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

Jeong-woo Cho and Yuming Jiang

Abstract—This paper discovers fundamental principles of thebackoff process that governs the performance of IEEE 802.11. Asimplistic principle founded upon regular variation theory is thatthe backoff time has a truncatedPareto-type tail distribution withan exponent of (log γ)/ logm (m is the multiplicative factorand γ is the collision probability). This reveals that the per-nodebackoff process is heavy-tailed in the strict sense forγ > 1/m2,and paves the way for the following unifying result.

The state-of-the-art theory on the superposition of the heavy-tailed processes is applied to establish adichotomy exhibitedby the aggregate backoff process, putting emphasis on theimportance of time-scales on which we view the backoff processes.While the aggregation on normal time-scales leads to a Poissonprocess, it is approximated by a new limiting process possess-ing long-range dependence (LRD) on coarse time-scales. Thisdichotomy turns out to be instrumental in formulating short -termfairness, extending existing formulas to arbitrary population, andto elucidate the absence of LRD in practical situations. A refinedwavelet analysis is conducted to strengthen this argument.

Index Terms—Point process theory, regular variation theory,mean field theory.

I. I NTRODUCTION

Since its introduction, the performance of IEEE 802.11has attracted a lot of research attention and the center ofthe attention has been thethroughput [6], [27]. Recently,other critical performance aspects of 802.11 also burst ontothe scene, which includeshort-term fairness[12], [26] anddelay [38]. It goes without saying that there has been aphenomenal growth of Skype and IPTV users [14], [15] and itis reported in [23] that an ever-increasing percentage of theseusers connects to the Internet through wireless connectionsin US. Remarkably, it is found in [15] thatjitter is morenegatively correlated with Skype call duration than delay,i.e.,Skype users tend to hang up their calls earlier with large jitters.This finding empirically testifies large jitter of access networksannoysSkype users, let alone QoS (quality of service). Thisquantified dissatisfaction of users provides a motivation fora thorough understanding of delay and jitter performance in802.11.

This work was supported in part by “Centre for Quantifiable Quality ofService in Communication Systems, Centre of Excellence” appointed byThe Research Council of Norway, and funded by The Research Council,NTNU and UNINETT. A part of this work was done when J. Cho was withEPFL, Switzerland. A preliminary abstract version of this work appeared atACM SIGMETRICS Workshop on Mathematical Performance Modeling andAnalysis (MAMA’09).

J. Cho and Y. Jiang are with the Centre for Quantifiable Qualityof Service in Communication Systems, Norwegian Universityof Sci-ence and Technology (NTNU), NO-7491 Trondheim, Norway (email:jeongwoo,jiang@q2s.ntnu.no).

For throughput analysis, Kumaret al., in the seminal paper[27], axiomized several remarkable observations based on afixed point equation(FPE), advancing the state of the artto more systematic models and paving the way for morecomprehensive understanding of 802.11. Above all, one of thekey findings of [27], already adopted in the field [28], [34],is that the full interference model1, also called the single-cell model [27], in 802.11 networks leads to thebackoffsynchrony property[31] which implies the backoff processcan be completely separated and analyzed through the FPEtechnique. Another observation in [27] was that if the collisionprobabilityγ is constant, one can derive the so-called Bianchi’sformula by appealing to renewal reward theorem [13], withoutthe Markov chain analysis in [6].

An intriguing notion, calledshort-term fairness, has beenintroduced in some recent works [5], [12], [26], definingP[z|ζ] as the probability that other nodes transmitz packetswhile a tagged node is transmittingζ packets. It can beeasily seen that this notion pertains to a purely backoff-relatedargument also owing to the backoff synchrony property inthe full interference model [27]. The two papers [5], [12], inthe course of deriving equations forP[z|ζ], assumed that thesummation of the backoff values generated per packet, whichwe denote byΩ, is uniformly and exponentially distributed,respectively. Specifically, despite the same situation where twonodes contend for the medium, the former [5] assumed thatΩis uniformlydistributed because the initial backoff is uniformlydistributed over the set0, 1, · · · , 2b0 − 1 where2b0 is theinitial contention window and observed in [5, Fig. 2] that thisassumption leads to a good match between the expressionP[z|ζ] derived under the uniform assumption onΩ and thetestbed data measured in their experiments, while the latter[12] also observed in [12, Fig. 5(a)] that the testbed datameasured in their experiments closely match the expressionP[Z|ζ] derived under the the exponential assumption onΩ:Q1:“What makes two different observations?”(to be an-

swered in Section III)In addition, the two works [5], [12] acquired the expression

of P[z|ζ] only for the two node case. A more general formulafor arbitrary number of nodes should deepen our appreciationof short-term fairness. It is natural to ask the followingpertinent questions:Q2:“Can we develop a general model for short-term fair-

ness?” (to be answered in Corollaries 1 & 2)In proportion as people take a growing interest in the

1In the full interference or single-cell model, every node interferes with therest of the nodes,i.e., its corresponding interference graph is fully connected.

2

delay performance of 802.11, the number of fundamentalquestions that we face increases. In [1], it was argued basedon simulation results that the access delay in 802.11 closelyfollows a Poisson distribution. They have shown that thenumber of successful packet transmissions by any node in thenetwork over a time interval has a probability distributionthatis close to Poisson by an upper bounded distribution distance.This raises an intriguing question:Q3:“Is there a Poissonian property? If yes, what is the

cause?” (to be answered in Theorem 1)Another case in point is found in a recent work [34] that

extends the access delay analysis in the seminal paper ofKwak et al. [28] and makes an attempt at analyzing higherorder moments by applying the FPE technique. One interestingfinding in [34] is that the access delay has awide-senseheavy-taileddistribution [34, Theorem 1] which means thatits moment generating function

∫∞0 etxf(x)dx is ∞, ∀t > 0,

where f(x) is the corresponding pdf (probability densityfunction) [32]. One should be careful in interpreting this find-ing because the wide-sense heavy-tailedness doesnot implystrict sense heavy-tailedness, which roughly means the ccdf(complementary cumulative distribution function) is of Pareto-type [17] with an exponent over(−2, 0). In fact, there arelots of distributions, namely, lognormal, Pareto, Cauchy andWeibull distributions, which belong to the class of wide-senseheavy-tailed distributions. Consequently, the discussion posesthe following challengewhich is undoubtedly a tantalizingquestion.Q4:“What is the distribution type of the delay-related vari-

ables?” (to be answered in Theorems 2 & 3)Finally, it is, perhaps, surprising that long-range dependence

of 802.11 has not been rigorously analyzed even for the singlenode case, not to mention the aggregate process of manynodes. One minor contribution of this paper is that we provein Theorem 3 that the individual arrival process (consistingof successful transmissions of one node) can be viewed asa renewal process with heavy-tailed inter-arrival times, whichimplies that the individual arrival process possesses long-rangedependence simply by appealing to [30].

However, for the superposition arrival process (consisting ofsuccessful transmissions of all nodes), there is no clear answer.For example, Tickoo and Sikdar [37] conjectured the absenceof long-range dependence of aggregate total load, which wecall superposition arrival process. It is remarkable that theabsence of long-range dependence has been also supportedthrough empirical analysis such as wavelet-based method [2]by Veres and Boda [39] in the context of TCP flows inwired networks. Since there is an analogy between the backoffmechanisms adopted by 802.11 and TCP (in wired networks)in that

1) both of them adopt backoff schemes (802.11) or retrans-mission scheme (TCP) where the probability of theseevents is either the collision probability (802.11) or thepacket drop probability in router buffers (TCP),

2) the mean of the backoff (contention window in 802.11)or retransmission time (timeout in TCP) doubles for eachbackoff or retransmission,

one might wonder if there is a fundamental reason thatelucidates these observations.Q5:“Does the aggregate transmission process possess long-

range dependence? If yes, why is it seldom observed?”(to beanswered in Theorem 4 and Section VII)

The focus of this paper is on the backoff process in 802.11,since it plays the central role in quantifying the performanceof 802.11 [27]. For example, to grasp the heart of the de-lay properties, the backoff value distribution in 802.11 DCF(distributed coordination function) mode can be used as asurrogatefor the access delay [28]. As discussed above, thethroughput performance and short-term fairness performancealso depend on the backoff process and are particularly af-fected by the backoff synchrony property. Essentially, once thebackoff distribution is obtained, various performance aspectscan be analyzed.

A. Contributions of this work

This paper discovers fundamental principles of the backoffprocess and provides answers to the open questions high-lighted above, which constitute the contributions of the paper.Particularly, it turns out that we find out the answers to mostaforementioned questionsQ2-Q5 in the course of deriving thefollowing two principles based on a new methodology,i.e.,point process approach.

• Power-tail principle : The per-packet backoff time distri-bution has a slowly-varying power-tail (Theorem 3).

• Dichotomy of aggregation: Depending on the time-scales on which the backoff processes are aggregated,the resultant process becomes either Poissonian or a newprocess (Theorems 1 & 4).

The power-tail principle, which is derivable only after weaccumulate a store of knowledge (Section III, Lemma 1and Theorem 2), characterizes the backoff distribution in atractable and simplistic way, owing to regular variation theory,answeringQ4. The dichotomy of aggregation implies that,when we view the aggregate process on normal time-scales,owing to the tendency of each component process to becomesparse as population grows, we observe only a Poissonian asits marginal distribution. However, viewed on coarse time-scales, the aggregate process is identified as a long-rangedependent process. This rigid dichotomy is instrumental infinding answers toQ2, Q3 and Q5, and expatiates upon thecoexistence of contrary properties suggested byQ3 and Q5.All the theorems in the paper areclosely linkedwith eachother, forming a solid framework for the performance analysisof 802.11. These results help us to get the complex detailsof the backoff process in 802.11 into perspective under oneframework.

