Performance analysis of feedback-free collision resolution NDMA protocol

Transcription

1 Lagen et al EURASIP Journal on Wreless Communcatons and Networkng (2018) 2018:45 RESEARCH Open Access Performance analyss of feedback-free collson resoluton NDMA protocol S Lagen 1,2*, A Agustn 1,JVdal 1 and J Garca 1 Abstract To support communcatons of a large number of deployed devces whle guaranteeng lmted sgnalng load, low energy consumpton, and hgh relablty, future cellular systems requre effcent random access protocols However, how to address the collson resoluton at the recever s stll the man bottleneck of these protocols The network-asssted dversty multple access (NDMA) protocol solves the ssue and attans the hghest potental throughput at the cost of keepng devces actve to acqure feedback and repeatng transmssons untl successful decodng In contrast, another potental approach s the feedback-free NDMA (FF-NDMA) protocol, n whch devces do repeat packets n a pre-defned number of consecutve tme slots wthout watng for feedback assocated wth repettons Here, we nvestgate the FF-NDMA protocol from a cellular network perspectve n order to elucdate under what crcumstances ths scheme s more energy effcent than NDMA We characterze analytcally the FF-NDMA protocol along wth the multpacket recepton model and a fnte Markov chan Analytc expressons for throughput, delay, capture probablty, energy, and energy effcency are derved Then, clues for system desgn are establshed accordng to the dfferent trade-offs studed Smulaton results show that FF-NDMA s more energy effcent than classcal NDMA and HARQ-NDMA at low sgnal-to-nose rato (SNR) and at medum SNR when the load ncreases Keywords: Slotted random access, Packet repetton, Multpacket recepton, Feedback-free NDMA, Energy effcency 1 Introducton The ffth generaton (5G) of cellular networks, set for avalablty around 2020, s expected to enable a fully moble and connected socety, characterzed by a massve growth n connectvty and an ncreased densty and volume of traffc Hence, a wde range of requrements arse, such as scalablty, rapd programmablty, hgh capacty, securty, relablty, avalablty, low latency, and long-lfe battery for devces [1] All these requrements pave the way for machne-type communcatons (MTC), whch enable the mplementaton of the Internet of Thngs (IoT) [2] Unlke typcal human-tohuman communcatons, MTC devces are equpped wth batteres of fnte lfetme and generate bursty and automatc data wthout or wth low human nterventon, so that traffc n the uplnk drecton s accentuated [3] MTC systems consder dfferent use cases that range *Correspondence: sandralagen@cttces 1 Sgnal Theory and Communcatons department, Unverstat Poltècnca de Catalunya (UPC), Barcelona, Span 2 Moble Networks department, Centre Tecnològc de Telecomuncacons de Catalunya, Castelldefels, Span from massve MTC, where the number of deployed devces s very hgh, to msson-crtcal MTC, where realtme and hgh-relablty communcaton needs have to be satsfed [4] To address such a massve number of low-powered devces generatng bursty traffc wth low latency requrements, smple medum access control (MAC)-layer random access protocols of ALOHA-type are preferred because they offer a relatvely straghtforward mplementaton and can accommodate bursty devces n a shared communcaton channel [4, 5] They are ndeed used n today s most advanced cellular networks (as the random access channel (RACH) n LTE) [6] and are beng consdered n dfferent MTC systems, such as LoRa [7], SgFox, enhanced MTC [8], narrowband (NB) LTE-M [9, 10], and NB-IoT [11 13] Basc ALOHA-type protocols are based on the collson model: a packet s receved error-free only when a sngle devce transmts Thus, the MAC layer and the physcal (PHY) layer are fully decoupled In [14], Guez et al made a fundamental change n the collson model and ntroduced the multpacket recepton (MPR) model: The Author(s) 2018 Open Access Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton 40 Internatonal Lcense ( whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded you gve approprate credt to the orgnal author(s) and the source, provde a lnk to the Creatve Commons lcense, and ndcate f changes were made

2 Lagen et al EURASIP Journal on Wreless Communcatons and Networkng (2018) 2018:45 Page 2 of 18 when there are smultaneous transmssons, nstead of assocatng collsons wth determnstc falures, recepton s descrbed by condtonal probabltes Therefore, sgnal processng technques enable a recever to decode smultaneous sgnals from dfferent devces and hence collsons can be resolved at the PHY layer As a result, a tghter nteracton between PHY and MAC layers s acheved [14 16] MPR can be realzed through many technques, whch are classfed accordng to three dfferent perspectves: transmtter, trans-recever, and recever (see [17] for detals) Among all of them, a promsng trans-recever approach based on random access for dfferent 5G servces s the network-asssted dversty multple access (NDMA) protocol NDMA was ntally presented n [18] for flat-fadng channels and, afterwards, extended to mult-path tme-dspersve channels n [19] The basc dea of NDMA s that the sgnals receved n collded transmssons are stored n memory and then they are combned wth future repettons at the recever so as to extract all collded packets wth a lnear detector In the sngle-antenna case and under the assumpton of perfect recepton, NDMA only requres the number of repettons to be equal to the number of collded packets [18] Thus, NDMA dramatcally enhances throughput and delay performance as compared to ALOHA-type protocols, but estmaton of the number of devces nvolved n a collson (eg, P devces) and a properly adjustment of the number of repettons (e P 1) s requred every tme a collson occurs Many NDMA protocols, and varatons of t, have been proposed and analyzed n the lterature, ncludng dfferent ways to determne the number of devces nvolved n a collson [18 22], nterference cancellaton recevers [23 25], and modfed protocols that use channel knowledge at the transmtter sde [26, 27] Stablty analyss of NDMA was addressed n [28 30] Fnally, the hybrd automatc repeat request (HARQ) concept was appled to NDMA n [31] (named H-NDMA) n order to deal wth recepton errors at low/medum SNR by forcng devces nvolved n a collson of P devces to transmt repettons more than P 1 tmes Ths way, packet recepton was sgnfcantly mproved at low SNR wth H-NDMA as compared to classcal NDMA One of the man drawbacks of NDMA protocols s, however, the overhead requred to dentfy collsons and adjust the number of repettons accordngly every tme a collson occurs (whch mples communcatng t to all the devces nvolved n the collson) [18] Indeed, devces need to decode control sgnalng at every tme slot to know f the subsequent tme slot s reserved for repettons or not, hence ncreasng the energy consumpton Ths aspect s crtcal for MTC devces wth fnte battery lfetme To cope wth these ssues, authors n [32] proposeda non-centralzed procedure for NDMA, coned feedbackfree NDMA (FF-NDMA), n whch the number of tme slots for repettons s kept constant to R (conformng a contenton perod (CP)) and s equal for all devces and transmssons See Fg 1 for R = 3 Accordngly, devces are only allowed to start transmsson at the begnnng of the CP and wll do so R tmes Ths way, collsons of up to R devces can be resolved n the sngle-antenna case wthout requrng the recever to communcate the collson multplcty to the devces every tme a collson occurs and avodng the sgnalng related to the state (reserved for repettons or not) of the subsequent tme slot The jont PHY-MAC performance analyss of FF- NDMA protocol was performed n [33] for the general case of MIMO systems 1 wth orthogonal space-tme block codng (OSTBC) Sgnfcant throughput and energy gans as compared to ALOHA-based schemes were reported wth a non-centralzed protocol that requres low overhead Nevertheless, t was assumed n [32, 33] that whenever a packet was receved n error at the recever then sad packet was lost, snce FF-NDMA was ntally desgned to address the broadcast protocol n ad hoc networks where no feedback s avalable Although NDMA and FF-NDMA were ntally proposed a decade ago, the emergng MTC systems (wth dfferent requrements than those of conventonal humanbased cellular networks) suggest revewng random access protocols wth MPR and analyzng ts applcablty to the uplnk communcaton n cellular networks [3], specally for scenaros characterzed by a large number of devces, lmted sgnalng load, low energy consumpton, and hgh relablty In partcular, NDMA-based protocols are hghly attractve for massve MTC NDMA has been deeply analyzed n the recent lterature wth dfferent protocols (eg, H-NDMA [31]) However, FF-NDMA msses such wde analyss whle t s sutable for massve MTC scenaros due to ts low assocated sgnalng load and reduced mplementaton complexty Indeed, t s worth mentonng that NB-IoT [11] and the new rado (NR) access technology desgn for 3GPP 5G systems [34] already consder a contenton-based transmsson mode wth a predefned number of packet repettons (known as uplnk grant-free access, n whch devces contend for resources, and multple predefned repettons are allowed, asspecfed n [34]) Such uplnk grant free access n NR targets at least for massve MTC and would allow the mplementaton of FF-NDMA In ths paper, we analyze the FF-NDMA protocol wth MIMO confguratons and OSTBC from a cellular network perspectve, n whch multple devces ntend to communcate wth a base staton (BS), as shown n Fg 1 The MIMO system defned and analyzed n the sequel carres over to a mult-cell scenaro where cell-edge termnals

