Fundamentals of the Backoff Process in 802.11: Dichotomy of the Aggregation

1 Fundamentals of the Back of f Process in 802.11 : Dichotomy of the Aggre gation Jeong-woo Cho and Y uming Jiang Abstract —This paper discov ers fun damental principles of the backoff process that gov erns th e performance of IEEE 802.11. A simplistic principle founded upon regular v ariation theory is that the b ackoff time has a tru ncated Pareto-type tail d istribution with an exponen t of (log γ ) / log m ( m is t he multip licativ e factor and γ is the collision probability). This rev eals t hat the per -node backoff proc ess is heav y-tailed in the strict sense f or γ > 1 /m 2 , and pa ves the way f or the following unifying r esult. The state-of-th e-art th eory on the superposition of the heav y- tailed processes is applied to establish a dichotomy exhibited by the aggregate backoff proc ess, p utting emphasis on the importance of time-sca les on which we view the backoff processes. While the aggregation on normal time-scales leads to a Poisson process, it is approximated by a new limitin g process possess- ing long-range dependence (LRD) on coarse time-scales. This dichotomy turns o ut to b e instrumental in f ormulating short-term fairness, extending existing f ormulas to arbitrary population, and to elucidate the ab sence of LRD in practical sit uations. A reﬁned wa velet analysis is co nducted to strengthen th is arg ument. Index T erms —Point p rocess theory , regular variation theory , mean ﬁeld theory . I . I N T RO D U C T I O N Since its introdu ction, the perform ance of IE EE 802. 11 has attr acted a lot of resear ch attention an d the center of the attention has been the thr oughp ut [ 6], [27]. Recently , other critical p erform ance aspects of 802.11 also burst onto the scen e, which include short-term fairness [12], [2 6] and delay [38]. It goes without saying that there h as been a pheno menal growth of Skype an d IPTV users [14], [ 15] and it is reported in [23] that an ever -increa sing perce ntage of these users conne cts to the In ternet through wireless connection s in US. Remarkab ly , it is fo und in [1 5] th at jitter is mor e negativ ely co rrelated with Skype c all du ration than delay , i.e. , Skype users tend to hang up their calls earlier with large jitters. This ﬁnding em pirically testiﬁes large jitter of access n etworks annoy s Skype users, let alone Qo S (qua lity of service). T his quantiﬁed dissatisf action of users p rovides a motiv atio n for a thoro ugh un derstandin g of dela y a nd jitter performan ce in 802.1 1. This work was supported in part by “Centre for Quantiﬁable Quality of Service in Communicati on Systems, Centre of Excellence” appointed by The Research Council of Norw ay , and funded by The Research Council, NTNU and UNINETT . A part of this work was done when J. Cho was with EPFL, Switzerlan d. A preliminary a bstract version of t his work appeared a t A CM SIGMETRICS W orkshop on Mathematical Performance Modeling and Analysis (MAMA ’09). J. Cho and Y . Jiang are with the Centre for Quantiﬁabl e Quality of Service in Communicati on Systems, Norwegian Uni versi ty of Sci- ence and T echnology (NTNU), NO-7491 Trondheim, Norway (email: { jeongw oo,jiang } @ q2s.ntnu.no). For throug hput analysis, K umar et al. , in the semin al pap er [27], axiomized sev eral remarkable observations based on a ﬁxed p oint equation ( FPE), ad vancing the state of the art to mo re systematic mo dels and p aving th e way fo r more compreh ensiv e under standing of 802.1 1. Above all, on e of the key ﬁnding s of [27], alre ady adop ted in the ﬁeld [28], [ 34], is th at th e full interfere nce mod el 1 , also called the single- cell mod el [27], in 802.1 1 networks leads to the backoff synchr ony pr operty [3 1] which implies the backoff process can be com pletely separated and ana lyzed throu gh the FPE technique . Another observation in [27] was that if the collision probab ility γ is constant, one can der iv e the s o-called Bianchi’ s formu la by appea ling to r enew al reward theorem [13], with out the Markov chain analysis in [6]. An intriguing notion, called short-term fairness , h as b een introdu ced in so me recent works [5], [ 12], [26], deﬁnin g P [ z | ζ ] as the prob ability that oth er no des transmit z packets while a tagged node is tran smitting ζ packets. It can be easily seen that this notion pertains to a purely b ackoff-related argument also owing to the backoff synchrony property in the full interfere nce model [27]. T he two papers [5], [12], in the co urse of deriving equations fo r P [ z | ζ ] , assumed that the summation of the ba ckoff values ge nerated p er packet, wh ich we deno te b y Ω , is un iformly and expon entially distributed, respectively . Speciﬁcally , d espite the same situation where two nodes c ontend for the m edium, the former [5] a ssumed that Ω is uniformly distributed because the initial backoff is unif ormly distributed over the set { 0 , 1 , · · · , 2 b 0 − 1 } where 2 b 0 is the initial co ntention wind ow and observed in [5, Fig. 2] that this assumption leads to a goo d match between the expression P [ z | ζ ] derived under the unifo rm assumption on Ω an d the testbed data measured in their experiments, while the latter [12] also obser ved in [ 12, Fig. 5 (a)] th at the testbe d data measured in their exper iments closely match the expression P [ Z | ζ ] derived un der the the exponential assumption on Ω : Q1 : “What makes two differ ent observations? ” (to be an- swered in Section III) In addition , the tw o works [ 5], [ 12] ac quired the expression of P [ z | ζ ] o nly fo r the two nod e ca se. A m ore genera l fo rmula for arbitrary numbe r of nodes sho uld dee pen ou r a ppreciation of sho rt-term fairness. It is n atural to ask the fo llowing pertinent questions: Q2 : “Can we develop a gene ral mod el for short-term fair- ness?” (to be answered in Coro llaries 1 & 2) In pr oportio n as people take a growing interest in the 1 In the full inte rference or single-cell model , e very node interfere s with th e rest of the nodes, i.e. , its correspon ding interfer ence graph is fully connect ed. 2 delay perf ormance of 8 02.11 , the nu mber o f fundam ental questions that we face increases. In [ 1], it was argued based on simu lation results that the acc ess delay in 8 02.11 closely follows a Po isson distribution. Th ey h av e shown that th e number of s uccessful packet transm issions by any nod e in the network over a time inter val h as a p robab ility distribution that is close to Po isson by an upper bou nded distribution d istance. This raises an intrig uing question: Q3 : “Is there a P oissonian pr operty? I f yes, what is the cause?” (to be answered in Theorem 1) Another case in po int is found in a recent work [34] th at extends the access delay analysis in th e semin al pap er of Kwak et al. [28] and m akes an attempt at an alyzing h igher order moments by applying the FPE technique. One interesting ﬁnding in [ 34] is that the access delay has a wide-sen se heavy-ta iled d istribution [3 4, Theorem 1] which m eans that its mom ent generating function R ∞ 0 e tx f ( x )d x is ∞ , ∀ t > 0 , where f ( x ) is the corr esponding pdf (probab ility density function ) [32]. One sho uld be car eful in interpreting this ﬁn d- ing because the wide-sense heavy-tailedness does not imply strict sense heavy-tailedness, wh ich rough ly means the ccdf (complem entary cumulativ e distribution functio n) is of Pareto- type [17] with an expo nent over ( − 2 , 0) . In fact, there ar e lots o f d istributions, nam ely , logn ormal, Pareto, Cauchy an d W eibull distributions, which b elong to th e class of wid e-sense heavy-tailed distrib utions. Consequen tly , the discussion poses the fo llowing challen ge which is und oubtedly a tan talizing question. Q4 : “What is th e distribution typ e of th e delay-related vari- ables?” (to be an swered in Theorem s 2 & 3) Finally , it is, p erhaps, surprising that long- range dependence of 80 2.11 has not b een r igorously analy zed even fo r the single node case, n ot to men tion the aggregate pro cess of many nodes. One minor con tribution of this pap er is tha t we prove in Theo rem 3 that the indi vidu al arr iv al pr ocess (con sisting of successful tran smissions of on e n ode) can be vie wed as a renew al process with heavy-tailed inter-arri val times, which implies that the ind i vidua l a rriv al process possesses long- range depend ence s imply by ap pealing to [30]. Howe ver , for th e superp osition arri val process (consisting of successful transmissions of all node s), there is no clear answer . For example, T ickoo an d Sik dar [37] con jectured the absence of long- range depend ence of agg regate total lo ad, which we call superposition arriv al process. It is rem arkable tha t th e absence of long-r ange d ependen ce has been also suppo rted throug h empirical an alysis such as wa velet-based me thod [2] by V eres an d Bod a [3 9] in the con text of T CP ﬂows in wired networks. Since there is an an alogy between the backoff mechanisms adopted by 8 02.11 and TCP (in wired networks) in that 1) b oth of the m adopt b ackoff schem es (802.11) or retrans- mission scheme (T CP) where the pro bability of these ev ents is eith er the collision pro bability (80 2.11) or the packet drop probab ility in router b uffers (TCP), 2) th e mean o f the backoff (con tention win dow in 802.1 1) or retr ansmission time (timeou t in TCP) dou bles fo r each backoff o r retransmission, one migh t wonder if there is a fund amental reason that elucidates these observations. Q5 : “Does th e aggr egate tr ansmission pr o cess possess long- range d epende nce? If yes, why is it seldom observed?” (to be answered in Theor em 4 and Section VII) The focus of this pape r is o n the backoff p rocess in 802.11, since it play s the central r ole in quan tifying the p erform ance of 802.11 [27]. For