Probabilistic sampling of finite renewal processes

0.82 MB·English

by  Nelson Antunes

Bernoulli 17(4), 2011, 1285–1326
DOI: 10.3150/10-BEJ321
Probabilistic sampling of finite
2
renewal processes
1
0
2
NELSON ANTUNES1,* and VLADAS PIPIRAS2,**
n
a 1University of Algarve/CEMAT, Faro, Portugal and INRIA Paris – Rocquencourt, Paris,
J
France. E-mail: [email protected]
5 2InstitutoSuperiorT´ecnico/CEMAT,Lisboa,PortugalandUniversityofNorthCarolina,Chapel

Hill, NC, USA. E-mail: [email protected]
]
T
S Considerafiniterenewalprocessinthesensethatinterrenewaltimesarepositivei.i.d.variables
. and the total number of renewals is a random variable, independent of interrenewal times. A
h
finite point process can be obtained by probabilistic sampling of the finite renewal process,
t
a where each renewal is sampled with a fixed probability and independently of other renewals.
m
The problem addressed in this work concerns statistical inference of the original distributions
[ of the total number of renewals and interrenewal times from a sample of i.i.d. finite point

processes obtained by sampling finite renewal processes. This problem is motivated by traffic
1
measurements in the Internet in order to characterize flows of packets (which can be seen as
v
finite renewal processes) and where the use of packet sampling is becoming prevalent due to
6
7 increasing link speeds and limited storage and processing capacities.
0
Keywords: IP flows; finiterenewal process; interrenewal times; numberof renewals; sampling;
1
thinning;asymptotic normality; decompounding
.
1
0
2 1. Introduction
1
:
v
1.1. Motivation
i
X
r Thestatisticalandprobabilisticproblemsconsideredinthisworkaremotivatedbyques-
a
tions arising from data traffic analysis in modern communication networks such as the
Internet. Over these networks, information is sent in the form of packets, and packets
are groupedinto flows. (A flow corresponds to a group of packets sharing commonchar-
acteristics.Ideally,aflowcanbe thoughtofasasetofpacketsthatarisesinthe network
through a remote terminal session or a web page download.) Each packet carries infor-
mation about the flow it belongs to and also whether it is the first or the last packet
*Currentaddress:CampusdeGambelas,8900Faro,Portugal.
**Currentaddress:Av.RoviscoPais,1049-001Lisboa,Portugal.
This is an electronic reprint of theoriginal article published by theISI/BS in Bernoulli,
2011, Vol. 17, No. 4, 1285–1326. This reprint differs from theoriginal in pagination and
typographic detail.
1350-7265 (cid:13)c 2011ISI/BS
1286 N. Antunes and V. Pipiras
in the flow. Examining each packet then allows all flows to be reconstructed. Knowing
thestructureofflows(flowlevelcharacteristics)helpsnetworkoperatorsandnetworking
researchersto understand and discover characteristics of data traffic.
Akeydifficulty withcapturingeachpacketisthatthisrapidlyleadsto ahugeamount
of data to store and analyze. For instance, capturing all traffic for a few hours on a
gigabit/sec.link at a medium load level yields severalhundred gigabytes of data. A way
to reduce the volume of data is by sampling packets. One of the simplest approaches is
probabilistic sampling, where each packet, independently of the others, is captured and
analyzed with a fixed probability q. The basic problem, known as the inversion problem
inthenetworktrafficliterature,isthentodeducethestructureoftheoriginalflowsfrom
sampled packets.
This problem has recently attracted much attention in the networking community,
with the focus almost exclusively on inference of the original distribution of flow sizes
(totalnumberofpackets);see[11,17,24].Inthiswork,wetakeacloserlookatstatistical
propertiesofthe previouslyconsideredestimatorofthe originaldistributionofflowsizes
and are also interested in inference of interarrival times between packets from sampled
data. This distribution allows traffic burstiness to be characterized and leads to other
major characteristics such as the duration of a flow.
