Forward Correction and Fountain codes in Delay Tolerant Networks

0.21 MB·English

by  Eitan Altman

Forward Correction and Fountain Codes in Delay Tolerant Networks
Eitan Altman Francesco De Pellegrini
INRIA, 2004 Route des Lucioles, CREATE-NET, via Alla Cascata 56 c,
06902 Sophia-Antipolis Cedex, France 38100 Trento, Italy
[email protected] [email protected]
Abstract—DelaytolerantAd-hocNetworksleveragethemobil- After fragmenting the message into smaller frames, it is
ityofrelaynodestocompensateforlackofpermanentconnectiv- convenientto better organizethe way informationis stored in
ityandthusenablecommunicationbetweennodesthatareoutof the relay nodes. We aim at improving the efficiency of the
9 range of each other. To decrease delivery delay, the information
DTN’s operation by letting the source distribute not only the
0 tobedeliveredisreplicatedinthenetwork.Ourobjectiveinthis
0 paper is to study a class of replication mechanisms that include original frames but also additional redundant packets. This
2 codinginorder toimprove theprobabilityof successfuldelivery results in a spatial coding of the distributed storage of the
n withinagiventimelimit.Weproposeananalyticalapproachthat frames. We consider coding based on either forward error
a allows to quantify tradeoffs between resources and performance correction techniques or on network coding approaches. Our
J measures (energy and delay). We study the effect of coding on maincontributionistoprovideacloseformexpressionforthe
9 theperformanceofthenetworkwhileoptimizingparametersthat performance of DTNs (in terms of transfer delay and energy
1 govern routing. Our results, based on fluid approximations, are
consumption) as a function of the coding that is used. Also,
compared to simulations which validate the model.1.
we derive scaling laws for the success probabilityof message
] IndexTerms—Forwardcorrection,fountaincodes,delaytoler-
I ant networks delivery.
N
The paper is organized as follows. In Sec. I we revise the
.
s state of art and outline the major contributions of the paper.
c I. INTRODUCTION In Sec. II we describethe modelof the system and in Sec. III
[
Delay tolerant Ad-hoc Networks make use of nodes’ mo- we derive the main results for the case of erasure codes.

3 bility to compensate for lack of instantaneous connectivity. Sec. IV is devoted to the analysis of fountain codes; for both
v Information sent by a source to a disconnected destination cases, we discover and then study an interesting phenomenon
7
can be forwarded and relayed by other mobile nodes. There of phase transition. Sec. III and IV involve also the design of
4
7 has been a growing interest in such networks as they have energy-aware forwarding policies where the source forwards
3 the potential of providing many popular distributed services packets with some fixed probability that we optimize. In
. [1]–[3]. Sec. V we study an alternative class of forwarding policies,
8
0 A naive approach to forward a file to the destination is by namely threshold type policies, that achieve the same energy
8 epidemic routing in which any mobile that has the message restrictions. The performances of the aforementioned coding
0 keeps on relaying it to any other mobile that falls within its techniques in the case threshold policies are then derived.
:
v radio range. This would minimize the delivery delay at the Sec. VI reports on simulation results in case of synthetic
i cost of inefficient use of network resources (e.g. in terms mobility and real-world traces. A concluding section ends
X
of the energy used for flooding the network). The need for this paper.
r
a more efficient use of network resources motivated the use of
less costly forwarding schemes such as the two-hops routing Related Works The idea to erasure code a message and
protocols. In two-hops routing the source transmits copies of distribute the generated code-blocks over a large number of
its message to all mobiles it encounters; relays transmit the relays in DTNs has been addressed first in [5] and [6]. The
message only if they come in contact with the destination. technique is meant to increase the efficiency of DTNs under
The two-hops protocol was originally introduced in [4]. uncertain mobility patterns. In [5] the performance gain is
In this paper we consider another aspect, i.e, the tradeoff comparedto simple replication,i.e. the techniqueof releasing
betweennetworkresourcesand delay.We assume thatthe file additionalcopiesof the same message. The benefitof erasure
to be transferred needs to be split into K smaller units: this codingis quantifiedin thatworkvia extensivesimulationsfor
happensdue to the finite durationof contactsbetween mobile various routing protocols, including two-hops routing.
nodes or when the file is large with respect to the buffering In [6], the case of non-uniform encounter patterns is ad-
capabilities of nodes. Such K smaller units (which we call dressed,showingthatthereisstrongdependenceoftheoptimal
chunks or frames) need to be forwarded independentlyof the successful delivery probability on the allocation of replicas
others. The message is considered to be well received only if over different paths. The authors evaluate several allocation
all K frames are received at the destination. techniques; also, the problem is proved to be NP–hard.
Generalnetworkcodingtechniques[7] have been proposed
1Aversionofthispaperappears intheproc.ofIEEEINFOCOM2009. for DTNs. In [8] ODE based models are proposed under epi-
2
Symbol Meaning
demic routing. Semi-analytical numerical results are reported
N numberofnodes(excluding thedestination)
describing the effect of finite buffers and contact times; a K numberofframescomposingthemessage
prioritization algorithm is also proposed. M number offrames needed todecode with success probability
The work in [9] addresses the use of network coding 1−δ,δ>0(fountain codes)
H numberofredundant frames
techniques for stateless routing protocols under intermittent
λ inter-meeting intensity
end-to-end connectivity. A forwarding algorithm based on τ timeout value
network coding is specified, showing a clear advantage over Xi(t) number of nodes having frame i at time t (excluding the
destination)
plain probabilistic routing in the delivery of multiple packets.