The rest of the paper is organized as follows. In SectionII, we revisit the Bianchi’s formula along with a survey ofrecent advances in mean field theory, with which the analysisof the backoff process at one node can be decoupled from othernodes. In Section III, we present the exact distribution of per-packet backoff. We establish in Section IV that the aggregatebackoff process can be approximated by a Poisson processunder the large population regime. In Section V, we extend

3

to asymptotic analysis and prove the power-tail principle.InSection VI, we first propose a new process approximation ona coarse time scale, which is then applied to formulate short-term fairness and to identify long-range dependence. Afterconducting a wavelet analysis on long-range dependence inSection VII, we conclude this paper.

II. B IANCHI ’ S FORMULA REVISITED

The backoff process in 802.11 is governed by a few rulesif the duration of per-stage backoff is taken to be exponential:(i) every node in backoff stagek attempts transmission withprobabilitypk for every time-slot; (ii) if it succeeds,k changesto 0; (iii) otherwise, k changes to(k + 1) mod (K + 1)whereK is the index of the highest backoff stage. Markovchain models, which have been widely used in describingcomplex systems including 802.11, however, very often leadto excessive complications as discussed in Section I. In thissection, we present a surrogate tool for the analysis,meanfield theory. It is noteworthy that the rules used in 802.11,i.e., (i)–(iii), closely resemble the mean field equations laidout below.

A. Basic Operation of DCF Mode

Time is slotted. Each node following the randomized accessprocedure of 802.11 distributed coordination function (DCF)generates abackoff valueafter receiving the Short Inter-FrameSpace (SIFS) if it has a packet to send. This backoff value isuniformly distributed over0, 1, · · · , 2b0−1 (or 1, 2, · · · ,2b0) where2b0 is the initial contention window.

Whenever the medium is idle for the duration of a Dis-tributed Inter-Frame Space (DIFS), a node unfreezes (starts)its countdown procedure of the backoff and decrements thebackoff by one per every time-slot. It freezes the countdownprocedure as soon as the medium becomes busy. There ex-ist K + 1 backoff stages whose indices belong to the set0, 1, · · · ,K where we assumeK > 0. If two or morewireless nodes finish their countdowns at the same time-slot,there occurs a collision between RTS (ready to send) packetsif the CSMA/CA (carrier sense multiple access with collisionavoidance) is implemented, otherwise two data packets collidewith each other. If there is a collision, each node who partici-pated in the collision multiplies its contention window by themultiplicative factorm. In other words, each node changes itsbackoff stage indexk to k + 1 and adopts a new contentionwindow 2bk+1 = 2mk+1 ·b0. If k+1 is greater than the indexof the highest backoff stage number,K, the node steps backinto the initial backoff stage whose contention window is setto 2b0. In the IEEE 802.11b standard,m = 2, K = 6 (7attempts per packet), and2b0 = 32 are used.

This work focuses on the performance ofsingle-cell802.11networks where it issufficientto analyze the backoff processin order to investigate the performance of single-cell networks

B. The Bianchi’s Formula

In performance analysis of 802.11, Bianchi’s formula andits many variants are probably the most known [6], [27],

[28], [34]. Assuming that there areN nodes, the Bianchi’sformula can be written compactly in a more generalfixed pointequation(FPE) form:

p =

∑Kk=0 γ

k

∑Kk=0

γk

qk

, (FPE1)

γ = 1− e−(N−1)p (FPE2)

wherep andγ respectively designate the average attempt rateand collision probability of every node at each time-slot. Theattempt probability in backoff stagek is denoted byqk anddefined as the inverse of the mean contention window,i.e.,qk = 2/(2bk − 1). It satisfies0 < qk ≤ 1 asbk ≥ 1. Note thatBianchi’s formula holds under the well-known assumption:

A.1 All the transmission queues of nodes are saturated.

Exactly under which condition (FPE1) holds is recentlybeing rediscovered with rigorous mathematical arguments [4],[11], [35], which, sometimes calledmean field approxima-tion. This fundamental approach was originally developed byBordenaveet al. [11] and Sharmaet al. [35]. Remarkably,Bordenaveet al. [11] adopted a generalized particle interactionmodel which encompasses Markovian evolution of the systemother than particles at the same time. Benaım and Le Boudec[4] overcame some limitations of the model [11], broadeningits applicability. The main result here is that, as the number ofparticles goes to infinity,i.e., N → ∞, the state distributionof every node evolves according to a set ofK+1 dimensionalnonlinear ordinarydifferential equations under an appropriatescaling of time. Benaım and Le Boudec [4] also observedthat decoupling approximation represented by (FPE1) doesnot hold if the differential equations does not have a uniqueglobally attractor.

Remarkably, Bordenaveet al. [11] have proven that thedifferential equations are globally stable ifK = ∞ and the re-scaled attempt probabilityQk := Nqk satisfiesQk+1 = Qk/2with Q0 < ln 2. In the meantime, Sharmaet al. [35] obtaineda result forK = 1 and mentioned the difficulty to go beyond.However, the case for other finiteK has remained to be proved[4, pp.833]. Recently, we solved this issue to a large extent[16]by proving that, for finiteK, a simplistic conditionQk ≤ 1(or qk ≤ 1/N ) for all k ∈ 0, · · · ,K guarantees the globalstability of the differential equations as well as the uniquenessof the solutions to the Bianchi’s formula. As we discussedin [16], there are still many outstanding problems upon thestability of the associated differential equations.

While one of the aims of these efforts [4], [11], [16], [35]is to identify the fundamental conditions under which thecollision probability is deterministic and time-invariant forlarge population (N = ∞), once we assume the collisionprobability is such forN < ∞, the demonstration of theformula (FPE1) is shown to be straightforward [27]. That isto say, we need to make the following simple assumption.

A.2 For each node, conditional upon its transmission attempt,the collision events form an i.i.d. sequence, which isindependent from other nodes.

The observation in [27] was that, under the above assumptions,one can easily derive (FPE1) by appealing to renewal reward

4

theorem [13], without the Markov chain analysis in [6]. Thusfrom now on, the attempt probability is given by (FPE1). Asa by-product, we can also see that the distribution of backoffstages, which we denote byφk, k ∈ 0, · · · ,K, takes thefollowing form

φk =γk

qk· 1∑K

k=0γk

qk

. (1)

The expression of the collision probability (FPE2) was firstused in [27, Section IV]. A similar expression was also usedin [4], [11] under the intensity scaling, which means that theattempt probability of every node in any backoff stage is of theorder of1/N . We use (FPE2) instead of its original version in[6] because, as argued in [4], [11], the approximation providedby (FPE2) is well founded on a mean field result. Lastly, it isalso noteworthy that the analysis in Section III and IV doesnot depend on whetherK is finite or not.

III. B ACKOFF ANALYSIS

The backoff value distribution and the backoff stage distri-bution should not be confused in meaning. While the latter isthe distribution of the backoff stage of a node, the former isthe distribution of the backoff value generated for initiating thebackoff countdown when the node has a packet to transmit.The backoff value distribution at backoff stagek has adiscreteuniformpdf (probability density function)fk(·) on the integers0, 1, · · · , 2bk−1 with mean1/qk = (2bk−1)/2 and variance

Vark =b2k3

− 1

12=

1

3· 2bk + 1

2bk − 1· 1

q2k=

v2kq2k

where the pre-factor is denoted byv2k to simplify the expositionin the current section. Note thatlimbk→∞ v2k = 1/3.

Let Ω andfΩ(·) respectively denote the sum of the backoffvalues generated for a packet, and its pdf. Also denote byΩits mean andσ2

Ω its variance. It should be clear that the sum ofthe backoff values generated for a packetΩ which we baptizein this paperper-packet backoffcan be formally defined as acompoundrandom variable:

Ω :=∑κ

k=0 Bk (2)

whereBk is a random variable denoting the backoff valuegenerated at thekth backoff stage, for a packet of a taggednode, andκ is also a random variable designating the highestbackoff stage reached by the packet.

The probability that thekth backoff stage is reached duringthe backoff duration for a packet can be computed asγk

irrespective of the backoff distribution at any backoff stage.Hence we have

P[κ = k] = γk − γk+1, ∀k ∈ 0, · · · ,K − 1,

andP[κ = K] = γK . From Bayes’ theorem,fΩ(·) becomes:

fΩ(x) =∑K

k=0 fΩ(x | κ = k) · P [κ = k] (3)

wherefΩ(x | κ = k) denotes the sum of the backoff valuesfrom 0th to kth stages for a givenk. Applying the fact that

the sum ofk random variables with pdfsf0(·), · · · , fk(·) hasa pdf of the convolution of the pdfs yields

fΩ(x) = f∗K(x)γK + (1 − γ)∑K−1

k=0 f∗k(x)γk (4)

wheref∗k(·) := (f0 ∗ · · · ∗ fk)(·) is the convolution ofk + 1functions. In a similar way,Ω can be computed from (2):

Ω = E [∑κ

k=0 Bk] =∑K

k=0 E[∑k

k′=0 Bk′

]· P[κ = k]

=∑K

k=0

(∑kk′=0

1qk′

)· P[κ = k]. (5)

By manipulating (5) combined with the expression ofP[κ =k], it is easy to see that

Ω =K∑

k=0

γk

qk. (6)

In addition, usingE[B2k] = (1 + v2k)/q

2k, the second moment

of Ω can be rearranged as

σ2Ω + Ω2 = E

[(∑κ

k=0 Bk)2]

=∑K

k=0 E[(∑k

k′=0 Bk′

)2]P[κ = k] (7)

=∑K

k=0

k∑

k′=0

1+v2k′

q2k′

+ 2

k∑

i=1

i−1∑

j=0

1qiqj

P[κ = k] (8)

=(∑K

k=0γk

q2k

(1 + v2k))+ 2

(∑Kk=1

γk

qk

∑k−1i=0

1qi

)

The above equalities can be easily verified by rearranging (7)and (8). Moreover, it is shown in Appendix A that, ifqk =2/(2b0m

k − 1) as in the standard,v2Ω := σ2Ω/Ω

2 simplifies to∑K

k=0 δk(∑Kk=0(b0m

k − 1/2)γk)2 − 1 (9)

where

δk =(b0m

k − 12

)γk(

m+1m−1 + v2k

) (b0m

k − 12

)− k − 2b0−1

m−1

.