1.2. Statement of problem and goals
Fromamathematicalstandpoint,flowsandtheirprobabilisticsamplingcanbedescribed
as follows. As suggestedby data traffic measurements (e.g., [18]), a flow can be modeled
as a finite renewal process, where the total number of renewals W is random and the
interrenewaltimes D , i=1,...,W 1,arepositive i.i.d. randomvariables,independent
i
−
of W. Suppose that the finite renewal process is sampled probabilistically with a fixed
probability q to obtain a sampled finite point process. (As shown below, the resulting
sampled point process is generally not a finite renewal process.) The sampled number
of renewals (which could be zero) and sampled interrenewal times (which are available
only when the number of sampled renewals is greater than 1) will be denoted by W
q
and D , i=1,...,W 1, respectively. We illustrate this notation in Figure 1, where
q,i q
−
allcircles (both empty and full) correspondto the finite renewalprocess,and full circles
correspond to the sampled finite point process.
Given N i.i.d. copiesof asampled finite point process(some of whichareempty when
the number of sampled renewals is zero), one of our main goals is to infer the original
Figure 1. Finite renewal process and sampled finitepoint process.
Probabilistic sampling of finite renewal processes 1287
distributionF ofinterrenewaltimesD .ThefocusisonnonparametricinferenceofF ,
D i D
statistical properties of the resulting estimator and its performance in simulations. The
estimator of F involves a nonparametric estimator of the original distribution f of
D W
the total number of renewals W, which has previously been considered in the network
trafficcontext[17].Statisticalpropertiesofthe estimatorof f willalsobe studiedhere
W
for the first time.
1.3. Inference procedure
To be able to make inference about F , we first need to relate it to F , where the
latter indicates the conditional distribDution function of D given W =Dqs,i|(swith i<s).
q,i q
As part of this work (see Theorem 3.1 in Section 3), we show that
∞
F = A F m, (1.1)
Dq,i|s s,m D∗
m=1
X
where m denotes the mth convolution. In particular, the right-hand side of (1.1), and
∗
hence F , does not depend i. We also note for later reference that the definition of
the sequDeqn,ic|es As = As,m m N involves the distribution fW. Somewhat independently
{ } ∈
of the main objectives above, we examine (1.1) for several underlying distributions f ,
W
such as geometric and heavy-tailed, that arise in network traffic studies (e.g., [6]). For
example, in the geometric case, the distribution F does not depend on s, although
thisseemstobeanexceptiontothegeneralrule.WDeq,ai|lssoprovidearesultsimilarto(1.1)
foramultidimensionalvector(D ,...,D ).This allows(conditional)dependencies to
q,1 q,n
be examinedamongsampledinterrenewaltimes. Forexample,whenthe distribution f
W
is geometric, the sampled interrenewal times turn out to be independent, in which case
the sampled finite point process is a finite renewal process. We should also note that
forms of conditioning other than on W =s are possible, such as W s; these will be
q q
≥
discussed briefly.
Having the relation (1.1), we would naturally expect that it can be inverted, in the
sense that
∞
FD= as,mFD∗mq,i|s, (1.2)
m=1
X
where as= as,m m N isobtainedby reversionof As= As,m m N,thatis,their respec-
tive z-trans{forma}tio∈ns (or formal power series) Gas(z){= ∞m}=∈1as,mzm and GAs(z)=
∞m=1As,mzm satisfy Gas(GAs(z))=GAs(Gas(z))=z. TPhe relation (1.2) suggests a
natural estimator of F , defined as (see Section 4.2)
D
P
∞
F = a F m , (1.3)
D s,m D∗q,i|s
m=1
X
b b b
where F is the empirical distribution function of D given W =s, and a is the
Dq,i|s q,i q s
reversion of the sequence A , where the latter is defined as A by replacing f in its
s s W
b b
b
1288 N. Antunes and V. Pipiras
definition by the empirical distribution f . Note that the estimator F is defined for
W D
fixed s and i.
b b
1.4. Contributions to the literature
Undertechnicalassumptions,wewillshowthatF isanasymptoticallynormalestimator
D
of F , namely,
D
b
√N(F F ) d X, (1.4)
D D
− →
d
where X is a limiting Gaussian processband the convergence in distribution holds in
→
a suitable space of functions (Theorem 4.3). The approach and some techniques behind
this result are closely related to the works in the so-called decompounding framework
by Buchmann and Gru¨bel [5], Bøgsted and Pitts [3] (see also [14, 15]). These authors
consideranalogousestimators,butwherethesequenceA in(1.1)isknownandhenceso
s
isitsreversionsequencea .Wethusdeviatefromtheseearlierworksbyassumingthata
s s
also needs to be estimated, which makes the analysis substantially more complex. More
specifically, the limit X in (1.4) can be written as
X=− ∞ (a(s1)∗(ζ◦as))nFD∗nq,i|s+ ∞ as,nnFD∗(qn,i−|s1)!∗Z, (1.5)
n=1 n=1
X X
where (ζ,Z) is a Gaussianprocess, is a composition operation, a(1) is the “derivative”
s
◦
of a and is the convolution.The secondterm in (1.5) is the term found in the decom-
s
∗
pounding literature when A and a are supposed to be known. The first term in (1.5)
s s
is new and accounts for the variations in a here.
