode can be caracterzed by te followng lemma. Lemma 8. Te uplnk SIR coverage probablty for te alfduplex parallel mode s gven by P(SIR P >T) Y p k L Ik (T P T )L Ik c (T P T ) kk H + Z L Ik (Tr P )L Ik c (Tr P )f Rk (r)dr. (9) e were calculatons of a (k) and te dstrbutons of {R zk } n ()-(3) are done over te set of actve stages and K H denotes te set of actve stages,.e., te even or te odd stages. For te alf-duplex mode, te rate coverage expresson n (8) canges n accordance wt (9), wc can be found by followng te steps n Lemma 7. Snce only alf of te stages are actve smultaneously, te overall rate s alf te rate of te actve stages. Tus, to aceve a rate tresold of, te rate tresold for te actve stages sould be set to. V. NUMERICAL RESULTS Te smulaton setup for te verfcaton of analytcal rate models developed n Sect. IV s as follows. Devce locatons are dstrbuted as PPP over a square regon of sze 5 5 sq. km. Total bandwdt s W = 0 5 Hz, =4, devce and BS denstes are = 0 3 and BS = per sq. km., and te devce payload s M = 00 bts. Note tat te densty of aggregators at stage k s k,.e., te number of aggregators decay geometrcally. Terefore, for g K values we need a regon wt muc larger area for te valdaton of te model, but scalng te regon ncreases te computatonal complexty exponentally. Terefore, we restrct ourselves to K =:3, and nvestgate te performance of te proposed model. Te rate coverages for te sequental and te full-duplex modes are llustrated n Fgs. and 3. Te results provde tgt approxmaton despte te nterstage ndependence assumpton n Sect. III-A, and te ndependent poweontrol assumpton n []. Te full-duplex mode does not offer ger rate coverage compared to sequental mode, wc s due to nterstage nterference as detaled n Sect. IV-B. Tus, n terms of rate coverage, sequental mode s preferable over full-duplex mode. Furtermore, te alf-duplex parallel mode as ger coverage tan full-duplex mode for te same K. Due to lmted space, we do not provde an llustraton for tat sceme. Consderng te operatng regme for MM devces and ter desgn smplcty, sequental and alf-duplex parallel modes are feasble tecnques and preferable as tey ave g coverage. Dfferent transmsson modes also ave dfferent energy demands. Investgaton of te coverage and te energy requrements for tese scemes can reveal te tradeoffs furter, wc s provded n te extended verson of ts paper [7]. Based on te numercal results, te take-away message s tat devces sould not do aggregaton,.e., drect transmsson provdes te gest rate. However, ts s true f we assume te fracton of aggregators,.e.,, s kept fxed for any number of stages. One of te reasons for mult-op transmssons s te dstance constrant between te devce and aggregator. If te devces are restrcted to transmt over sorter dstances to compensate te transmt poweonstrant, ten sould ncrease, wc sfts ter rate coverage curves towards rgt. Altoug our aggregaton model s SIR-based, ncludng nose s a natural way to lmt te lnk dstances and decrease te coverage. Terefore, nose wll not ave a strong nfluence on te analyss and ouonclusons wll not be effected muc. VI. CONCLUSIONS We study a general mult-op-based uplnk communcaton sceme for MM communcaton, and develop SIR and rate coverage models for MM devces for dfferent transmsson scemes, usng tools from stocastc geometry. Consderng te operatng regme of nterest for MM devces, sequental and alf-duplex parallel modes are more feasble compared to full-duplex mode. Interestng extensons would nclude te mnmzaton of te energy expendture troug jont optmzaton of te optmal number of mult-op stages and fracton of aggregators. Strateges for syncronzaton of transmssons s also mportant to prevent mult-op delays and save energy.