X(t) sumoftheXis
Finally, an architecture supporting random linear coding in X(t) sumoftheXiswhenui(t)=1,∀i=1,2,...,K
challenged wireless networks is reported in [10]. E(t) energyexpenditure bythewholenetworkin[0,t]
x maximumnumberofcopies duetoenergyconstraint
z :=X(0)
Novel contributions ε energyperframe
The main contribution of this paper is the closed form ui(t) forwarding policyforframei
description of the performanceof Delay tolerant Ad-hoc Net- pi static forwarding policyforframei;p=(p1,p2,...,pK)
p sumofthepis
works under the two-hops relaying protocol when a message Di(τ) probability ofsuccessfuldelivery offrameibytimeτ
is split into multiple frames. Our fluid model accounts both Ps(τ) probability of successful delivery of the message by time τ;
for the overhead of the forwarding mechanism, captured in Ps(τ,K,H)isusedtostressthedependence onK andH
the form of a given bound on energy, and the probability of TABLEI
successful delivery of the entire message to the destination MAINNOTATIONUSEDTHROUGHOUTTHEPAPER
within a certain deadline. The effect of coding is included
in the model and both erasure codes and fountain codes are
accounted for in closed form. The two coding strategies are
The source has a message that contains K frames. If at
characterized in the case of static probabilistic forwarding
timetitencountersamobilewhichdoesnothaveanyframe,it
policies and in the case of threshold policies.
givesitframeiwithprobabilityu ,andweletu= u 1
Leveraging the model, the asymptotic properties of the i i i ≤
(we shall consider both the case where u depends on t and
i
system are derived in the form of scaling laws. In particular, P
thecasewhereitdoesnot).Forthemessagetoberelevant,all
there exists a threshold law ruling the success probability
K framesshouldarriveatthedestinationbytimeτ.LetX (t)
i
which ties together the main parameters of the system.
bethenumberofthemobilenodes(excludingthedestination)
Tothebestofthe authors’knowledge,theresultscontained
that have at time t a copy of frame i. Denote by D (τ) the
i
in this work represent the first description in closed form of
probabilityofasuccessfuldeliveryofframeibytimeτ.Then,
thebehavioroferasurecodesandfountaincodesinchallenged
giventhe processX (forwhich a fluid approximationwill be
i
networks.
used), we have
II. THEMODEL D (τ) =1 exp λ τ X (s)ds
i − − 0 i
Fortheeaseofreading,themainsymbolsusedinthepaper
This expression has been deriv(cid:0)ed inR[13] for th(cid:1)e fluid model.
are reported in Tab. I.
The probability of a successful delivery of the message by
Consider a network that contains N + 1 mobile nodes.
time τ is thus
We assume that two nodes are able to communicate when
they are within reciprocal radio range, and communications K K τ
are bidirectional. We also assume that contact intervals are Ps(τ)= Di(τ)= 1 exp λ Xi(s)ds
− −
sufficient to exchange all frames: this let us consider nodes iY=1 iY=1(cid:20) (cid:18) Z0 (cid:19)(cid:21)
meeting times only, i.e., time instants at which a pair of not where we assumed that the success probability of a given
connected nodes fall within reciprocal radio range. frame is independent of the success probability of other
Also, let the time between contacts of pairs of nodes be frames; this decoupling assumption is confirmed by our nu-
exponentiallydistributed with given inter-meetingintensity λ. merical experiments.
The validity of this model been discussed in [11], and its
A. Fluid Approximations
accuracy has been shown for a number of mobility models
(Random Walker, Random Direction, Random Waypoint). 2 Let X(t) = K X (t). Then we introduce the following
i=1 i
We assume that the transmitted message is relevant during standard fluid approximation (based on mean field analysis)
P
some time τ. We do not assume any feedback that allows the [14]
source or other mobiles to know whether the messages has dXi(t) =u (t)λ(N X(t)) (1)
made it successfully to the destination within time τ. dt i −
Takingthe sum overall i, we obtainthe separabledifferential
2Werecallthatstudiesbasedontracescollectedfromreal-lifemobility[2]
equation
arguethatinter-contacttimesmayfollowapower-lawdistribution,butrecently
the authors of [12] have shown that these traces and many others exhibit dX(t)
=u(t)λ(N X(t)) (2)
exponential tails afteracutoffpoint. dt −
3
whose solution is C. Non-constrained problem.
The success probability when using p is
X(t)=N +(z N)e−λR0tu(v)dv, X(0)=z
− K τ
P (τ,p)= 1 exp( λ X (v)dv)
s i
Thus, X (t) is given by the solution of − −
i Yi=1(cid:16) Z0 (cid:17)
dXdit(t) =−ui(t)λ(z−N)e−λR0tu(v)dv (3) = iY=K1h1−exp(cid:16)− λp Z0τNpi(cid:16)1−e−λpv(cid:17)dv(cid:17)i
K
Constant policies
= Z(p ) (6)
i
In the case of constant policies, we let ui(t) = pi, p := iY=1
(p ,p ,...,p ), and p= p . Hence it follows
1 2 K i i where Z(pi):=1 exp L(τ,p)pi and (7)
−
P p (cid:16) (cid:17)
Xi(t)=Xi(0)+(N −z)pi 1−e−λpt (4) L(τ,p):= N 1 λpτ e−λpτ
p2 − −
(cid:2) (cid:3) (cid:16) (cid:17)
For fixed ratios p /p, P (τ,p) is increasing in p and is
Let us assume X (0)=0 for i=1,2,...,K: hence, i s
i
∀ maximized at p=1.
p Let P∗(τ) be the optimal delivery probability for the
X (t)=N i 1 e−λpt (5) s
i problem of maximizing P (τ) with p fixed. The proof of the
p − s
(cid:16) (cid:17) following can be found in the Appendix.
