On the Design of Universal Schemes for Massive Uncoordinated Multiple Access

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1 On he Design of Universal Schemes for Massive Uncoordinaed Muliple Access Ausin Taghavi, Avinash Vem, Jean-Francois Chamberland, Krishna R. Narayanan Deparmen of Elecrical and Compuer Engineering Texas A&M Universiy, College Saion, TX Absrac Fuure wireless access poins may have o suppor sporadic ransmissions from a massive number of unaended machines. Recenly, here has been a lo of ineres in he design of massive uncoordinaed muliple access schemes for such sysems based on clever enhancemens o sloed ALOHA. A close connecion has been esablished beween he design of he muliple access scheme and he design of low densiy generaor marix codes. Based on his connecion, opimal muliple access schemes have been designed based on sloed ALOHA and successive inerference cancellaion, assuming ha he number of users in he nework is known a he ransmiers. In his paper, we exend his work and consider he design of universal uncoordinaed muliple access schemes ha are agnosic o he number of users in he nework. We design Markov chain based ransmission policies and numerical resuls show ha subsanial improvemen o sloed ALOHA is possible. I. INTRODUCTION The emergence of machine-driven wireless communicaion is poised o aler he wireless landscape wihin he nex few years. In addiion o radiional raffic generaed by individuals who inerac wih personal phones and mobile compuers, a sizable porion of wireless resources will be consumed by legions of unaended devices ha seek o disseminae informaion sporadically. This is an imporan paradigm shif, as i changes he profile of subscribers and may impac he performance of currenly deployed proocols. I requires he design of muliple access schemes where a massive number of users need o ac in an uncoordinaed fashion. To address his challenge, several researchers have shifed heir ineres from informaion-cenric resource allocaion algorihms and scheduling schemes o random access proocols. While he performance of radiional random access such as sloed ALOHA scales poorly wih he number of users (he hroughpu is limied by 1/e 0.37 [1]), recen resuls have shown ha by using clever reransmission sraegies and successive inerference cancellaion, he performance of sloed ALOHA can be subsanially improved [2], [3], [4]. By esablishing a connecion beween successive inerference cancellaion and message-passing decoding on biparie graphs, Liva e al. opimized he ransmission sraegy numerically and showed ha subsanial improvemen over sloed ALOHA is possible [2]. In [3], he opimal ransmission sraegy is derived analyically and i is shown ha uncoordinaed largescale sloed ALOHA can be as efficien as coordinaed access. Exising resuls on improving he performance of sloed ALOHA assume ha he number of users is known a he ransmiers in order o implemen he random access sraegy. We poin ou ha, in [5], he auhors consider he seing of unknown number of users and design a proocol which uses muliple rounds of ransmission where esimaion of he number of users and resoluion of he ransmied packes are performed joinly. In his paper, we consider a single round of ransmission and consider he imporan quesion of wheher i is possible o design universal, adapive massive uncoordinaed random access schemes ha are agnosic o he number of users in he nework. Our conribuion is hreefold. Firs, we inroduce a novel formulaion of he sloed muliple access problem wih successive inerference cancellaion. In our formulaion, he access poin does no need o esimae he number of acive devices in he sysem prior o a communicaion round, and he number of ime slos wihin a round is deermined dynamically. Second, we propose a Markov framework for slo selecion by devices. Wihin his framework, every device independenly elecs o send a copy of is own message during a slo based on wo variables: he ime elapsed since he beginning of he communicaion round, and he number of imes he device has ransmied