Remark 1 R1.1 The result puts forward analternativeviewpoint. We can view the backoff processreflecting thecollision effect among nodes as if there isno collision at alland the per-packet backoff for every node has a distributionwith meanΩ and CVvΩ (or equivalently varianceσ2

Ω).R1.2 [Answer to Q1] Consider the caseN = 2. It can

be computed from (9) thatΩ is approximatelyuniformlydistributed in 802.11b while it isexponentiallydistributed in802.11a/g in the sense thatvΩ ≈ 0.7 (though slightly largerthan1/

√3) andvΩ ≈ 1.0, respectively, mainly due to different

initial contention windows (2b0 = 32 in 802.11b and2b0 = 16in 802.11a/g). This is the reason why they [5], [12] observedthat their testbed data closely match the expressions of inter-transmission probabilityP[Z|ζ], which were derived undertheir respective assumptions. Note that we communicated withthe first author of [12] to verify the protocol (802.11g) usedintheir testbed. We will formally defineP[Z|ζ] in Section VI-A.

To verify the analysis, simulations have been conducted. Wehave usedns-2 version 2.33 with its built-in 802.11 moduleand the parameter set of 802.11b,i.e., m = 2 and2b0 = 32,

5

0 0.1 0.2 0.3 0.4 0.5

100

101

102

K=∞

K=15

K=6

collision probability (γ)

per−

pack

et b

acko

ff C

V (v Ω

)

↑N=2

N=5

N=10

N=20

N=40

N=60

N=80

N=100

analysis: contours of Kanalysis: contours of Nsimulation: K=15simulation: K=6error

Fig. 1. Per-packet backoff CVvΩ vs. collision probabilityγ for K =6, 15,∞; andN = 2, · · · , 100.

except thatK is varied to observe the asymptotic property. Allsimulations use a3000s warm-up period and all quantities aremeasured over the next320, 000s (≈ 90h).

Fig. 1 presents the per-packet CVvΩ, computed from (9),(FPE1) and (FPE2), and compared with the simulation results.The figure shows a good match between them. In the figure,the intersecting points of contours ofK andN at each leveldecidevΩ and γ simultaneously. As is predicted by (9),vΩgoes to∞ as K goes to∞ for γ ≥ 1/m2 = 0.25. It isremarkable that for a givenN ≥ 9 (N ≥ 5 for 802.11a/g),vΩis extremely sensitive toK, forming a striking contrast withthe insensitivity ofγ to K.

The discrepancy between analysis and simulation study ispartly due toreduced contention effect, which is a less-knownsubtlety of DCF behavior discovered by Bianchiet al. [7] andis shown through simulations to be a factor of error by Sakuraiand Vu [34].

IV. POINT PROCESSAPPROACH: POISSONIAN INSIGHTS

A basic property of per-packet backoffΩ discovered byKwak et al. [28, Theorem 1] and later strengthened by Kumaret al. [27, Theorem 7.2] is that the mean of per-packet backoffis proportional to the population,i.e., Ω = Θ(N). This turnsout to play a key role in our point process approach in thissection.

A. Justification of Point Process Approach

In order to justify our point process approach, we needto show that the backoff process of each node has nonzerointensity, i.e., Ω = E[Ω] is finite. Though, for finiteK, thisis self-evident from the form of (6), we need to assume thefollowing to proveΩ < ∞ for K = ∞.

A.3 qk = 2/(2b0mk−1) for all k ∈ 0, · · · ,K, andm > 1.

Under this assumption we can prove the following lemmawhich assures us thatΩ is finite whetherK is finite or not.We also would like to point out that a part of the proof of[27, Theorem 7.2], which corresponds to the caseK = ∞ of

Lemma 1 in our work, has a flaw because they should haveprovenγ < 1/m before using

∑∞k=0(mγ)k = 1/(1−mγ).

Lemma 1 (Mean Exists)Under the above assumption, there exists a finiteK0 such thatγ < 1/m andγ is decreasing inK. This implies:

• there existK0 such thatγ < 1/m for all K ≥ K0

includingK = ∞,• the meanΩ = E[Ω] exists forK = ∞.

Proof: Supposeγ ≥ 1/m. Then we have from (FPE1)and qk = 2/(2b0m

k − 1) that, for anyǫ > 0, there existsK1 such thatp < ǫ for all K ≥ K1. In the meantime, from1 − e−x ≤ x, we also haveγ ≤ (N − 1)p < (N − 1)ǫ. Thiscontradictsγ ≥ 1/m, implying that there must existK0 suchthat γ < 1/m for K = K0.

Denote the right-hand side of (FPE1) byP(K). Since theright-hand side of (FPE2) is increasing inp and P(K) isnonincreasing inγ from [27, Lemma 5.1],1 − e−(N−1)P(K)

is nonincreasing inγ. Therefore, itsufficesto show that

1− e−(N−1)P(K0+1) ≤ 1− e−(N−1)P(K0),

or equivalentlyP(K0 + 1) ≤ P(K0), for all γ < 1/m. Aftersome manipulation and some intricate factorization, it canbeverified thatP(K0)−P(K0 + 1) takes the form:

b0γK0+1

∑K0

k=0 γk(mK0+1 −mk

)∑K0

k=0

(b0mk − 1

2

)γk∑K0+1

k=0

(b0mk − 1

2

)γk

which is greater than zero form > 1, implying that thesolutionγ∗ of (FPE1) and (FPE2) forK = K0 +1 is smallerthan that forK = K0. Applying mathematical inductioncompletes the proof. Also note that this impliesγ < 1/mfor anyK ≥ K0.

For the caseK = ∞, since we have shown thatγ < 1/mis decreasing inK for all K ≥ K0, it follows from [33,Theorem 3.14] that asK goes to infinity,γ should convergeto γ < 1/m. The existence ofE[Ω] follows from (6).

Since m > 1 guarantees that there existsK0 such thatγ < 1/m for all K ≥ K0, it can be seen from (1) that,for the case ofK = ∞, m > 1 is also a sufficient conditionfor the existence ofκ such thatφk > φk+1 for all k ≥ κ,i.e., the average number of nodes in backoff stagek is largerthan that in backoff stagek + 1. This corresponds to thetightness condition ofφk, which prevents a node from escapingto infinite backoff stage [8]. The fact that the conditionm > 1prevents a node from escaping to infinite backoff stage appearsto be in best agreement with our usual intuition.

B. Essential Assumption

To establish Poisson limit result in Theorem 1 and to justifypoint process approach in the remaining sections, we need thefollowing essential assumption.

A.4 Per-stage backoff distributionfk(·) is a uniform continu-ous function. It also meansvk = 1/

√3.

Recall that fΩ(·) is expressed by (4), hence now it is aweighted sum of convolutions of continuous pdfsfk(·) wherethe weight for each convolution functionf∗k(·) = (f0 ∗ · · · ∗

6

fk)(·) is (1− γ)γk, which is a function ofγ. As we noted inRemark 1,fΩ(·) reflects the collision effect throughγ whichdetermines how muchfΩ(·) is dispersed.On continuity assumption: Denote byDn(t) the number ofcumulative per-node successful transmissions until time-slot t.Formally,Dn(t) is discrete-timerenewal process that countsthe number of arrivals during the interval[0, t] where theinter-arrival times are i.i.d. copies of discrete random variableΩ. Consider superposition processD(t) :=

∑Nn=1 D

n(t). Asubtletyin 802.11 is that there may be no intervening backofftime-slot between two consecutive successful transmissions.More precisely, at the beginning of a backoff time-slot,if the transmission attempts of nodes lead to a successfultransmission, the time-slot is rendered unused, meaning thatthe time-slot is reused after the successful transmission.Thesame subtlety applies to collision events. Simply suppose theprobability that a successful transmission (or a collisionevent)occurs at the beginning of a time-slot converges toPS (or PC )asN → ∞. Putting

P (x) := P[limN→∞ D(t+ 1)−D(t) = x], x ∈ 0, 1, · · · ,we can see from the subtlety that

P (x + 1) = P (x) ·∑∞i=0 P

iCPS = PS

1−PCP (x).

Because∑∞

x=0 P (x) = 1, we have a geometric distribution

P (x) =(1− PS

1−PC

)(PS

1−PC

)x, x ∈ 0, 1, · · ·

hence the limiting (asN → ∞) distribution of cumulativeprocessD(t) for arbitrary integert takes aPascal (negativebinomial) distribution2. This fact can be exploited for a moreaccurate approximation. A simpler approximation at the costof accuracy is to be presented in Theorem 1.