3 Lagen et al EURASIP Journal on Wreless Communcatons and Networkng (2018) 2018:45 Page 3 of 18 Fg 1 Slotted random access asssted by retransmsson dversty and MPR for FF-NDMA wth R = 3 The frame s composed of contenton perods (CPs), each contanng R consecutve tme slots Devces access the shared channel whenever they have a packet to transmt at the CP start As an example, two and four devces transmt n the frst and second CPs, respectvely experence smlar average SNR to an x-number of BSs that are able to receve and decode packets n a dstrbuted way wth an x-fold number of receved antennas Dfferently from [32, 33], n whch no feedback was consdered and whenever a packet was receved n error then the packet was dscarded, we use a general model n whch packets are not dscarded To do so, we consder a fnte-user slotted random access system where devces can be ether transmttng, thnkng (e, there s no packet to transmt), decodng, or backlogged (e, packet transmsson was erroneous and the devce s watng for a new transmsson opportunty) and we assume that each devce s equpped wth a sngle-packet buffer 2 Therefore, FF-NDMA s feedback-free n the sense that t s not needed to broadcast nformaton related to the number of repettons and to the state of the forthcomng tme slots (as n NDMA or H-NDMA) but, n contrast to [32, 33], ACK feedback to acknowledge a correct detecton of the devces packets per CP s assumed In ths context, the man contrbutons of ths paper are summarzed as follows: we develop a jont PHY-MAC analyss of the FF-NDMA protocol by usng the MPR model and, then, characterze the system through a fnte Markov chan, for whch the system state probabltes and the transton probabltes among them are obtaned n closed-form we characterze analytcally the FF-NDMA protocol n terms of throughput, delay, capture probablty (e, probablty of a successful transmsson or, equvalently, relablty of the protocol), energy, and energy effcency (e, effcency of the protocol, whch s measured through a throughput-energy rato) Also, we propose two crtera to analyze the stablty of fnte-user random access wth sngle-packet buffer 3 we nvestgate the system performance of FF-NDMA as a functon of the CP length (R), for dfferent SNR and load condtons, and we compare FF-NDMA wth S-ALOHA, classcal NDMA [18], and H-NDMA 4 [31] As we wll see, the energy consumpton s reduced wth FF-NDMA as compared to H-NDMA n certan stuatons due to the lower control sgnalng to be decoded To address the throughput-energy trade-off, we use the energy-effcency metrc and focus on determnng the crcumstances n whch FF-NDMA s more energy effcent than H-NDMA Organzaton: The paper s organzed as follows: n Secton 2, we assess the dfferences between FF-NDMA and other NDMA-based protocols (ncludng NDMA and H-NDMA) and then we present the system model and the man features of the FF-NDMA protocol Secton 3 establshes the MPR model and characterzes the system by usng a fnte Markov chan, for whch the system state probabltes (related to the backlog state) are derved Then, n Secton 4, based on the obtaned system state probabltes, expressons for throughput, delay, capture probablty, energy, and energy effcency are developed and two stablty condtons are set