copies of is packe hus far. Finally, we derive necessary and sufficien condiions for a sequence of probabiliy disribuions on he number of packes ransmied wihin he firs slos o be achievable, and we show ha suiable families of disribuions lie wihin he feasible se. Alogeher, our framework yields a universal access scheme where devices gracefully adap o curren condiions wihou having o collec and broadcas informaion prior o every communicaion round. II. SYSTEM MODEL Figure 1 depics he main componens of random muliple access over wireless infrasrucures, as discussed in his paper. The communicaion process akes place on he uplink, and ransmission opporuniies are divided ino rounds. One communicaion round feaures a collecion of synchronous ime slos. In he envisioned framework, he access poin broadcass a beacon on he downlink ha is used for coarse slo synchronizaion by acive devices. A he onse of every round, he access poin indicaes he sar of a communicaion cycle and invies devices o begin ransmiing messages. Every

2 slo 1 slo 2 slo 3 slo 4 slo 5 m 4 m 4 m 4 m 3 m 3 m 3 m 2 m 2 m 2 m 1 device 1 device 2 device 3 device 4 Fig. 1. This figure depics a communicaion process where several devices seek o concurrenly ransmi messages o an access poin. A he conclusion of he round, he access poin firs decodes insances where a single codeword appears wihin a ime slo, and hen applies successive inerference cancellaion o ieraively decode he remaining codewords. acive device repeas is message over a random subse of available slos. The messages are encoded wih a code a he physical layer such ha when a packe is received a he access poin wihou inerference, i is assumed o be decodable. We also assume ha he header of each packe conains he seed of a random number generaor ha is used by he user o deermine which slos o ransmi in. Once a packe is decoded, he access poin can hen deermine all he ime slos where he user will ransmi. Successive inerference cancellaion is hen applied o slos where collisions occurred, enabling he ieraive decoding of addiional codewords. This process coninues unil he access poins signals he end of he round on he downlink, poenially also informing devices abou successfully decoded packes. As he round evolves, he access poin can obain a good esimae of he number of acive devices, bu we do no discuss hese specifics herein. We focus on uplink raffic, and we adop a performance objecive based on he number of successfully decoded packes as a funcion of ime and device densiy. In spiri, his mechanism is a connecionless ransmission proocol, wih no guaranee of delivery. Daa inegriy is achieved hrough he judicious selecion of error correcing codes a he nework layer. Issues peraining o susained connecions, packe ordering, and duplicae are relegaed o higher layers. As menioned in he inroducion, his packe ransmission mechanism is aimed a sporadic raffic generaed by a massive amoun of sensors and machines. I is no suiable for real-ime raffic or sreaming applicaions. The seing considered earlier in [2], [3], [4] falls wihin he general framework considered here. An imporan difference is ha he number of acive devices is esimaed prior o he beginning of every cycle and he access poin broadcass he slo coun, M, for he upcoming communicaion round. For sysems wih a large number of acive devices, near opimal performance can be achieved by having individual devices selec he cardinaliy of heir ransmission subses according o a solion disribuion, and hen choose specific slos uniformly from he se {1, 2,..., M} [3]. In he absence of knowledge of M, such a policy canno be implemened. To he bes of our knowledge, good esimaes on he achievable performance in he presen case are no known. III. ABSTRACT FRAMEWORK An uncoordinaed ransmission policy akes he elapsed ime and he number of repeaed ransmissions of he packe since he beginning of he round as is argumens, and i oupus one or zero as an indicaion of wheher o ransmi or no during he upcoming slo. We are ineresed in scenarios where