example, to grasp the heart of the de- lay prope rties, the b ackoff value distribution in 8 02.11 DCF (distributed coordina tion fun ction) mode can be u sed as a surr ogate for th e access delay [28]. A s discussed above, th e throug hput p erforman ce an d shor t-term fairness per forman ce also depen d on the b ackoff proce ss and are pa rticularly af- fected by the b ackoff synchrony proper ty . Essentially , once the backoff distribution is o btained, various perf ormance aspects can be analyzed. A. Con trib ution s of this work This paper discovers fundamental principles of the backof f process an d provide s an swers to the open question s high- lighted ab ove, wh ich c onstitute the contributions of the paper . Particularly , it turns out that we ﬁn d out the answers to mo st aforemen tioned question s Q2 - Q5 in the c ourse of deriving the following two p rinciples based o n a n ew m ethodolo gy , i.e. , point proc ess approach. • Power -tail principle : The per-packet backoff time distri- bution has a slo wly-varying po wer-tail ( Theorem 3). • D ichotomy o f ag gregation : Depen ding on th e tim e- scales on wh ich the bac koff pro cesses are agg regated, the resultan t process b ecomes either Poissonian or a ne w process (Theo rems 1 & 4). The p ower -tail princ iple, which is deriv able o nly af ter we accumulate a stor e of k nowledge (Section III, Lemm a 1 and Theorem 2 ), characterizes the b ackoff distribution in a tractable and simp listic way , owing to regular variation theo ry , answering Q4 . The dichotomy of aggr egation implies that, when we view th e ag gregate proce ss on no rmal time-scales, owing to the tendency of each componen t pr ocess to become sparse as popu lation grows, we observe only a Poissonian as its ma rginal distribution. Howe ver , viewed on co arse time- scales, the ag gregate pr ocess is iden tiﬁed as a lon g-rang e depend ent p rocess. This rig id dichoto my is instrumental in ﬁnding answer s to Q2 , Q 3 and Q 5 , an d expatiates upo n th e coexistence of c ontrary p roperties sugg ested by Q3 an d Q5 . All the theorems in the p aper are closely lin ked with eac h other, forming a solid framework for the p erforma nce analysis of 802. 11. These results help us to get the co mplex d etails of the backoff proc ess in 8 02.11 in to persp ectiv e under o ne framework. The rest of the pa per is organized as follo ws. In Section II, we re visit the Bianchi’ s f ormula along with a sur vey of recent adv ances in mea n ﬁeld theo ry , with which the analysis of the b ackoff process at one node can be decoup led from other nodes. In Section III, we present th e exact distribution of per - packet backof f. W e establish in Section IV that the aggr egate backoff process can be a pprox imated by a Poisson p rocess under the large popu lation regime. In Section V, we extend 3 to asymptotic ana lysis and prove the p ower -tail prin ciple. In Section VI, we ﬁrst propose a ne w process approxim ation on a coarse time scale, which is th en applied to fo rmulate short- term fairn ess and to ide ntify long -range dep endence. After condu cting a wav elet analy sis on lon g-rang e d ependen ce in Section VII, we con clude this paper . I I . B I A N C H I ’ S F O R M U L A R E V I S I T E D The backoff pr ocess in 802 .11 is governed b y a few rules if the d uration of pe r-stage backoff is taken to be expo nential: (i) every no de in ba ckoff stage k attempts transmission with probab ility p k for every time-slot; (ii) if it succeeds, k changes to 0 ; (iii) o therwise, k cha nges to ( k + 1) mod ( K + 1 ) where K is the index o f th e highest backoff stage. Markov chain m odels, which have been widely u sed in d escribing complex systems inc luding 802.1 1, howe ver , very of ten lead to excessiv e complicatio ns as discussed in Section I. In th is section, we present a surrogate tool for the analysis, mean ﬁeld the ory . It is no tew orthy that the rules used in 8 02.11 , i.e. , (i)–(iii), closely r esemble the me an ﬁeld equa tions laid out below . A. Ba sic Operation of DCF Mode T ime is slotted . Each n ode fo llowing the rand omized access proced ure of 802.1 1 d istributed coordin ation fu nction (DCF) generates a backof f va lue af ter rec ei ving the Short Inter-Frame Space (SIFS) if it has a packet to send. This backof f v alue is uniformly distributed over { 0, 1, · · · , 2 b 0 − 1 } (or { 1 , 2, · · · , 2 b 0 } ) wher e 2 b 0 is the initial con tention window . Whenever the medium is idle for the duration of a Dis- tributed In ter-Frame Space (DIFS), a node unfree zes (starts) its countd own pr ocedur e of the backoff and d ecrements the backoff by one p er every time-slot. It fr eezes the c ountdown proced ure as soo n as the medium beco mes busy . Ther e ex- ist K + 1 backoff stages who se in dices belong to the set { 0 , 1 , · · · , K } whe re we assume K > 0 . I f two or more wireless no des ﬁnish their c ountdowns at the sam e time- slot, there occurs a collision between R TS (rea dy to sen d) packets if the CSMA/CA (carrier sense multiple acce ss with collision av o idance) is im plemented , otherwise two d ata packets collide with ea ch other . If there is a collision, ea ch nod e who partici- pated in the collision multiplies its contention win dow by the multiplicative factor m . In o ther words, each nod e chang es its backoff stage in dex k to k + 1 and ad opts a new conten tion window 2 b k +1 = 2 m k +1 · b 0 . If k + 1 is greater than the index of the hig hest backoff stage number, K , th e no de steps back into the initial b ackoff stage whose c ontention wind ow is set to 2 b 0 . In th e IEEE 8 02.11b stan dard, m = 2 , K = 6 ( 7 attempts per packet), and 2 b 0 = 32 are used . This work focu ses on the perf ormance of single- cell 802 .11 networks where it is sufﬁcient to ana lyze the backoff process in orde r to in vestigate the performan ce of single- cell networks B. The Bianchi’ s F o rmula In perfo rmance ana lysis of 802.11 , Bianchi’ s form ula and its many variants are p robably the mo st known [6], [27], [28], [3 4]. Assuming that there are N n odes, the Bianch i’ s formu la can be written compactly in a more general ﬁxed point equation (FPE) form : ¯ p = P K k =0 γ k P K k =0 γ k q k , (FPE1) γ = 1 − e − ( N − 1) ¯ p (FPE2) where ¯ p and γ respectively design ate the a vera ge attempt rate and collision probability o f e very node at each time-slot. The attempt pr obability in bac koff stage k is denoted by q k and deﬁned as the in verse of the mean contention window , i.e. , q k = 2 / (2 b k − 1) . It satisﬁes 0 < q k ≤ 1 as b k ≥ 1 . Note tha t Bianchi’ s form ula holds under the well-known assumption: A.1 All the transmission queues of nod es are saturated. Exactly und er which co ndition (FPE1) h olds is recen tly being rediscovered with rigorous math ematical arguments [4] , [11], [35], which, som etimes called mean ﬁ eld app r oxima- tion . This funda mental approach was originally de veloped by Bordenave et al. [11] and Sharma et al. [35]. Remark ably , Bordenave et a l. [11] adopted a generalized particle interaction model which encom passes Mar kovian ev olution of the system other than particles at the same time. Bena¨ ım and Le Boud ec [4] overcame s ome limitations of the mod el [11], broaden ing its app licability . The ma in result here is th at, as the nu mber of particles go es to in ﬁnity , i.e. , N → ∞ , the state distribution of every node evolves accord ing to a set of K + 1 dimension al nonlinear ordinary differ ential equations under an appropriate scaling of time. Bena¨ ım and L e Boud ec [4] also o bserved that decoup ling appr oximation rep resented by ( FPE1) does not h old if the differential equation s do es not h av e a uniqu e globally attractor . Remarkably , Bordenave et a l. [11] have proven that the differential equations are globally stable if K = ∞ an d the r e- scaled attempt prob ability Q k := N q k satisﬁes Q k +1 = Q k / 2 with Q 0 < ln 2 . I n the meantime , Sharma et a l. [35 ] obtained a re sult for K = 1 an d men tioned the d ifﬁculty to go beyond. Howe ver , the case for other ﬁnite K has remained to be proved [4, pp.833]. Recen tly , we s olved this issue to a large extent [1 6] by p roving that, fo r ﬁnite K , a simp listic con dition Q k ≤ 1 (or q k ≤ 1 / N ) f or all k ∈ { 0 , · · · , K } guar antees th e glo bal stability of the differential eq uations as well as the uniqu eness of the solution s to the Bianchi’ s formula. As we discussed in [16], there are still many outstanding problems upon the stability of the associated differential equations. While on e of th e aims o f these efforts [4], [11], [1 6], [3 5] is to id entify the fundam ental co nditions u nder wh ich the collision proba bility is determ inistic and time -in variant f or large popula tion ( N = ∞ ), once we assume the collision probab ility is su ch fo r N < ∞ , the demonstra tion of the formu la (FPE1) is shown to be straightf orward [ 27]. Th at is to say , we need to m ake the following simp le assumption . A.2 For each node, cond itional u pon its transmission attempt, the collision events f orm an i.i.d. sequence , which is indepen dent from other nodes. The observation in [2 7] was that, under the above assum ptions, one can easily der i ve (FPE1) by appealing to renewal reward 4 theorem [13], without the Markov ch ain analysis in [6]. Thus from now on, th e attempt p robability is given by (FPE1). As a by-p roduct, we can also see that the d istribution o f backoff stages, which we denote by φ k , k ∈ { 0 , · · · , K } , takes the following form φ k = γ k q k · 1 P K k =0 γ k q k . (1) The expression o f the collision proba bility (FPE2) w as ﬁr st used in [27, Section IV]. A similar expr ession was also used in [4], [11] und er the intensity scaling, which mea ns that the attempt pro bability of ev ery node in a ny backoff stage is o f the order of 1 / N . W e use (FPE2) instead of its o