4 Lagen et al EURASIP Journal on Wreless Communcatons and Networkng (2018) 2018:45 Page 4 of 18 Secton 5 presents the smulaton results by usng dfferent SNR and dfferent offered loads and system desgn clues are extracted Fnally, conclusons are drawn n Secton 6 Notaton: In ths paper, scalars are denoted by talc letters Boldface lower-case and upper-case letters denote vectors and matrces, respectvely For gven real-valued scalars a and b, Pr(a b), Pr(a=b), Pr(a=b C), a, and log 2 (a), denote the probablty of a beng smaller than b, the probablty of a beng equal to b, the probablty of a beng equal to b gven condton C, the celng functon of a, and the base 2 logarthm of a, ( respectvely ) For gven a postve nteger scalars a and b, refers to the bno- b mal coeffcent and a! denotes the factoral of a Fora gven vector a, a T stands for the vector transpose Q() refers to the Q-functon (e, the ntegral of a Gaussan densty) R m n, R m n +,andcm n denote an m by n dmensonal real space, real postve space, and complex space, respectvely 2 Systemmodel In ths secton, we frst compare FF-NDMA protocol wth classcal NDMA [18] and H-NDMA[31], and then present the system model for FF-NDMA 21 Comparson of NDMA protocols Fgure 2 shows the protocol dfferences between NDMA 5 and FF-NDMA wth R = 3 To perform a far protocol comparson, we assume that each tme slot contans a data part for data transmsson and a control part for feedback from BS (whch s not always used n FF-NDMA) In FF-NDMA, transmssons are attempted at the CP startandthenumberofrepettonssfxedtothecp length (R repettons) ndependently of the number of devces that collde In contrast, n NDMA, transmssons are attempted at the tme slot scale and the number of repettons s dynamcally adapted accordng to the number of collded packets H-NDMA follows classcal NDMA operaton but, at low/medum SNR, the BS mght ask for addtonal repettons on a HARQ bass to mprove packet recepton For these reasons, the throughput of FF- NDMAcannotbeaslargeasthatofclasscalNDMA at hgh SNR and as that of H-NDMA at any SNR range However, load sgnalng, mplementaton complexty, and energy consumpton are reduced wth FF-NDMA Under NDMA, recevng and decodng control sgnalng from the BS s requred at every tme slot for dfferent purposes: to know f the subsequent tme slot s ether busy or free (e, reserved for repettons of collded packets or not),toreceveackncaseapacketwastransmtted,and to know the number of repettons to be performed n case a packet was transmtted but not successfully decoded due to collson[18] H-NDMA requres extra sgnalng load from the BS towards devces to request addtonal repettons on a HARQ bass [31], once the repettons of NDMA have been completed On the other hand, n FF-NDMA, control sgnalng s only needed to receve ACK at those CPs n whch a packet was transmtted Ths makes the applcaton of FF-NDMA to MTC systems hghly attractve because the energy consumpton for control sgnalng decodng s reduced The dfference n the control sgnalng to be decoded wth FF-NDMA and NDMA s llustrated n Fg 2 n orange color To summarze, the throughput of FF-NDMA s gong to be lower than the throughput of H-NDMA, but the energy Fg 2 Slotted random access for NDMA and FF-NDMA wth R = 3 In NDMA, decodng control sgnalng at every tme slot s needed In FF-NDMA, control sgnalng has to be decoded only at those CPs n whch packets were transmtted

5 Lagen et al EURASIP Journal on Wreless Communcatons and Networkng (2018) 2018:45 Page 5 of 18 consumpton can be reduced wth FF-NDMA In ths lne, n Secton 52, weusetheenergyeffcencyasasutable metrc to address the throughput-energy trade-offs between FF-NDMA and H-NDMA and, hence, determne whch protocol s more energy effcent under dfferent crcumstances In addton, due to the lower control sgnalng to be decoded wth FF-NDMA, ts mplementaton complexty s also sgnfcantly reduced as compared to NDMA or H- NDMA, because devces do not need to decode control sgnalng from the BS at every tme slot and can enter nto sleep mode Wth FF-NDMA, decodng of a sngle-control sgnalng per CP n whch transmsson was attempted s requred Wth NDMA or H-NDMA, decodng of control sgnalng at every tme slot whle data s n the buffer s needed to know f transmsson can be attempted and to get the feedback Fnally, t s mportant to emphasze that NDMA and H-NDMA requre a self-contaned tme slot, as shown n Fg 2, n whch the feedback for repettons s receved just after the packet transmsson and devces can attempt a repetton at the subsequent tme slot However, conventonal repettons processes (eg, HARQ) mght take some tme slots between obtanng the feedback and retransmttng agan [35] In ths stuaton, FF-NDMA avods the addtonal delay that appears n NDMA and H-NDMA under non-deal repetton processes owng to the fact that FF-NDMA does not rely on feedback to perform repettons Both the energy savngs (due to lower control sgnalng to decode) and the delay reductons (under non-deal repetton processes) are evaluated n Secton System model for FF-NDMA Consder a wreless cellular system composed of one BS wth N receve antennas and a deployment of K devces that wll transmt packets to the BS through a slotted random access network, as shown n Fg 1 Everydevce s equpped wth M transmt antennas and has a snglepacket buffer A frame composed of tme slots s adopted Each tme slot contans a data part for data transmsson from devces to BS and a control part for feedback from BS to devces (whch s not always used), see Fg 2 Tmeslots are grouped nto contenton perods (CPs) of R tme slots We assume that each devce s CP- and slot-synchronous wth the BS Devces transmt whenever they have a packet n ther buffer at the begnnng of the CP, and packet repettons are performed durng the CP, so that devces transmt ther packets R tmes usng the data plane After the R repettons, the BS acknowledges recepton of the correctly receved packets through the control channel, so that devces known f transmsson was successful or not Note that the maxmum number of packets that can be smultaneously decoded at a BS wth N antennas and R repettons s R = NR In ths scenaro, collsons come up and every devce can be n one of four dfferent devce states: thnkng, transmttng, decodng, orbacklogged The devce state dagram s shown n Fg 3 In the thnkng state, the devce does not have a packet n ts buffer and does not partcpate n any schedulng actvty In ths devce state, a devce generates a packet wth probablty σ Once a packet s generated, ts transmsson s attempted at the begnnng of the next CP and repeated durng R tme slots (whch corresponds to the transmttng state) After transmsson, the devce decodes an acknowledgment of recept message from the BS If the transmsson succeeds (e, ACK feedback s receved), the devce remans n the thnkng state Otherwse, the devce moves nto the backlogged state and retransmts the packet wth probablty υwhen the packet s fnally successfully decoded at the BS, the devce moves back to the thnkng state and the process restarts agan We follow classcal NDMA [18] andh-ndma[31] assumpton that unform average power from every devce s receved at the BS Ths s possble thanks to the uplnk slow power control mechansm [36] Accordngly, all devces are receved at the BS wth the same average SNR (γ ) The use of uplnk power control has the beneft that the scenaro s termnal-wse symmetrc (n terms of average SNR) and the MPR model can be thus appled, as t wll be shown n Secton Sgnal model To explot transmt dversty wth no channel knowledge at the termnal sde 6, transmsson of each devce s done through an OSTBC wth Q complex symbols that are spread n tme and space over T channel uses and M transmt antennas Therefore, the transmtted sgnal matrx for the kth devce, X k C M T, s expressed as [37] Q X k = α k, q A q + jβ k, q B q, (1) q=1 where α k,q and β k,q refer to the real and magnary parts of the qth complex symbol at the kth devce, respectvely, and A q, B q R M T denote the par of real-valued code Fg 3 State dagram of the devce operaton