Theorem 2.1: p∗ = (p/K,...,p/K) is the unique solution
B. Taking Erasures into Account
to the problem of maximizing P (τ) s.t. p =p, p 0.
s i i i ≥
So far we have assumed that the transmission of a frame is
D. Constrained problem. P
always successful. Assume that this is not the case and that
the transmission of a frame fails with some probability q. We DenotebyE(t)theenergyconsumedbythewholenetwork
assume that the process describing whether packets transmis- for the transmission of the message during the time interval
sions are successful or not is i.i.d. We assume moreover that [0,t]. It is proportionalto X(t) X(0) since we assume that
−
a packet that suffers from unrecoverable transmission errors themessageistransmittedonlytomobilesthatdonothavethe
at a mobile is discarded so that it does not occupy memory message,andthusthenumberoftransmissionsofthemessage
space in the relay node; this ensures that such a mobile node during[0,t]plusthenumberofmobilesthathaditattimezero
canstillactasarelayatthenextmeetingwithanothermobile equalstothenumberofmobilesthathaveit.Also,letε>0be
having a packet to be transmitted. theenergyspentto forwarda frameduringa contact.We thus
haveE(t)=ε(X(t) X(0)). In the followingwe will denote
Losses such as those just described do not need an extra
−
x as the maximumnumberof copies that can be released due
modeling:wemayreplacetherateλofinter-meetingsbetween
to energy constraint.
twonodesbyλ(1 q),i.e.,therateofthepotentiallysuccessful
− We compute in particular the optimal probability of suc-
inter-meetingsofthe nodes.Thiscan beused inthe equations
cessful delivery of the message by some time τ under the
that we derived in describing the dynamics of the system and
constraint that the energy consumption till time τ is bounded
its performance measures.
by some positive constant.
An additional type of loss may occur at the destination:
Define X(t) to be the solution of (2) when u(t)=1, i.e.
this is the case when it is not mobile and it is connected
to an external network (possibly a wired one): in this case X(t)=N +(z N)e−λt
losses may occur in that part of the network. Assume that a −
loss there occurs with probability q′. We do not assume any Note that X(t)=X(pt).
feedbackthatwouldallowtheDTNtoknowabouteventsthat −1
Denote σ(z) := X (x +z) given X(0) = z, which is
occur at the external network. In order to be able to recover
the time elapsed until x extra nodes (in addition to the initial
from such losses we assume that the destination may keep
z ones) receive the message in the uncontrolled system. We
receiving copies of the same frame. In particular, a mobile
have
that has transmitted a frame to the destination will keep the 1 N x z
copy and could try to retransmitit to the destination at future σ(z)= log − − (8)
−λ N z
inter-meeting occasions. This additional type of loss process (cid:18) − (cid:19)
does not alter the fluid dynamics of X (t). Its impact on the For p=σ(z)/τ we obtain the expression
i
performance is by replacing λ in (1) by λ(1 q′).
− N x x
L(τ,p)= log 1 + (9)
p2 − N z N z
(cid:16) (cid:16) − (cid:17) − (cid:17)
4
Theorem 2.2: Consider the problem of maximizing P (τ) (iii) If X(τ) > z + x > X(u τ) (or equivalently,
s min
subject to a constraint on the energy E(τ) εx. τ > σ(z) > u τ), then the best control policy is given by
min
≤
(i)IfX(τ) x+z (orequivalently,τ σ(z)),thenacontrol pi =p∗/(K +H) where p∗ is given in (10) and the optimal
≤ ≤
policy u is optimal if and only if p =1/K for all i. value is
i
(ii) If X(uminτ) > z +x (or equivalently, uminτ > σ(z)), K+H
then there is no feasible control strategy. P∗(τ,K,H)= B(p,K+H,j)
s
(iii) If X(τ) > z + x > X(uminτ) (or equivalently, τ > jX=K
σ(z)>uminτ), then the best control policy is given by pi = where b
p∗/K where
p=1 exp(L(p∗,τ)p∗/(K+H))
σ(z) −
p∗ = (10)
τ B. Properties and Approximations
b
and the optimal value is We now derive further characterizations for the optimal
success probability; these results will provide both bounds
P∗(τ)= 1 1 x p∗NK exp N x K for the case of block coding and help also in the analysis
s − − N z − p∗K N z of fountain codes.
h (cid:16) − (cid:17) (cid:16) − (cid:17)i
Corollary 3.1: P∗(τ,K,H) is increasing with H.
Proof:Part(i)and(ii)areobvious.Part(iii)followsfrom s
Proof: We make the following observation. Fix K and
the fact that X σ(z)τ = X(σ(z)) = x + z, so that the
τ H and a vector (p ,...,p ) whose entries sum up to p∗
1 K+H
energy bound is(cid:0)attaine(cid:1)d for p∗ = σ(τz); also, the expression (given in (10)). Then the success probability is the same as
forP∗(τ) followsfromanimmediateapplicationoftheresult whenincreasingtheredundancytoH+1andusingthevector
s
in Thm. 2.1 to (7). (p ,...,p ,0). By definition, the latter is strictly smaller
1 K+H
thanP∗(τ,K,H+1)(whichistheoptimalsuccessprobability
III. ADDING FIXED AMOUNT OF REDUNDANCY s
with H +1 redundant packets).