riginal version in [6] bec ause, as argued in [ 4], [1 1], the appr oximation provided by (FPE2) is well foun ded on a mea n ﬁeld result. L astly , i t is also n otew orthy that the ana lysis in Section III an d IV doe s not depend on wheth er K is ﬁnite or n ot. I I I . B AC KO FF A N A L Y S I S The backoff value distribution and th e backoff stage distri- bution shou ld n ot b e confused in meaning . While the latter is the distribution o f the b ackoff stage of a nod e, the for mer is the distribution of the backoff value generated for initiating the backoff countd own when the no de has a packet to transmit. The backoff value distrib ution at b ackoff s tage k h as a d iscr ete uniform pdf (proba bility density functio n) f k ( · ) on the integers { 0 , 1 , · · · , 2 b k − 1 } with mean 1 /q k = (2 b k − 1) / 2 and variance V ar k = b 2 k 3 − 1 12 = 1 3 · 2 b k + 1 2 b k − 1 · 1 q 2 k = v 2 k q 2 k where the pre-factor is denoted by v 2 k to simplify th e exposition in the curre nt section. Note that lim b k →∞ v 2 k = 1 / 3 . Let Ω and f Ω ( · ) r espectively deno te the sum of the b ackoff values generated f or a packet, and its p df. Also deno te by ¯ Ω its m ean and σ 2 Ω its variance. It shou ld be clear th at the sum o f the backoff values ge nerated for a packet Ω which we baptize in th is paper per-pack et backoff can be f ormally deﬁne d as a compou nd random v ariable: Ω := P κ k =0 B k (2) where B k is a random variable denoting the backoff value generated at the k th backoff stage, for a packet of a tag ged node, and κ is also a rando m variable design ating the highest backoff stag e reached by the packet. The prob ability that the k th b ackoff stage is reac hed d uring the backoff du ration for a packet can be computed as γ k irrespective of the b ackoff d istribution at any backoff stage. Hence we have P [ κ = k ] = γ k − γ k +1 , ∀ k ∈ { 0 , · · · , K − 1 } , and P [ κ = K ] = γ K . From Bayes’ theor em, f Ω ( · ) beco mes: f Ω ( x ) = P K k =0 f Ω ( x | κ = k ) · P [ κ = k ] (3) where f Ω ( x | κ = k ) deno tes th e sum of the backoff values from 0 th to k th stages for a given k . Applying the fact that the sum of k rando m variables with pd fs f 0 ( · ) , · · · , f k ( · ) has a pdf of the conv olution of the pdf s yields f Ω ( x ) = f ∗ K ( x ) γ K + (1 − γ ) P K − 1 k =0 f ∗ k ( x ) γ k (4) where f ∗ k ( · ) := ( f 0 ∗ · · · ∗ f k )( · ) is the conv olution of k + 1 function s. In a similar w ay , ¯ Ω can be co mputed from (2): ¯ Ω = E [ P κ k =0 B k ] = P K k =0 E h P k k ′ =0 B k ′ i · P [ κ = k ] = P K k =0  P k k ′ =0 1 q k ′  · P [ κ = k ] . (5) By man ipulating (5) combin ed with the expression of P [ κ = k ] , it is easy to see that ¯ Ω = K X k =0 γ k q k . (6) In addition , using E [ B 2 k ] = (1 + v 2 k ) /q 2 k , the second mo ment of Ω can be re arranged as σ 2 Ω + ¯ Ω 2 = E h ( P κ k =0 B k ) 2 i = P K k =0 E   P k k ′ =0 B k ′  2  P [ κ = k ] (7) = P K k =0   k X k ′ =0 1+ v 2 k ′ q 2 k ′ + 2 k X i =1 i − 1 X j =0 1 q i q j   P [ κ = k ] (8) =  P K k =0 γ k q 2 k (1 + v 2 k )  + 2  P K k =1 γ k q k P k − 1 i =0 1 q i  The abov e equalities can be easily veriﬁed by re arrangin g ( 7) and (8). Moreover, it is shown in Appen dix A that, if q k = 2 / (2 b 0 m k − 1) as in th e standard, v 2 Ω := σ 2 Ω / ¯ Ω 2 simpliﬁes to P K k =0 δ k  P K k =0 ( b 0 m k − 1 / 2) γ k  2 − 1 (9) where δ k =  b 0 m k − 1 2  γ k n m +1 m − 1 + v 2 k   b 0 m k − 1 2  − k − 2 b 0 − 1 m − 1 o . Remark 1 R1.1 The resu lt pu ts fo rward an alternative viewpoint. W e can view th e backoff pro cess reﬂecting the collision effect amo ng no des as if th ere is no collision at all and th e per-packet backoff for ev ery node has a d istribution with mean ¯ Ω and CV v Ω (or equiv alently v ariance σ 2 Ω ). R1.2 [An swer to Q1 ] Consider the case N = 2 . It can be compu ted f rom (9) th at Ω is app roximately uniformly distributed in 80 2.11b while it is exponentially distributed in 802.1 1a/g in th e sense that v Ω ≈ 0 . 7 ( though slig htly larger than 1 / √ 3 ) and v Ω ≈ 1 . 0 , respectiv ely , mainly due to dif ferent initial conten tion windows ( 2 b 0 = 32 in 8 02.11b and 2 b 0 = 16 in 8 02.11 a/g). This is the reason why they [5], [12 ] observed that their testb ed data closely match th e expression s of inter- transmission prob ability P [ Z | ζ ] , which were derived und er their respective assumptions. Note that we com municated with the ﬁrst autho r of [ 12] to verify th e pro tocol (802 .11g) used in their testbed . W e will formally deﬁn e P [ Z | ζ ] in Section VI-A. T o verify the an alysis, simulations have been conduc ted. W e have used ns-2 versio n 2.3 3 with its built-in 802.1 1 mo dule and the p arameter set of 802 .11b, i.e. , m = 2 and 2 b 0 = 3 2 , 5 0 0.1 0.2 0.3 0.4 0.5 10 0 10 1 10 2 K= ∞ K= 15 K= 6 collision probability ( γ ) per−packet backoff CV ( v Ω ) ↑ N= 2 N= 5 N= 10 N= 20 N= 40 N= 60 N= 80 N= 100 analysis: contours of K analysis: contours of N simulation: K= 15 simulation: K= 6 error Fig. 1. Per -pack et back off CV v Ω vs. collisi on probability γ for K = 6 , 15 , ∞ ; and N = 2 , · · · , 100 . except that K is varied to obser ve the asymptotic pro perty . All simulations u se a 30 00 s warm-u p p eriod and all qu antities are measured over the next 320 , 000 s ( ≈ 90 h ). Fig. 1 p resents the per-packet CV v Ω , co mputed from (9), (FPE1) and (FPE2), an d co mpared with th e simulation r esults. The ﬁgure shows a g ood m atch b etween them . In the ﬁgu re, the in tersecting po ints of co ntours o f K and N at each lev el decide v Ω and γ simu ltaneously . As is predicted b y (9), v Ω goes to ∞ as K goes to ∞ for γ ≥ 1 /m 2 = 0 . 25 . It is remarkab le that f or a given N ≥ 9 ( N ≥ 5 f or 80 2.11a/g) , v Ω is extremely sensiti ve to K , formin g a striking co ntrast with the insensitivity of γ to K . The discrepan cy between a nalysis an d simulatio n study is partly due to r educed co ntention effect , which is a less -kn own subtlety of DCF behavior discovered by Bian chi e t al. [7] and is shown throu gh simulations to be a factor of e rror by Saku rai and V u [34]. I V . P O I N T P RO C E S S A P P RO A C H : P O I S S O N I A N I N S I G H T S A basic proper ty o f pe r-packet bac koff Ω discovered by Kwak et al. [28, The orem 1] and later stren gthened by Kumar et al. [27, Th eorem 7 .2] is that th e mean o f per-packet backoff is p ropor tional to th e po pulation, i.e. , ¯ Ω = Θ( N ) . Th is turns out to play a key ro le in our poin t pr ocess a pproach in this section. A. Justiﬁcation of P o int Pr ocess Ap pr oa ch In o rder to justify o ur poin t pro cess appro ach, we need to show that the b ackoff p rocess of each nod e h as nonzer o intensity , i.e. , ¯ Ω = E [Ω] is ﬁnite. Thou gh, for ﬁnite K , this is self-evident fro m the for m of (6), we need to assume the following to prove ¯ Ω < ∞ for K = ∞ . A.3 q k = 2 / (2 b 0 m k − 1) fo r all k ∈ { 0 , · · · , K } , a nd m > 1 . Under this assumption we can prove the following lemma which assur es us that ¯ Ω is ﬁn ite wh ether K is ﬁnite or n ot. W e also would like to point out that a p art of the pr oof o f [27, Theorem 7.2], which correspond s to the case K = ∞ o f Lemma 1 in our work, has a ﬂaw because th ey should h av e proven γ < 1 / m before using P ∞ k =0 ( mγ ) k = 1 / (1 − mγ ) . Lemma 1 (Mean Exists) Under the above assumption , ther e exists a ﬁnite K 0 such tha t γ < 1 /m and γ is decr easing in K . Th is implies: • th ere exist K 0 such th at γ < 1 /m for all K ≥ K 0 including K = ∞ , • th e mean ¯ Ω = E [Ω] exists for K = ∞ . Pr oo f: Su ppose γ ≥ 1 /m . Then we have from (FPE1) and q k = 2 / (2 b 0 m k − 1) that, for any ǫ > 0 , there exists K 1 such that ¯ p < ǫ fo r all K ≥ K 1 . In the meantim e, from 1 − e − x ≤ x , we also h av e γ ≤ ( N − 1) ¯ p < ( N − 1) ǫ . This contradicts γ ≥ 1 /m , implying th at there must e xist K 0 such that γ < 1 /m for K = K 0 . Denote the right-h and side of (FPE1) b y P ( K ) . Since the right-ha nd side of (FPE2) is in creasing in ¯ p and P ( K ) is nonincr easing in γ fr om [27, Lem ma 5 .1], 1 − e − ( N − 1) P ( K ) is nonincr easing in γ . Th erefore, it sufﬁces to sho w that 1 − e − ( N − 1) P ( K 0 +1) ≤ 1 − e − ( N − 1) P ( K 0 ) , or equiv alently P ( K 0 + 1) ≤ P ( K 0 ) , fo r all γ < 1 /m . After some manipula tion a nd some intricate factoriza tion, it can be veriﬁed that P ( K 0 ) − P ( K 0 + 1) takes the form: b 0 γ K 0 +1 P K 0 k =0 γ k  m K 0 +1 − m k  n P K 0 k =0  b 0 m k − 1 2  γ k o n P K 0 +1 k =0  b 0 m k − 1 2  γ k o which is greater than zero fo r m > 1 , implying tha t the solution γ ∗ of (FPE1) and (FPE2) f or K = K 0 + 1 is smaller than that f or K = K 0 . App lying m athematical inductio n completes the pr oof. Also no te tha t this imp lies γ < 1 /m for any K ≥ K 0 . For the ca se K = ∞ , since we h av e shown that γ < 1 /m is d ecreasing in K for all K ≥ K 0 , it follows from [3 3, Theorem 3 .14] th at as K goes to inﬁnity , γ should conv erge to ˆ γ < 1 /m . The existence of E [Ω] fo llows fro m (6). Since m > 1 guarantees th at there exists K 0 such th at γ < 1 / m for all K ≥ K 0 , it can b e seen fro m (1) that, for the case of K = ∞ , m > 1 is also a sufﬁcient condition for the existence of κ such th at φ k > φ k +1 for all k ≥ κ , i.e. , the a verage number of nodes in backoff stage k is larger than that in backoff stage k + 1 . This cor respond s to the tightness condition of φ k , which p rev ents a node f rom escaping to inﬁnite backoff stage [8]. The fact th at th e cond ition m > 1 prevents a nod e from escaping to inﬁnite backoff stage appears to be in be st agreement with our usual intuition . B. Essentia l Assumption T o establish Poisson limit result in Theorem 1 and to justify point p rocess appro ach in th e re maining sections, we need the following essential assumption. A.4 Per-stage backof f distribution f k ( · ) is a uniform con tinu- ous fun ction. It also means v k = 1 / √ 3 . Recall th at f Ω ( · ) is expressed by (4), h ence now it is a weighted sum of conv olutions of con tinuous pdfs f k ( · ) w here the weight fo r each conv olution function f ∗ k ( · ) = ( f 0 ∗ · · · ∗ 6 f k )( · ) is (1 − γ ) γ k , which is a function of γ . As we noted in Remark 1, f Ω ( · ) reﬂects the collision effect through γ which determines how much f Ω ( · ) is dispersed . On continuity assumption : Den ote by D n ( t ) the num ber of cumulative per-node successful tr ansmissions un til tim e-slot t . Formally , D n ( t ) is discr ete-time r enew al process that counts the num ber of a rriv als d uring the in terval [0 , t ] wher e the inter-arriv al times are i.i.d. copies of discrete ran dom variable Ω . Consider superposition pro cess D ( t ) := P N n =1 D n ( t ) . A subtlety in 802.1 1 is that there may be n o interven ing backoff time-slot between two consecutive successful transmissions. More p recisely , at the beginnin g of a bac koff time-slot, if th e transmission attempts of nodes lead to a successful transmission, the time-slot is r endered unu sed, m eaning that the time-slo t is reu sed after the successful tran smission. The same subtlety app lies to collision events. Simply suppose the probab ility that a successful tr ansmission (or a co llision event) occurs at the beginning of a time- slot converges to P S (or P C ) as N → ∞ . Putting P ( x ) := P [lim N →∞ D ( t + 1) − D ( t ) = x ] , x ∈ { 0 , 1 , · · · } , we can see fro m the subtlety that P ( x + 1) = P ( x ) · P ∞ i =0 P i C P S = P S 1 − P C P ( x ) . Because P ∞ x =0 P ( x ) = 1 , we hav e a geom etric distrib ution P ( x ) =  1 − P S 1 − P C   P S 1 − P C  x , x ∈ { 0 , 1 , · · · } hence th e limitin g ( as N → ∞ ) d istribution of cumulative process D ( t ) f or arb itrary integer t takes a Pascal (negative binomial) distribution 2 . This fact can be e xploited for a more accurate appro ximation. A simp ler appro ximation at the cost of accuracy is to be pr esented in Theorem 1. Once again, the continuity as sumptio n turns out unavoidable in Section V beca use r egular variation theor y [10] exploited by Th eorem 3 is not well de veloped for discrete functions. The un iform distribution assump tion of f k ( · ) was made only to simplify the exposition of Theorems 2 and 3 in Section V. C. P oisson Pr ocess Appr oximation W e can now view th e bac koff p rocedur e of n ode n as a stationary simp le r enewal pr o cess A n ( t ) that c ounts the number of arriv als during the interval (0 , t ] where the j th inter- arriv al times, T n j − T n j − 1 , are given by the i.i.d. copies of the continuo us rand om variable Ω . Th en the backoff proced ure of all nodes can be regarded as a s uperpo sition of N statistically identical renewal processes, i.e . , A ( t ) := P N n =1 A n ( t ) . It should be remarked that, if one or more compo nent pr o- cesses are not Poisson, th e super position process A ( t ) is not rene wal , an d even if the inter-arriv al times of A ( t ) are identically distributed, they are not independen t [3]. In th e following, we pr esent a novel way to tackle this analytical difﬁculty caused by the dependence among the inter - arriv al times o f the sup erposition pr ocess. The key observation 2 Sakurai and V u [34, Section III-B] assumed D ( t ) is a Bernoulli process. This simpliﬁcati on w as justiﬁed by the reduced contention effect [7]. is that the en tr op y of the sup erposition point process A ( t ) increases with N , wh ich is implied by the following known result [18, Propo sition 11.2.VI]. Lemma 2 (Poisson Limit for Superposition) Let Ξ( t ) d enote the po int process ob tained b y super posing M independent replicates B m ( t ) , m ∈ { 1 , · · · , m } , of a simple stationary po int pr ocess with intensity λ and dilatin g the time- scale by a factor M . Formally speaking, Ξ( t ) = P M m =1 B m ( t/ M ) . (10) Then as M → ∞ , Ξ( t ) converges weak ly to a Poisson pro cess with the intensity λ . Now it follows from the b asic proper ty [27, Theorem 7.2] for K = ∞ that the mean inter-arrival time of A n ( t ) , ¯ Ω , is of order N . Therefo re, there must e xist a po int process B n ( t ) := lim N →∞ A n ( N t ) with intensity λ = lim N →∞ N / ¯ Ω where intensity λ does not scale with N and we have B n ( t/ N ) ≈ A n ( t ) as N goes to ∞ . This in tur n implies P N n =1 A n ( t ) ≈ P N n =1 B n ( t/ N ) which has the same f orm of (10). Applying Lem ma 2 to the above equation leads to the following theo rem. Theorem 1 (Dichoto my of Aggregatio n: First Part) Suppose ¯ Ω = Θ( N ) . Then the superpo sition pro cess P N n =1 A n ( t ) conv erges weakly to a Poisson p rocess as N → ∞ . Remark 2 This result states that the Poissonian natu re is inherent in the backoff p rocess of 802.1 1 and provides an answer to Q3 . R2.1 The reason we do not require K = ∞ : Recalling our discussion at the b eginning of this sectio n, we can see that K = ∞ [27, Theorem 7.2] = ⇒ ¯ Ω = Θ ( N ) Theorem 1 = ⇒ Poisson . If we require K = ∞ instead of ¯ Ω = Θ( N ) , th e a bove theorem would look simple r , but it would n ot be applicable for the ca se K < ∞ . Even if K is ﬁnite , the crucial scaling condition ¯ Ω = Θ( N ) ho lds for a wide rang e of N , as h inted by pr evious works ( See the simulation resu lt with a practical parameter set in [34 , Figures 2 and 5 ]). Ho wever , for extremely large N , the scaling b ecomes ¯ Ω = Θ (1) . R2.2 From a different ang le, the ba ckoff p rocedu re of 802.1 1 alo ng with its setting K = 6 is inten tionally designed so that the successful a ttempt intensity of each n ode 1 / ¯ Ω is kept b eing of th e o rder o f 1 / N fo r a wide rang e of N , b y allowing en ough numb er o f backoffs for each packet. What is t he premise of Poisson limit? : The q uestion rem ains whether the approximatio n is precise even f or t = ∞ . As Whitt discussed in [41, Chapter 9 .8], the under lying assump tion of the Po isson limit theo rem (Le mma 2 ) is that t is ﬁnite. In th e meantime, the basic premise of the Poisson limit theorem is that th e comp onent proc ess A n ( t ) shou ld beco me sparse ( ¯ Ω = Θ( N ) ) [40, pp.8 3]. If we allow t → ∞ at th e same time as N → ∞ , A n ( t ) may not remain sparse. This is essentially why we mu st ado pt an another app roximation in Sectio n V I wh ere 7 t = Θ( N ) . In the light of these points, the above theo rem provides a natural appr oximation of the backoff pr ocesses on normal time-scale, as co mpared with the other approx imation in Section VI on coar se time-scales. V . A S Y M P T OT I C A N A LY S I S A stochastic p rocess with inﬁnite variance and self- similarity exhibits pheno mena called No ah effect and Joseph effect , respectiv ely , in Mandelbr ot’ s terminolog y [36], [41]. Noah and Joseph effects refer to the biblical ﬁgures No ah, who experienced an extreme ﬂood – exception ally large v alues – and, Joseph, wh o exp erienced long periods of plenty and famine – self-similarity or strong positive d ependen ce. This section lifts th e veil to discover these effects an d to explain their inﬂu ences on the backoff process in 802.11. W e have not assumed K = ∞ because all results derived so far are applicable if either of ﬁnite and inﬁnite K is used (See Remark 2 also). However , all results deriv ed in this section require K = ∞ , h ence we form ally assum e the following. A.5 T here are inﬁnite backoff st ages, i.e. , K = ∞ . A. Mome nt Analysis W e in troduce the notio n of a wide-sense heavy-tailed dis- tribution borrowed from [32]. W e call a p df f ( x ) wide-sense heavy-ta iled if its moment g enerating function is inﬁnite, i.e. , R ∞ 0 e tx f ( x )d x = ∞ , ∀ t > 0 . W e now characterize the existence of all fraction al moments of Ω . Let us d eﬁne α := − (log γ ) / lo g m where α > 1 is satisﬁed by Lemma 1. Also it is remark able that Sakura i and V u [34] established a similar r esult for integer moments. Note howe ver that we ca nnot prove T heorem 3 without the fo llowing extended result fo r f ractional moments. Theorem 2 (Existence of Fract ional Moments) The per-packet b ackoff Ω has a wide-sen se h eavy-tailed d is- tribution. In addition, its c th moment E [Ω c ] is • inﬁnite if c ≥ α , • and ﬁnite if 0 ≤ c < α . Pr oo f: First we n ote α = − (lo g γ ) / log m is eq uiv alent to m α γ = 1 . It also follows from Lem ma 1 that α > 1 . Letting c be any real n umber such that c ≥ α , we have m c γ ≥ 1 . Then the c th mo ment of Ω , E [Ω c ] , can be co mputed as P ∞ k =0 E h P k k ′ =0 B k ′  c i · P [ κ = k ] ≥ P ∞ k =0  E h P k k ′ =0 B k ′ i c · P [ κ = k ] = P ∞ k =0  P k k ′ =0 ( b 0 m k ′ − 1 2 )  c · P [ κ = k ] ≥ P ∞ k =0 P k k ′ =0 ( b 0 m k ′ − 1 2 ) c · P [ κ = k ] = P ∞ k =0 ( b 0 m k − 1 2 ) c γ k where the ﬁrst inequ ality holds by H ¨ older’ s inequality for expectations, i.e. , ( E [ X ]) c ≤ E [ X c ] , an d the seco nd ineq uality follows fr om c > 1 . Hence, fro m the last expression, we ha ve E [Ω c ] → ∞ as K → ∞ . Note that c is real. Since there e xist inﬁnite moments, Ω h as a wide-sense heavy-tailed distribution. Now consider the c th moment for 1 < c < α . E  ( P κ k =0 B k ) c  = P ∞ k =0 E h P k k ′ =0 B k ′  c i · P [ κ = k ] ≤ P ∞ k =0 E h ( k + 1) c − 1 P k k ′ =0 ( B k ′ ) c i · P [ κ = k ] (11) = P ∞ k =0 ( k + 1 ) c − 1 P k k ′ =0 (2 b 0 m k ′ − 1) c ( c +1) · P [ κ = k ] (12) ≤ (2 b 0 ) c ( c +1) P ∞ k =0 ( k + 1) c − 1 ( m c ) k +1 − 1 m c − 1 · P [ κ = k ] ≤ (2 b 0 ) c ( c +1) P ∞ k =0 ( k + 1) c − 1 ( m c ) k +1 γ k m c − 1 (13) = (2 b 0 m ) c ( c +1)( m c − 1) P ∞ k =0 ( k + 1) c − 1 ( m c γ ) k (14) where (1 1) can be o btained by app lying origin al H ¨ older’ s inequality , i.e. ,  P k k ′ =0 1 · b k ′  ≤  P k k ′ =0 1 c c − 1  c − 1 c  P k k ′ =0 ( b k ′ ) c  1 c . (12) can be veriﬁed by co mputing R b c f k ′ ( b )d b wh ere f k ′ ( b ) is a un iform p df with me an b 0 m k ′ − 1 / 2 . (1 3) follows fr om P [ κ = k ] ≤ γ k . Then it sufﬁces to show that