6 Lagen et al EURASIP Journal on Wreless Communcatons and Networkng (2018) 2018:45 Page 6 of 18 matrces that defne the OSTBC [38] We assume that the transmtted symbols are m-qam 7 Consderng a flat fadng channel constant over the tme slot and that k devces are transmttng, the receved sgnal at the N antennas of the BS over T channel uses n the rth tme slot, Y r C N T,sgvenby[39] Y r = k k=1 P k H k,r X k + W r, (2) ML k where P k stands for the transmtted power of the kth devce, L k refers to the slow propagaton losses (ncludng pathloss and shadowng) between the kth devce and the BS, H k,r C N M s the Raylegh flat-fadng channel matrx between the antennas at the kth devce and the BS durng the rth tme slot that contans zero mean complex Gaussan components, and W r C N T denotes the receved nose that s composed of zero mean complex Gaussan components wth varance σw 2 The average receved SNR s gven by γ = P k and s unform among devces due to L k σw 2 the uplnk slow power control mechansm (whch adjusts the uplnk power P k accordng to the slow propagaton losses L k at every devce) The BS combnes the receved sgnals n a CP of R tme slots to perform mult-user detecton We assume that the channel s constant on one tme slot but uncorrelated between tme slots (fast-fadng channel assumpton) 8 Accordngly, assumng that k devces are present, therecevedsgnalnacpcanbearrangednvectorform by separatng the real and magnary parts as (see [37], Secton 71): y = = y 1 y R = γσw 2 2M γσ 2 w 2M Hx + w H 1,1 H k,1 H 1,R H k,r x 1 x k + w 1 w R, (3) where y r R 2NT 1 and w r R 2NT 1 contan the real and magnary parts of the receved sgnal and the nose samples n the rth tme slot (see (2)), x k =[ α k,1 α k,q β k,1 β k,q ] T R 2Q 1 contans the 2Q real and magnary parts of the complex symbols transmtted by the kth devce (see (1)), and H k,r R 2NT 2Q denotes the equvalent channel matrx for the kth devce durng the rth tme slot The equvalent channel matrx H k,r depends on the Raylegh flat-fadng channel matrx (H k,r n (2)) and the par of real-valued code matrces (A q, B q n (1)) (see detals n [33], Appendx) Accordng to ths, y, w R 2NTR 1, x R 2Q k 1, and H R 2NTR 2Q k Note that to perform decodng of the contendng sgnals, the recever (BS) has to get the dentty of the contendng devces to estmate the channel matrces from them In ths regard, we assume that all devces have orthogonal plot sgnals and that channels are perfectly acqured at the recever sde The effect of a lmted number of orthogonal plot sgnals, non-orthogonal plot sgnals, and mperfectly acqured channels s out of the scope of the paper and s left as nterestng future work 222 Packet error rate By usng a decorrelatng recever at the BS that combnes the repettons of devces attemptng transmsson wthn acpofr tme slots (see (3)), the multple access nterference s vanshed and the bt error rate (BER) s nvarant to the ampltudes of the nterferng sgnals [33] Therefore, for m-qam, the BER of devce k gven that k devces are transmttng s gven by [40, 41] BER k,k = ( m ) log 2 (m) Q 3χ k,k γ, k K, 2M(m 1) where Q() refers to the Q-functon (the ntegral of a Gaussan densty) and χ k,k s a ch-square dstrbuted random varable wth dof k degrees of freedom for any OSTBC wth M=T: (4) dof k = 2(RNM Q k + Q) (5) ( χ k,k γ 2M For 4-QAM (QPSK), the ) BER expresson n (4) s reduced to BER k,k =Q Incasethatm-PSK was consdered, the BER expresson n (4) should be modfed accordng to [40] and the whole forthcomng analyss would apply as well In (4), we have assumed fxed power spent at devces per tme slot Ths wll allow us to compare the FF-NDMA protocol wth classcal NDMA [18]andH-NDMA[31], n whch constant power per tme slot s used snce devces do not know the number of repettons to be performed untl a collson occurs and the BS communcates so Note that whle R s a value to be fxed by the network, the value of k s random n each CP and depends on K, σ,andυ So, the BER n a CP depends not only on the

7 Lagen et al EURASIP Journal on Wreless Communcatons and Networkng (2018) 2018:45 Page 7 of 18 average SNR (γ )butalsoontheactualnumberofdevces that are transmttng ( k) As n [33], we assume that a packet s n error whenever the BER n (4) s above a certan threshold ω Therefore, an upper bound of the packet error rate (PER) for devce k gven that k devces are transmttng can be found as PER k,k Pr(BER k,k ω) Accordng to ths and (4), we get PER k,k Pr χ k,k Q 1 ω log 2 (m) 2M(m 1) 4(1 1, m ) 3γ whch can be computed accordng to the cumulatve functon of the ch dstrbuton n closed-form as PER k,k 1 F k Q 1 ω log 2 (m) 2M(m 1) 4(1 1, m ) 3γ where (6) (7) NDMA, the MPR matrx C R R ( R+1) + wth R = NR s gven by C 1,0 C 1,1 0 0 C 2,0 C 2,1 C 2,2 0 C =, (10) C R,0 C R,1 C R,2 C R, R where C x,y,1 x R, and0 y x denotes the probablty that, gven x transmttng devces, y out of x transmssons are successful The number of non-zero rows of the MPR matrx s gven by the maxmum number of packets that can be smultaneously decoded, e, R As γ s assumed equal for all K devces, we do not need to dstngush among specfc devces so that the element C x,y of the MPR matrx C contans the product of PERs correspondng to the combnatons of x devces for whch y transmssons are successful and x y are not Accordng to (9), the elements of the MPR matrx n (10)(e,C x,y for 1 x R,0 y x)aregvenby C x,y = ( ) x (PER y x ) x y (1 PER x ) y, (11) F k(z) = e z2 /2 I (z 2 /2) l, I = dof k l! 2 l=0 1 (8) and C x,y =0fory>x Thus, we can complete the MPR matrx that characterzes the FF-NDMA protocol, C n (10), usng (7), (9), and (11) It s mportant to recall that, as γ s equal for all devces, dstncton among specfc devces s not necessary and the followng condton s fulflled (see (7)): PER k = PER k,k = PER k,j, j, k (9) 3 Markov model for FF-NDMA Analytc characterzaton of the performance and stablty of the FF-NDMA protocol wth MPR requres the use of a Markov model that ncorporates dfferent states of the system and the transton probabltes between them In ths regard, n ths secton we frst set up the MPR model for the FF-NDMA protocol, whch wll allow us to work wth condtonal probabltes nstead of assocatng collsons or erroneous receptons wth determnstc falures Then, accordng to the MPR model, we derve analytc expressons for the system state probabltes of the fnte Markov chan that represents the FF-NDMA protocol 31 MPR matrx The MPR model s characterzed by an MPR matrx that contans condtonal probabltes, see [14] Under FF- 32 Markov chan for the system states Let random varable B(s) denote the number of backlogged devces at the begnnng of CP s B(s) s referred to as the system state, whch depends on the prevous system state (e, B(s 1))aswellasonthe numberofdevces whose state has changed durng CP s Hence,theprocess can be modeled by a fnte Markov chan snce B(s) K Fgure 4 shows the Markov chan for a smplfed scenaro wth K = 3 Fg 4 Markov chan of the system state (number of backlogged devces) n FF-NDMA protocol for K=3

