Analyzing the Performance of LRU Caches under Non-Stationary Traffic Patterns

0.18 MB·English

by  Mohamed Ahmed

Analyzing the Performance of LRU Caches under
Non-Stationary Traffic Patterns
Mohamed Ahmed∗, Stefano Traverso †, Paolo Giaccone†, Emilio Leonardi† and Saverio Niccolini∗
∗ NEC Laboratories Europe, Heidelberg, Germany – {firstname.lastname}@neclab.eu
† Department of Electronics and Telecommunications, Politecnico di Torino, Torino, Italy – {lastname}@tlc.polito.it
Abstract—Thiswork presents,tothebestof ourknowledgeof to their different static popularity distributions - typically
theliterature,thefirstanalyticmodeltoaddresstheperformance assumed to be Zipf.
3
ofanLRU(LeastRecentlyUsed)implementingcacheundernon-
In reality however, the popularity of contents varies over
1
stationary traffic conditions, i.e., when the popularity of content
0 time and differentcontentsexhibita wide range of popularity
evolves with time. We validate the accuracy of the model using
2 Monte Carlo simulations. We show that the model is capable evolution patterns [6], [7]. Contents tend to differentiate in i)
n of accurately estimating the cache hit probability, when the whentheystarttoattractuserattention,ii)howmuchattention
a popularity of content is non-stationary. they attract and iii) how long they sustain the attraction. For
J We find that there exists a dependency between the perfor- instance,contentsrelatedsportingorgeo-politicaleventssuch
1 mance of an LRU implementing cache and i) the lifetime of as Olympic videos tend to enjoy a very short lifetime [2],
contentinasystem, ii)thevolumeof requestsassociated withit,
2
reflecting users’ immediate interest in the topic. In contrast,
iii)thedistributionof content requestvolumesand iv) theshape

] of the popularity profile over time. someYOUTUBEmusicvideoskeeponattractinguserattention
I many months after being released [7]. Clearly, this observed
N
I. INTRODUCTION behaviourisincompatiblewithtime-invariantpopularity mod-
.
s els, and raises the need for more accurate tools that take into
c Content caching is today a primitive network management consideration the evolution of the popularity of contents over
[
operation, while the operation and performance of Content time.