s
Sincea involvestheestimatorf viaA ,wealsoneedtheasymptoticnormalityresult
s W s
for fW. Although fW alreadyappearsin tbhe networktraffic literature [17], its statistical
propertibeshavenotbeenstudiedbbeforetobthebestofourknowledge,andtheasymptotic
normbality result isbalso derived here. We show that, under suitable assumptions,
√N(f f ) d S(ξ), (1.6)
W W
− →
whereS(ξ)= S(ξ)w w N isa Gaussiabnprocess(Theorem4.1andProposition4.5).Itis
{ } ∈
interesting to note that the result (1.6) is shown under technical assumptions which do
notcoverdistributionswithheaviertails(suchasheavy-taileddistributions).Itisproved,
however, that the assumptions can be thought of as sharp (in the sense of Proposition
4.4 of Section 4.1). What the asymptotics of f are beyond these assumptions remains
W
an interesting open question.
We wouldliketomakeseveralotherrelatedbcomments.First,itiswell knownfromes-
timationofflowsizesinthenetworktrafficcontextthattheperformanceoftheestimator
f degrades rapidly as the sampling probability q decreases. For example, for q=0.1,
W
the inference becomes impractical for reasonable sample sizes N. We shall discuss this
b
Probabilistic sampling of finite renewal processes 1289
fact in light of the derived asymptotic normality result for f (Sections 4.1 and 5.1) by
W
indicating two different regimes for the estimator performance in terms of the limiting
asymptotic variance. The regimes are defined as: b
stable, if supES(ξ)2 < ;
• w N w ∞
∈ (1.7)
explosive, if supES(ξ)2 = ,
• w N w ∞
∈
whereS(ξ)isthelimitingGaussianprocessin(1.6)andES(ξ)2 isexpressedintermsofq,
w
andf orf .Theperformanceoftheestimatorf issatisfactoryinthestableregime,
W Wq W
butpoorintheexplosiveregime.Forsmallq,f typicallybelongstotheexplosiveregime
W
and hence to the case of poor performance. Analobgous difficulties for smaller q remain
when using the estimator F . Because of these practical considerations, in the network
D
traffic context, alternative sampling schemes have been sought and considered, such as
the sample-and-hold methobd [7, 12], and the dual sampling technique [22]. We plan to
study inference of interrenewal times in one of these frameworks and postpone any real
dataapplicationtofuturework.Thisworkwillthereforebelimitedtoasimulationstudy
(Section 5).
Inanotherdirection,theresultsoncharacterizingthesamplednumberofrenewalsand
sampled interrenewal times, such as the relation (1.1), contribute to a substantial liter-
ature on sampling of point processes. Sampling (also known as thinning) is discussed in
severalmanuscriptssuchas [8, 9, 19].To the bestof ourknowledge,sampling (thinning)
of finite renewal processes has been largely unexplored. The results of our work (Sec-
tion 3) show that sampling of finite renewal processes does not generally lead to finite
renewal processes (in that, conditionally on the number of sampled renewals, sampled
interrenewal times are generally dependent).
1.5. Structure of the paper
The rest of the paper is organized as follows. In Section 2 we collect the notation used
throughouttheworkandincludeotherpreliminaries.Section3concernspropertiesofthe
finite point process obtained from sampling the finite renewal process. Inference of the
originaldistributionsofthetotalnumberofrenewalsandinterrenewaltimesisstudiedin
Section 4. A simulation study can be found in Section 5. For better readability, most of
the proofsare deferredto Appendix A. The main body of the article contains the proofs
of only those results which we consider key in individual sections. Finally, Appendix B
containsatechnicalresultusedinSection4.2,withboundsonremaindertermsinTaylor
expansions of compositions of formal power series.