all acive devices employ a same sochasic policy. We denoe his policy by µ = (µ 0 ( ), µ 1 ( ),...), where µ (m) represens he decision rule a ime as a funcion of message coun m. In general, he number of imes a packe is ransmied by a specific device is a random process X and i is governed by he recursive form X +1 = X + µ (X ) 0. (1) Under a fixed policy µ, we can use p o denoe he disribuion of he ensuing process a ime. All devices share a same sequence of marginal disribuions in he symmeric scenario, and heir packe couns are independen from one anoher. The recursive form of (1) naurally leads o a firs-order Markov chain inerpreaion for X. The sae of he chain corresponds o he number of packes ransmied so far and, as such, X 0 = 0. The chain can eiher say in is curren sae or jump one level up because devices send a mos one message per ime slo. Transiion probabiliies are given by Pr(X +1 = m X = m) = Pr(µ (m) = 0) Pr(X +1 = m + 1 X = m) = Pr(µ (m) = 1). A message is sen ou during slo whenever X X 1 = 1. Using his inerpreaion, p can be viewed as he probabiliy disribuion of Markov chain X a ime. Under hese wo equivalen represenaions, i is possible o design uncoordinaed ransmission policies by working direcly wih µ, or by consrucing a Markov chain X wih suiable aribues and subsequenly deermining µ from he saisical properies of he chain. We cener our aenion on he laer approach. To beer undersand how o selec X and appropriaely shape he sequence of disribuions (p 0, p 1,...), we briefly review he connecion beween successive inerference cancellaion and he ieraive decoding of codewords on erasure channels [2], [4]. In his analogy, i is insighful o absrac he ransmission and decoding processes as aking place on a biparie graph wih wo ses of nodes (V, C), where V = {v 1,..., v K } denoes he collecion of acive devices and C = {c 1,..., c M } represens he aggregae signals received during individual ime slos. An edge appears beween v k and c whenever device k ransmis during slo. Figure 1 also shows a sample communicaion process and is associaed biparie graph. The larger circle nodes porray variable nodes (messages) ha need o be recovered. The smaller nodes denoe he check nodes (or, generaor nodes [6]), and hey emphasize he fac ha received signals {c } are sums of codewords ransmied during individual ime slos. A graph consruced in his manner is idenical o a Tanner graph for

3 a low-densiy generaor-marix (LDGM) code. In he presen case, he sum operaion occurs in he space of signals, whereas in a ypical LDGM code, he sum akes place over a finie field. Sill, his discrepancy does no affec he decoding sraegy nor is analysis. A ime, he uncoordinaed policy resuls in an ensemble of LDGM code equivalens. For his ensemble, we denoe he variable-node degree disribuions from he node and edge perspecive by L(x) = i L ix i and λ(x) = i λ ix i 1. By consrucion, L i = p (i) and, necessarily, λ(x) = L (x) L (1). The check-node disribuions are somewha more involved o derive in he presen scenario. Firs, we noe ha he uncondiioned probabiliy ha a device ransmi during slo is governed by he Wassersein (earh mover s) disance beween disribuions p and p +1, Pr(X +1 = X + 1) = W 1 (p, p +1 ) = inf d 1 (m, n)dγ(m, n) γ Γ(p,p +1) N 0 N 0 where Γ(p, p +1 ) is he collecion of join probabiliy measures on N 0 N 0 wih marginals p and p +1 on heir firs and second facors, respecively. Thus, under his sochasic sraegy, he uncondiioned probabiliy of C, he number of packes ransmied during slo, is equal o Pr(C = k) ( ) K = (W 1 (p, p +1 )) k (1 W 1 (p, p +1 )) K k. k For large value of K and small value of W 1 (p, p +1 ), his expression is amenable o a Poisson approximaion Pr(C = k) (KW 1(p, p +1 )) k exp( KW 1 (p, p +1 )). k! IV. MARKOV CHAINS AND DISTRIBUTION SHAPING The framework of Secion III offers a valuable plaform in he design of Markov processes and probabiliy disribuions for efficien decoding in massive uncoordinaed muliple access. Sill, he evoluion of X is governed by physical consrains. This