We addH redundantframesandconsiderthenew message The above is in particular true when taking p =
i
that now contains K + H frames. If at time t the source p∗/(K + H) (which maximizes P (τ,K,H)), and hence
s
encounters a mobile which does not have any frame, it gives P∗(τ,K,H)<P∗(τ,K,H +1).
s s
it frame i with probability p. Now we provide two useful bounds for P∗(τ,H,K).
s
Let Sn,p be a binomially distributed r.v. with parameters n First, it is possible to derive the asymptotic approximation
and p, i.e. of P∗(τ,H,K) for H . For the sake of notation, let
s → ∞
V =L(p,τ)p; the following holds
n
P(S =m)=B(p,n,m):= pm(1 p)n−m
n,p
m − H+K
The probability of a successful del(cid:18)ivery(cid:19)of the message by Ps∗(τ,H,K)=eV H +s K e−HV+K −1 s
time τ is thus sX=K (cid:18) (cid:19)(cid:16) (cid:17)
K−1
H +K V
K+H =1 eV − +o((H +K)−s)
− s (H +K)s
Ps(τ,K,H)= B(Di(τ),K +H,j), Xs=0 (cid:18) (cid:19)(cid:16) (cid:17)
j=K
X For large values of H +K, the s-th term of the right-hand
where D (τ)=1 exp( λ τX (s)ds). summation writes
i − − 0 i
A. Main Result R H +K −V s = (H +K)! (−V)s
s H +K s!(H +K s)!(H +K)s
Lemma 3.1: The maximum of Ps(τ,K,H) over pm (cid:18) (cid:19)(cid:16) (cid:17) −
0,m = 1,...,K + H} under the constraint Ki=+1H{pi =≥p ∼ (−s!Ves)s 1 H+K ∼ (−sV!)s
is achieved at pm =p/(K+H). 1 s
P − H+K
The proof is delayed to the Appendix.
(cid:16) (cid:17)
UsingthesameargumentsasthosethatledtoTheorem2.2, where the Stirling formula applies, n! √2πn (n)n, with
together with Lemma 3.1 yields the following: ∼ e
f g meaning lim f/g =1.
H→∞
Theorem 3.1: Consider the problem of maximizing ∼
Corollary 3.2: For K 1 and τ 0,
P (τ,K,H) over p ,i=1,...,K+H, subject to a constraint ≥ ≥
s i
on the energy E(τ) εx. K−1[ L(p,τ))p]s
≤ lim P∗(τ,H,K)=1 eL(p,τ)p −
(i)IfX(τ)≤x+z (orequivalently,τ ≤σ(z)),thenacontrol H→∞ s − s!
policy u is optimal if and only if pi =1/(K+H) for all i. Xs=0
(ii) If X(u τ) > z +x (or equivalently, u τ > σ(z)), We note that in sight of the monotonicity stated in Cor. 3.1,
min min
then there is no feasible control strategy. the limit is reached from below.
5
K=25, N=300, τ=3000 s N=300, τ=3000 s τ=3000 s
K=10
a) 1 λ1 b) 1 c) 1 Lower Bound
Optimal prob. Upper Bound
Asymptotic Bound
0.8 0.8 K=25 Binomial Bound 0.8
Success probability00..46 OABsipnytoimmmpaitalo lp tBircoo bBu.onudnλd2 Success probability00..46 K=60 Success probability00..46 N =50,125,300,500,5000
0.2 0.2 0.2
λ3
0 0 0
0 50 100 150 200 250 0 50 100 150 0 0.05 0.1 0.15
Number of redundant frames H Number of redundant frames H Normalized number of frames K/N
Fig. 1. a) Success probability for N = 300, K = 25, p = 1 under different values of λ; λ1 = 0.22·10−03s−1, λ2 = 0.091·10−03s−1 λ3 =
0.065·10−03s−1 b) Success probability for λ1, N = 300, p = 1 under different values of K c) Upper and lower bound p = 1, δ =0.02 as given in
Cor.4.1,forvarious values ofN =50,125,300,500,5000,versusnormalized numberoftransmitted framesK/N.
Also, the binomial bound [15, Thm 1.1] applies. For 1 the limit x:=lim x(N)/N >0 exists 3.
N→∞
≤
m n 1, and u>1, define u through m= upn . Then Proposition 3.1: Introduce the threshold
≤ − ⌈ ⌉
b
n1/2 u1−upn 1 p (1−up)n Γ :=λτ 1+ x ,
P(Sn,p ≥m)≤ (2πm(n−m))1/2 1−u (cid:18)1−−up(cid:19) 0 (cid:16) log(1b−x)(cid:17)
then, the following holds:
In Fig 1a) we reported the numerical comparison of the b
optimaldeliveryprobabilityasafunctionofthenumberofre- 0 if K+H ≤Γ0
disunrdepanotrtferdamaess,afnorhaopriazrotinctuallardoset-tstilnagsh.eTdheliansey;mtphteotbicinboomuinadl limN→∞Ps(τ,K,H)=8>>><0 if Kb +Hb >Γ0 and K >Γ0
bound is reported with slashed line. We notice that the bi- 1 if Kb +Hb >Γ0 and Kb <Γ0
nomial boundprovesa good approximationfor lower success The proof is in the Appen>>>:dix. Webconcblude that thbere exist a
probabilities,i.e., at smaller valuesof λ. The relative increase phase-transitioneffect.ItsthresholdΓ is the same as thatwe
0
of the successprobabilityundererasure codesis apparent:for shall obtain later for the fountain codes.
instance, the success probability with λ = 0.22 10−03s−1 The way we interpretsuch result is that, in order to deliver
·
increases from 0.12 for H = 0 to one with a few redundant withhighprobabilitythemessage,forlargeN,theK message
frames (H = 12); the maximum attainable improvement, frames should not exceed such threshold, but the sum of the
though, is dictated by the upper bound. encoded ones should.
In Fig 1b) the optimal delivery probability is depicted as a
IV. FOUNTAIN CODES
function of the number of redundant frames, at the increase
of K; even in this case the binomial bound proves a better Each time the source meets a node, it sends to it (with
approximationfor lowersuccess probabilities,asit appearsin probability p) a packet obtained by generating a new random
the case of K =60. linear combination of the K original packets. Using fountain
codes,weknowthatforanyδinorderforthedestinationtobe
C. Phase transition able to decode the original message with probability at least
1 δ,ithastoreceiveatleastM :=Klog(K/δ)packets[16,
InwhatfollowsweelaboratebasedonresultfromThm.3.1, −
Chap 50]. A useful expression for large K will be used later:
and we study the case of large values of N. Let us assume
if we write M = K(1+α) then α that guarantees that the
that the total number of frames, H +K, grows as a function
destination can decode the original message with probability
of N, i.e., H +K =(H +K)(N).
of at least 1 δ is given by
The question we would like to answer is, in the asymptotic −
regime, what is the effect of redundancy onto the delivery (log(K/δ))2
α=
probability, that is, how the number of redundant frames √K
should scale with respect to the total number of frames.