8 Lagen et al EURASIP Journal on Wreless Communcatons and Networkng (2018) 2018:45 Page 8 of 18 The steady-state probablty of the system beng n state (π )sthusgvenby π = lm s Pr (B(s) = ), (12) and the transton probablty from system state to j (p j,0, j K)sdefnedas[42] p,j = lm s Pr ( B(s) = j B(s 1) = ) (13) Notce that, under conventonal slotted ALOHA, downward transtons are only possble from system state to j= 1, snce a sngle packet can be decoded at a tme, and p 0,1 =0 In contrast, under FF-NDMA, downward transtons are possble from system state to j R as long as j 0 In Fg 4, all downward transtons have been represented; however, only those from system state to j R are possble, e, are such that p,j =0 Now we focus on obtanng the transton probabltes p,j n (13), whch depend on the MPR matrx C n (10), the generaton probablty σ, and the retransmsson probablty υ To do so, let us defne the followng parameters Defne φ m,n as the probablty that m 0 backlogged devces transmt and n 0 new packets are generated by thnkng devces gven that the system state s (e, there are devces n the backlog and K devces n the thnkng state) Snce packet generaton and packet retransmsson are ndependent events, φ m,n s obtaned as ( ) ( ) φ m,n = υ m (1 υ) m K σ n (1 σ) K n m n (14) Smlarly, defne ϕ m,n as the probablty that more than m backlogged devces transmt and n 0 new packets are generated by thnkng devces gven that the system state s ( ϕ m,n = 1 m l=0 ( ) ) ( ) υ l (1 υ) l K σ n (1 σ) K n l n (15) Ths way, the transton probabltes p,j n (13) for R j + R can be found by performng the followng operaton: + p, R p, 2 p, 1 p, p,+1 p,+2 p,+ R φ 1,0 = φ 1,0 φ 0,1 φ 0, φ 2,0 0 φ 2,0 φ 1,1 C 2,0 φ 1,1 φ 0,2 C 2,1 φ 0,2 0 C 2, φ 2,0 φ 1,1 φ 0,2 [ C1,0 0 0 φ R,0 φ R 1,1 0 0 φ R 2,2 0 φ R,0 φ R,0 φ R 1,1 φ R 2,2 φ R 1,1 φ 0, R φ R 2,2 0 φ 0, R 0 φ 0, R C 1,1 ] + C R,0 C R,1 C R,2 C R, R + φ 0, ϕ R,0 ϕ R 1,1 ϕ R 2,2 ϕ 0, R (16) ( φ 1,0 The left-hand-sde vector n (16) ncludes all transton probabltes from system state to states n between R and + R For llustratve purposes, let us explan how, for nstance, p, n (16) s computed (e, the probablty of remanng n state ) Then, by takng each row of the MPR matrx, we consder all the possble cases where from 1 to R packets are transmtted The frst rght-handsde matrx product takes nto account the case where 1 packet s transmtted In ths case, two events can happen: a backlogged packet s transmtted but t s not successfully decoded C 1,0 ), or a new packet s generated and ( t s successfully decoded φ 0,1 C 1,1 ) In both stuatons, the state of the backlog does not change The rest of terms n (16) account for the cases n whch 2, 3,, R packets were transmtted, and we have obtaned them by extrapolatng the aforementoned reasonng In ths partcular

9 Lagen et al EURASIP Journal on Wreless Communcatons and Networkng (2018) 2018:45 Page 9 of 18 case, where the system state remans unchanged, ( ) the probablty of not transmttng any packet e, φ 0,0 as well as the case( where more than R backlogged packets are transmtted e, ϕ R,0 ) have to be consdered (see last rght-hand-sde vector n (16)) The transton probabltes p,j n (16) for R j + R canalsobeobtanedncompactform,asshownnnext Eq (17) The expresson n (17) for R j + R has been obtaned by compactng (16) Let us recall that φ m,n and ϕ m,n are gven by (14) and(15), respectvely, for m 0 and n 0, but take value 0 otherwse The remanng transton probabltes p,j for j< R and j>+ R are ncluded n (17) and are obtaned as follows: downwards transtons from system state towards states j< R are mpossble because at most R packets can be successfully decoded, and therefore p,j =0for j< R Upwards transtons from system state towards states j>+ R happen when j thnkng devces have generatedpacketsandcollded(theactvtyofthebacklogged devces s mmateral n ths case because they do not alter the backlog state, so collson s generated by thnkng devces alone), and are thus gven by thelastequatonn(17) It consders all the combnatons n whch, among the K devces that were thnkng, j thnkng devces have generated packets and K j have not To sum up, transton probabltes are gven by 0, j < R, R x C x,y φ x (y+j ),y+j + x=1 y=0 p,j = ϕ R+ j,j + φ j,j, R j + R, ( K j ) σ j (1 σ) K j, j > + R (17) Once we have all the transton probabltes p,j by usng (17), we can focus on obtanng the steady-state probabltes π n (12) By arrangng all the transton probabltes p,j n a matrx P R (K+1) (K+1) + ( as row ndex, and j as column ndex) and all the steady-state probabltes π n avectorπ R (K+1) 1 +, the steady-state vector must satsfy [43]: π = Pπ and K =0 π = 1 Therefore, π can be obtaned as the normalzed sngle egenvector assocated wth the unt egenvalue of P 4 Performance analyss of FF-NDMA In ths secton, we derve throughput, delay, capture probablty, energy, and energy effcency for FF- NDMA by usng the steady-state probabltes obtaned n Secton 32 Then, two stablty crtera are proposed 41 Throughput The throughput (S) s defned as the average number of correctly decoded packets per tme slot It s gven by the product of the steady-state probabltes and the assocated throughput on each state (S ), e, S = 1 K S π [packets/slot], (18) R =0 where the R 1 penalty arses because devces do repeat the same packet R tmes wthn the CP S n (18) denotes the throughput obtaned n system state and consders the dfferent cases where successful decodng takes place (e, the elements of the MPR matrx C x,y such that 1 x R and 1 y x, each wth ts assocated throughput of y successfully decoded packets): S = R x yc x,y x=1 y=1 m+n=x φ m,n, (19) where m+n=x φm,n denotes the probablty that exactly x packets are transmtted (whch can come from backlog and/or thnkng states) For example, wth R=2, the throughput assocated wth each system state (S n (19), =0,, K) results: ( ) ( ) S = C 1,1 φ 0,1 +φ 1,0 + C 2,1 φ 1,1 + φ 2,0 + φ 02 ( ) (20) + 2C 2,2 φ 1,1 +φ 2,0 +φ 0,2 42 Delay The mean delay (D) s the average number of tme slots requred for a successful packet transmsson, whch ncludes the mean backlog delay, the duraton of packet transmsson, and the watng tme untl a transmsson opportunty (e, CP start) To derve D, we frst compute the mean backlog delay, e, the mean tme a devce spends n the backlog [42], as follows: let B denote the mean number of devces n the backlog that s smply gven by K B = π (21) =0 If devces jon the backlog at a rate b, byusnglttle s formula [44],themeantmespentnthebacklogs B/b Afracton(S b)/s of the packets are never backlogged and thus have a (3R 1)/2 mean delay, whch comes from the duraton of a packet transmsson (e, R tme slots) plus the mean watng tme untl the CP starts (e, (R 1)/2) Contrarly, the packets whose fracton s b/s wll experence the mean backlog delay (e, B/b) plusa (3R 1)/2delay Therefore, the mean delay D (measured n number of tme slots) s gven by the weghted sum of delays assoc-

