4.3 Performance Evaluation
4.3.1 Case 1: no QoS constraints
In each scenario, we compare different allocation functionsρ(n) =ck·nk, where ck = EcM
p[nk] is a normalization factor such that the constraint Ep[ρ(n)] = cM is satisfied.
In Fig. 4 we present simulation results in scenarios for theTVCM(Fig. 4(a)), SLAW (Fig. 4(b)),Cabspotting(Fig. 4(c)) andInfocom(Fig. 4(d)) traces. Content popularity (Pp(n)) follows aZipf distribution with n ≤ 30 and exponent α = {1,2,3}. The availability constraint is set tocM = 10. It can be seen that the
4The difference here is that the constraintρn≥0is active.
−1 0 0.5 1 2
Figure 4: Content access delay E[TM] of different allocation policies ρ(n) = ck·nk, whereck= EcM
p[nk].
optimal delay E[TM]is achieved fork = 0.5, as Result 5 predicts (despite the fact that we used the expression of the lower bound as an approximation for the expected delayE[TM]).
4.3.2 Case 2:QoSconstraints
To evaluate the performance of the allocation functionρ(n)that follows after solv-ing Eq. (26) (i.e. optimal allocation), we compare the offloading ratio Rof f it achieves with the offloading ratios of the following policies:
Random: We randomly select a content and give a copy of it to a node. We repeat M·cMtimes.
Square Root: We selectρ(n)∝√
n(i.e. the allocation that achieves the minimum expected delayE[TM]).
Log: We selectρ(n)∝logn.
Total copies = 50 Total copies =100
Total copies = 50 Total copies =100 0
Figure 5: Offloading RatioRof f.of different allocation policiesρ(n).
Random policy has been used in related work as a baseline [9] and square rootpolicy is the optimal policy when the metric of interest is the content access delay (Section 4.1). Finally, we observed that theoptimalpolicy (Eq. (26)), in the scenarios considered, allocated copies only to the10%−20%highest popularity contents. Thelogpolicy allocates in a similar manner the copies (e.g. no copies to contents with low popularity).
Simulation results on theSLAWandInfocomscenarios are presented in Fig. 5(a) and 5(b), respectively. The parameters in these scenarios are: M = 50messages, Pp ∼Zipf withn∈ [1,30]andα = 1, total copiesM ·cM ={50,100}. As it can be seen ouroptimalpolicy (leftmost bar) achieves the highest offloading ratio Rof f.. The random policy is clearly inferior than the others. Betweensquare root andlogpolicies, it is the latter that achieves better performance. These results indi-cate that, to maximizeRof f., it is better to allocate the available resources only for popular contents, and serve the non-popular exclusively through the infrastructure.
5 Related Work
Content-centric applications were introduced in opportunistic networking under thepublish - subscribeparadigm [2, 17, 16, 10], for which several data dissemi-nation techniques have been proposed. In [2], authors propose a mechanism that identifies social communities and the nodes-“hubs”, and builds an overlay network between them in order to efficiently disseminate data. SocialCast [16] based on information about nodes interests, social relationships and movement predictions, selects the set of holders. Similarly to the above approaches, ContentPlace [10]
uses both community detection and nodes social relationships information, to im-prove the performance of the content distribution.
Under a different setting, [1, 11] study content sharing mechanisms with lim-ited resources (e.g. buffer sizes, number of holders). In [1], authors analyti-cally investigate the data dissemination cost-effectiveness tradeoffs, and propose
techniques based on contact patterns (i.e. λij) and nodes interests. Similarly, CEDO [11] aims at maximizing the total content delivery rate: by maintaining a utility per content, nodes make appropriate drop and scheduling decisions.
Recently, further novel content-centric application have been proposed, like location-based applications [3, 4] and mobile data offloading [7, 8, 9]. The latter category, due to the rapid increase of mobile data demand, has attracted a lot of attention. In the setting of [7], content copies are initially distributed (through the infrastructure) to a subset of mobile nodes, which then start propagating the contents epidemically. Differently, in [8] the authors consider a limited number of holders, and study how to select the best holders-target-set for each message.