d’Alembert’ s ratio of the series (14) is less than one. Recalling that m c γ < m α γ = 1 , we can see th at lim k →∞ ( k + 2 ) c − 1 ( m c γ ) k +1 ( k + 1 ) c − 1 ( m c γ ) k = m c γ < 1 . This establishes (14) is ﬁnite f or K = ∞ , and co mpletes the proof . Remark 3 [Answer to Q4 ] This theor em reveals that Ω is wide-sense heavy-tailed in the sense that not all of its momen ts exist, as Sakurai and V u [34, Theor em 1] ﬁrst noted. As shown in Fig. 1, the variance σ 2 Ω in 802.11b is not very large. Nev ertheless, the statistics of Ω certain ly con tain precurso rs o f in ﬁnite-variance d istributions, as shown in the next section. B. S trict-Sense Heavy-T ailedn ess: T auberian Insights Although there h as been som e work to pr ove the wide-sense heavy-tailedness of the delay or b ackoff duratio n [3 4] and th e power -law like behavior of access de lays was iden tiﬁed on ly throug h simulations in a few works [3 4], [3 7], to the best of our k nowledge, n one o f them proved th at th e delay or backoff duration has a p ower -law tail. Th is q uite intu iti ve p roperty has not been established ma inly due to the theo retical difﬁculties underlin ing the p roof. It is im portant to no te that this th eorem is a prerequ isite for mathematical a nalysis of Noah effect, which implies strict-sense heavy-tailedn ess. W e would like to p lace par ticular emphasis on the f ollowing theorem for an other reason. W e n ote that some work [2 1], [39] considered the question whether a sin gle lo ng-lived TCP ﬂo w can gen erate trafﬁc that exhibits long- range depend ence (or, equiv alently , asympto tical second- order self-similarity). It is signiﬁcant that long-ran ge dependence is a p roperty which is automatica lly implied by heavy-tailed inter-arri val times [30] for the single ﬂo w (or node) case, ir respective o f the context. 8 That is, e ven a r enew al pro cess (no correlation of inter -arriv al times) with heavy-tail distributed inter-arri val times generates long-r ange dep endence in th e co unting process. In the light of this point, one do n ot n eed to con duct analyses o f tremendo us trafﬁc traces if there is a solid m athematical work that can settle this kind of dispu te. In the following theor em, we pr ove that the p er-packet backoff distribution has a power tail by ligh ting up on the fact that the mome nt generating function has a recur sive relation , and by applying the the ory of r e gula r variation [10] and th e less-known mod iﬁed T au berian theo r em o f Bingh am & Doney [9]. For your own good, n ote that this theorem r equires o nly K = ∞ , n othing about N . Theorem 3 (Power T ail P rinciple 3 ) The p er-packet backoff Ω has a Pareto-typ e tail with an exponent of − α . Formally , F c Ω ( x ) := Z ∞ x f Ω ( x ) dx ∼ x − α ℓ ( x ) . (15) The no tation f ( x ) ∼ g ( x ) me ans lim x →∞ f ( x ) / g ( x ) = 1 , and ℓ ( x ) is slo wly varying 4 . Remark 4 Th is p rinciple, f ormulated in terms of the ccd f F c Ω ( · ) , not o nly deﬁn es a fun damental chara cteristic of delay but also lays the groun dwork for fu rther an alysis u sing r egular variation theory . R4.1 [An swer to Q4 ] This clear-cut and simp le result reveals the statistical attribute of Ω f or any popu lation N . It has a Pareto-type distrib ution whose exponent parameter is − α . Theorem 3 proves the strict -sense heavy-tailedness o f Ω for α < 2 , and puts an end to the discussion s in Section I. R4.2 Th is theo rem dispe nses the complica ted conv olution expression (4 ) and lead s us to a simpler conclusion. T he most representative distribution of backoff times Ω is a tru ncated Pareto-type d istribution (tho ugh it m ust be slo wly-varying), rather than uniform or exponential as o bserved in the simula- tion studies of [5], [ 12]. R4.3 Th e simplistic term ℓ ( · ) in (15) is ir r eplaceab le with any other expressions, implying its piv otal role. For instance, Final V alue Theore m tells nothing b ut lim x →∞ f Ω ( x ) = 0 . The ccdf of Ω ob tained through n s-2 simulations is plotted in Fig. 2 on a log-log scale where the estimated slopes ˆ α are compare d with the an alytical f ormulae α = − (lo g γ ) / log m , (FPE1) an d (FPE2). Observe that these simple for mulae along with (15) pr ovide a p recise estimate for the tail distribution. Remarkably , even fo r K = 6 , i.e. , the value adopted in 802.1 1b, the ccdf of Ω c an b e accurately ap proxim ated by a truncated power-la w tail. 3 The proof i n f act requires α to be not an integer . For the complicated case when α is an i ntege r , we refer to [19] and [10, Theore m 8.1.6]. Howe ver , since an i ntege r α can be approximate d for an y small ǫ > 0 by a real number ˜ α such that | α − ˜ α | < ǫ , we expect the result of T heorem 3 to be va lid for all α > 0 . 4 A function f ( x ) is called re gularly varying [10] at inﬁnity of index ρ if f lim x →∞ f ( λx ) /f ( x ) = λ ρ , ∀ λ > 0 . F or the special case ρ = 0 , it is calle d slowly v arying an d usual ly denot ed by ℓ ( x ) . F or example, a positiv e constant , (log x ) ǫ for any real number ǫ is a slowly varyi ng function. A slo wly va rying function ℓ ( x ) is dominated by an y posit iv e powe r function, i.e. , lim x →∞ ℓ ( x ) /x ǫ = 0 , ∀ ǫ > 0 . 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 per−packet backoff ( x ) ccdf ( F Ω c (x) ) ւ ˆ α = 1 . 0 1 α = 1 . 07 ւ ˆ α = 1 . 1 5 α = 1 . 18 ւ ˆ α = 1 . 7 8 α = 1 . 83 ˆ α = 1 . 7 6 α = 1 . 81 ր simulation: K =6, N =10 simulation: K =15, N =10 simulation: K =6, N =40 simulation: K =15, N =40 Fig. 2. Complement ary cumulat iv e distributi on functio n F c Ω ( x ) for K = 6 , 15 ; and N = 10 , 40 . V I . S H O RT - T E R M F A I R N E S S A N A LY S I S First of all, we cancel the assumption K = ∞ we m ade in Section V because we presen t in th is section a new approx imation f or the superposition p rocess and sho rt-term fairness analysis, both of which will be app licable to both cases K < ∞ an d K = ∞ . A. In ter -T ransmission Pr o bability The no tion of short-term fairness [ 5], [1 2], [26], deﬁned as the distribution of successful transmissions of nod es for a ﬁnite time, has be en g etting the limelight du e to its central role in quantif ying th e behavior o f rand om access protocols over sho rt time- scales and its close link to ac cess delay s. Among the set o f nod es { 1 , · · · , N } , we tag nod e N , without loss of gen erality . Assume that the tagg ed no de successfully transmitted a pac ket at time t = 0 . Denote by Z n the number of packets successfu lly transmitted by n ode n while the tag ged node transm its ζ packets. Recalling that A n ( t ) counts the arriv als during the interval (0 , t ] , we can see Z n := A n ( t ′ ) wher e t ′ = min { t : A N ( t ) = ζ } . It is clear that Z N = ζ from th e above deﬁnition. For sho rt- term fairness analysis, we con sider Z = P N − 1 n =1 Z n . For the sake of con venience, we denote P [ Z = z | Z N = ζ ] by P N [ z | ζ ] . W e call the cond itional pr obability P N [ z | ζ ] inter - transmission pr obab ility . In term s o f the point pro cesses A n ( t ) , it is equ iv a lent to P N [ z | ζ ] = P h P N − 1 n =1 A n  P ζ j =1 Ω j  = z i where Ω j denotes th e per-packet backoff for each j th packet of the tagg ed node N and are i. i.d. copies of Ω . 9 B. In termediate T elecom Pr ocess on Coarse T ime S cales The pr emise does not hold : Look in to the above superposition process P N − 1 n =1 A n ( t ) where t = P ζ j =1 Ω j . Recall the basic premise of Po isson limit theo rem (Lemma 2) is that each compon ent process must become spa rse as N grows. It is easy to see that this premise d oes not hold any longer here because t = P ζ j =1 Ω j is of o rder of ζ · ¯ Ω in the sense that E [ t ] = Θ( ζ ¯ Ω) and ¯ Ω is of ord er of N in mo st c ases (See Remar k 2). Therefo re, we need a n ew approxim ation of the superposition process on coarse time-scales such that t = Θ ( N ) . Before that, we ep itomize theo ry o f stable law [41, Chapter 4] brieﬂy only for the case α ∈ (1 , 2] . Denote by S α ( σ , β , µ ) L ´ evy α -stable laws whose four parameters are: the index α ; th e scale parameter σ ; the skewness p arameter β ; and the mean µ . If X 1 , · · · , X n are i.i.d. copies o f S α ( σ , β , µ ) , they satisfy the stability prop erty which takes the following form P m i =1 ( X i − µ ) d = m 1 α ( X 1 − µ ) where the notation d = m eans equ ality in distrib ution. Th e case α = 2 is singular because we have S 2 ( σ , β , µ ) = N ( µ, 2 σ 2 ) where β p lays no role. Howe ver , for the rest of cases α ∈ (1 , 2) , there is no closed form expression for its pdf . Since Lelan d et al. [29] c reated a wa ve o f intere st in th e self-similarity in the In ternet, the probab ilistic comm unity has be en concer ned with the limit processes of agg regate renewal p rocesses under different limit regimes. Here a point at issue was the orde r of limit operation s, i.e. , t → ∞ an d N → ∞ . Recently , Kaj et a l. [ 22], [24], [2 5] have estab lished a fun damental connectio n b etween Noah effect and Joseph effect, elucidating the above issue as well. Aggregate Pr ocess on Coarse Time Scales : A premise of [24, Theorem 1] is that each compo nent p rocess should not become sparse as N grows, i.e. , inter-arri val times no t scaling with N . T his premise is fu lly satisﬁed when we consider A n ( ¯ Ω t ) instead of A n ( t ) . In other words, we n ow view A n ( τ ) on coarse time- scales τ = ¯ Ω t . Also note that E [ A n ( ¯ Ω t )] = t . Then apply ing [24, Theorem 1] yields to the following result which is ap plicable to various cases K = ∞ , K < ∞ , ﬁnite time (which must be large enou gh though), and inﬁnite time. Theorem 4 (Dichoto my of Agg regation: Second Part 5 ) Suppose, f or K = ∞ , the inter-arri val time s of A n ( ¯ Ω t ) has ccd f F c Ω ( ¯ Ω x ) in (1 5) which do es not v ary with N . For K < ∞ , noth ing is as sumed. Deﬁne the centred super position process