The number of packets that reach the destination during the
We assume throughout that the following limits exist:
time interval [0,τ] has a Poisson distribution with parameter
L(τ,p∗)p∗ where p∗ is given in (10) and where L(τ,p∗) is
K := lim K(N)/N and H := lim H(N)/N −
N→∞ N→∞ given in (9).
Noticebthat this implies of coursebthat the energy constraint 3Inwhatfollowswewillexcludethetrivialcasexb=0,whichcorresponds
shouldgrowatmostlinearlywith N and we thusassume that tothecasewhennorelaying isallowed.
6
The probabilitythat less than M packets reach the destina- We notice that P(N)(τ) = Pr X K(N)(1+α) 1 ,
M { N ≤ − }
tion is given by where X is a Poisson r.v. From the Strong Law of Large
N
Numbers we have lim X /N =Γ P-a.s. Thus
M−1( L(τ,p∗)p∗)i N→∞ N 0
PM(τ)= Xi=0 − i! exp(L(τ,p∗)p∗) (11) Nl→im∞PM(N)(τ)=1−Nl→im∞Pr(XN ≥K(N))
We conclude that P∗(τ) 1 δ P (τ).
Finally, we can lesverag≥e Co−r. 3−.2 anMd obtain = 1, if K >Γ0
Corollary 4.1: Given p∗ as in (10), (0, if K <Γ0
b
1 δ P (τ) P∗(τ) 1 P (τ) from which we can deduce
− − M ≤ s ≤ − M b
We notice that the boundon the right-endof the inequality is 1 δ, if K <Γ0
lim P (τ) ≥ −
obtained by using redundancy as in the previous section for N→∞ s (=0, if K >Γ0
H M, then taking the limit of H . b
≥ →∞ where we used Cor. 4.1.
b
A. Numerical examples: a phase transition This demonstrates the effect of phase transition observed
In Fig 1c) we reportedthe representationof the boundsde- numerically in Fig 1c).
scribedabove:thetwoboundsprovideatightcharacterization
V. THRESHOLD POLICIES
oftheperformancesoffountaincodes;inparticular,wenotice
that P (τ), which in fact is the CDF of a Poisson r.v., tends The way the energy constraints are handled so far is by
M
to one as K increases: this causes the success probability to using only a fraction of the transmission opportunities. This
tend to zero as K increases. The intuition is that, for a large was done uniformly in time by transmitting at a probability
number of transmitted frames, the probability of receiving all thatistimeindependent;wewillrefertothisstrategyasstatic
ofthemsuccessfullywithinthegivendeadlinedecreasesfaster policy. In this section we consider an alternative way to dis-
thanthegainobtainedbyaddingredundancythroughfountain tribute the transmissions: we use every possible transmission
codes. opportunity till some time limit and then stop transmitting.
But, the numerical insight of the model says more of the This is motivated by the ”spray and wait” policy [17] that is
performance attained by fountain codes. In fact, in Fig 1c) knownto tradeoffveryefficientlymessage delayandnumber
we reported P∗ versus the normalized number of transmitted of replicas in case of a single message.
s
frames K/N, for increasing N. Interestingly, the success In addition we need to specify the values of p : the proba-
i
probabilitybecomesmoreandmoreclosetoastepfunctionat bility packet i when there is an opportunity to transmit (and
the increase of N. We thus observe a phase transition: above the time limit has not yet elapsed). We have K p =1.
i=1 i
a given number of transmitted frames, the success probability In the lack of redundancy, we recall that the success
vanishes, and below the same threshold success occurs with probability writes P
probability one.
We shall study this phenomenon analytically in the next K τ
P (τ,p)= 1 exp( λ X (v)dv)
subsection. s i
− −
iY=1(cid:16) Z0 (cid:17)
B. Analysis Asymptotic behavior
Clearly, transmitting always when there is an opportunity
We now want to understand the behavior of the foun-
to transmit is optimal if by doing so the energyconstraint are
tain codes for large values of N in the asymptotic regime
not violated. This can be considered to be a trivial time limit
K = K(N) N; in what follows, we will assume that
≤ policy with a limit of r = τ. Otherwise, the optimal value r
K := limN→∞K(N)/N exists. This implies of course that for a thresholdis the one thatachievesthe energyconstraints,
theenergyconstraintshouldgrowlinearlywithN andwethus or equivalently, the one for which the expected number of
abssume that the limit x := limN→∞x(N)/N exists. We can transmissions during time interval [0,r] by the source is x;
rewrite L(τ,p∗)p∗ =N Γ(0), where we obtain as before, x is related to the total constraint on the energy
− b · 0 through the constraint E(τ) εx. The value r of the time
1 x ≤
Γ(N) = 1 λp∗τ e−λp∗τ =λτ 1+ N−z limit is then given by σ(z), see eq. (8).