Markov chain is monoone increasing and, a every poin, i can eiher remain a is curren level or jump up by a uni sep. These oucomes correspond o he device elecing o ransmi is message during he nex slo or choosing o remain silen. From a disribuion viewpoin, he realm of possibiliies is deermined by sochasic dominance. Le X be a random variable wih probabiliy disribuion p X ; and Y, a random variable wih probabiliy disribuion p Y. We use X Y or, equivalenly, p X p Y o denoe firsorder sochasic dominance. By definiion, we wrie X Y whenever Pr(X > l) Pr(Y > l) for all l R. As discussed below, his parial order underlies our abiliy o achieve sequences of disribuions hrough monoone increasing, Markov chains wih a mos uni incremens [7], [8]. In he conex of uncoordinaed muliple access, we are especially ineresed in ime-inhomogeneous, monoone increasing Markov chains over he non-negaive inegers, which we represen by N 0. As menioned above, prospecive chains should only feaure self-ransiions and ransiions o neares neighbors on he righ. Proposiion 1: Suppose ha (p 0, p 1,...) is a sequence of discree probabiliy disribuions over N 0. This sequence of disribuions can be achieved hrough a monoone increasing Markov chain conaining solely self-ransiions and ransiions o neares neighbors on he righ if and only if p p +1 and p +1 Sp N 0 (2) where S denoes he sandard righ shif operaor acing on one-sided infinie sequences of numbers. Proof: We esablish his resul in a consrucive manner, building a Markov chain {X } wih he desired properies. We emphasize ha he resuling Markov chain can be and is ofen ime-inhomogeneous. To begin, his Markov chain should have disribuion p 0 a ime zero. Also, {X } should only feaure wo ypes of ransiions: self-ransiions and jumps o neares neighbors on he righ. Based on hese requiremens, we can analyze he progression of he chain over ime. On he lef boundary, a 0, we have p +1 (0) = Pr(X +1 = 0 X = 0)p (0) ( 1 ) 0 p (0). Furhermore, for any sae m > 0, we can wrie p +1 (m) = Pr(X +1 = m X = m 1)p (m 1) + Pr(X +1 = m X = m)p (m) m 1 p (m 1) + ( 1 γ m () ) p (m). Rearranging hese expressions, we obain generic condiions in he form of a sysem of linear equaions, m p (m) = p (m) p +1 (m) + m 1 p (m 1) m m = p (l) p +1 (l) where m akes value in N 0. Since γ m () can be inerpreed as a condiional probabiliy, i mus lie wihin [0, 1]. This ranslaes ino wo necessary and sufficien condiions. The firs one corresponds o γ m () being non-negaive. This is achieved by having he difference beween he wo cumulaive sums in (5) remain non-negaive for all m N 0 or, equivalenly, p p +1. When his is saisfied, he second requiremen ensures ha γ m () is no greaer han one. We can express his requiremen as m m p (l) p +1 (l) p (m) (6) which yields m 1 p (l) m p +1(l) afer reorganizing erms yields. Using he sandard righ shif operaor acing on one-sided infinie sequences of numbers, we ge he compac expression p +1 Sp. Thus, whenever he condiions in (2) are saisfied, i is possible o define (ime-dependen) ransiion probabiliies for {X } such ha he desired disribuions are (3) (4) (5)

4 achieved a every sep. Conversely, given a Markov chain wih admissible ransiion probabiliies, i.e., self-ransiions and ransiions o neares neighbors on he righ, he corresponding sequence of discree disribuions (p 0, p 1,...) mus saisfy he sochasic dominance condiions in (2). Alogeher, he parial order induced by sochasic dominance is key in being able o idenify which sequences of disribuions are realizable using firs-order Markov chains. When he parial orders in (2) are fulfilled, explici form for he ransiion probabiliies can be inferred from Proposiion 1. Definiion 1 (Uncoordinaed Markov Sraegy): Suppose ha (p 0, p 1,...) is a sequence of discree probabiliy disribuions over N 0 such ha he condiions in (2) are saisfied. For inegers m, N 0, define consans µ (m) = { m p(l) m p+1(l) p (m) p (m) > 0 0 p (m) = 0. Le {X } be a firs-order Markov chain wih iniial probabiliy disribuion p 0, and ransiion probabiliies