1 DeliveryNetworks(CDNs) is predicatedon understandingon This work presents the first steps in this direction. We
v how users consume content. extend the results of Che et al. [8] to take the time-variant
9
This argument is given urgency by two main factors. First, popularity of contents explicitly into account, and present an
0
9 the continued growth of Internet traffic, especially in the approximatedmodelofanLRU(LeastRecentlyUsed)caching
4 mobile context, increases the demand on limited network under non-stationary traffic conditions. The accuracy of our
. resources[1],[2].Second,thedominanceofmultimediatraffic modelisvalidatedagainstMonteCarlosimulationsandshows
1
0 is today de-facto [3]. Recent studies [2] show that in the US, that the non-stationarity of content popularity has a dramatic
3 services offering real-time video and audio streaming occupy effecton caching performance.We find that even when cache
1 62.5%and 54.7%of peak-perioddownstreamtraffic forfixed sizes are small as in the context of ICNs, i.e., when cache
v: and mobile networks respectively. Similarly in Europe, real- dynamics change much faster than the popularity evolution,
i time multimedia traffic accounts for 33.5-50%of peak-period the performanceof the cache is largelysensitive to popularity
X
downstream traffic in fixed networks, while globally, video dynamics.
r traffic alone is project to account for 55% of all “consumer
a
Internet traffic” by 2016 [1]. II. LRUUNDER NON-STATIONARYCONDITIONS
This traffic profile poses unique challenges to providing In this section, we first present the assumptions that unpin
users with a reliable Quality of Service (QoS). For instance, ourmodel(Sec.II-A).Thisisfollowedbythederivationofthe
Liuet. al.[3] reportthat 20%ofusersexperiencere-buffering cache hit probability under the non-stationary traffic scenario
whenstreamingcontents,while14%ofuserssuffersignificant (Sec. II-B), with specific reference to scenarios with large
delays before videosstart to play. Furthermore,new network- (Sec. II-C) and small cache sizes (Sec. II-D).
ing paradigmssuch as ICN (Information-CentricNetworking)
are built on the implicit assumption of ubiquitous content A. A simple non-stationary traffic model
caching [4], [5], such that small caches are co-located with We start by assuming that contents are introduced into
routers in order to offset traffic latency. a catalogue (i.e. uploaded on some server) at random. For
Therefore, improving the effectiveness of content caching simplicity, this is taken to be according to a homogeneous
is paramount in aiding to address the problem of scaling the Poisson process with rate γ. Furthermore, it is assumed that
networkwhileprovidingthenecessaryQoStousers.However, individualcontentpopularityevolves(overtime) accordingto
the vast majority of the studies on caching assume traffic some predetermined profile. Initially, it is assumed that all
patterns to be time-invariant, i.e., users browse through a contents follow the same popularity profile, in Sec. IV we
largestaticcatalogueofcontentsandmakerequestsaccording show that this assumption can be relaxed.
Let us now consider a generic content m, introduced into Λ(t) = V λ(t −τ ). Extending Che’s approximation,
m m m
the catalogue at time τ , and whose popularity evolves over the hit probability is given by:
m
P
time according to:
∞ φ′ − TC λ(τ −θ)dθ
V 0
λm(t)=Vmλ(t−τm) phit =1−Z0 λ(τ) (cid:16) R E[V] (cid:17)dτ (1)
where λ(t) representsthe popularityprofileand V a random
m where T is the solution to the equation:
mark (i.e. a random quantity) associated to content m. Popu- C
larity (λm(t)) in this contextrepresentsthe instantaneousrate ∞ TC
at which requests for a given content m arrive at the cache. C =γ 1−φV − λ(τ −θ)dθ dτ (2)
Requests are assumed to form an independent time- Z0 Z0 !
inhomogeneous Poisson processes and λ(t) is taken to be
and γ is the rate at which new contents are introduced into
an arbitrary function satisfying the following conditions: i)
the catalogue.
(positiveness) λ(t) ≥ 0 ∀t with λ(0+) > 0, ii) (causality)
Proof: Proceeding along the same lines as for the sta-
λ(t) = 0 ∀t < 0, iii) (smoothness) λ(t) continuous almost
tionarycase (recalledin AppendixA), we considera constant
∞
everywhere, iv) (integrability) λ(t)dt = 1. The average
0 ∞ cache eviction time TC. We can now start evaluating the
content lifetime can be computed as L= tλ(t)dt.
R 0 probability of finding a given content in the cache at time
Observe that Vm represents the expected total number of t, conditional on the time it has been introduced into the
R
requests (volume) induced by content m during its whole life catalogue (τ) and its request volume (V). This corresponds
in the system. More specifically, by construction, the total
to the event in which one or more requests for the content
number of requests for content m is given by a Poisson occur in the time interval [t−T ,t]. This probability can be
C
distribution with an average of Vm. We assume that volumes evaluated as 1: as:
of requests for different contents form an i.i.d. sequence of
random variables distributed around some reference V. We pin(t|τ,V)=1−e−VRtt−TCλ(θ−τ)dθ (3)
denoteφ (x)=E[exV]tobethemomentgeneratingfunction
V
of V and φ′ (x) its first derivative. Now, unconditioningwith respect to V in (3), we obtain:
V
Finally, the aggregate process of requests arriving to the
cache is now by construction a Cox process [9] whose pin(t|τ)=EV 1−e−V Rtt−TCλ(θ−τ)dθ =
stochastic intensity is given by: Λ(t)= V λ(t−τ ).
m m m h ti
1−φ − λ(θ−τ)dθ
V
B. Cache hit probability P (cid:18) Zt−TC (cid:19)
InthissectionweextendtheLRUapproximationofCheet. To evaluate the probabilityof finding the given content in the
al. [8] to our non-stationarytraffic model in orderto estimate cache at time t, we unconditionwith respect to τ and obtain:
the cache hit probability, i.e. the probability that a generic
1 t t