0 −p∗(cid:16) − − (cid:17) (cid:16) log(1− Nx−z)(cid:17) Since we considered a time limit policy, Xi(v) first grows
till the time limit r is reached and then it stays unchanged
x
Γ := lim Γ(N) =λτ 1+ during the interval (r,τ]. For the non-trivial case where r =
0 N→∞ 0 log(1 x)
(cid:16) − (cid:17) σ(z) we thus have:
b
We have
b
K(N)(1+α)−1(N Γ(N))i K σ(z)
P(N)(τ)= · 0 exp( N Γ(N)) P (τ,p)= 1 exp λ Np 1 e−λv dv
M i! − · 0 s − − i −
Xi=0 iY=1h (cid:16) Z0 (cid:16) (cid:17)
7
x=70, K=15,30, N=300, τ=3000 s x=70, δ=0.05, N=300, τ=3000,7000 s
a) 1 b) 1
K =15 Static policy
0.8 0.8 Threshold policy
Success probability00..46 K =30 Static policy Success probability00..46 τ =7000s
0.2 Threshold policy 0.2 τ =3000s
0 0
0 10 20 30 40 50 60 70 0 10 20 30 40 50
Number of redundant frames H Number of frames K
Fig. 2. A comparison ofsuccess probability witherasure codes whenusingthreshold andstatic policies; N =300, x=70,τ =3000,7000: a)Erasure
Codes,K=15,30b)Fountaincodes,δ=0.05.
K
Here again, we used the fact that the success probability is
λ(τ σ(z))X (σ(z)) = Z˜(p )
− − i i maximized for equal pi’s, i.e.
(cid:17)i iY=1
1/(H +K) if t σ(z)
whereZ˜(p ):=1 exp p L˜(τ) ,p L˜(τ):= λ τ X (v)dv p∗(t)= ≤
i − i i − 0 i i (0 if t>σ(z)
and where we have by e(cid:16)q. (5) a(cid:17)nd (8) R
and P∗(τ,H,K) is given as in eq. (13). The proof follows
s
Np x
X (σ(z))=Np 1 e−λσ(z) = i the same lines as that of Lemma 3.1.
i i
− N z Asafinalremark,considerthetrivialcasewheretheenergy
(cid:16) (cid:17) −
constraintisnotactive:τ <σ(z),i.e.,itisoptimaltotransmit
Also, the integral can be expressed as
all the x packets up to time τ; in this case results from
σ(z) Np x x Thm. 3.1 hold.
i
X (v)dv = +log 1 Now we want to specialize the use of threshold policies
i
− λ −N z − N z
Z0 (cid:20) − (cid:16) − (cid:17)(cid:21) in the case of fountain codes when the energy constraint is
In particular, the following holds active:τ σ(z).Inthiscase,thenumberofpacketsthatreach
≥
the destination during the time interval [0,τ] has a Poisson
L˜(τ)= NN−xz +Nlog 1− Nx−z −λ(τ −σ(z))NN−xz distributionwithparameterΛ=−L˜(τ)= NN−xzλτ+β(z);the
probability that less than M = Klog(K) packets reach the
“ ”
destination is given by
Nx
=− λτ −β(z), where
N−z
M−1
Nx 1 Nx i
P (τ)=exp λτ β(z) λτ+β(z)
M
−N z − i! N z
β(z)= Nx N 1 x log 1 x 0. (cid:18) − (cid:19) Xi=0 (cid:16) − (cid:17)
−N z − − N z − N z ≥ andthe statementofCor. 4.1 holdsaccordingly(noticethat
− (cid:16) − (cid:17) (cid:16) − (cid:17)
when the energy constraint is not active, we fall back to the
Finally, the p ’s are selected to be all equal (and to sum to
i original form of Cor 4.1).
1) due to the same argumentsas in the proof Theorem 2.1 as
here too, log(Z˜) is concave in its argument. A. Comparison with static policies
In the case of redundancy, the calculations are similar to
Here we would like to see what is the relative performance
those in eq. (12), where
of static and threshold policies, both for fountain codes and
erasurecodes.InFig.2a)andFig.2b)wereportedonthecase
H+K
Ps(τ,H,K)= H+K Z˜s(pi)(1−Z˜(pi))N−s (12) aboundonenergyexistsx=70;insuchcase,σ(z)=1000s.
s
s=K ! Asconcernserasurecodes,whenthenumberofframesishigh
X
(30) the usage of a large number of redundant frames proves
Hence, given H 0, it holds
much more effective compared to static policies. Conversely,
≥
a for a lower number of frames, the advantage of threshold
Nx λτ +β(z)
P∗(τ,H,K)=exp N−z (13) policies is less marked.
s − H +K ! In the case of fountain codes, threshold policies are again
H+K H +K Nx λτ +β(z) s more efficient than static policies, and the effect is more
exp N−z 1 relevant for large values of the time constraint τ. We recall
× s=K (cid:18) s (cid:19)" H +K !− # that we refer to optimal static policies.
X
8
a) N=200, τ=80000, x=70, H=10,20 b) N=200, τ=80000, x=70, H=10,20 c) N=200, τ=80000, x=70
1 Theory 1 Theory 1 Theory
Simulation Simulation Simulation
Success Probability0000....2468 H =10 (cid:0)(cid:18) (cid:0)(cid:9)H =20 Success Probability (thr. policy)0000....2468 H =10 (cid:0)(cid:18) (cid:0)(cid:9)H =20 Success Probability0000....2468
0 0 0
0 10 20 30 40 0 10 20 30 40 0 5 10 15 20
K K K
Fig.3. a)Simulationresultsforthesuccessprobabilityoferasurecodedmessages,staticpolicy,H=10,20,N =200,τ =80000s,x=70.b)Sameas
a)butunderthreshold policies c)Simulation results forthesuccess probability offountain coded messages,static policy, N =200,τ =80000s,x=70,
δ=0.02.