10 Lagen et al EURASIP Journal on Wreless Communcatons and Networkng (2018) 2018:45 Page 10 of 18 ated wth packets that are never backlogged and packets that are backlogged: ( )( ) S b 3R 1 D = + b ( B S 2 S b + 3R 1 ) 2 = (3R 1) + B (22) 2 S [slots] Note that although b has been defned to derve D, the fnal expresson of D n (22) does not depend on t 43 Capture probablty The capture probablty (P cap ) s the probablty of a successful packet transmsson gven that a packet has been transmtted It measures the relablty of the transmsson scheme [4] P cap can be computed by consderng the weghted average for all system states of the probablty that the transmsson s successful gven that a packet s transmtted and the system state s ( e, P cap ) : P cap P cap = K =0 π P cap (23) s obtaned by consderng all the cases where a successful transmsson takes place (e, k=1,, R) In each case, t s gven by the product of the probablty of a successful decodng gven that k devces transmt (e, (1 PER k)) tmes the probablty that k 1 devces transmt P cap = ( e, R k=1 m+n= k 1 φm,n (1 PER k) where PER k sshownn(9) m+n= k 1 ) Thus, t results to be φ m,n, (24) 44 Energy To compute the mean energy consumpton (E) for a successfully packet transmsson, we consder that each devce can be n four dfferent devce states (beng each one assocated wth a dfferent power consumpton level: P 0, P 1, P 2,andP 3,measurednWatts)(seeFg3): thnkng (or dle) state (P 0 ): there s no data to transmt, transmttng state (P 1 ): the devce s transmttng, decodng state (P 2 ): the devce s lstenng to the BS sgnalng and decodng the acknowledgement, or watng state (P 3 ): there s a packet to transmt but there s no transmsson opportunty 9 For FF-NDMA, the mean number of tme slots that a devce spends on every devce state (T 0, T 1, T 2, T 3 )s T 0 = 1/σ, T 1 = τrn tx, T 2 = (1 τ)n tx, T 3 = D T 1 T 2 (25) The number of tme slots n the thnkng state (T 0 )s gven by the nverse of the packet generaton probablty (σ ) The number of tme slots for the transmttng state (T 1 ) depends on the number of transmssons requred for a successful transmsson (denoted by N tx,andgven n next Eq (26)), the fact that wthn a CP the packet s repeated R tmes, and the fracton of a tme slot that s devoted for data transmsson (τ) For the decodng state, T 2 depends on N tx and the fracton of a tme slot that s reserved to receve feedback from the BS (1 τ) Recall that only one decodng per CP n whch a packet was transmtted s needed n FF-NDMA Fnally, The number of tme slots n the watng state (T 3 ) s determned by the average delay D n (22) mnus the mean transmttng and decodng tmes, hence, ncludng the watng tme n the backlog and the watng tme for the CP to start The number of transmssons requred for a successful transmsson (N tx n (25)) s gven by the nverse of the capture probablty P cap shown n (23): N tx = n=1 np cap (1 P cap ) n 1 = 1 Pcap (26) Note that the number of transmssons n (26) doesnot consder the number of repettons wthn a CP, t s rather gven by the number of tmes the devce accesses the channel Therefore, the mean consumed energy E (measured n Watts slot) s gven by the product of the tme that devces spend on each state by the power spent on each devce state: E = T 0 P 0 + T 1 P 1 + T 2 P 2 + T 3 P 3 [Watts slots] (27) 45 Energy effcency The energy effcency (EE) s a beneft-cost rato that measures the effcency of a protocol [45] It s defned as the amount of data (beneft) that can be relably transmtted per Joule of consumed energy (cost) Thus, t s measured n bts/joule or, equvalently, n packets/slot/watt (accordng to the defntons n prevous sectons) The energy effcency s a hghly relevant metrc n low-powered and fnte battery lfetme MTC devces [4] Based on the model presented n Secton 44, themean power consumpton for a successful packet transmsson (measured n Watts) s gven by P = T 0P 0 + T 1 P 1 + T 2 P 2 + T 3 P 3 E = T 0 + T 1 + T 2 + T 3 T 0 + D [Watts] (28) Accordngly, EE (n packets/slot/watt) s gven by the ratobetweenthethroughputs n (18) andthemean consumed power P n (28):