Pr(X +1 = m X = m) = 1 m Pr(X +1 = m + 1 X = m) = m where m, N 0. Necessarily, Pr(X +1 = n X = m) = 0 whenever n / {m, m + 1}. Then {X }, or equivalenly µ, forms an admissible uncoordinaed ransmission policy. Corollary 1: The discree random process {X } specified in Definiion 1 is a valid Markov chain and i possesses discree disribuion p a ime. Proof: This resul follows from he proof of Proposiion 1. Firs, p p +1 guaranees ha he consans defined in (7) are non-negaive. Second, p +1 Sp ensures ha he consans remain bounded by one. Random process {X } is hen characerized as a firs-order Markov chain hrough is iniial disribuions and ransiion probabiliies. Furhermore, by consrucion, i only feaures self-ransiions and ransiions o neares neighbors on he righ. Finally, hrough he recursive equaions (3) and (4), we deduce ha he probabiliy disribuion of he Markov process a ime is p, as desired. As a firs example for he proposed mehodology, we consider he solion sequence of disribuions. Solion disribuions have played an insrumenal role in he developmen of raeless codes. Through dualiy, hey arise naurally in he radiional seing for uncoordinaed muliple access where K is known beforehand. As seen below, i is possible o creae a Markov chain whose disribuion a every ime insan falls wihin he solion family. Example 1 (Solion Disribuions): The solion disribuions fulfill he requiremens of Proposiion 1 and, as such, he corresponding sequence can serve as a basis for an uncoordinaed Markov sraegy. Recall ha, for N, he solion disribuion is given by { 1/ 1 p sol() (m) = 1/((m 1)m) 2,.... (7) Checking he firs condiion in Proposiion 1, we ge m m p (l) p +1 (l) = = 1 for 0,..., ; he difference vanishes for m +1. Thus, p sol() p sol(+1). Likewise, verifying he second condiion in Proposiion 1, we sar wih m m 1 p +1 (l) p (l) = for 1; moving forward, we have m m 1 1 p +1 (l) p (l) = (m 1)m 1 0 for 2,...,. Again, he difference vanishes for m +1 and hence p sol(+1) Sp sol(). Joinly, hese resuls imply ha he solion probabiliy disribuions ordered according o parameer form an admissible sequence for an uncoordinaed Markov sraegy. Specifically, consider he sequence p 0 = e 0, p 1 = p sol(1),..., p = p sol(),... where e 0 is he canonical uni vecor wih uni mass a zero. We noe briefly ha e 0 p 1 and p 1 = Se 0, which preserve he admissibiliy of he sequence. By applying Corollary 1 o hese disribuions, we gaher ha he ransiion probabiliies for he Markov sraegy become (m 1)m (+1) 2,..., 0 oherwise. for N. Explicily, he ransiion probabiliies associaed wih ime-inhomogeneous Markov chain {X } for N are given by (m 1)m Pr(X +1 = m X = m) = 1 (m 1)m Pr(X +1 = m + 1 X = m) = for 2,..., ; and Pr(X +1 = 2 X = 1) = Pr(X +1 = 1 X = 1) = + 1. A ime zero, he Markov chain sars in sae X 0 = 0, and deerminisically ransiions o X 1 = 1. The emission probabiliy for a paricular device is governed by i p sol() (i) = 1 for i=1 While he solion disribuion is opimal for he case when K is known a he ransmiers, he sequence of disribuions does no yield an efficien scheme for universal uncoordinaed muliple access. One issue ha degrades performance wih he

5 solion Markov sraegy is he fac ha devices are collecively more likely o send packes a he onse of he communicaion round, as heir behavior is correlaed; his fails o induce a suiable righ degree disribuion. To miigae his issue, we inroduce wo new classes of disribuions, and we adop a mixure of hese disribuions as a basis for universal access. Example 2 (Saeless Disribuions): The saeless family of disribuions is characerized by a device using emission probabiliies deermined solely by he ime elapsed since he onse of he communicaion round. Iniially, he saeless disribuion is simply p 0 = e 0. Given p, he disribuion a he nex ime sep is governed by p +1 = ( 1 ) p + Sp, where S is he sandard righ shif operaor. For he problem a hand, candidaes for he emission probabiliies include = c ( ) c log(ɛ) and = 1 exp where c is a uning parameer. The firs funcion maximizes he