Once again, the continuity assumption turns out unavoidablein Section V because regular variation theory [10] exploitedby Theorem 3 is not well developed for discrete functions.The uniform distribution assumption offk(·) was made onlyto simplify the exposition of Theorems 2 and 3 in Section V.

C. Poisson Process Approximation

We can now view the backoff procedure of noden asa stationary simplerenewal processAn(t) that counts thenumber of arrivals during the interval(0, t] where thejth inter-arrival times,T n

j − T nj−1, are given by the i.i.d. copies of the

continuous random variableΩ. Then the backoff procedure ofall nodes can be regarded as asuperpositionof N statisticallyidentical renewal processes,i.e.,

A(t) :=∑N

n=1 An(t).

It should be remarked that, if one or more component pro-cesses are not Poisson, the superposition processA(t) isnot renewal, and even if the inter-arrival times ofA(t) areidentically distributed, they arenot independent[3].

In the following, we present a novel way to tackle thisanalytical difficulty caused by the dependence among the inter-arrival times of the superposition process. The key observation

2Sakurai and Vu [34, Section III-B] assumedD(t) is a Bernoulli process.This simplification was justified by the reduced contention effect [7].

is that theentropy of the superposition point processA(t)increases withN , which is implied by the following knownresult [18, Proposition 11.2.VI].

Lemma 2 (Poisson Limit for Superposition)Let Ξ(t) denote the point process obtained by superposingMindependent replicatesBm(t), m ∈ 1, · · · ,m, of a simplestationary point process with intensityλ and dilating the time-scale by a factorM . Formally speaking,

Ξ(t) =∑M

m=1 Bm(t/M). (10)

Then asM → ∞, Ξ(t) converges weakly to a Poisson processwith the intensityλ.

Now it follows from the basic property [27, Theorem 7.2]for K = ∞ that the mean inter-arrival time ofAn(t), Ω, is oforderN . Therefore, there must exist a point process

Bn(t) := limN→∞

An(Nt) with intensityλ = limN→∞

N/Ω

where intensityλ does not scale with N and we haveBn(t/N) ≈ An(t) asN goes to∞. This in turn implies

∑Nn=1 A

n(t) ≈ ∑Nn=1 B

n(t/N)

which has the same form of (10). Applying Lemma 2 to theabove equation leads to the following theorem.

Theorem 1 (Dichotomy of Aggregation: First Part)Suppose Ω = Θ(N). Then the superposition process∑N

n=1 An(t) converges weakly to a Poisson process asN →

∞.

Remark 2 This result states that the Poissonian nature isinherent in the backoff process of 802.11 and provides ananswer toQ3.

R2.1 The reason we do not requireK = ∞: Recallingour discussion at the beginning of this section, we can see that

K = ∞ [27, Theorem 7.2]=⇒ Ω = Θ(N)

Theorem 1=⇒ Poisson.

If we require K = ∞ instead of Ω = Θ(N), the abovetheorem would look simpler, but it would not be applicablefor the caseK < ∞. Even if K is finite, the crucial scalingconditionΩ = Θ(N) holds for a wide range ofN , as hintedby previous works (See the simulation result with a practicalparameter set in [34, Figures 2 and 5]). However, for extremelylargeN , the scaling becomesΩ = Θ(1).

R2.2 From a different angle, the backoff procedure of802.11 along with its settingK = 6 is intentionally designedso that thesuccessfulattempt intensity of each node1/Ω iskept being of the order of1/N for a wide range ofN , byallowing enough number of backoffs for each packet.

What is the premise of Poisson limit?: The question remainswhether the approximation is precise even fort = ∞. As Whittdiscussed in [41, Chapter 9.8], the underlying assumption ofthe Poisson limit theorem (Lemma 2) is thatt is finite. In themeantime, the basic premise of the Poisson limit theorem isthat the component processAn(t) should become sparse (Ω =Θ(N)) [40, pp.83]. If we allowt → ∞ at the same time asN → ∞, An(t) may not remain sparse. This is essentially whywe mustadopt an another approximation in Section VI where

7

t = Θ(N). In the light of these points, the above theoremprovides a natural approximation of the backoff processes onnormal time-scale, as compared with the other approximationin Section VI on coarse time-scales.

V. A SYMPTOTIC ANALYSIS

A stochastic process with infinite variance and self-similarity exhibits phenomena calledNoah effectand Josepheffect, respectively, in Mandelbrot’s terminology [36], [41].Noah and Joseph effects refer to the biblical figures Noah,who experienced an extreme flood – exceptionally large values– and, Joseph, who experienced long periods of plenty andfamine – self-similarity or strong positive dependence. Thissection lifts the veil to discover these effects and to explaintheir influences on the backoff process in 802.11. We havenot assumedK = ∞ because all results derived so far areapplicable if either of finite and infiniteK is used (See Remark2 also). However, all results derived in this section requireK = ∞, hence we formally assume the following.

A.5 There areinfinite backoff stages,i.e., K = ∞.

A. Moment Analysis

We introduce the notion of a wide-sense heavy-tailed dis-tribution borrowed from [32]. We call a pdff(x) wide-senseheavy-tailedif its moment generating function is infinite,i.e.,

∫∞0

etxf(x)dx = ∞, ∀t > 0.

We now characterize the existence of all fractional momentsof Ω. Let us define

α := −(log γ)/logm

whereα > 1 is satisfied by Lemma 1. Also it is remarkablethat Sakurai and Vu [34] established a similar result for integermoments. Note however that wecannot prove Theorem 3without the following extended result for fractional moments.

Theorem 2 (Existence of Fractional Moments)The per-packet backoffΩ has a wide-sense heavy-tailed dis-tribution. In addition, itscth momentE[Ωc] is

• infinite if c ≥ α,• andfinite if 0 ≤ c < α.

Proof: First we noteα = −(log γ)/logm is equivalent tomαγ = 1. It also follows from Lemma 1 thatα > 1. Lettingc be any real number such thatc ≥ α, we havemcγ ≥ 1.Then thecth moment ofΩ, E[Ωc], can be computed as

∑∞k=0 E

[(∑kk′=0 Bk′

)c]· P[κ = k]

≥∑∞k=0

(E[∑k

k′=0 Bk′

])c· P[κ = k]

=∑∞

k=0

(∑kk′=0(b0m

k′ − 12 ))c

· P[κ = k]

≥∑∞k=0

∑kk′=0(b0m

k′ − 12 )

c · P[κ = k]

=∑∞

k=0(b0mk − 1

2 )cγk

where the first inequality holds by Holder’s inequality forexpectations,i.e., (E[X ])c ≤ E[Xc], and the second inequality

follows from c > 1. Hence, from the last expression, we haveE[Ωc] → ∞ asK → ∞. Note thatc is real. Since there existinfinite moments,Ω has a wide-sense heavy-tailed distribution.Now consider thecth moment for1 < c < α.

E[(∑κ

k=0 Bk)c]

=∑∞

k=0 E[(∑k

k′=0 Bk′

)c]· P[κ = k]

≤∑∞k=0 E

[(k + 1)c−1

∑kk′=0 (Bk′)

c]· P[κ = k] (11)

=∑∞

k=0(k + 1)c−1∑k

k′=0(2b0m

k′−1)c

(c+1) · P[κ = k] (12)

≤ (2b0)c

(c+1)

∑∞k=0(k + 1)c−1 (mc)k+1−1

mc−1 · P[κ = k]

≤ (2b0)c

(c+1)

∑∞k=0(k + 1)c−1 (mc)k+1γk

mc−1 (13)

= (2b0m)c

(c+1)(mc−1)

∑∞k=0(k + 1)c−1(mcγ)k (14)

where (11) can be obtained by applying original Holder’sinequality,i.e.,(∑k

k′=0 1 · bk′

)≤(∑k

k′=0 1c

c−1

) c−1c(∑k

k′=0(bk′)c) 1

c

.

(12) can be verified by computing∫bcfk′(b)db wherefk′(b)

is a uniform pdf with meanb0mk′ − 1/2. (13) follows fromP[κ = k] ≤ γk. Then it suffices to show that d’Alembert’sratio of the series (14) is less than one. Recalling thatmcγ <mαγ = 1, we can see that

limk→∞

(k + 2)c−1(mcγ)k+1

(k + 1)c−1(mcγ)k= mcγ < 1.

This establishes (14) is finite forK = ∞, and completes theproof.

Remark 3 [Answer to Q4] This theorem reveals thatΩ iswide-sense heavy-tailed in the sense thatnotall of its momentsexist, as Sakurai and Vu [34, Theorem 1] first noted.

As shown in Fig. 1, the varianceσ2Ω in 802.11b is not

very large. Nevertheless, the statistics ofΩ certainly containprecursors of infinite-variance distributions, as shown inthenext section.

B. Strict-Sense Heavy-Tailedness: Tauberian Insights

Although there has been some work to prove the wide-senseheavy-tailedness of the delay or backoff duration [34] and thepower-law like behavior of access delays wasidentifiedonlythrough simulations in a few works [34], [37], to the best ofour knowledge,noneof them proved that the delay or backoffduration has a power-law tail. This quite intuitive property hasnot been established mainly due to the theoretical difficultiesunderlining the proof. It is important to note that this theoremis a prerequisite for mathematical analysis of Noah effect,which implies strict-sense heavy-tailedness.

We would like to place particular emphasis on the followingtheorem for another reason. We note that some work [21], [39]considered the question whether asingle long-lived TCP flowcan generate traffic that exhibits long-range dependence (or,equivalently, asymptotical second-order self-similarity). It issignificant that long-range dependence is a property which isautomaticallyimplied by heavy-tailed inter-arrival times [30]for the single flow (or node) case, irrespective of the context.