e A ( t ) := n P N n =1 A n ( ζ ¯ Ω t ) o − N ζ t. Then, as ζ → ∞ and N → ∞ , we have e A ( t ) ζ weakly − → − c · Y α  t c  , for K = ∞ , α ∈ (1 , 2) , (16) e A ( t ) √ N ζ weakly − → v Ω · B ( t ) ,  for K < ∞ , for K = ∞ , α ∈ (2 , ∞ ) , (17) where th e scaling co nstant c := { N ¯ Ω − α ℓ ( ζ ¯ Ω) } 1 / ( α − 1) /ζ , B ( · ) is a standar d Brownian motio n, and Y α ( · ) belong s to the family o f Intermed iate T elecom p r ocess [25] of index α whose cgf takes the form log E  e θ Y α ( τ )  = τ 1 − α α − 1  e θ τ − 1 − θ τ  + R τ 0  e θ x − 1 − θ x   ατ x − α − 1 + (2 − α ) x − α  d x. (18) Pr oo f: First, for K = ∞ , the cc df of inter-arriv al times of A n ( ¯ Ω t ) now satisﬁes F c Ω ( ¯ Ω x ) ∼ x − α ¯ Ω − α ℓ  ¯ Ω x  due to its scaling. From E [ A n ( ¯ Ω t )] = t , the mean inter-arri val time is one . It follows from th e underlined assumption that ¯ Ω − α ℓ  ¯ Ω x  does not scale with N and it is a slowly-varying function o f x . Ap plying [2 4, Th eorem 1] yie lds that e A ( t ) /ζ weakly conver ges to the process in (16). For the rest of cases, (i) K < ∞ and (ii) K = ∞ and α ∈ (2 , ∞ ) , we do not need any assumptio n because E [Ω 2 ] < ∞ holds bo th f or (i) and (ii) by ap pealing to Theore m 2. These ﬁnite variance cases were analy zed in [36, Section 2 .1.3(a) ] whose ‘ON/OFF sour ce m odel’ redu ces to our model if we use µ 1 = 1 and µ 2 ≫ 1 . Remark that it is discussed in [36, Section 2.3] that the ord er of limit oper ations do es no t matter in these cases. The phrase ‘as ζ → ∞ and N → ∞ ’ is pregnant with meaning. The fund amental stren gth of the above theorem for th e case of (16) is in that its result is not su bject to the orde r o f limit op erations. Instead, the scaling structure between ζ and N , represented b y c , determine s the kind of the app roximation in the sense that, as c → 0 and c → ∞ , c 1 /α Y α ( t c ) and c H Y α ( t c ) respectiv ely con verges to Λ α ( t ) ( α - stable L ´ evy mo tion) and B H ( t ) ( fractional Brownian motio n of index H = (3 − α ) / 2 ), up to constants [22]. For ﬁn ite c ∈ (0 , ∞ ) , Y α ( t c ) become s an in-betwee n pro cess. For the case of (17), even this scaling structure does not matter . It is sign iﬁcant that c → 0 and c → ∞ r espectively equiv alent to lim N →∞ lim ζ →∞ and lim ζ →∞ lim N →∞ in the literature. Therefo re, th e essenc e o f th e a dvance [24, Theo rem 1] is that it has emancipated the limit form of the su per- position process fr om the or der of the two limit operations , widening the applicability of th e theory . Remark 5 Tho ugh, fo r K = ∞ , the un derlined phr ase makes a strong assumption which is not reason able in vie w of α = − (log γ ) / lo g m which hea vily depends on N , the above theorem deserves it s result in the sense that it suggests a possible approximatio n of the b ackoff process in 8 02.11 , based on th e state-o f-the-art theory . W e will come b ack to the preciseness of the approximation later in Remark 7 where we observe that α is require d to be not too close to 1 . R5.1 As we have discu ssed in Footnote 5 an d [ 41, Chapter 9] as well as at th e b eginning o f this section, Poisson approx- imation in Theor em 1 is poor on coa rse time -scales, i.e. , large time. Th erefore, fo r short- term fairness an alysis, the f ollowing 5 Consistenc y between (17) and Theorem 1: Suppose K = ∞ and α ∈ (2 , ∞ ) (which is very unlik ely as N must be large ). Then assume the superposition process A ( ζ ¯ Ω t ) is Poisson . For large ζ ¯ Ω , this Poisson process should hav e a Gaussian marginal distribut ion with mean N ζ t and v ariance N ζ t , where as the process (17) has mean N ζ t and v ariance v 2 Ω N ζ t . Therefore, Theorem 1 is inconsisten t with (17) for v Ω 6 = 1 . The inconsistenc y is due to the pre mise of Theorem 1, i.e . , ﬁnite time. A similar remark is giv en in [41, Remark 9.8.1]. 10 approx imations inspired by (16) and (17) are essential: e A ( t ) ≈ − ζ · c · Y α  t c  , for K = ∞ , α ∈ (1 , 2) , ( 19) e A ( t ) ≈ p N ζ · v Ω · B ( t ) , otherwise . ( 20) R5.2 [An swer to Q5 ] I t turns out that f or K = ∞ and α ∈ (1 , 2) , th e superp osition proce ss A ( ζ ¯ Ω t ) = P N n =1 A n ( ζ ¯ Ω t ) exhibits long-ra nge dependence due to the heavy power tail of inter-arriv al times Ω . This p rocess is non- Gaussian and non- stable and h as stationary , but str o ngly d ependen t , increm ents in the sense that it h as the same covariance as a mu ltiple of fractional Brownian motion o f index H = (3 − α ) / 2 [22]. It is also shown in [2 2] that this p rocess is (both lo cally and globally) asymptotically self-similar though not self-similar . W e b eliev e that network ing commu nity has been longing for a m athematical evidence which makes extensive simulations in [37] less n ecessary . T u rning b ack to the discussion of inter -transmission proba- bility P N [ z | ζ ] in Section VI -A, we demon strate the strength of the above appr oximations in the fo llowing corollaries where ζ is now taken to be num ber of p ackets transmitted b y the tagged node. Corollary 1 (Asymp. Inter-T ransmission Probability) Suppose ζ ≫ 1 and N ≫ 1 . If K = ∞ alo ng with α ∈ (1 , 2) , we have P N [ Z = z | ζ ] ≈ Z ∞ −∞ Z q + ( τ ( y )) q − ( τ ( y )) Tc τ ( y ) /c ( x )d x · Lv ( y )d y (21) where q ± ( τ ( y )) := − { z ∓ δ − ( N − 1 ) ζ · τ ( y ) } / ( ζ c ) , δ = 1 / 2 , τ ( y ) := 1 + ζ (1 − α ) /α ℓ 0 ( ζ ) · y . Here ℓ 0 ( · ) is slo wly varying at inﬁn ity , Tc τ ( · ) is the pdf of Y α ( τ ) whose cgf is given by (18), and Lv ( · ) is the pdf of S α (1 , 1 , 0) wh ose index is α = − (log γ ) / lo g m . Pr oo f: Under th e assump tion ζ ≫ 1 an d N ≫ 1 , it follows from Th eorem 4 that P N − 1 n =1 A n ( ζ ¯ Ω t ) can be approx- imated by an Intermediate T eleco m process s o that its marginal distribution takes the form P h P N − 1 n =1 A n  ζ ¯ Ω t  = z i ≈ P [( N − 1) ζ t − ζ c Y α ( t/c ) ∈ ( z − δ, z + δ )] = P [ Y α ( t/c ) ∈ ( q − ( t ) , q + ( t ))] = P h R q + ( t ) q − ( t ) Tc t/c ( x )d x i . (22) In the meantime, it follows fro m the deﬁnition of ske wn ess β and f Ω ( − x ) = 0 , ∀ x > 0 that β := lim x →∞ 2 F c Ω ( x ) F c Ω ( x )+ R − x −∞ f Ω ( x ) dx − 1 = 1 . Put t = P ζ j =1 Ω j / ( ζ ¯ Ω) . Ap plying the lesser-known stable- law central limit theorem [41, Th eorem 4.5.1] to the power tailedness result of Theorem 3, taken together with the fact β = 1 , it follows that, for ζ ≫ 1 , t d ≈ 1 + ζ (1 − α ) /α · ℓ 0 ( ζ ) · S α (1 , 1 , 0) . Plugging this line in to (22) yields (21). Corollary 2 (Inter-T ransmission Probability) Suppose ζ ≫ 1 and N ≫ 1 . If K < ∞ , or K = ∞ along with α ∈ (2 , ∞ ) , we have P N [ z | ζ ] ≈ Nm  z − ( N − 1) ζ ( N − 1) √ ζ v Ω  (23) where the CV v Ω is given b y (9), and Nm ( x ) := 1 √ 2 π e − x 2 2 . Pr oo f: Like wise, we have P h P N − 1 n =1 A n  ζ ¯ Ω t  = z i ≈ P  ( N − 1) ζ t + N  0 , v 2 Ω ( N − 1) ζ t  ∈ ( z − δ, z + δ )  . ≈ P " Nm z − ( N − 1) ζ t v Ω p ( N − 1) ζ t !# (24) where N  µ, σ 2  is th e Gau ssian ran dom variable with mean µ and variance σ 2 . Puttin g t = P ζ j =1 Ω j / ( ζ ¯ Ω) , t is approxi- mated by t d ≈ 1 ζ · N  ζ , v 2 Ω ζ  d = 1 + v Ω √ ζ · N (0 , 1) for ζ ≫ 1 . Thus (24) becomes Z ∞ −∞ Nm   z − ( N − 1)  ζ + √ ζ v Ω x  v Ω q ( N − 1)  ζ + √ ζ v Ω x    Nm ( x )d x which is ap proxim ated as (23) because the d enominato r v Ω ( N − 1) 1 / 2 ( ζ + √ ζ v Ω x ) 1 / 2 is very large so that th e ﬁrst pdf of the integrand is con centrated around z = ( N − 1)  ζ + √ ζ v Ω x  . Remark 6 The derived equations provid e us several pen etrat- ing insig hts an d answers to Q2 as well. Note that the mea n and variance of (21) are giv en by ¯ Z := P ∞ z =0 z · P N [ z | ζ ] ≈ ( N − 1) ζ σ 2 Z :=  P ∞ z =0 z 2 · P N [ z | ζ ]  − ¯ Z 2 ≈ ∞ , (25) while those of (23) are giv en by ¯ Z ≈ ( N − 1) ζ , σ 2 Z = ( N − 1) 2 ζ · v 2 Ω . (26) R6.1 For the case of (23), we can say tha t Z is approxi- mately Gaussian for large ζ an d N : Z d ≈ N  ( N − 1) ζ , ( N − 1) 2 ζ v 2 Ω  (27) whereup on the C V o f Z can b e computed from (26) as v Z := σ Z / ¯ Z ≈ v Ω / p ζ . (28) Remarkably , we have der i ved the most gene ral expression of the inter-transmission prob ability P N [ z | ζ ] wh ile [ 5], [1 2] derived the expression s of P N [ z | ζ ] only for N = 2 . R6.2 (2 1) can not b e sim pliﬁed in gen eral. H owe ver, for for very large ζ , hen ce very small c , it can be easily seen that Z has a L ´ ev y α -stable distrib ution. Apply ing [22, Prop osition 2] to the right-h and side of (16) yields that it is negligible, implying tha t the inner integral of (21) can be re moved. Then Z becomes ap proximate ly L ´ evian and is expr essed in the f orm Z d ≈ S α (( N − 1 ) ζ 1 α ℓ 0 ( ζ ) , 1 , ( N − 1 ) ζ ) . 