VI. NUMERICAL VALIDATION Haggle: in [2] and related works, the authors report exten-
sive experimentation conducted in order to trace the meeting
In this section we provide a numerical validation of the
pattern of mobile users. A version of iMotes, equipped with
model. Our experiments are trace based; message delivery is
a Bluetooth radio interface, was distributed to a number of
simulatedbyaMatlab(cid:13)R scriptreceivingasinputpre-recorded
people, each device collecting the time epoch of meetings
contact traces.
with other Bluetooth devices. In the case of Haggle traces,
due to the presence of several spurious contacts with erratic
Synthetic Mobility
Bluetoothdevices,werestrictedthecontactstoasubsethaving
We considered first a Random Waypoint (RWP) mobility experienced at least 50 contacts, resulting in 19 active nodes.
model[18].WeregisteredacontacttraceusingOmnet++with
N = 200 nodes moving on a squared playground of side 5 CN: the CN dataset has been obtained by monitoring 21
Kms. The communication range is R = 15 m, the mobile employeewithinCreate-Netandworkingondifferentfloorsof
speed is v = 5 m/s and the system starts in steady-state the same building during a 4-week period. Employees volun-
conditions in order to avoid transient effects [19]. The time teered to carry a mobile runninga Java applicationrelyingon
limit is set to τ = 80000 s, which corresponds roughly to 1 Bluetooth connectivity. The application periodically triggers
day operations, and the constraint on the maximum number (every60seconds)aBluetoothnodediscovery;detectednodes
of copies is x = 70. With this first set of measurements, we are recorded via their Bluetooth address, together with the
want to check the fit of the model for the erasure codes and currenttimestamponthedevicestorageforalaterprocessing.
fountaincodes.Inthecaseoferasurecodes,wefixedH =10
Fig 4 a) and b) depictthe results of experimentsperformed
and 20 and increased the number of message frames K. We
with these data sets. A major impact is played by the non-
selected at random pairs of source and destination nodes and
stationarity of traces. This is mainly due to “holes” appearing
registered the sample probability that the message is received
in the trace which impose unavoidable cutoff effects on the
at the destination by time τ. As seen in Fig. 3a) the fit with
success probability. In particular, in the case of fountain
the model is rather tight and an abrupt transition from high
coding, for K > 1, i.e., when the original file is actually
success probability to zero is visible. Also, in Fig. 3b) we
fragmented, the performance is rather poor. This is due to
reported on the results obtained in case of threshold policies;
the increase with K of the number of frames M required for
thefitissimilartowhatobtainedforstaticpolicies,confirming
decoding,exacerbatedbythereducedsizeofthenetwork.For
the gain of performance with respect to static policies.
example, for K = 3 and δ = 0.05 it holds M = 13, i.e. M
We repeated the same experiment in the case of fountain
is close to N for the Haggle trace and larger than N/2 for
codes, as reported in Fig. 3c) for static policies; in this case
the CN trace;thismeansthat,in orderto deliverthe message,
thecodespecificparameterisδ =0.02andagainweincreased
basically almost all nodes should meet the destination and
K.Eveninthiscaseweseethatthethresholdeffectpredicted
deliver a frame within τ.
by the model is apparent.
With erasure codes, bothtraces show the characteristic cut-
Real World traces
off of performances at the increase of K. The decay, though,
Our model captures the behavior of a sparse mobile ad depends on the trace considered; in particular, a close look
hoc network under some assumptions: the most stringent is to the two data sets showed that the Haggle trace has higher
the uniformity and the stationarity of intermeeting intensities. averageinter-meetingintensitycomparedtotheCNtraceover
We would like now to understand what is the impact of non- the considered interval; the shape seen in Fig 4a) resembles
uniformandnon-stationaryencounterpatterns.We considered more closely the theoretical sigmoid cutoff predicted by the
two sets of experimental contact traces: model.
9
a) N=17, τ=80000, x=15, Haggle trace b) N=21, τ=80000, x=19, CN trace
1 Erasure H=3 1 Erasure H=3
Fountain δ=0.05 Fountain δ=0.05
0.8 0.8
Suc. Probability00..46 Suc. Probability00..46
0.2 0.2
0 0
0 2 4 6 8 10 12 14 0 5 10 15
K K
Fig.4. a)Simulationresultsforthesuccessprobability ofstaticpolicies, Haggletrace,N =17,τ =80000s,x=17andδ=0.05b)Simulationresults
forthesuccess probability ofstaticpolicy, CNtrace, N =21,τ =80000s,x=17andδ=0.05.
VII. CONCLUSIONS AND REMARKS [8] Y. Lin, B. Liang, and B. Li, “Performance modeling of network
coding inepidemic routing,” inProc.ofMobiSys workshop onMobile
We have considered in this paper the tradeoff between
opportunistic networking (MobiOpp). San Juan, Puerto Rico: ACM,
energy and probability of successful delivery in presence of June112007,pp.67–74.
finite duration of contacts or limited storage capacity at a [9] J.WidmerandJ.-Y.L.Boudec, “Network codingforefficient commu-
nication in extreme networks,” in Proc. ofthe ACM SIGCOMM work-
node: it can store only part (a frame) of a file that is to
shoponDelay-tolerantnetworking(WDTN),Philadelphia,Pennsylvania,
be transferred. To improve performance we considered the
USA,August262005,pp.284–291.
generation of erasure codes at the source which allows the [10] A. E. Fawal, K. Salamatian, D. C. Y. Sasson, and J. L. Boudec, “A
DTN to gain in spatial storage diversity. Both fixed erasure frameworkfornetworkcodinginchallengedwirelessnetwork,”inProc.
ofMobiSys. Uppsala,Sweden:ACM,June19-222006.
codes (Reed Solomon type codes) as well as rateless fountain
[11] R. Groenevelt and P. Nain, “Message delay in MANETs,” in Proc. of
codes have been studied. SIGMETRICS. Banff, Canada: ACM, June 62005, pp. 412–413, see
Fountain codes can be viewed as special case of general also R. Groenevelt, Stochastic Models for Mobile Ad Hoc Networks.