probabiliy of geing a check node of degree one owards he end of he round, whereas he second funcion ses he probabiliy of an empy slo a he end of a round o ɛ. I is oo possible o define an alernae version of his disribuion where he number of ransmied packes a ime is upper bounded by a prescribed maximum τ. Example 3 (Skewed Disribuions): In conras o he previous example, he skewed family of disribuions favors nodes ha have ransmied several packes in he pas. Again, a he onse of he communicaion process p 0 = e 0. Given an average emission arge, he sae ransiion probabiliies a ime become m 0, i=0 p (i) < 1 1, i=m+1 p(i) p (m) i=m p (i) oherwise. A slighly differen version of he skewed disribuions limis he number of packes a device can send by ime. In boh cases, hese disribuions seek o radeoff he probabiliy of geing a single packe per ime slo owards he end of a round wih he likelihood of empy slos. The former class favors ransmission behavior where he number of packes sen by a device rapidly concenraes around he mean. The laer classes feaures a heavier ail wih a selec few devices being responsible for a large porion of he raffic. No oo surprisingly, raffic shaping can affec he decoding process significanly. V. NUMERICAL RESULTS AND DISCUSSION The resuls in Fig. 2 showcase mean hroughpu, measured as he average number of decoded packe per ime slo for he Markov access schemes based on a mixure of he saeless and skewed disribuions. The mean emission probabiliy is employed o conrol he degree of check nodes as a funcion of ime. The weighing facor beween hese wo disribuions acs Aveage Number of Messages per Slo c = 1.0 c = 1.1 c = 1.2 c = 1.3 c = ,000 1,500 2,000 Number of Time Slos (N) Fig. 2. This graph shows hroughpu efficiency defined as he average number of decoded packes divided by slo coun. Every line corresponds o a unique parameer. Opimal performance is achieved a c = 1.2 and N = as a means o conrol he variance in he degree disribuion of he variable nodes. Graphs are shown for K = 1000 devices wih a weighing facor of 0.85 in favor of he saeless disribuions; he various curves correspond o differen uning parameers. The maximum efficiency is approximaely equal o 69 percen, and i is achieved wih N = 1250 and c = 1.2, This level of performance is encouraging, as i subsanially exceeds he performance of radiional ALOHA. Similar performance is observed under he same Markov access schemes, bu wih differen device coun K. Wih perfec knowledge of he number of users, close o 80 percen efficiency can be obained and, hence, here appears o be a penaly for no having precise knowledge of he number of users a he ransmiers. Ye, igh upper bounds on he average number of messages per slo are no known for universal uncoordinaed muliple access. I is an ineresing open quesion o undersand he behavior of universal muliple access in he limi of large K. REFERENCES [1] R. Gallager, A perspecive on muliaccess channels, IEEE Trans. Inf. Theory, vol. 31, no. 2, pp , [2] G. Liva, Graph-based analysis and opimizaion of conenion resoluion diversiy sloed ALOHA, IEEE Trans. Commun., vol. 59, no. 2, pp , [3] K. R. Narayanan and H. D. Pfiser, Ieraive collision resoluion for sloed ALOHA: An opimal uncoordinaed ransmission policy, in Inern. Symp. Turbo Codes & Ier. Inf. Proc. (ISTC). IEEE, 2012, pp [4] E. Paolini, C. Sefanovic, G. Liva, and P. Popovski, Coded random access: applying codes on graphs o design random access proocols, IEEE Commun. Mag., vol. 53, no. 6, pp , [5] C. Sefanovic, K. F. Trilingsgaard, N. K. Praas, and P. Popovski, Join esimaion and conenion-resoluion proocol for wireless random access, in IEEE In. Conf. Commun., 2013, pp [6] T. Richardson and R. Urbanke, Modern Coding Theory. Cambridge Universiy Press, [7] J. R. Norris, Markov Chains, ser. Saisical and Probabilisic Mahemaics. Cambridge Universiy Press, 1997, no. 2. [8] P. Brémaud, Markov Chains: Gibbs Fields, Mone Carlo Simulaion, and Queues, ser. Texs in Applied Mahemaics. Springer Science & Business Media, 2008, vol. 31.