11 Lagen et al EURASIP Journal on Wreless Communcatons and Networkng (2018) 2018:45 Page 11 of 18 EE = S P = S E (T 0 + D) [packets/slot/watt] (29) Note that the energy effcency EE captures the trade-offs n throughput and energy consumpton that mght arse wth dfferent NDMA-based protocols 46 Stablty crtera Stablty analyss s usually performed for nfnte-user random access (see [14]) or for fnte-user buffered random access (see [46] and references theren), where devces are equpped wth a buffer of nfnte sze In the former case, the system s unstable when the number of devces n the backlog grows to nfnty whle, n the later, the system s unstable when the buffer sze grows to nfnty For fnte-user random access wth sngle-packet buffer, stablty has not been defned However, t can be addressed f a sensble defnton related to undesred states of the system s done In ths sense, we here set two stablty crtera for fnte-user random access wth sngle-packet buffer Stablty based on the probablty of beng n the last system state The system s sad to be stable f the probablty of beng n system state K s below a certan threshold, e, f π K α, (30) where 0 <α<1 Stablty based on the mean number of devces that are n the backlog The system s sad to be stable f the mean number of devces n the backlog s below a certan threshold, e, f B β, (31) wth 0 <β<k 5 Results and system desgn clues In ths secton, we evaluate the FF-NDMA protocol n terms of throughput, delay, capture probablty, energy, and energy effcency so as to devse the most sutable CP length (R) as a functon of the the offered load (G=σ K) and the SNR (γ ) K=30 devces are consdered A symmetrc scenaro wth an equal average SNR (e, γ )for all devces s used γ s determned by devces n worst propagaton condtons, and so γ wll be vared through smulatons to emulate the dfferent propagaton condtons The retransmsson probablty s set equal to the generaton probablty, e, υ=σ The 2 2 MIMO wth Alamout OSTBC s consdered (e, two antennas at devces and two antennas at BS, M=N=T=Q=2) The transmtted symbols are QPSK (e, m=4n(4)) The BER threshold s equal to ω=0001 For the power consumpton, P 0 =001 mw, P 1 =200 mw (e, 23 dbm as transmt power at devces), P 2 =150 mw, P 3 =10 mw, and τ = 08 are used (accordng to [47] and [48]) The performance of FF-NDMA s compared to classcal slotted ALOHA (S-ALOHA), classcal NDMA [18], and H-NDMA [31], all wth MIMO confguratons S-ALOHA corresponds to the case of R=1 To emulate NDMA and H-NDMA under the same condtons, the proposed framework n ths work can be appled wth some slght but mportant modfcatons For H-NDMA, we denote as R h the number of addtonal repettons that the BS may requestonaharqbassascomparedtondma,ths reduces the PER but mght ncrease the energy consumed n devces for data decodng For smulatons, we use up to R h =4 Therefore, the modfcatons requred to emulate NDMA and H-NDMA are The degrees of freedom dof k n (5) are equal to the followng: ( NDMA : dof k = 2 kn ) NM Q k + Q, (( ) ) H-NDMA : dof k = 2 kn +R h NM Q k+q, (32) snce the number of repettons s adjusted at each collson accordng to the number of collded packets k H-NDMA mght have more degrees of freedom than NDMA, and thus a lower PER (see (7)), whch s benefcal at low SNR NDMA and H-NDMA protocols can (deally) decode R= k packets (e, all collded packets) 10 by settng kn 1 repettons n NDMA and up to kn 1+R h repettons n H-NDMA Therefore, the MPR matrx C n (10) has a sze of K (K+1), snce collsons of up to K devces can be resolved n both protocols The throughput S for NDMAand H-NDMAcanbe computed as follows: S = 1 l K S π, (33) =0 where l s the average number of repettons: l = K l π, l = =0 The mean delays D are: K (m + n)φ m,n (34) m=0 n=0 NDMA : D = (3l 1) 2 + B S, H-NDMA : D = (3(l+R h) 1) 2 + B (35) S The mean energy consumpton E n (27) s also dependent on the average number of repettons n

12 Lagen et al EURASIP Journal on Wreless Communcatons and Networkng (2018) 2018:45 Page 12 of 18 (34), as l mpacts on the mean transmttng, decodng, and watng tmes: a T 1 = τln tx, T 2 = (1 τ)d, T 3 = D T 1 T 2 (36) Note that, n NDMA and H-NDMA, decodng of control sgnalng from the BS at devces s requred n every tme slot, as shown n Fg 2 ThssreflectednT 2,see (36) Conversely, wth FF-NDMA, decodng s needed per CP (e, 1 decodng every R tme slots) to receve ACK only at those CPs n whch a packet has been transmtted (see T 2 n (25)) Fnally, n order to take nto account practcal mplementaton ssues, we defne parameter d retx as the delay (n number of tme slots) between recepton of feedback and the next repetton n NDMA and H-NDMA protocols (see explanaton n Secton 21) In the deal case, d retx =0 Otherwse, delay n (35) s modfed as follows: D retx =D+(l 1)d retx, e, each repetton has an assocated delay of d retx tme slots In ths case, the mean watng tme s gven by T 3 =D retx T 1 T 2 So, the mean watng tme T 3 ncreases as d retx ncreases For smulatons, we consder the deal case wth d retx =0andthecaseof d retx =4 (whch do affect the delay and energy metrcs of NDMA and H-NDMA) 51 Performance In ths secton, we evaluate the FF-NDMA protocol n terms of throughput (S), delay (D), capture probablty (P cap ), and energy (E), by followng the expressons n (18), (22), (23), and (27), respectvely, as a functon of the offered load (G=σ K) for an average SNR (γ )of10and 0 db under dfferent R values (ndcated n the legends) Fgures 5 and 6 show the performance results for γ =10 db and γ =0 db, respectvely For γ =10 db (see Fg 5), deal NDMA and deal H- NDMA wth d retx =0 provde the largest performance (n terms of throughput, energy, and delay) because they are able to adapt the number of repettons dynamcally to the number of collded packets At medum/hgh SNR, H-NDMA s equvalent to NDMA, snce no addtonal repettons on a HARQ bass are requred Dfferently, for γ =0 db(seefg6), the performance of deal NDMA vanshes because the system s lmted by the erroneous detectons rather than by the number of collded packets Ths stuaton s resolved wth deal H-NDMA, whch provdes the largest performance gans (n terms of throughput, energy, and delay) at low SNR when d retx =0, snce t can cope wth the erroneous packet receptons through addtonal repettons, mprovng as well the relablty Remark 1 At low SNR regme, FF-NDMA outperforms deal NDMA protocol (d retx =0) n terms of throughput, b c d Fg 5 Performance vs offered load (G = σ K) for γ = 10 db a Throughput, b delay, c capture probablty, d energy