2. Notation and other preliminaries
Here,weintroducenotationandmakeanumberofassumptionsthatwillbeusedthrough-
out the paper. As in Section 1, we consider a finite renewalprocess consisting of a finite
1290 N. Antunes and V. Pipiras
but random number W (with W 1) of renewals and positive i.i.d. interrenewal times
≥
D , i=1,...,W 1,independentof W.The terminologyoffinite renewalprocesseshere
i
−
follows that of [9], Example 5.3(b), page 125. We denote by D the variable with a com-
mon distribution of D . We also let V denote the total duration of the finite renewal
i
process defined by V = iW=−11Di (if W =1, then V =0). Suppose now that each re-
newal,independently of the others, is sampled with a fixed probability q (0,1) to form
P ∈
a sampled finite point process.We let W denote the total number of sampled renewals,
q
Dq,i, i=1,...,Wq −1, the sampled interrenewal times and Vq = Wi=q1−1Dq,i the total
duration of the sampled finite point process.
P
The following notation will be used many times throughout the paper. For a discrete
randomvariableX,its distributionorprobabilitymassfunction(p.m.f.) willbe denoted
byf (x)=P(X=x).Forexample,weshallwritef (w), f (w),etc.Foracontinuous
X W Wq
random variable Y, its distribution function will be denoted by F (y)=P(Y y) and
Y
≤
its Laplace transform (LT) will be denoted by F (v)= e vyF (dy), v 0 (v R ).
Y − Y +
≥ ∈
We shall also use the conditional distribution functions F (y)=P(Y y W =s) and
Y s q
R | ≤ |
F (y)=P(Y y W s). So, for example, wee shall write F , F , F , etc. In
seYv|esr+al instances,≤we|shqa≥ll use analogous notation, but where Y Dis repDlacedDqb,iy|sa multi-
dimensional. e
beFrso,rwaesdeeqnuoetneceitsaf=or{maanl}np∈oNw0e,rwsehreirees Nor0i=tsNz-∪tr{a0n}sfoarnmdaNtioins tbhyeGseat(zo)f=natu∞nr=a0lannuzmn-.
Conversely, any such formal power series is associated with a sequence. When a stands
P
for a p.m.f. f , we shall also write G in place of G . Since a sequence a will not
X X fX
amn+neecn=essis|oaannrai|l,lyapnbodewGenrao(nszen)rei=egsaGtGiava+e((,zz1w),e=..s.Ph,za∞nml=l)0a=|lason|zu∞nns1e.=I0tnh·s·ee·vneor∞ntamalt=iino0snatananz+c1ne1=s·,·w{·azem+nnsm}hnaa∈llsNsu0os,cediaaetmfienduelwdtiidtaihs-
a= an n Nm and n=(n1,...,nm).
{ } ∈ 0 P P
We shall also use the following common operations on sequences (or their formal
power series) a = {an}n∈N0 and b = {bn}n∈N0. The composition of a and b will be
denoted a b and is defined by its formal power series as G (z)=G (G (z)). The
a b a b
Gktah(k)d(ezr)iv=at◦divkeGoaf(za)/=dz{kan=}n∈N∞n0=kwailnlnb(end−en1o)t·e·d·(nby−ak(k+) =1){zan(n−kk)}.◦nT∈hNe0 raenvderissiodnefionfeadsbey-
quence a= an n N will be defined as a sequence b= bn n N with its formal power
{ } ∈ P { } ∈
series satisfying G (G (z))=G (G (z))=z. The reversion is defined for any sequence
b a a b
a= an n N with a1 =0. As usual, the symbol will stand for convolution in either
{ } ∈ 6 ∗
the discrete or continuous setting. With all these notions for sequences and their formal
power series, we follow the nice monograph of [16].
3. From finite renewal process to sampled finite point
process
Inthissection,westudyseveralcharacteristicsofthesampledfinitepointprocessinterms
of the original finite renewal process. We first briefly consider the number of sampled
renewals and then turn to sampled interrenewal times.
Probabilistic sampling of finite renewal processes 1291
3.1. Number of sampled renewals
The relation between the probability mass functions of the number of sampled renewals
and the number of original renewals is given by
∞
f (s)= P(W =sW =w)P(W =w)
Wq q |
w=s
X (3.1)
∞ w
= qs(1 q)w sf (w), s 0.