In [9], the same problem is considered, and (centralized) optimization algorithms are proposed that take into account more information about the network: namely, size and lifetimes of different contents, and interests, privacy policies and buffer sizes of each node.
In the majority of previous studies, although node interests and content pop-ularity are taken into account, the focus has been on the algorithms and the ap-plications themselves. We believe that our study complements existing work, by providing a common analytical framework for a number of these approaches that can be used both for predicting the performance of proposed schemes, as well as proposing improved ones.
6 Conclusion
The increasing number of mobile devices and traffic demand, renders content-centric applications through opportunistic communication very promising. Hence, motivated by the lack of a common analytical framework, we modeled and ana-lyzed the effects of content popularity / availability patterns in the performance of content-centric mechanisms.
As a part of future work we intend to study, in more detail, extensions of our model and to investigate further characteristics of content traffic patterns, like traf-fic localityin location based social networks, and their performance effects.
A Minimum of Pareto distributed random variables
For the random variableTM= min
i∈Ca(M){Tij}, where eachTij is a random vari-able distributed with a Pareto distribution with scale parametert0 and shape pa-rameterαij, it holds that:
P{TM> t}= Y
i∈Ca(M)
P{Tij > t}= Y
i∈Ca(M)
µt0 t
¶αij
= µt0
t
¶P
i∈C(M) a
αij
(27)
which means thatTMfollows a Pareto distribution with scale and shape parameters t0andAM=P
i∈Ca(M)αij, respectively
B Proofs for the performance metrics expressions of the Pareto case
B.1 Content Access Delay
The expectation of a Pareto random variable (pareto(t0, αij)) is α−1t0α. Hence, in the derivation of Eq. (11), one only needs to change the integral in the last equality as:
E[TM] =X
m
Z x·t0
x−1·fmα(x)dx·Pareq.(m) (28) SubstitutingPareq.(m)from Lemma 3.2 and proceeding as in the exponential case, we subsequently get: which the exact expression forE[TM]in Table 3.
ApplyingJensen’s inequalityfor the convex functionh(x) = x−11 , gives:
Emα where for the last line we appliedJensen’s inequalityfor the expectationEgh
1 m·µα−1
i .
B.2 Content Access Probability
In the Pareto case, the integral in Eq. (17) changes as: R ¡ t0
T T L
¢x
·fmα(x)dx, for T T L≥ t0, because for a Pareto random variablex ∼ pareto(t0, α)it holds that P{x ≤ T T L} = 1−¡ t0
T T L
¢α
. Following the same methodology as before and observing that the functionh(x) =¡ t0
T T L
¢α
is convex, the expressions of Table 3 follow similarly.
C Proof of Result 3
Proof. To calculate the average performance, we need to modify the previous anal-ysis as following: Consider a contentMof initial popularityNp(M)(0) = nand availabilityNa(M)(0) =m, i.e. initiallynnodes are looking for the content andm nodes hold the content. When the first requester access the content, the number of holders will increase tom+ 1and the remaining requesters will ben−1. Building a Markov Chain as in Fig 1, where each state denotes the number of holders, it can be shown for the expected delay of moving from statem+kto statem+k+ 1, k∈[0,1], that it holds
E[Tk,k+1]≈ 1
(m+k)·(n−k)·µλ (32) wherem+kare the nodes holding the content,n−kthe remaining requesters and µλthe mean contact rate.
From the above analysis, it follows straightforward that the expected time till the first requester to access the message is E[T1] = E[T0,1]and till the ℓth re-quester to access it is
E[Tℓ] =
ℓ−1
X
k=0
E[Tk,k+1] (33)
Let us now define the sum of delays E[Tℓ](i.e. delivery delays for each re-quester) for a messageMwith initial availabilityNa(M)(0) =mand initial popu-larityNp(M)(0) =n, as:
S(TM|m, n) =
n
X
ℓ=1
E[Tℓ|m, n] (34)
From Eq. (32) and Eq. (33), we can write forS(TM|m, n):
and using the approximation of the harmonic sum5, we get S(TM|m, n)≈ 1 Averaging over all the content in the network, we can write for the expected content access delay:
or, since (i) (by definition) there areM·Pp(n)contents in the network, and (ii) we do not differentiate between contents with the same popularity/availability:
E[TM] =
where in the last line we substituted from Eq. (36).