8

That is, even a renewal process (no correlation of inter-arrivaltimes) with heavy-tail distributed inter-arrival times generateslong-range dependence in the counting process. In the lightofthis point, one do not need to conduct analyses of tremendoustraffic traces if there is a solid mathematical work that cansettle this kind of dispute.

In the following theorem, we prove that the per-packetbackoff distribution has a power tail by lighting upon the factthat the moment generating function has arecursive relation,and by applying the theory ofregular variation [10] and theless-knownmodified Tauberian theoremof Bingham & Doney[9]. For your own good, note that this theorem requires onlyK = ∞, nothing aboutN .

Theorem 3 (Power Tail Principle3)The per-packet backoffΩ has a Pareto-type tail with anexponent of−α. Formally,

F cΩ(x) :=

∫ ∞

x

fΩ(x)dx ∼ x−αℓ (x) . (15)

The notationf(x) ∼ g(x) meanslimx→∞ f(x)/g(x) = 1,andℓ (x) is slowly varying4.

Remark 4 This principle, formulated in terms of the ccdfF cΩ(·), not only defines a fundamental characteristic of delay

but also lays the groundwork for further analysis using regularvariation theory.

R4.1 [Answer to Q4] This clear-cut and simple resultreveals the statistical attribute ofΩ for any populationN . Ithas a Pareto-type distribution whose exponent parameter is−α. Theorem 3 proves thestrict-sense heavy-tailednessofΩ for α < 2, and puts an end to the discussions in Section I.

R4.2 This theorem dispenses thecomplicatedconvolutionexpression (4) and leads us to asimplerconclusion. The mostrepresentative distribution of backoff timesΩ is a truncatedPareto-type distribution (though it must be slowly-varying),rather than uniform or exponential as observed in the simula-tion studies of [5], [12].

R4.3 The simplistic termℓ(·) in (15) is irreplaceablewithany other expressions, implying its pivotal role. For instance,Final Value Theorem tells nothing butlimx→∞ fΩ(x) = 0.

The ccdf ofΩ obtained through ns-2 simulations is plottedin Fig. 2 on a log-log scale where the estimated slopesα arecompared with the analytical formulaeα = −(log γ)/ logm,(FPE1) and (FPE2). Observe that these simple formulae alongwith (15) provide a precise estimate for the tail distribution.Remarkably,even for K = 6, i.e., the value adopted in802.11b, the ccdf ofΩ can be accurately approximated bya truncated power-law tail.

3The proof in fact requiresα to be not an integer. For the complicated casewhen α is an integer, we refer to [19] and [10, Theorem 8.1.6]. However,since an integerα can be approximated for any smallǫ > 0 by a real numberα such that|α− α| < ǫ, we expect the result of Theorem 3 to be valid forall α > 0.

4A function f(x) is called regularly varying [10] at infinity of index ρiff limx→∞ f(λx)/f(x) = λρ,∀λ > 0. For the special caseρ = 0, it iscalled slowly varyingand usually denoted byℓ(x). For example, a positiveconstant,(log x)ǫ for any real numberǫ is a slowly varying function. Aslowly varying functionℓ(x) is dominated by any positive power function,i.e., limx→∞ ℓ(x)/xǫ = 0, ∀ǫ > 0.

100

101

102

103

104

105

106

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

per−packet backoff (x)

ccdf

(FΩc

(x))

ւα = 1.01

α = 1.07

ւα = 1.15

α = 1.18

ւα = 1.78

α = 1.83

α = 1.76

α = 1.81ր

simulation: K=6, N=10simulation: K=15, N=10simulation: K=6, N=40simulation: K=15, N=40

Fig. 2. Complementary cumulative distribution functionF cΩ(x) for K =

6, 15; andN = 10, 40.

VI. SHORT-TERM FAIRNESSANALYSIS

First of all, we cancel the assumptionK = ∞ we madein Section V because we present in this section a newapproximation for the superposition process and short-termfairness analysis, both of which will be applicable to bothcasesK < ∞ andK = ∞.

A. Inter-Transmission Probability

The notion ofshort-term fairness[5], [12], [26], definedas the distribution of successful transmissions of nodes for afinite time, has been getting the limelight due to its centralrole in quantifying the behavior of random access protocolsover short time-scales and its close link to access delays.Among the set of nodes1, · · · , N, we tag nodeN , withoutloss of generality. Assume that the tagged node successfullytransmitted a packet at timet = 0. Denote byZn the numberof packets successfully transmitted by noden while the taggednode transmitsζ packets. Recalling thatAn(t) counts thearrivals during the interval(0, t], we can see

Zn := An(t′) wheret′ = mint : AN (t) = ζ.

It is clear thatZN = ζ from the above definition. For short-term fairness analysis, we consider

Z =∑N−1

n=1 Zn .

For the sake of convenience, we denoteP[Z = z|ZN = ζ] byPN [z|ζ]. We call the conditional probabilityPN [z|ζ] inter-transmission probability. In terms of the point processesAn(t), it is equivalent to

PN [z|ζ] = P[∑N−1

n=1 An(∑ζ

j=1 Ωj

)= z]

whereΩj denotes the per-packet backoff for eachjth packetof the tagged nodeN and are i.i.d. copies ofΩ.

9

B. Intermediate Telecom Process on Coarse Time Scales

The premise does not hold: Look into the above superpositionprocess

∑N−1n=1 An(t) where t =

∑ζj=1 Ωj. Recall the basic

premise of Poisson limit theorem (Lemma 2) is that eachcomponent process must become sparse asN grows. It iseasy to see that this premise does not hold any longer herebecauset =

∑ζj=1 Ωj is of order of ζ · Ω in the sense

that E[t] = Θ(ζΩ) and Ω is of order ofN in most cases(See Remark 2). Therefore, we need a new approximationof the superposition process oncoarse time-scalessuch thatt = Θ(N).

Before that, we epitomizetheory of stable law[41, Chapter4] briefly only for the caseα ∈ (1, 2]. Denote bySα(σ, β, µ)Levyα-stable lawswhose four parameters are: theindexα; thescaleparameterσ; the skewnessparameterβ; and the meanµ. If X1, · · · , Xn are i.i.d. copies ofSα(σ, β, µ), they satisfythe stability property which takes the following form

∑mi=1(Xi − µ)

d= m

1α (X1 − µ)

where the notationd= means equality in distribution. The case

α = 2 is singular because we haveS2(σ, β, µ) = N(µ, 2σ2)whereβ plays no role. However, for the rest of casesα ∈(1, 2), there is no closed form expression for its pdf.

Since Lelandet al. [29] created a wave of interest in theself-similarity in the Internet, the probabilistic communityhas been concerned with the limit processes of aggregaterenewal processes under different limit regimes. Here a pointat issue was the order of limit operations,i.e., t → ∞ andN → ∞. Recently, Kajet al. [22], [24], [25] have establisheda fundamental connection between Noah effect and Josepheffect, elucidating the above issue as well.Aggregate Process on Coarse Time Scales: A premise of [24,Theorem 1] is that each component process shouldnot becomesparse asN grows, i.e., inter-arrival times not scaling withN . This premise is fully satisfied when we considerAn(Ωt)instead ofAn(t). In other words, we now viewAn(τ) oncoarse time-scalesτ = Ωt. Also note thatE[An(Ωt)] = t.Then applying [24, Theorem 1] yields to the following resultwhich is applicable to various casesK = ∞, K < ∞, finitetime (which must be large enough though), and infinite time.

Theorem 4 (Dichotomy of Aggregation: Second Part5)Suppose, forK = ∞, the inter-arrival times ofAn(Ωt)has ccdfF c

Ω(Ωx) in (15) which does not vary withN . ForK < ∞, nothing is assumed. Define the centred superpositionprocess

A(t) :=∑N

n=1 An(ζΩt)

−Nζt.

Then, asζ → ∞ andN → ∞, we have

A(t)

ζ

weakly−→ −c ·Yα

(t

c

), for K = ∞, α ∈ (1, 2), (16)

A(t)√Nζ

weakly−→ vΩ ·B(t),

for K < ∞,for K = ∞, α ∈ (2,∞),

(17)

where the scaling constantc := N Ω−αℓ(ζΩ)1/(α−1)/ζ,B(·) is a standard Brownian motion, andYα(·) belongs to

the family of Intermediate Telecom process[25] of index αwhose cgf takes the form

logE[eθYα(τ)

]= τ1−α

α−1

(eθτ − 1− θτ

)

+∫ τ

0

(eθx − 1− θx

) (ατx−α−1 + (2 − α)x−α

)dx. (18)

Proof: First, for K = ∞, the ccdf of inter-arrival timesof An(Ωt) now satisfiesF c

Ω(Ωx) ∼ x−αΩ−αℓ(Ωx)

dueto its scaling. FromE[An(Ωt)] = t, the mean inter-arrivaltime is one. It follows from the underlined assumption thatΩ−αℓ

(Ωx)

does not scale withN and it is a slowly-varyingfunction of x. Applying [24, Theorem 1] yields thatA(t)/ζweakly converges to the process in (16).

For the rest of cases, (i)K < ∞ and (ii) K = ∞ andα ∈(2,∞), we do not need any assumption becauseE[Ω2] < ∞holds both for (i) and (ii) by appealing to Theorem 2. Thesefinite variance cases were analyzed in [36, Section 2.1.3(a)]whose ‘ON/OFF source model’ reduces to our model if weuseµ1 = 1 andµ2 ≫ 1. Remark that it is discussed in [36,Section 2.3] that the order of limit operations does not matterin these cases.