− W
s − ≥
w=s(cid:18) (cid:19)
X
The function G (z) is given by
Wq
∞ ∞ w
G (z)= zs qs(1 q)w sf (w)
Wq s − − W
s=0 w=s(cid:18) (cid:19)
X X
∞ w
= f (w)(zq+(1 q)) (3.2)
W
−
w=1
X
=G (zq+(1 q)), z <1.
W
− | |
The same relations (3.1) and (3.2) also appear in [17] and will be used in Section 4.1 to
obtain inversion results.
3.2. Sampled interrenewal times
Giventhat s renewalshavebeensampled,the nextresultcharacterizesthe LTofthe ith
(i<s) sampled interrenewal time in terms of the LT of the original interrenewal times.
Theorem 3.1. The LT of D given W =s can be expressed as
q,i q
F (v)=G (F (v)), 1 i<s,v 0, (3.3)
Dq,i|s As D ≤ ≥
where As ={As,m}m∈eN and As,m deneotes the probability that the number of original
renewals between the ith and (i+1)th sampled renewals is equal to m 1 given W =s,
q
−
qs ∞ w m
As,m= fW(w) − (1 q)w−s. (3.4)
f (s) s 1 −
Wq w=s+m 1 (cid:18) − (cid:19)
X−
Proof. For 1 i<s and t 0, we have
≤ ≥
∞
P(D t,W =s)= f (w)P(D t,W =sW =w)
q,i q W q,i q
≤ ≤ |
w=s
X (3.5)
∞
= f (w)P(W =sW =w)P(D tW =w,W =s).
W q q,i q
| ≤ |
w=s
X
1292 N. Antunes and V. Pipiras
LetM denotethenumberoforiginalrenewalsnotsampledbetweenthe ithand(i+1)th
sampledrenewalsplus 1.If the number oforiginalrenewalsis W =w andthe number of
sampled renewals is W =s, then M can take the values 1,2,...,w s+1, and
q
−
w s+1
−
P(D tW =w,W =s)= F m(t)P(M =mW =w,W =s). (3.6)
q,i≤ | q D∗ | q
m=1
X
By considering the location of the ith sampled renewal in the total number of renewals
w along with the possible distinct locations of the sampled renewals before the ith and
after the (i+1)th sampled renewals, we obtain that
w m (s i 1)
− − −− l 1 w m l w
P(M =mW =w,W =s)= − − −
q
| i 1 s i 1 s
l=i (cid:18) − (cid:19)(cid:18) − − (cid:19)(cid:30)(cid:18) (cid:19)
X (3.7)
w m w
= − ,
s 1 s
(cid:18) − (cid:19)(cid:30)(cid:18) (cid:19)
where l is the location of the ith sampled renewaland the last equality follows from the
identity
x b
− l 1 x l x
− − = , a 1,b 0,a+b x. (3.8)
a 1 b a+b ≥ ≥ ≤
l=a(cid:18) − (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
X
By using (3.5)–(3.7), we deduce that
P(D t,W =s)
q,i q
F (t)= ≤
Dq,i|s fWq(s)
w s+1
1 ∞ − w m
= f (s) fW(w)qs(1−q)w−s s−1 FD∗m(t) (3.9)
Wq w=s m=1 (cid:18) − (cid:19)
X X
∞
= A F m(t),
s,m D∗
m=1
X
where A is given by (3.4). The relation (3.3) follows by taking the LT in (3.9). (cid:3)
s,m
Theorem 3.1 implies that, when conditioning on W =s, the distribution of the ith
q
(i<s) sampled interrenewal time depends, in general, only on s and not on i, and
hence that the times between consecutive sampled renewals are identically distributed
conditionally on W . In contrast to the original finite renewal process which assumes
q
independence of W and D , the sampled quantities W and D are dependent.
i q q,i
Probabilistic sampling of finite renewal processes 1293
Other forms of conditioning on the number of sampled renewals are possible. For
example, from (3.3), the LT of D given W s can be expressed as
q,i q
≥
∞
FDq,i|s+(v)= FDq,i|s′(v)P(Wq=s′|Wq≥s)
s′=s
X
e e
1 ∞ ∞
= P(W s) As′,mFD(v)mfWq(s′) (3.10)
q≥ s′=sm=1
X X
e
=G (F (v)),
As+ D
where As+ ={As+,m}m∈N and As+,me is the probability that the number of original
renewalsbetweenthe ithand (i+1)thsampledrenewalsis equalto m 1 givenW s,
q
− ≥
As+,m= P(W1 s) ∞ ∞ fW(w) ws−m1 qs′(1−q)w−s′
q≥ s′=sw=s′+m 1 (cid:18) ′− (cid:19)
X X − (3.11)
= q(1−q)m−1 ∞ fW(w) 1 s−2 w−m qs′(1 q)w−m−s′ .