We can further use Jensen’s inequality (since the functionh(x) = ln¡ 1 +nx¢ is convex) or the respective approximation, and finally write:
E[TM]≈ 1 which proves the result.
5PN
k=1≈ln(N) +γ+O`1
N
´, whereγis theEuler-Mascheroniconstant.
D Proof of Result 4 and Example
Proof. Def. 5 says thatwho holds a content andwho is interested in it is not in-dependent of their mobility patters. The contact rates between the requester of a content and the holders of it, are not distributed with the contact rates distribution fλ(λ), since the requesters-holders contact rates are mobility dependent. It can be shown that the requesters-holders contact rates are distributed as [27]
fπ(λ) = 1
Eλ[π(λ)]·π(λ)·fλ(λ) (40) Hence, Eq. (9) and Eq. (10) need to be modified as:
XM ∼fmπ(x) = (fπ∗fπ· · · ∗fπ)m (41) and
E[XM|Na(M) =m] =Emπ[x] =m·Eλ[λ·π(λ)]
Eλ[π(λ)] =m·µ(π)λ (42) Then, it can be easily seen that following the same analysis, we get the same ex-pressions as in Theorems 3.3 and 3.4 and Result 3 where, now, the mean contact rateµλis replaced by themean mobility dependent requesters-holders contact rate µ(π)λ .
Example Scenario For each contentM, its holders are selected taking into ac-count their contact rates with the requesters with the following mechanism: Each nodeicandidate to be a holder is assigned a weightwi =Q
j∈C(M)p λij. Then, each of them is selected to be one of theNa(M) holders with probability pi = Pwi
iwi. Now, for the node pair{i, j}(i∈ Ca(M),j∈ Cp(M)) it holds that
πij = wi P
iwi
= Q
k∈Cp(M)λik P
i
Q
k∈C(pM)λik
=
λij·Q
k∈Cp(M)\{j}λik P
i
Q
k∈C(pM)λik
(43)
for which, when the node popularityNp(M)=|Cp(M)|is large enough, we can write πij ≈ λij ·c1
c2 (44)
wherec1, c2 take approximately the same value∀i, j, i.e. π(λ) = c·λ, c = cc1
2. Substitutingπ(λ)in Result 4, gives
µ(π)λ = Eλ[λ·π(λ)]
Eλ[π(λ)] = Eλ[λ2]
Eλ[λ] =µλ·(1 +CVλ2) (45)
E Proof of Expressions in Table 2
Meeting rates are distributed with a Gamma distribution fλ(x) = Γ(x;α, β) = βα
Γ(α) ·xα−1·e−β·x
whereΓ(α)is thegamma functionand the parametersα, βrelate to the expectation µλand coefficient of variationCVλ of the meeting rates, as following:
α= 1
CVλ2 (46)
β= 1
µλ·CVλ2 (47)
(48) The sum ofmrandom variablesλij ∼fλ(x) = Γ(x;α, β)is a random variable and is distributed with a gamma distribution with parametersm·αandβ, i.e.
fmλ= Γ(x;m·α, β) = βm·α
Γ(m·α) ·xm·α−1·e−β·x (49) Moreover, when
Pp(n)∼P areto(x;n0, α= 2) = 2·n20
x3 , f or x≥n0 (50) the mean value of the popularity is
Ep[n] = 2·n0
2−1 = 2·n0 (51)
E.1 Delivery DelayE[TM]
To calculate the delivery delay, given by Result 1, we need first to calculate the quantityEmλ
because it holds that (∗)
To calculate the expectationEph Finally, substituting Eq. (51) and Eq. (54) in Eq. (53), we get the expression of Table 2.
E.2 Delivery ProbabilityP{TM ≤T T L}