13 Lagen et al EURASIP Journal on Wreless Communcatons and Networkng (2018) 2018:45 Page 13 of 18 a b c delay, and energy wthout the need of nvokng HARQ processes (whch are needed for H-NDMA and mght nvolve larger delays n case a delay between the packet transmsson, the feedback, and the repetton s consdered, e, d retx >0) FF-NDMA performance can get close to deal H- NDMA for dfferent SNR ranges when choosng a sutable fxed CP length accordng to the offered load of the system It can be observed that a maxmum throughput level at low loads s acheved but as the load ncreases the throughput dmnshes (see Fgs 5a and 6a) Ths s because no backoff polcy s consdered at all (υ=σ ), and the system gets saturated for hgh-offered loads Usng larger R ncreases the value of the maxmum throughput and ts decay wth the load starts later (e, the stablty regon s enlarged) Hence, f σ takes hgh values, t mght be wser to use larger R Also, t s mportant to note that the load pont n whch the network should swtch towards alargerr s reduced for low SNR regons and, hence, the use of a larger CP length starts to be relevant for lower loads (see Fg 6a) The delay and the energy grow rapdly to nfnty as the load ncreases (see Fgs 5b d and 6b d) By usng larger R, the delay and energy are reduced and mantaned for a wde range of offered loads The system relablty s larger wth large R (see Fgs 5c and 6c), snce a larger CP length allows mprovng the PER (see (5)) Remark 2 In FF-NDMA, the optmal R for maxmum throughput, mnmum delay, or mnmum energy, depends on the offered load As the load ncreases, hgher R can provde larger throughput gans, delay reductons, and energy consumpton savngs, due to the effectve capablty for packet collson resoluton of FF-NDMA d Fg 6 Performance vs offered load (G = σ K) for γ = 0dB a Throughput, b delay, c capture probablty, d energy Remark 3 In FF-NDMA, the optmal R for maxmum relablty s provded wth a large R, snce the system can operate n a wde range of offered loads whle mantanng the capture probablty at ts maxmal value It s mportant to note that, for γ =10 db, the capture probablty s mproved wth FF-NDMA as compared to S-ALOHA, NDMA, and H-NDMA (see Fg 5c) Ths s due to the fact that the addtonal repettons provded by a fxed CP length allow the reducng of the PER (see (7)) and, hence, enhancng the system relablty, e, the probablty of a successful transmsson, as compared to NDMA and H-NDMA n whch the repettons are set manly to resolve collsons Dfferently, for γ =0 db,the relablty wth H-NDMA s also hgh because the HARQ mechansm starts to play a key role for successful packet recepton (see Fg 6c) Regardng the energy consumpton, at medum/hgh SNR, FF-NDMA provdes smlar energy consumpton

14 Lagen et al EURASIP Journal on Wreless Communcatons and Networkng (2018) 2018:45 Page 14 of 18 levels as compared to deal H-NDMA wth d retx =0(see Fg 5d) At low SNR (see Fg 6d), the energy consumpton can even be reduced wth FF-NDMA as compared to deal H-NDMA due to the energy savngs provded by a lower amount of control sgnalng to be decoded Remark 4 The performance of FF-NDMA s not far from the deal H-NDMA (d retx =0) and t gets closer as the SNR s reduced, whle much less sgnalng overhead and mplementaton complexty s requred The lower control sgnalng to be decoded s reflected n a reduced energy consumpton of FF-NDMA as compared to deal H-NDMA ether at low SNRs (see Fg 6d) or at hgh loads (see Fg 5d) When non-deal feedback for the repetton process s consdered (eg, d retx =4), the FF-NDMA protocol obtans a sgnfcantly reduced delay and lower energy consumpton as compared to NDMA and H-NDMA schemes The non-deal feedback for repettons has a detrmental mpact on delay of NDMA and H-NDMA protocols at any SNR range, as shown n Fgs 5b and 6b for d retx =4 Instead, FF-NDMA s not affected by the non-deal feedback repetton process Thus, delay reductons of up to 70% are obtaned wth FF-NDMA as compared to H- NDMA for d retx =4 Ths s because devces have to wat for repettons wth H-NDMA whle n FF-NDMA the repetton procedure s fxed to the CP length The nondeal feedback process also cause a reduced energy consumpton wth FF-NDMA as compared to NDMA and H-NDMA because devces spent less tme n the watng state to successfully complete a packet transmsson The energy s reduced wth FF-NDMA n two stuatons: () at low SNRs (see Fg 6d, for whch energy savngs of 5 20% are obtaned) and () when the load ncreases at medum/hgh SNRs (see Fg 5d, for whch energy savngs up to 10% are reported) In both cases, the addtonal control sgnalng to be decoded wth H-NDMA becomes relevant n terms of energy consumpton because devces are actve more tme to successfully transmt a packet Remark 5 Non-deal feedback repetton processes (e, d retx >0) have a detrmental effect over NDMA and H- NDMA In ths condtons, FF-NDMA provdes sgnfcant delay reductons for any SNR range and load condton Also, energy savngs are reported at low SNR and at medum/hgh SNR wth hgh load condtons To summarze, Table 1 ncludes the SNR regons n whch FF-NDMA protocol outperforms the benchmarked protocols (deal NDMA, non-deal NDMA, deal H- NDMA, and non-deal H-NDMA) n terms of throughput S, delayd, energye, and capture probablty P cap, separately Table 1 SNR Regons where FF-NDMA outperforms NDMA and H-NDMA protocols S D E P cap FF-NDMA > deal NDMA FF-NDMA > non-deal NDMA FF-NDMA > deal H-NDMA FF-NDMA > non-deal H-NDMA Low SNR Low SNR Low SNR SNR Low SNR SNR Low SNR SNR Low SNR Med/hgh SNR SNR Low SNR Med/hgh SNR 52 Energy effcency In ths secton, we evaluate the energy effcency (EE) of FF-NDMA n (29) and of deal H-NDMA (d retx =0) as a functon of the average SNR (γ ) Let us recall that, at hgh SNR, EE H-NDMA =EE NDMA snce both approaches are equvalent However, at low SNR, EE H-NDMA s hgher than EE NDMA For FF-NDMA, EE FF-NDMA s computed by adoptng the best R for each load and SNR condton Let us note that n ths secton, we use an deal scenaro for H-NDMA (e, d retx =0),soalltheenergyeffcency gans of FF-NDMA over deal H-NDMA that are reported come due to the lower control sgnalng to be decoded wth FF-NDMA Note also that the EE s a useful metrc to capture the throughput/energy trade-offs that have been observed n the prevous secton nto a sngle fgure of mert As t was shown n Secton 45, the energy effcency depends on the power consumpton levels assocated wth the dfferent devce states (P 0, P 1, P 2, P 3 ) So, to llustrate the effect of the addtonal control sgnalng to be decoded wth H-NDMA, we use dfferent power decodng values P 2 ={150, 200, 250} mw whle keepng fxed the transmt power to P 1 =200 mw Fgure 7 dsplays the energy effcency for two offered load condtons G={10, 18} packets/slot and dfferent power decodng values (P 2,ndcated n the legends) As t s expected, varyng the P 2 value has a hgher mpact on H-NDMA than FF-NDMA, snce FF-NDMA only needs to decode ACK feedback whle H-NDMA needs to decode ACK feedback, feedback assocated wth repettons, and feedback related to the state of the forthcomng tme slots A larger P 2 value ncreases the power consumpton and, hence, reduces the EE Table 2 summarzes the SNR regons n whch FF- NDMA scheme s more energy effcent than deal H-NDMA protocol for the dfferent G and P 2 values dsplayed n Fg 7 By consderng the nterval [ 5, 5] db as low SNR, [ 5, 15] db as medum SNR, and [ 15, + ]dbas hgh SNR (recall we are usng QPSK symbols), we can conclude the followng from Fg 7 and Table 2 FF-NDMA s more energy effcent than deal H-NDMA at low SNR for any load condton At medum SNR, FF-NDMA scheme