The phrase ‘asζ → ∞ and N → ∞’ is pregnant withmeaning. The fundamental strength of the above theoremfor the case of (16) is in that its result is not subject tothe order of limit operations. Instead, thescaling structurebetweenζ andN , represented byc, determines the kind ofthe approximation in the sense that, asc → 0 and c → ∞,c1/αYα(

tc ) andcHYα(

tc) respectively converges toΛα(t) (α-

stable Levy motion) andBH(t) (fractional Brownian motionof index H = (3 − α)/2), up to constants [22]. For finitec ∈ (0,∞), Yα(

tc) becomes an in-between process. For the

case of (17), even this scaling structure does not matter.It is significant thatc → 0 and c → ∞ respectively

equivalent tolimN→∞ limζ→∞ and limζ→∞ limN→∞ in theliterature. Therefore, the essence of the advance [24, Theorem1] is that it hasemancipated the limit form of the super-position processfrom the order of the two limit operations,widening the applicability of the theory.

Remark 5 Though, for K = ∞, the underlined phrasemakes a strong assumption which is not reasonable in viewof α = −(log γ)/ logm which heavily depends onN , theabove theorem deserves its result in the sense that it suggestsa possible approximation of the backoff process in 802.11,based on the state-of-the-art theory. We will come back to thepreciseness of the approximation later in Remark 7 where weobserve thatα is required to be not too close to1.

R5.1 As we have discussed in Footnote 5 and [41, Chapter9] as well as at the beginning of this section, Poisson approx-imation in Theorem 1 is poor on coarse time-scales,i.e., largetime. Therefore, for short-term fairness analysis, the following

5Consistency between (17) and Theorem 1: SupposeK = ∞ andα ∈ (2,∞) (which is very unlikely asN must be large). Then assumethe superposition processA(ζΩt) is Poisson. For large ζΩ, this Poissonprocess should have a Gaussian marginal distribution with meanNζt andvarianceNζt, whereas the process (17) has meanNζt and variancev2

ΩNζt.

Therefore, Theorem 1 is inconsistent with (17) forvΩ 6= 1. The inconsistencyis due to the premise of Theorem 1,i.e., finite time. A similar remark is givenin [41, Remark 9.8.1].

10

approximations inspired by (16) and (17) are essential:

A(t) ≈ −ζ · c ·Yα

(t

c

), for K = ∞, α ∈ (1, 2), (19)

A(t) ≈√Nζ · vΩ ·B(t), otherwise. (20)

R5.2 [Answer toQ5] It turns out that forK = ∞ andα ∈(1, 2), the superposition processA(ζΩt) =

∑Nn=1 A

n(ζΩt)exhibits long-range dependencedue to theheavypower tailof inter-arrival timesΩ. This process is non-Gaussian and non-stable and has stationary, butstrongly dependent, incrementsin the sense that it has the same covariance as a multiple offractional Brownian motion of indexH = (3 − α)/2 [22].It is also shown in [22] that this process is (both locally andglobally) asymptotically self-similarthough not self-similar.We believe that networking community has been longing fora mathematical evidence which makes extensive simulationsin [37] less necessary.

Turning back to the discussion of inter-transmission proba-bility PN [z|ζ] in Section VI-A, we demonstrate the strength ofthe above approximations in the following corollaries whereζ is now taken to be number of packets transmitted by thetagged node.

Corollary 1 (Asymp. Inter-Transmission Probability)Supposeζ ≫ 1 andN ≫ 1. If K = ∞ along withα ∈ (1, 2),we have

PN [Z = z|ζ] ≈∫ ∞

−∞

∫ q+(τ(y))

q−(τ(y))

Tcτ(y)/c(x)dx · Lv(y)dy

(21)

where q±(τ(y)) := −z ∓ δ − (N − 1)ζ · τ(y) /(ζc), δ =1/2, τ(y) := 1+ζ(1−α)/αℓ0(ζ)·y. Hereℓ0(·) is slowly varyingat infinity, Tc

τ (·) is the pdf ofYα(τ) whose cgf is givenby (18), andLv(·) is the pdf ofSα(1, 1, 0) whoseindex isα = −(log γ)/logm.

Proof: Under the assumptionζ ≫ 1 and N ≫ 1, itfollows from Theorem 4 that

∑N−1n=1 An(ζΩt) can be approx-

imated by an Intermediate Telecom process so that its marginaldistribution takes the form

P[∑N−1

n=1 An(ζΩt

)= z]

≈ P [(N − 1)ζt− ζcYα(t/c) ∈ (z − δ, z + δ)]

= P [Yα(t/c) ∈ (q−(t), q+(t))]

= P[∫ q+(t)

q−(t) Tct/c(x)dx

]. (22)

In the meantime, it follows from the definition of skewnessβandfΩ(−x) = 0, ∀x > 0 that

β := limx→∞2F c

Ω(x)

F cΩ(x)+

∫ −x

−∞fΩ(x)dx

− 1 = 1.

Put t =∑ζ

j=1 Ωj/(ζΩ). Applying the lesser-known stable-law central limit theorem [41, Theorem 4.5.1] to the powertailedness result of Theorem 3, taken together with the factβ = 1, it follows that, forζ ≫ 1,

td≈ 1 + ζ(1−α)/α · ℓ0(ζ) · Sα(1, 1, 0).

Plugging this line into (22) yields (21).

Corollary 2 (Inter-Transmission Probability)Supposeζ ≫ 1 andN ≫ 1. If K < ∞, or K = ∞ alongwith α ∈ (2,∞), we have

PN [z|ζ] ≈ Nm

(z − (N − 1)ζ

(N − 1)√ζvΩ

)(23)

where the CVvΩ is given by (9), andNm(x) := 1√2π

e−x2

2 .

Proof: Likewise, we have

P[∑N−1

n=1 An(ζΩt

)= z]

≈ P[(N − 1)ζt+N

(0, v2Ω(N − 1)ζt

)∈ (z − δ, z + δ)

].

≈ P

[Nm

(z − (N − 1)ζt

vΩ√(N − 1)ζt

)](24)

whereN(µ, σ2

)is the Gaussian random variable with mean

µ and varianceσ2. Puttingt =∑ζ

j=1 Ωj/(ζΩ), t is approxi-mated by

td≈ 1

ζ·N(ζ, v2Ωζ

) d= 1 +

vΩ√ζ·N (0, 1)

for ζ ≫ 1. Thus (24) becomes

∫ ∞

−∞Nm

z − (N − 1)

(ζ +

√ζvΩx

)

vΩ

√(N − 1)

(ζ +

√ζvΩx

)

Nm(x)dx

which is approximated as (23) because the denominatorvΩ(N − 1)1/2(ζ +

√ζvΩx)

1/2 is very large so that the firstpdf of the integrand is concentrated aroundz = (N −1)(ζ +

√ζvΩx

).

Remark 6 The derived equations provide us several penetrat-ing insights and answers toQ2 as well. Note that the meanand variance of (21) are given by

Z :=∑∞

z=0 z · PN [z|ζ] ≈ (N − 1)ζ

σ2Z :=

(∑∞z=0 z

2 · PN [z|ζ])− Z2 ≈ ∞, (25)

while those of (23) are given by

Z ≈ (N − 1)ζ, σ2Z = (N − 1)2ζ · v2Ω. (26)

R6.1 For the case of (23), we can say thatZ is approxi-mately Gaussian for largeζ andN :

Zd≈ N

((N − 1)ζ, (N − 1)2ζv2Ω

)(27)

whereupon the CV ofZ can be computed from (26) as

vZ := σZ/Z ≈ vΩ/√ζ. (28)

Remarkably, we have derived the most general expressionof the inter-transmission probabilityPN [z|ζ] while [5], [12]derived the expressions ofPN [z|ζ] only for N = 2.

R6.2 (21) cannot be simplified in general. However, for forvery largeζ, hence very smallc, it can be easily seen thatZ has a Levyα-stable distribution. Applying [22, Proposition2] to the right-hand side of (16) yields that it is negligible,implying that the inner integral of (21) can be removed. ThenZ becomes approximatelyLevianand is expressed in the form

Zd≈ Sα((N − 1)ζ

1α ℓ0(ζ), 1, (N − 1)ζ).

11

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1

2

3

4

5x 10

−4

K=6 Z = 3900 Z = 5900

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1

2

3

4

5x 10

−4

number of transmissions by N−1 nodes (z)

inte

r−tr

ansm

issi

on p

roba

bilit

y P N[z

|ζ]

K=15 simulation: N=40simulation: N=60analysis: N=40analysis: N=60

Fig. 3. Inter-transmission probabilityPN [z|ζ] for ζ = 100; K = 6, 15;andN = 40, 60.

This manifeststhe heavy-tail ofZ, i.e.,

P[Z > x] ≈ ζ(N − 1)ℓ0(ζ)αCα · x−α (29)

whereℓ0(·) is the same function used inτ(y) in (21) and

Cα = (α− 1)/ (Γ(2− α) sin(π(α − 1)/2)) .

R6.3 For the case ofK = ∞ andα ∈ (1, 2), the representa-tion (29) reveals the striking similarity between the ccdfsof ΩandZ. In terms of regular variation theory, both areregularlyvarying of index−α, and in Mandelbrot’s terminology,Noaheffectof Ω infiltrates into Z.