PhDthesis,University ofNice-Sophia Antipolis, April2005.
networkcodingsuchasthosestudiedin[7],[8],[10].Wenote
[12] T. Karagiannis, J.-Y. L. Boudec, and M. Vojnovic´, “Power law and
however that in order to go beyond fountain codes (in which
exponential decay of inter contact times between mobile devices,” in
coding is done at the source) and consider general network MobiCom. Montre´al, Que´bec, Canada: ACM, September 9–14 2007,
coding (where coding is also done in the relay nodes), one pp.183–194.
[13] E. Altman, T. Bas¸ar, and F. De Pellegrini, “Optimal monotone for-
needs not only change the coding approach but one should
warding policies in delay tolerant mobile ad hoc networks,” in Proc.
allow storage of several frames at each relay node. Also,
ofACM/ICSTInter-Perf. Athens,Greece: ACM,October242008.
interesting hints come from the presence of non-stationary [14] X.Zhang,G.Neglia,J.Kurose,andD.Towsley,“Performancemodeling
patternsarising in real world traces. New tradeoffissues arise ofepidemic routing,” Elsevier Computer Networks, vol. 51, pp. 2867–
2891,July2007.
which we shall study in future work.
[15] B.Bolloba´s, RandomGraphs. CambridgeUniversity Press,2001.
[16] D.J.MacKay,InformationTheory,Inference,andLearningAlgorithms.
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primer,”SIGCOMMComput.Commun.Rev.,vol.36,no.1,pp.63–68, P∗(τ) = max P (τ)= max Z(p )
s s i
2006. p:Pipi=p p:Pipi=pi=1
Y
10
p∗ is optimal if and only if it maximizes g(p) := Notice also that,
K log(Z(p )) s.t. p = p, p 0. Note that L(τ) is
noni=-p1ositive.loigZ turnsiouit to be aic≥oncavefunction.It then E[SH+K,p] = (H +K)p=(H +K) 1 e−NHΓ+(0NK)
Pfollows from Jensen’sPinequality that b −
(cid:16) (cid:17)
NΓ(N) 1
K = (H +K)b 0 +o
1 H +K H +K
g(p) = log(Z(p ))
K i (cid:16) (cid:0) (cid:1)(cid:17)
i=1 from which it follows that
X
1 K p E[SH+K,p]
≤ log Z K pi =log Z K (14) Nl→im∞ N b = Nl→im∞(H +K)p/N =Γ0
(cid:16) (cid:16) Xi=1 (cid:17)(cid:17) (cid:16) (cid:16) (cid:17)(cid:17)
Notice that the binomial r.v. S can be interpreted
where equality is attained for p = p/K. This concludes the H+K,p b
i as the sum of H +K independent binabry random variables
proof.
i with mean p and variance p(1 p). Also, the series of
n
−
B. Proof of Lemma 3.1 the normalized variances σ2 = 1 p(1 p) 1 is finite.
in n2 − ≤ n2
Proof: Let A(K,H) be the set of subsets h Thus, the Stronbg Law of Largbe Numbbers [20] ensures that
⊂
1,...,K+H thatcontainatleastK elements.Forexample, limN→∞SH+K,p/N =Γ0 P-a.s. b b
{1,2,...,K } A(K,H). Fix p such that K+Hp = p. Now we can dberive
{ } ∈ i i=1 i
Then the probabilityof successful deliveryby time τ is given
0, if K+H Γ
P 0
by lim P S H +K = ≤
H+K,p
N→∞ { b≤ } (1, if K >Γ0
Ps(τ,K,H)= Z(pi) b b
h∈AX(H,K)Yi∈h and, in a similar way, b
For any i and j in 1,...,K+H we can write
0, if K <Γ
{ } 0
lim P S <K =
H+K,p
Ps(τ,K,H)=Z(pi)Z(pj)g1+(Z(p1)+Z(p2))g2+g3 N→∞ { b } (1, if K Γ0
b ≥
where g1, g2 and g3 are nonnegative functions of Finally, by enumeration, b
Z(p ),m=i,m=j . For example,
m
{ 6 6 }
0 if K+H Γ
0
g = Z(p ) ≤
1 m
lim P (τ,K,H)=0 if K+H >Γ and K Γ
h∈A{iX,j}(H,K)mm∈6=hYi,m6=j N→∞ s  b b 0 ≥ 0
1 if K+H >Γ and K <Γ
whereA (K,H)isthesetofsubsetsh 1,...,K+H that b b 0 b 0
v
contain at least K elements and such th⊂at{v ⊂h. } This concludes the proof. b b b
Now consider maximizing P (τ,K,H) over p and p that
s i j
satisfy p′ =p +p for a given p′ p. Since Z() is strictly
i j
≤ ·
concave, it follows by Jensen’s inequality that Z(p )+Z(p )
i j
has a unique maximum at p = p = p′/2. This is also the
i j
unique maximum of the product Z(p )Z(p ) (using the same
i j
argument as in eq. (14)), and hence of P (τ,K,H).
s
Since this holds for any i and j and for any p′ p, this
≤
implies the Lemma.
C. Proof of Proposition 3.1
Proof: From Thm. 3.1,
P∗(τ,K,H)=P S H +K P S <K ,
s { H+K,pb≤ }− { H+K,pb }
where p=1 exp L(p∗,τ)p∗ .
− H+K
We can rewrite (cid:16)L(τ,p∗)p∗(cid:17)=N Γ(0), where we obtain
b − · 0
x
1
Γ(N) = 1 λp∗τ e−λp∗τ =λτ 1+ N−z
0 −p∗ − − log(1 x )
(cid:16) (cid:17) (cid:16) − N−z (cid:17)
and we define
x
(N)
Γ := lim Γ =λτ 1+
0 N→∞ 0 log(1 x)
(cid:16) − (cid:17)
b
b