6 Rate Coverage K= Smulaton K= Analyss K= Smulaton K= Analyss K=3 Smulaton K=3 Analyss Rate Tresold (bps) Fg. : Rate coverage for sequental mode, K = {,, 3}. Rate Coverage K= Analyss K= Smulaton K= Analyss K= Smulaton K= 3 Analyss K= 3 Smulaton Rate Tresold (bps) Fg. 3: Rate coverage for full-duplex mode, K = {,, 3}. A. Proof of Lemma APPENDIX Te Laplace transform of te nterference can be wrtten as L Ir (s) (a) = E Y E Rz +smn{p Tmax, P T Rz }Dz (b) e (c) = e e R R R y>0 z u\{x} E Rz y>0 pe Rz +s mn{p Tmax,P T Rz }y Rz<y u(dy) +s P T R z y Rz<y,Rz<rc u(dy) y>0 ( p)e Rz (d) = e ape Rz R R z R e a( p)e Rz R z (e) +s P Tmax y +s P T = e ape Rz[R Rz<rc] z e a( p)e Rz (f) = e e Rz<y,Rz>rc u(dy) dt t/ Rz<rc +s P Tmax t/ Rz sp T C (sp T ) dt R z> R z sp Tmax R z C (sp Tmax R z ( e a (+ ar c ))P T C (sp T ) ( p) ap Tmax E Rz[R z C (sp Tmax R ) R z> z ) R z>], were (a) follows from te d nature of {g z } and te ndependence of {R z } (see Assumpton ). Te process u s not a PPP but a Posson-Vorono perturbed lattce and ence te functonal form of te nterference (or te Laplace functonal of u) s not tractable [3]. Autors n [3] propose an approxmaton to caracterze te correspondng process as an nomogeneous PPP wt ntensty measure functon u (dy) = a y(dy). Hence, (b) follows from te defnton of probablty generatng functonal (PGFL) of te PPP [8], and te ndependent pat loss between te devce and ts servng aggregator [3],.e., Rz s are ndependent, (c) follows from te maxmum poweonstrant, were p = exp a rc denotes te probablty tat R z <, (d) follows from cange of varables t =(y/r z ),(e) follows usng te defnton of Gauss-Hypergeometrc functon, yeldng R dt = +s t / F,,, s = C (s), (f) follows from Assumpton for R z. Hence, Rz s exponentally dstrbuted wt rate parameter /( )= c (+ ar c ) a tat yelds pe Rz R e z R z < = ar a. Condtoned on te dstance between te devce and ts assocated aggregator, P(SIR > T r < ) = E Ir P g>tp T I r r< L Ir (T P T ), and P(SIR > T r > ) = E Ir P g>tr P I r r> L Ir (Tr P ). Hence, te uplnk SIR coverage s obtaned as P(T ) pl Ir (T P T )+ R L Ir (Tr P )f R (r)dr, were R Rayleg( ) wt = p /( a ). REFERENCES [] Csco vsual networkng ndex: Global moble data traffc forecast update, 04-09, wte paper, Csco, Feb. 05. [] H. S. Dllon, H. Huang, and H. Vswanatan, Wde-area wreless communcaton callenges for te Internet of Tngs, submtted, arxv preprnt arxv: , 05. [3] H. S. Dllon, H. C. Huang, H. Vswanatan, and R. A. Valenzuela, Power-effcent system desgn foellular-based macne-to-macne communcatons, IEEE Trans. Wreless Commun., vol., no., pp , Nov. 03. [4], Fundamentals of trougput maxmzaton wt random arrvals for MM communcatons, IEEE Trans. Commun., vol. 6, no., pp , Nov. 04. [5] A. Ajaz, M. Tsangn, M. Naka, X. Cu, and H. Agvam, Energyeffcent uplnk resource allocaton n LTE networks wt MM/HH co-exstence under statstcal QoS guarantees, IEEE Trans. Commun., vol. 6, pp , 04. [6] S. Bandyopadyay and E. J. Coyle, An energy effcent erarccal clusterng algortm for wreless sensor networks. n Proc., IEEE Infocom, vol. 3, Apr. 003, pp [7] V. Rodoplu and T. Meng, Mnmum energy moble wreless networks, IEEE J. Sel. Areas Commun., vol. 7, 999. [8] F. Baccell and S. Zuyev, Posson-Vorono spannng trees wt applcatons to te optmzaton of communcaton networks, Operatons Researc, vol. 47, no. 4, pp , 999. [9] S. J. Baek, G. de Vecana, and X. Su, Mnmzng energy consumpton n large-scale sensor networks troug dstrbuted data compresson and erarccal aggregaton, IEEE J. Sel. Areas Commun., vol., no. 6, pp , Aug [0] M. Haengg and D. Puccnell, Routng n ad oc networks: A case for long ops, IEEE Commun. Mag., vol. 43, 005. [] A. Laya, L. Alonso, and J. Alonso-Zarate, Is te random access cannel of LTE and LTE-A sutable for MM communcatons? A survey of alternatves, IEEE Commun. Surveys Tuts., vol. 6, pp. 3, 04. [] T. D. Novlan, H. S. Dllon, and J. G. Andrews, Analytcal modelng of uplnk cellular networks, IEEE Trans. Wreless Commun., vol., no. 6, pp , Jun. 03. [3] S. Sng, X. Zang, and J. G. Andrews, Jont rate and SINR coverage analyss for decoupled uplnk-downlnk based cell assocatons n HetNets, IEEE Trans. Wreless Commun., vol. 4, no. 0, Oct. 05. [4] F. Baccell and B. Błaszczyszyn, Stocastc Geometry and Wreless Networks, Volume I Teory, ser. Foundatons and Trends n Networkng. NoW Publsers, 009, vol. 3, no [5] J.-S. Ferenc and Z. Neda, On te sze-dstrbuton of Posson Vorono cells, Pysca A: Statstcal Mecancs and ts Applcatons, vol. 385, no., Nov [6] S. Sng, H. Dllon, and J. Andrews, Offloadng n eterogeneous networks: Modelng, analyss, and desgn nsgts, IEEE Trans. Wreless Commun., vol., no. 5, pp , 03. [7] D. Malak, H. S. Dllon, and J. G. Andrews, Optmzng data aggregaton for uplnk macne-to-macne communcaton networks, to appear, IEEE Trans. Commun., arxv preprnt arxv: v, 06. [8] D. Stoyan, W. Kendall, and J. Mecke, Stocastc Geometry and Its Applcatons, nd ed. Jon Wley and Sons, 996.