R6.4 For the case ofK = ∞, the inter-transmissionprobability bifurcates into two different categories atα = 2(or γ = 1/m2). Plainly speaking, ifγ < 1/m2, Z can still beapproximated by the Gaussian distribution in (27), otherwise802.11 suffers from extreme unfairness containing precursorsof power-tailed characteristics such as infinite variancesandthe skewness(β = 1).

R6.5 The skewness inducesleaning tendencyand direc-tional unfairness. The leaning tendency implies the distribu-tion is heavily leaning to the left, and the tendency increasesasα decreases. The directional unfairness6implies that whilethe right part of the inter-transmission probabilityz ∈ (Z,∞)has a heavy power tail given by (29), its left partz ∈ (−∞, Z)decaysfaster than exponentially [41, pp.113].

We conjecture based on extensive simulations thatℓ(·) in(15) is approximately a constant, implying thatℓ0(·) in τ(y)in (21) is also a constant. Then it follows that the constantℓcorresponds to they-intercept of the straight line obtained bytaking logarithms of (15), and can be estimated from Fig. 2.After manipulation akin to [41, Theorem 4.5.2], we can showa simple relation between them:

ℓ0 = (ℓ/Cα)1/α/Ω

6The Levyα-stable law used in this work has support on the entire realline becauseα ∈ (1, 2).

which implies that we need to estimate onlyℓ andα.In Fig. 3, the inter-transmission probability obtained through

ns-2 simulations is compared with the derived formulae ofCorollaries 1 and 2 forζ = 100. It is significant that, forK =6, PN [Z = z|ζ] is well approximated by Gaussian formula(23) along with (9), (FPE1) and (FPE2) for largeN . Thisforms a striking contrast with the caseK = 15 where thedistribution (21) is leaning to the left and its peak isfar apartfrom its mean,i.e., Z = (N − 1)ζ, meaning that there areeven heaviertails on the right part. Our extensive simulationsattested to the inevitability of complicated form (21).

Remark 7 Preciseness of the approximation(19): remains aquestion due to the underlined assumption of Theorem 4. Notethat N is determined byα = −(log γ)/ logm provided thatα is fixed, whereas [24, Theorem 1] demands thatN → ∞provided thatα is fixed. Through extensive simulations, wehave found out that the approximation (19) becomes pooras α → 1 (or as N → ∞). Under the above simulationsetting, ifN > 80, the approximation appears not reasonable.A thorough theory addressing this dependence betweenN andα is left for future work.

VII. WAVELET ANALYSIS OF LONG-RANGE DEPENDENCE

We provide simulation results to support the argument overthe long-range dependence in Section VI-B under the assump-tion K = ∞. Recall from Theorem 4 that the time-scaledversion of the superposition arrival process is approximately

A(ζΩt) =∑N

n=1 An(ζΩt) ≈ Nζt− ζc ·Yα

(tc

)

which holds forN such thatα = −(log γ)/ logm < 2. Notethat suchN is to ensureΩ is strict-sense heavy-tailed(SeeTheorem 3). Then by appealing to [22], one can show thatA(ζΩt) has long-range dependent increments in the sense that

• A(ζΩt) has the same covariance as a multiple of frac-tional Brownian motion of indexH := (3− α)/2.

It is easy to see that1/2 < H < 1 due to1 < α < 2.All simulations obtained fromns-2 simulator use a437h

warm-up period, after which we collected728h-long traces.To analyze these traces, we use the latest addition to the toolkitof inference techniques for long-range dependence,i.e., therefinedwavelet-based method usingDaubechieswavelets withM vanishing moments which was proposed by Abryet al. [2]They proposed the first unbiased estimatoryj taking the form

E[yj ] = log2(E[d2j])

,

considering the complication presented by the propertyE[log(·)] 6= log(E[·]) wheredj is called detail processesofthe wavelet transform.

The estimatesyj of the wavelet spectra over all time-scalesj, called octaves, are shown in Fig. 4 forK = 6, K = 15andK = 25. Here we fix the other parameters asN = 40and M = 2. Though we present here only the simulationresults using Daubechies wavelets withM = 2, we obtainedsimilar results using Daubechies wavelets withM > 2 andDiscrete Meyerwavelets. To quantify the integrity of themethod, Gaussian95% confidence intervals corresponding to

12

2 4 6 8 10 12 14 16 185

6

7

8

9

10

11

Logscale Diagram, [(j1, j2)= (5, 18), s Est.=0.00, H Est.=0.50]

Octave j (time−scale 2j+7 backoff time−slots)

y j

(a) K = 6, N = 40

2 4 6 8 10 12 14 16 185

6

7

8

9

10

11Logscale Diagram

Octave j (time−scale 2j+7 backoff time−slots)

y j

(b) K = 15, N = 40

2 4 6 8 10 12 14 16 185

6

7

8

9

10

11

Logscale Diagram, [(j1, j2)= (12, 18), s Est.=0.66, H Est.=0.83]

Octave j (time−scale 2j+7 backoff time−slots)

y j

(c) K = 25, N = 40

Fig. 4. Wavelet spectra using Daubechies wavelets withM = 2.

the variability of yj are also shown as the vertical segmentscentered on the estimatesyj. Then the measurement of indexH , called Hurst parameter, is reduced to the identificationof region of alignment, the determination of the its lowerand upper cutoff octaves,j1 and j2, respectively, and thedetermination of the slope over the alignment region whichwe denote bys. From the slope estimates, we can obtain theestimates ofH from the formula

H := (1 + s)/2.

Fig. 4(c) demonstrates that, for the caseK = 25, thesuperposition arrival process possesses a sustained correlationstructure over a broad range of time-scalesj ∈ [1, 18] wheresj converges to0.66 at octavej = 18, whereas, for the caseK = 6, it shows a weaker correlation structure over a narrowrangej ∈ [1, 5] as shown in Fig. 4(a). The estimate ofHfor K = 25 over the alignment region(j1, j2) = (12, 18)approachesH = 0.83 around(16, 17) which approximatelymatches with analytical formulaH = (3 − α)/2 = 0.90whereα is obtained fromα = (log γ)/ logm, Eqs. (FPE1)and (FPE2). The slope estimate over the alignment region forK = 6 is computed asH = 0.50, implying that long-rangedependence is not observed. A striking observation that canbe made by comparing Figs. 4(a) and 4(b) with Fig. 4(c) isthat the per-octave slopesj increases as octavej increases andconvergent only ifK is large enough as in Fig. 4(c).

Observation 1 (LRD over coarse times scales)Long-range dependence of the superposition process is con-spicuous only over coarse time-scales.

Remark 8 Essentially, there are two reasons behind this phe-nomenon which also give us answers toQ5.

R8.1 Per-node process slows down: It is important torecall that, forK = ∞, we first established Poisson processapproximation for the superposition process in Theorem 1,meaning that wecannot observe long-range dependence onnormal time-scales. As is the constant intensity of the super-position process for Theorem 1, the constant intensity of thecomponentprocess is essential for Theorem 4. To satisfy thelatter, we had to considerAn(ζΩt) instead ofAn(t) becauseAn(t) becomes sparser asN → ∞. That being said, wemust view the superposition process overcoarsetime-scalesζΩt instead oft to satisfy the premise of Theorem 4, which

explains long-range dependence.R8.2 Additional scaling of time: Another assumption of

the limit regime considered in [24] isζ → ∞ at the same timeasN → ∞. This implies we need additional scaling of timeto compensate for the scaling of space.

R8.3 In practical terms, if the wireless link capacity isshared by many nodes, the aggregate transmission processis highly invulnerable to long-range dependence for mostpracticalK values, essentially due toreduced per-node rateandadditional time scaling.

We also conjecture that the above coarser time scalingscaused the empirical analyses of Veres and Boda [39] (inthe context of TCP) and Tickoo and Sikdar [37] (in thecontext of 802.11) not to support long-range dependence of thesuperposition arrival process of TCP sources — they observedthat H ≈ 0.5 (or s = 0), implying short-range dependence.This is because both 802.11 nodes accessing a common basestation and TCP flows traversing a common bottleneck link(i) have similar backoff mechanisms and (ii) reduce (or slowdown) their transmission rates to share the given capacity asthe population increases.

VIII. C ONCLUDING REMARKS

Beginning with derivation of per-packet backoff distribution,based on which we studied its coefficient of variation that playsa key role in formulating short-term fairness in later sections,we have conducted a rigorous analysis of the backoff processin 802.11 and provided answers to several open questions.

The power-tail principle states that the per-packet backoffhas a truncatedPareto-type tail distribution, a simplistic de-scription elucidating existing works. This in turn indicates thatits heavy-tailedness in the strict-sense inherits fromcollisionand paves the way for the rest of analysis. Thedichotomyof aggregation, proven with the aids of a recent advance [24,Theorem 1] in probabilistic community, now tells the wholestory of contrary limits of the superposition process,i.e., Pois-son process and Intermediate Telecom process, emphasizingthe importance of time-scales on which we view the backoffprocesses. Thanks to the applicability of [24] widened by theorder-free scaling operations of time (ζ) and population (N ),we identified long-range dependencein 802.11 and discov-ered that the inter-transmission probability bifurcates into two

13

categories: either approximately Gaussian or a complicateddistribution which, under a limiting condition, simplifiestoLevy α-stable distribution withα ∈ (1, 2) possessing strongpower-tail characteristics.

Though we have also conducted empirical analysis usingwavelet-based method to support long-range dependence be-havior inherent in 802.11, since we are with Willingeret al.[42] on the point — of cardinal importance is to advanceour genuine physical understanding applicable to many othersystems, we believe that the essence of our analysis of long-range dependence lies in its mathematical explanation for thebehavior. That is, the heavy-tailed inter-arrival time of eachper-node transmission process causes long-range dependenceof the aggregate transmission process at the base stationthough this dependence is seldom observed.