11 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 1 2 3 4 5 x 10 −4 K= 6 ¯ Z = 3 90 0 ¯ Z = 5 90 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 1 2 3 4 5 x 10 −4 number of transmissions by N −1 nodes ( z ) inter−transmission probability P N [ z | ζ ] K= 15 simulation: N =40 simulation: N =60 analysis: N =40 analysis: N =60 Fig. 3. Inter -transmission probabilit y P N [ z | ζ ] for ζ = 100 ; K = 6 , 15 ; and N = 40 , 60 . This manifests the heavy-tail of Z , i. e. , P [ Z > x ] ≈ ζ { ( N − 1) ℓ 0 ( ζ ) } α C α · x − α (29) where ℓ 0 ( · ) is the same fu nction used in τ ( y ) in (21) and C α = ( α − 1 ) / (Γ(2 − α ) sin( π ( α − 1 ) / 2)) . R6.3 For the case of K = ∞ and α ∈ (1 , 2) , the representa- tion (2 9) reveals the strik ing similarity between the ccdfs o f Ω and Z . In terms of r egular variation theo ry , bo th a re re gula rly varying of index − α , and in Mandelbrot’ s terminology , Noah effect of Ω inﬁltrates into Z . R6.4 For the case of K = ∞ , the inter-transmission probab ility bifu rcates in to two different categories at α = 2 (or γ = 1 /m 2 ). P lainly speaking, if γ < 1 /m 2 , Z can still b e approx imated by the Gaussian distribution in (2 7), oth erwise 802.1 1 suf fers from extreme unfairness containin g p recursors of power-tailed ch aracteristics such as inﬁnite variances and the skewness ( β = 1 ). R6.5 Th e skewness induce s leaning ten dency and direc- tional u nfairness . Th e lea ning ten dency implies the d istribu- tion is heavily leaning to the left, and the tendency increases as α decreases. The directio nal unfairne ss 6 implies that while the righ t part o f the in ter-transmission p robability z ∈ ( ¯ Z , ∞ ) has a h eavy po wer tail given by (2 9), its left part z ∈ ( −∞ , ¯ Z ) decays faster than expon entially [41, pp.113]. W e conjec ture ba sed on extensi ve simulatio ns that ℓ ( · ) in (15) is approx imately a con stant, implyin g th at ℓ 0 ( · ) in τ ( y ) in ( 21) is also a con stant. Then it follows that the co nstant ℓ correspo nds to the y -interce pt of the straight li ne obtained by taking logarithm s of (15), and can be estimated fr om Fig. 2. After manipulation akin to [41, Theorem 4.5.2 ], we can show a simple relation between th em: ℓ 0 = ( ℓ/C α ) 1 /α / ¯ Ω 6 The L ´ evy α - stable la w used in this work has support on t he enti re real line because α ∈ (1 , 2) . which implies that we n eed to estimate only ℓ and α . In Fig. 3, the i nter-transmission probability obtained through ns-2 simulations is compare d with the derived fo rmulae of Corollaries 1 an d 2 fo r ζ = 100 . It is signiﬁcan t that, for K = 6 , P N [ Z = z | ζ ] is well app roximated by Gau ssian fo rmula (23) along with (9), (FPE1) and (FPE2) for large N . This forms a striking co ntrast with the case K = 15 where the distribution ( 21) is lean ing to the left and its peak is far apart from its mean, i.e. , ¯ Z = ( N − 1) ζ , meaning that there are even heavier tails o n the right pa rt. Ou r extensive simu lations attested to the inevitability of complicated form (21). Remark 7 Preciseness of the appr oximat ion (19): remains a question due to the un derlined assumption of Theor em 4. No te that N is determine d by α = − (log γ ) / log m pr ovided that α is ﬁxed, whereas [24, Th eorem 1] d emands that N → ∞ provided that α is ﬁxed. T hroug h extensive simulatio ns, we have foun d ou t that the ap proximatio n (19) b ecomes poor as α → 1 ( or as N → ∞ ). Under the above simulation setting, if N > 80 , th e approximation appear s not reasonable. A tho rough theo ry add ressing this depen dence between N and α is left for fu ture work. V I I . W A V E L E T A N A LYS I S O F L O N G - R A N G E D E P E N D E N C E W e provide simu lation results to support the argument over the lon g-rang e dependenc e in Sectio n VI-B under the assump- tion K = ∞ . Recall from Theo rem 4 that the time- scaled version of the superpo sition ar riv al process is approxim ately A ( ζ ¯ Ω t ) = P N n =1 A n ( ζ ¯ Ω t ) ≈ N ζ t − ζ c · Y α  t c  which ho lds for N such that α = − (log γ ) / log m < 2 . Note that such N is to ensure Ω is strict-sense hea vy-tailed (See Theorem 3). Then by ap pealing to [22], one can show that A ( ζ ¯ Ω t ) h as lo ng-ran ge depende nt incremen ts in the sense that • A ( ζ ¯ Ω t ) has the same covariance as a mu ltiple of frac- tional Brownian motion of index H := (3 − α ) / 2 . It is easy to see that 1 / 2 < H < 1 due to 1 < α < 2 . All simulations obta ined from ns-2 simulator use a 437 h warm-up perio d, after wh ich we collected 728 h -lo ng trace s. T o an alyze these traces, we use the latest addition to the toolkit of inf erence techniq ues for long-rang e depen dence, i.e. , the r eﬁned wav elet-based metho d using Da ubechies wa velets with M vanishing mo ments which was pr oposed by Abry et al. [2] They pro posed th e ﬁrst unbiased estimator y j taking th e fo rm E [ y j ] = lo g 2  E  d 2 j  , considerin g the comp lication presented by the property E [log( · )] 6 = log( E [ · ]) whe re d j is called detail p r ocesses of the wa velet transform . The estimates y j of the wavelet spectra o ver all time- scales j , called octaves , a re shown in Fig. 4 f or K = 6 , K = 15 and K = 25 . Here we ﬁx the other par ameters as N = 40 and M = 2 . Thou gh we pr esent here only th e simulation results using Daubechies wa velets with M = 2 , we o btained similar results using Daubechies wa velets with M > 2 and Discr e te Meyer wa velets. T o quantif y the in tegrity of the method, Gau ssian 95 % con ﬁdence intervals correspon ding to 12 2 4 6 8 10 12 14 16 18 5 6 7 8 9 10 11 Logscale Diagram, [( j 1 , j 2 )= (5, 18), s Est.=0.00, H Est.=0.50] Octave j (time−scale 2 j+7 backoff time−slots) y j (a) K = 6 , N = 40 2 4 6 8 10 12 14 16 18 5 6 7 8 9 10 11 Logscale Diagram Octave j (time−scale 2 j+7 backoff time−slots) y j (b) K = 15 , N = 40 2 4 6 8 10 12 14 16 18 5 6 7 8 9 10 11 Logscale Diagram, [( j 1 , j 2 )= (12, 18), s Est.=0.66, H Est.=0.83] Octave j (time−scale 2 j+7 backoff time−slots) y j (c) K = 25 , N = 40 Fig. 4. W avel et spectra using Daubechies wav elets with M = 2 . the variability of y j are also shown as the vertical segments centered on the estimates y j . Then the measurement of index H , ca lled Hurst param eter , is reduced to the iden tiﬁcation of region of alignment , the de termination of the its lower and upper cu toff octav es, j 1 and j 2 , r espectiv ely , and the determinatio n of the slop e over the alignment region wh ich we d enote by ¯ s . Fr om the slope estimate ¯ s , we can obtain the estimates of H from the for mula ¯ H := (1 + ¯ s ) / 2 . Fig. 4(c) demonstrates that, for the case K = 2 5 , the superpo sition a rriv al process possesses a sustained corr elation structure over a br oad ran ge of time-scales j ∈ [1 , 18] wh ere s j conv erges to 0 . 66 at octave j = 18 , wh ereas, for the case K = 6 , it shows a weaker correlation structu re ov er a nar row range j ∈ [1 , 5 ] as shown in Fig. 4(a). The estimate of H for K = 25 over the alignment region ( j 1 , j 2 ) = (12 , 1 8) approa ches ¯ H = 0 . 83 ar ound (16 , 17) which approximate ly matches with ana lytical formula H = (3 − α ) / 2 = 0 . 90 where α is obtained from α = (log γ ) / log m , Eqs. (FPE1) and (FPE2). Th e slope estimate o ver the alignment re gion for K = 6 is com puted as ¯ H = 0 . 50 , im plying that lon g-rang e depend ence is not observed. A striking o bservation that can be made by co mparing Fig s. 4( a) and 4 (b) with Fig. 4(c) is that th e per-octav e slope s j increases a s octave j inc reases and conv ergent on ly if K is large eno ugh as in Fig. 4(c). Observation 1 (LRD over co arse t imes sca les) Long- range dep endence of the superpo sition pro cess is co n- spicuous only over coarse time-scales. Remark 8 Essen tially , ther e a re two r easons behind this ph e- nomeno n which also gi ve us answers to Q5 . R8.1 Per -node process slows down : It is imp ortant to recall that, for K = ∞ , we ﬁrst established Poisson proc ess approx imation f or the sup erposition pro cess in Theorem 1, meaning that we cannot o bserve long -range d ependen ce o n normal time-scales. As is th e co nstant intensity of th e sup er- position process for Theorem 1, the constant intensity of the component process is essential for T heorem 4. T o satisfy the latter , we had to con sider A n ( ζ ¯ Ω t ) instead of A n ( t ) becau se A n ( t ) beco mes sparser as N → ∞ . Th at being said, we must view the superp osition proce ss over coarse time- scales ζ ¯ Ω t instead of t to satisfy th e premise of Th eorem 4, which explains long-range dependen ce. R8.2 Additiona l scaling of time : Another a ssumption of the limit regime con sidered in [2 4] is ζ → ∞ at th e same time as N → ∞ . Th is im plies we need addition al scaling of tim e to compensate for the scaling o f space. R8.3 In pr actical ter ms, if the wireless link capacity is shar ed b y many nod es, the agg regate transmission process is hig hly invulnerable to long -range dependen ce for mo st practical K values, essentially due to r educed per-node rate and additio nal time scaling . W e also conjectur e that the above coarser time scaling s caused the em pirical analyses of V ere s and Boda [ 39] (in the context o f TCP) and Tickoo and Sik dar [ 37] (in the context of 802.1 1) not to support long-r ange dependen ce o f the superpo sition arriv al p rocess of TCP sources — they ob served that ˆ H ≈ 0 . 5 (or ˆ s = 0 ), imp lying short- range dep endence. This is becau se both 802.11 nod es accessing a comm on base station and T CP ﬂows traversing a common bottleneck link (i) h av e similar backoff mechanisms and (ii) re duce (o r slow down) their transmission r ates to share the g iv en cap acity as the populatio n increases. V I I I . C O N C L U D I N G R E M A R K S Beginning with deriv ation of per-packet backoff d istribution, based on which we s tudied its coef ﬁcient o f v ariation that plays a ke y role in formu lating shor t-term fairness in later s ections, we ha ve conducted a rigorou s analy sis o f the b ackoff p rocess in 802.11 and pr ovided answers to se veral open questions. The power-tail principle states that the p er-packet backoff has a truncated Pareto-type tail distribution, a simplistic de- scription elu cidating existing work s. This in turn in dicates that its heavy-tailedn ess in the strict-sense inh erits from collision and paves the way for the rest o f analysis. The dichotomy of aggregation , proven with the aids of a rece nt advance [ 24, Theorem 1 ] in pr obabilistic co mmunity , now tells the whole story of contrary limits of th e super position proce ss, i.e. , Pois- son process an d Inte rmediate T elecom process, emphasizing the im portance of tim e-scales on which we v iew the backoff processes. Thanks to the applicability of [2 4] widened by th e order -free scaling op erations of time ( ζ ) and population ( N ), we id entiﬁed lon g-range de penden ce in 802.11 and discov- ered that the in ter-transmission pr obability bifurca tes into tw o 13 categories: either ap proximate ly Gaussian or a complica ted distribution wh ich, under a limiting cond ition, simpliﬁes to L ´ evy α - stable distribution with α ∈ (1 , 2) p ossessing strong power -tail characteristics. Thoug h we have also conducted empiric al analysis using wa velet-based m ethod to supp ort lon g-rang e de pendenc e be- havior inh erent in 802 .11, since we are with Willinger et al. [42] o n the point — of cardinal imp ortance is to advance our g enuine phy sical under standing applicable to many other systems, we believe that the essence o f our analysis o f lo ng- range depend ence lies in its mathematical explanation f or the behavior . Th at is, the heavy-tailed inter-arri val time of each per-node transmission proc ess cau ses long- range d ependen ce of the aggregate tran smission p rocess at the base station though this dep endence is seldom observed. These results explore the fundamental principles character- izing the backoff pro cess in 80 2.11. Som e o f them re call to our mind the b eauty o f simplicity , governing the asym ptotic dynamics o f 802.11 , and the oth ers for m the the oretical groun dwork of sho rt-term fairness. A C K N O W L E D G M E N T The auth ors would like to thank Jean-Yves Le Boudec for helpful discussions and penetrating com ments which hav e improved the quality of this work. W e also would like to than k Ingemar Kaj fo r elab orating upon his work. Lastly , we w ould like to than k Hans Alm ˚ asbak k for m aking it p ossible to run very long simulations on a cluster of a do zen processor s. R E F E R E N C E S [1] A. Abdrabou and W . Zhuang. Service time approximat ion in IEEE 802.11 single-hop ad hoc networks. IEEE T rans. W ir eless Commun. , 7(1):305–3 13, Ja n. 2008. [2] P . Abry , P . Flandrin, M. T aqqu, and D. V eitch. Se lf-similari ty and long- range de pendence through the wa velet lens. Theory and Applications of Long-Range Dependenc e, Birkh ¨ auser Boston, pp. 527–556, 2003. [3] S. L. Albin. Approximating a point proc ess by a rene wal process, II: Superposit ion arri va l processes to queues. Ope r . R es. , 32(5):1133–1162, 1984. [4] M. Bena¨ ım and J.-Y . 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T aqqu. Long-ra nge depe n- dence and data network traf ﬁc. Theory and applicat ions of long-rang e depende nce, Birkh ¨ auser Boston, pp. 373–407, 2003. 14 A P P E N D I X A. Derivatio n of (9) Plugging b k = m k b 0 into v Ω = σ Ω / ¯ Ω yields v Ω = r Υ  P k ∈ K ( b 0 m k − 1 / 2) γ k  2 − 1 . ( 30) Here the nom inator inside the square root is simpliﬁed as Υ :=  P K k =0 ( b 0 m k − 1 / 2) 2 γ k (1 + v 2 k )  + 2 K X k =1 ( b 0 m k − 1 / 2) γ k  b 0 m k − 1 m − 1 − k / 2  = K X k =0  m +1 m − 1 + v 2 k  ( b 0 m k − 1 / 2) 2 γ k ! − 2 m − 1 ( b 0 − 1 / 2) 2 − K X k =1 ( b 0 m k − 1 / 2) γ k ( k + 2 b 0 − 1 m − 1 ) = P K k =0  m +1 m − 1 + v 2 k   b 0 m k − 1 2  2 γ k −  k + 2 b 0 − 1 m − 1   b 0 m k − 1 2  γ k . Plugging the last line into (30) yields (9). B. Pr oof of Theo r em 3 Throu ghout the proof , we denote th e sets o f real nu mbers (positive real n umbers) , integers (positi ve integers), and ra- tional nu mbers by R ( R + ), Z ( Z + ) an d Q , to simplify the exposition. Denotin g the LST of f i ( b ) by F i ( s ) , we begin the proof by consid ering the LST of (4): F Ω ( s ) = 1 − γ γ ∞ X k =0 ( k Y i =0 γ F i ( s ) ) | {z } G ( s ) . (31) This is an in ﬁnite sum of the prod ucts of F i ( s ) = 1 − exp  − (2 b 0 m i − 1) s  (2 b 0 m i − 1) s (32) that is the L ST of the unifo rm distribution with mean b 0 m i − 1 / 2 . For n otational simplicity , we adop t the chang e of variable x := 2 b 0 s such that x also b elongs to R + . Thus we have G ( x ) = ∞ X k =0 ( k Y i =0 γ 1 − exp  − ( m i − 1 / (2 b 0 )) x  ( m i − 1 / (2 b 0 )) x ) . (33) Since F i ( x ) < 1 for x ∈ R + , it is easy to see that G ( x ) is conv ergent o n R + . Then it follows from Bernstein’ s Theo r e m [20, pp .439] that G ( x ) is completely mon otone . That is, G ( x ) > 0 and it has der i vati ves o f all orders, which satisfy ( − 1) i d i G d x i ( x ) > 0 , ∀ i ∈ Z + , (34) which im plies th at the i th derivati ve of G ( x ) is strictly monoton e fo r all i ∈ Z + . Step 1: Recursiv e relation in G ( · ) The cruc ial observation that p av es the way for ap plying the theory of regular variation [10] is the following r ecursive r elation hidden in the un derbrac ed term of (31): G ( x ) = γ F 0 ( x ) { 1 + H ( mx ) } (35) where H ( x ) = ∞ X k =0 ( k Y i =0 γ 1 − exp  − ( m i − 1 / (2 b 0 m )) x  ( m i − 1 / (2 b 0 m )) x ) . (3 6) Let α = − (log γ ) / lo g m ∈ R + and z := ⌈ α ⌉ ∈ Z + which designates the smallest integer n ot less than α . It follows fro m α > 0 that z ≥ 1 . App ealing to Th eorem 2 and the basic proper ty of the L ST , i.e. , lim s → 0 + d z F Ω d s z = ( − 1) z E [Ω z ] for z ∈ Z + , it follows that lim x → 0 + d z G d x z ( x ) = 1 (2 b 0 ) z lim s → 0 + d z G d s z ( s ) = γ (2 b 0 ) z (1 − γ ) ( − 1) z E [Ω z ] = ( − 1) z · ∞ . Recall lim x → 0 + d i G d x i ( x ) is ﬁn ite for i < z by Th eorem 2. Like wise, it is easy to see that lim x → 0 + d i H d x i ( x ) is ﬁnite for i < z a nd inﬁnite for i ≥ z . Mor eover , since b oth the nominato r and denom inator o f the limit lim x → 0 + d z G d x z ( x ) d z H d x z ( x ) (37) are inﬁnite, we may remove a rbitrary nu mber of produ cts whose z th deriv atives at x = 0 + are ﬁnite f rom both of G ( x ) and H ( x ) , implying in turn that we may replace the summation operation P ∞ k =0 in (33) and (3 6) by P ∞ k = k ′ for any k ′ ≥ 0 . Formally speaking, we have d z d x z G ( x ) =    d z d x z k ′ − 1 X k =0 k Y i =0 γ F i ( x )    +    d z d x z k ′ − 1 Y i =0 γ F i ( x )    ( ∞ X k = k ′ k Y i = k ′ γ F i ( x ) ) +    k ′ − 1 Y i =0 γ F i ( x )    ( d z d x z ∞ X k = k ′ k Y i = k ′ γ F i ( x ) ) (38) where only th e third ter m (38) be comes inﬁnite as x → 0 + . Therefo re, we can easily see that the difference betwee n G ( x ) and H ( x ) vanishes as k ′ increases and h ence (37) must be 1 . T ak ing d eriv ati ves of bo th side s of (35) z time s and after some manipulatio n, it b ecomes clear that it is sufﬁcient to consider on ly inﬁnite ter ms wh ich are related to each other in the following form: h ( m ) := lim x → 0 + d z G d x z ( mx ) d z G d x z ( x ) = lim x → 0 + d z H d x z ( mx ) d z G d x z ( x ) lim x → 0 + d z G d x z ( mx ) d z H d x z ( mx ) = m − z γ − 1 = m α − z (39) where we also exploited the fact tha t (3 7) is 1 . Because the conv ergence of ( 39) hold s for any r eal sequen ces of x k → 0 + , we have that h ( y ) = y α − z for y ∈ M where M := { m i | i ∈ Z } is a co untably inﬁnite set tha t is no wher e dense in R + . Th e set on wh ich the relation h ( y ) = y α − z holds is o ften baptized quantiﬁ er s et in r egular v ariation theory . 15 Step 2: Quantiﬁer set is dense in R + W e will show that h ( y ) = y α − z holds on a dense subset L of R + . Deﬁne a set L := { λ ∈ R + | (log λ ) / log m ∈ R \ Q } where R \ Q is the set o f irration al n umbers. It s hould b e clear that M and L are disjoint, i.e. , M ∩ L = ∅ and the set L is dense in R + because it can be rewritten as L = { m y ∈ R + | y ∈ R \ Q } . Deﬁning Υ( y , x ) := d z G d x z ( y x ) / d z G d x z ( x ) , we can see that Υ( y , x ) is strictly decreasing in y b ecause it follows from (34), i.e. , complete monotonicity , that dΥ( y ,x ) d y = d( y x ) d y d z +1 G d x z +1 ( y x ) / d z G d x z ( x ) < 0 , ∀ z ∈ Z + . Pick λ ∈ L in the in terval ( m i , m i +1 ) for an y i ∈ Z . Since Υ( y , x ) > 0 is strictly decreasing in y , it is upper-bou nded by m i ( α − z ) as x → 0 + , meanin g that Υ( y , x ) is ultimate ly bound ed in x . From its series expansion , it is easy to see that it is ultim ately monoto ne in x as x → 0 + . Then we can ap ply [33, Theorem 3.14 ] to show tha t there exists ˜ α such that h ( λ ) = lim x → 0 + Υ( λ, x ) = λ ˜ α − z , (40) which in tur n im plies that h ( λ j ) = λ j ( ˜ α − z ) , ∀ j ∈ Z , as (39) did. Assume that ˜ α 6 = α . Because Υ( y , x ) is strictly decreasing in y , irre spectiv e of z , we have m α − z ≤ lim x → 0 + Υ( y , x ) ≤ 1 (41) for y ∈ (1 , m ) . Put ˆ y := m −⌊ j (log λ ) / log m ⌋ λ j for j ∈ Z . This can be rearr anged as ˆ y = m j (log λ ) / log m −⌊ j ( log λ ) / log m ⌋ and (log λ ) / log m is irration al, hence its exponen t is o n (0 , 1) and ˆ y is on the interval (1 , m ) . W e n ow have from (39) and (40) that lim x → 0 + Υ( ˆ y, x ) = lim x → 0 + d z G d x z ( m −⌊ j log λ log m ⌋ λ j x ) d z G d x z ( λ j x ) · d z G d x z ( λ j x ) d z G d x z ( x ) = m −⌊ j log λ log m ⌋ ( α − z ) λ j ( ˜ α − z ) = m −⌊ j log λ log m ⌋ ( α − z ) + j log λ log m ( ˜ α − z ) = m ( j log λ log m −⌊ j log λ log m ⌋ ) ( α − z ) · λ j ( ˜ α − α ) . where the key point is that the second equality follows from M ∩ L = ∅ . Since the last term b elongs to the closed interv al I ( j ) := [ m α − z λ j ( ˜ α − α ) , λ j ( ˜ α − α ) ] and ˜ α 6 = α , we mu st be ab le to pick j ∈ Z such th at I ( j ) do es not overlap with [ m α − z , 1] . Th is proves b y contradictio n that h ( λ ) = λ α − z holds for λ ∈ L that is den se in R + . Step 3: A pplying regular variatio n theor y Applying the ‘Karamata Theo rem for mon otone functions’ [10, T heorem 1. 10.2] to the conclusion we obtained i n Step 2 establishes that d z G d x z ( x ) is re gularly varying (o n th e right) at the o rigin x = 0 with index α − z . Formally speakin g, G ( s ) satisﬁes d z G d s z ( s ) ∼ s α − z ℓ ∗  1 s  as s → 0 + , (42) where ℓ ∗ ( x ) is slowly varying at inﬁnity x = ∞ , i.e. , lim x →∞ ℓ ∗ ( y x ) / ℓ ∗ ( x ) = 1 for all y ∈ R + . Note th at the original Karamata T auberian Theorem in [1 0, Theo rem 1.7.1 ] and [20, p p.445 ] ca nnot be app lied due to the fact α − z ≤ 0 . T hese theor ems are compleme nted by the mo diﬁed Karamata T aube rian Theorem in [10, T heorem 8.1.6 ] and [9], wh ich we ap ply to (4 2) to show (15) .

Fundamentals of the Backoff Process in 802.11: Dichotomy of the Aggregation

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