P(Wq≥s) w=m+s 1 −s′=0(cid:18) s′ (cid:19) − !
X− X
In the rest of the paper, we shall focus only on the conditioning W =s, but similar
q
results can be derived for other forms of conditioning such as W s.
q
≥
The next result gives the joint LT for a finite number of sampled interrenewal times.
We shall use this generalresult below to investigate dependence between sampled inter-
renewal times (see Section 3.2.1); see Appendix A for the proof.
Theorem 3.2. The joint LT of D =(D ,...,D ) with 1 i < <i given
q,n q,i1 q,in ≤ 1 ··· n
W =s, s>i , can be expressed as
q n
FDq,n|s(v)=GBs(FD(v1),...,FD(vn)), v=(v1,...,vn)∈Rn+, (3.12)
where Bse= Bs,m m Nn weith m=(me1,...,mn) and Bs,m denotes the probability that
{ } ∈
thenumberofrenewalsbetweenthei thand(i +1)thsampledrenewalsisequaltom 1,
j j j
−
1 j n, given W =s:
q
≤ ≤
qs ∞ w m
Bs,m= fW(w) − (1 q)w−s (3.13)
f (s) s n −
Wq w=s+m n (cid:18) − (cid:19)
X−
with m=m + +m .
1 n
···
ByTheorem3.2,the jointdistributionofasubsetofsampledinterrenewaltimes given
W =s depends only on s and not on the indices of the sampled times considered.
q
Another relation of interest is between the duration of the sampled finite point process
and the original interrenewal times; see Appendix A for the proof.
1294 N. Antunes and V. Pipiras
Proposition 3.1. The LT of V given that W 2 can be expressed as
q q
≥
F (v)=G (F (v)), v 0, (3.14)
Vq|2+ C D ≥
where C= Cm m N and Cem is the probabielity that the number of renewals between the
{ } ∈
first and last sampled renewals is equal to m 1, given W 2:
q
− ≥
q2 ∞
C = f (w)(w m)(1 q)w m 1. (3.15)
m W − −
P(W 2) − −
q
≥ w=m+1
X
SincetheLTofthedurationoftheoriginalfiniterenewalprocessV = iW=−11Di,given
W 2, can be expressed as
≥ P
1 ∞
F (v)= f (w+1)F (v)w,
V|W≥2 P(W 2) W D
≥ w=1
X
e e
we expect that
∞
F (v)= D F (v)n, (3.16)
D n V W 2
| ≥
n=1
X
where D= Dn n N is the reveersion of fW(we +1)/P(W 2) w N (defined in Section
{ } ∈ { ≥ } ∈
2).Byplugging(3.16)into(3.14),wecanobtainanexpressionforthedurationV ofthe
q
sampled finite point process in terms of the duration V of the finite renewal process.
Also, note that, equivalently, the relations (3.3), (3.12) and (3.14) can be expressedin
termsofthedistributionsfunctions(see(3.9),andrelations(A.4)and(A.7)inAppendix
A, respectively) or via the characteristic functions.
3.2.1. Examples
In the following, we examine characteristics of the sampled finite point process for par-
ticular distributions of W.
Geometric distribution
Suppose that W is a geometric random variable with parameter c, that is, f (w)=
W
cw 1(1 c),w 1,c (0,1).Substitutingthisf (w)into(3.4),weobtain,afterstraight-
− W
− ≥ ∈
forward calculations, that
As,m=(c(1 q))m−1(1 c(1 q)), m 1.
− − − ≥
Therefore, A is the p.m.f. of a geometric distribution with parameter c(1 q) which
s
−
does not depend on s. From (3.3) and using the generating function of the geometric
distribution, we have
(1 c(1 q))F (v)
D
F (v)= − − .
Dq,i|s 1 c(1 q)F (v)
D
− − e
e
e