These results explore the fundamental principles character-izing the backoff process in 802.11. Some of them recall toour mind the beauty of simplicity, governing the asymptoticdynamics of 802.11, and the others form the theoreticalgroundwork of short-term fairness.

ACKNOWLEDGMENT

The authors would like to thank Jean-Yves Le Boudecfor helpful discussions and penetrating comments which haveimproved the quality of this work. We also would like to thankIngemar Kaj for elaborating upon his work. Lastly, we wouldlike to thank Hans Almasbakk for making it possible to runvery long simulations on a cluster of a dozen processors.

REFERENCES

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14

APPENDIX

A. Derivation of (9)

Pluggingbk = mkb0 into vΩ = σΩ/Ω yields

vΩ =√

Υ(

∑

k∈K(b0m

k − 1/2)γk)2 − 1. (30)

Here the nominator inside the square root is simplified as

Υ :=(∑K

k=0(b0mk − 1/2)2γk(1 + v2k)

)

+ 2K∑

k=1

(b0mk − 1/2)γk

(b0mk − 1

m− 1− k/2

)

=

(K∑

k=0

(m+1m−1 + v2k

)(b0m

k − 1/2)2γk

)

− 2m−1 (b0 − 1/2)2 −

K∑

k=1

(b0mk − 1/2)γk(k + 2b0−1

m−1 )

=∑K

k=0

(m+1m−1 + v2k

) (b0m

k − 12

)2γk

−(k + 2b0−1

m−1

) (b0m

k − 12

)γk.

Plugging the last line into (30) yields (9).

B. Proof of Theorem 3

Throughout the proof, we denote the sets of real numbers(positive real numbers), integers (positive integers), and ra-tional numbers byR (R+), Z (Z+) and Q, to simplify theexposition. Denoting the LST offi(b) by Fi(s), we begin theproof by considering the LST of (4):

FΩ(s) =1−γγ

∞∑

k=0

k∏

i=0

γFi(s)

︸ ︷︷ ︸G(s)

. (31)

This is an infinite sum of the products of

Fi(s) =1− exp

(−(2b0m

i − 1)s)

(2b0mi − 1)s(32)

that is the LST of the uniform distribution with meanb0mi −1/2. For notational simplicity, we adopt the change of variablex := 2b0s such thatx also belongs toR+. Thus we have

G(x) =

∞∑

k=0

k∏

i=0

γ1− exp

(−(mi − 1/(2b0))x

)

(mi − 1/(2b0))x

. (33)

SinceFi(x) < 1 for x ∈ R+, it is easy to see thatG(x) isconvergent onR+. Then it follows fromBernstein’s Theorem[20, pp.439] thatG(x) is completely monotone. That is,G(x) > 0 and it has derivatives of all orders, which satisfy

(−1)i diG

dxi (x) > 0, ∀i ∈ Z+, (34)

which implies that theith derivative of G(x) is strictlymonotonefor all i ∈ Z+.

Step 1: Recursive relation inG(·)

The crucial observation that paves the way for applyingthe theory of regular variation [10] is the followingrecursiverelation hidden in the underbraced term of (31):

G(x) = γF0(x) 1 +H(mx) (35)

where

H(x) =

∞∑

k=0

k∏

i=0

γ1− exp

(−(mi − 1/(2b0m))x

)

(mi − 1/(2b0m))x

. (36)

Let α = −(log γ)/logm ∈ R+ and z := ⌈α⌉ ∈ Z+ whichdesignates the smallest integer not less thanα. It follows fromα > 0 that z ≥ 1. Appealing to Theorem 2 and the basicproperty of the LST,i.e., lims→0+

dzFΩ

dsz = (−1)zE[Ωz] forz ∈ Z+, it follows that

limx→0+dzGdxz (x) =

1(2b0)z

lims→0+dzGdsz (s)

= γ(2b0)z(1−γ)(−1)zE[Ωz] = (−1)z · ∞.

Recall limx→0+diGdxi (x) is finite for i < z by Theorem 2.

Likewise, it is easy to see thatlimx→0+diHdxi (x) is finite for i <

z and infinite fori ≥ z. Moreover, since both the nominatorand denominator of the limit

limx→0+

dzGdxz (x)dzHdxz (x)

(37)

are infinite, we may remove arbitrary number of productswhosezth derivatives atx = 0+ are finite from both ofG(x)andH(x), implying in turn that we may replace the summationoperation

∑∞k=0 in (33) and (36) by

∑∞k=k′ for any k′ ≥ 0.

Formally speaking, we have

dz

dxzG(x) =

dz

dxz

k′−1∑

k=0

k∏

i=0

γFi(x)

+

dz

dxz

k′−1∏

i=0

γFi(x)

∞∑

k=k′

k∏

i=k′

γFi(x)

+

k′−1∏

i=0

γFi(x)

dz

dxz

∞∑

k=k′

k∏

i=k′

γFi(x)

(38)

where only the third term (38) becomes infinite asx → 0+.Therefore, we can easily see that the difference betweenG(x)andH(x) vanishes ask′ increases and hence (37) must be1.

Taking derivatives of both sides of (35)z times and aftersome manipulation, it becomes clear that it is sufficient toconsider only infinite terms which are related to each other inthe following form:

h(m) := limx→0+

dzGdxz (mx)dzGdxz (x)

= limx→0+

dzHdxz (mx)dzGdxz (x)

limx→0+

dzGdxz (mx)dzHdxz (mx)

= m−zγ−1 = mα−z (39)

where we also exploited the fact that (37) is1. Because theconvergence of (39) holds for any real sequences ofxk → 0+,we have thath(y) = yα−z for y ∈ M whereM := mi | i ∈Z is a countably infinite set that isnowheredense inR+. Theset on which the relationh(y) = yα−z holds is often baptizedquantifier setin regular variation theory.

15

Step 2: Quantifier set is dense inR+

We will show thath(y) = yα−z holds on a dense subsetL

of R+. Define a set

L := λ ∈ R+ | (log λ)/ logm ∈ R\QwhereR\Q is the set of irrational numbers. It should be clearthat M and L are disjoint, i.e., M ∩ L = ∅ and the setLis densein R+ because it can be rewritten asL = my ∈R+ | y ∈ R\Q. Defining

Υ(y, x) := dzGdxz (yx)/

dzGdxz (x),

we can see thatΥ(y, x) is strictly decreasing iny because itfollows from (34), i.e., complete monotonicity, that

dΥ(y,x)dy = d(yx)

dydz+1Gdxz+1 (yx)/

dzGdxz (x) < 0, ∀z ∈ Z+.

Pick λ ∈ L in the interval(mi,mi+1) for any i ∈ Z. SinceΥ(y, x) > 0 is strictly decreasing iny, it is upper-boundedby mi(α−z) as x → 0+, meaning thatΥ(y, x) is ultimatelybounded inx. From its series expansion, it is easy to see thatit is ultimately monotone inx asx → 0+. Then we can apply[33, Theorem 3.14] to show that there existsα such that

h(λ) = limx→0+ Υ(λ, x) = λα−z, (40)

which in turn implies thath(λj) = λj(α−z), ∀j ∈ Z, as (39)did. Assume thatα 6= α. BecauseΥ(y, x) is strictly decreasingin y, irrespective ofz, we have

mα−z ≤ limx→0+ Υ(y, x) ≤ 1 (41)

for y ∈ (1,m). Put y := m−⌊j(log λ)/ logm⌋λj for j ∈ Z. Thiscan be rearranged as

y = mj(log λ)/ logm−⌊j(log λ)/ logm⌋

and(log λ)/ logm is irrational, hence its exponent is on(0, 1)and y is on the interval(1,m). We now have from (39) and(40) that

limx→0+ Υ(y, x) = limx→0+dzGdxz (m

−⌊j log λlog m

⌋λjx)

dzGdxz (λjx)

·dzGdxz (λjx)dzGdxz (x)

= m−⌊ j log λlog m

⌋(α−z)λj(α−z)

= m−⌊ j log λlog m

⌋(α−z)+ j log λlog m

(α−z)

= m( j log λlog m

−⌊ j log λlog m

⌋)(α−z) · λj(α−α).

where the key point is that the second equality follows fromM ∩ L = ∅. Since the last term belongs to the closed interval

I(j) := [mα−zλj(α−α), λj(α−α)]

andα 6= α, we must be able to pickj ∈ Z such thatI(j) doesnot overlap with[mα−z , 1]. This proves by contradiction thath(λ) = λα−z holds forλ ∈ L that is dense inR+.

Step 3: Applying regular variation theoryApplying the ‘Karamata Theorem for monotone functions’

[10, Theorem 1.10.2] to the conclusion we obtained in Step 2establishes thatd

zGdxz (x) is regularly varying(on the right) at

the originx = 0 with indexα − z. Formally speaking,G(s)satisfies

dzGdsz (s) ∼ sα−zℓ∗

(1s

)ass → 0+, (42)

whereℓ∗ (x) is slowly varying at infinityx = ∞, i.e.,

limx→∞

ℓ∗ (yx) /ℓ∗ (x) = 1

for all y ∈ R+. Note that the original Karamata TauberianTheorem in [10, Theorem 1.7.1] and [20, pp.445] cannotbe applied due to the factα − z ≤ 0. These theorems arecomplemented by the modified Karamata Tauberian Theoremin [10, Theorem 8.1.6] and [9], which we apply to (42) toshow (15).