request finds the content in the cache.
p (t)= 1−φ − λ(θ−τ)dθ dτ (4)
in V
ConsideracachecapableofstoringC distinctcontents.Let t Z0 (cid:18) Zt−TC (cid:19)
T (m) be the time neededfor C distinct contentsnotinclud-
C This result exploits the elementary property of Poisson pro-
ing m to be requested by users. T (m) therefore represents
C cesses that, when a point falls within a specified interval of
the cache eviction time for content m, i.e. after which point
time, its distribution is uniform over the considered interval.
contentmwillbeevictedfromthecache.Che’sapproximation
Now, as in the case of the stationary popularity scenario,
is centred on assuming that the cache eviction time (T (m))
C for a sufficiently large t, the cache is completely filled with
is deterministic and independent from the selected content
contents introduced to the catalogue before t and the number
(m).Thisassumptionhasbeengivenatheoreticaljustification
of contents in the cache is exactly equal to the size of the
in [10], where it is shown that, under a Zipf-like static pop-
cache. We can therefore claim:
ularity distribution, the coefficient of variation of the random
variablerepresenting TC(m) tendsto vanishas the cache size C = [I I ]
grows.Furthermore,thedependenceoftheevictiontimeon m {contentmincache|τm≤t} τm≤t
m
becomes negligible when the content catalogue is sufficiently X
where τ is the time at which content m is introduced into
large. The arguments given in [10] are easily extended to our m
thecatalogue,andthesumextendsoverallthecontentsinthe
non-stationary traffic model when γ and C are sufficiently
infinite content catalogue. Averaging both terms we obtain:
large.
Returning to our non-stationary traffic model, we can now
C = E[I I ]=p (t) E[I ]
state our main result: {contentmincache|τm≤t} τm≤t in τm≤t
m m
Theorem 1: Consider a cache of size C implementing X X (5)
LRUpolicy,operatingunderanon-stationarypopularitymodel
(as introduced in Sec. II-A) with total stochastic intensity: 1similarly to(19)obtained understationary popularity
Since the average number of contents introduced to the cat- and
alogue at any time interval of size t is γ, by combining (4) 2
E[V2] ∞ TC
with (5), we can evaluate the size of the cache C as: C ≥γE[V]T −γ λ(τ −θ)dθ dτ (9)
C
2
E[I ] t t Z0 Z0 !
C = τm<t 1−φV − λ(θ−τ)dθ dτ = Proof:The inequalityin (8) is derivedfor(2) by exploit-
t
m !Z0 (cid:18) Zt−TC (cid:19) ing the inequality 1+x ≤ ex. In particular we exploit the
X
t t previousinequality to lower bound φ (− TCλ(τ −θ)dθ)=
γZ0 1−φV (cid:18)−Zt−TC λ(θ−τ)dθ(cid:19)dτ (6) E[e−R0TCλ(τ−θ)dθ)] with 1−E[V] 0TVCλ(τR0−θ)dθ inside the
Equation (6) proves (2), and must be solved (numerically) to integral appearing in (2).
R
evaluate the eviction time (T ) given a cache size of C. Similarly, the inequality in (9) is obtained by exploiting
C
Having defined T , we now return to evaluating the hit 1−x+ x2 ≥e−x for every x≥0.
C 2
probabilityforagivencontentwiththeparameters(τ ,V ).By Note that the upperboundin (8) has a simple meaning;as-
0 0
definition, the request for a given content at time t, generates sumingthatallthe requestsarereferringto differentcontents,
a hit at the cache iff the content is located on the cache. the cache size is bounded by the overall number of requests
Therefore, the probability of a hit is given by: γE[V]duringthetimeintervalT .InSec.II-D,wewillshow
C
thatthisboundisagoodapproximationforC whencachesize
p (t|τ ,V )=p (t|τ ,V )
hit 0 0 in 0 0 is small.
Now, to uncondition p (t | τ ,V ) with respect to V To gather more insights on the impact of different param-
hit 0 0 0
and τ , we have to consider that the probability with which eters on p , we now derive a simplified expression for the
0 hit
contents are requested by users is biased toward contents two extreme scenario regimes of large cache and small cache
with higher instantaneouspopularity. Let N(V ,∆V,τ ,∆τ ) sizes.
0 0 0
be the average number of contents that have been generated
C. Large-cache regime
during the interval [τ ,τ + ∆τ ) with request volume in
0 0 0
[V ,V + ∆V ). Now the probability that a request arrives A closed formexpressionforthe asymptotichit probability
0 0 0
for one of the contents defined above is (p ) when the cache C → ∞ can be derived from (1) by
hit,∞
N(V0,∆V,τ0,∆τ0) V0λ(t−τ0) making TC →∞.
×
γt E[V] Corollary 2: For large cache sizes,
where the second term represents the instantaneous rate orig- 1−φ (−1) 1 E[e−V]
V
p =1− =1− + (10)
inated by every considered content. Thus, recalling (3) we hit,∞ E[V] E[V] E[V]
have:
Proof: Consider the limit as T → ∞ in the integral
C
Vλ(t−τ) within (1); it holds that:
p (t)=E p (t|τ,V) =
hit τ,V E[V] in
(cid:20) (cid:21) ∞ ∞
EV t VλE(t[V−]τ) 1−e−VRtt−TCλ(θ−τ)dθ dτ = Z0 λ(τ)φ′V (cid:18)−Z0 λ(τ −θ)dθ(cid:19)dτ =
tλZ(0t−(cid:20)τ)EV EV[V(cid:16)] − Ve−V RttE−T[VC]λ(θ−τ)d(cid:17)θ(cid:21) dτ = Z0∞λ(τ)φ′V (cid:18)−Z0τ λ(α)dα(cid:19)dτ (11)
Z0 ! Now if we define Λ(τ) = τ λ(α)dα (by construction, Λ(α)
0
t φ′ − t λ(θ−τ)dθ is also the primitive of λ(α)) and β = Λ(τ), by substituting
V t−TC R
λ(t−τ) 1− dτ β into (11), we obtain:
Z0  (cid:16) R E[V] (cid:17)
∞ 1
By substituting α=t−τ and β =t−θ:  λ(τ)φ′ (−Λ(τ))dτ = φ′ (−β)dβ =
V V
Z0 Z0
φ (0)−φ (−1)=1−φ (−1) (12)
t φ′ − TC λ(α−β)dβ V V V
V 0
p (t)= λ(α) 1− dα Finally, (10) is obtained by using (12) within (1).
hit Z0  (cid:16) R E[V] (cid:17) Observethat(10)dependsheavilyonthedistributionofthe
  (7) contentrequestvolumes(V),andiscompletelyindependentof
Thankstotheintegrabilitypropertyof λ(t),(1)isobtainedby the temporalprofileof the popularity(λ(t)). Thisis expected,
letting t→∞ in (7). when we consider that as C and T grow large, contents are
C
The following corollary sheds some light on the relation never evicted from the cache. In effect, only the first request
between C and TC: for every content will lead to a cache miss, independently of
Corollary 1: The variables C and TC satisfy: the arrival request pattern.
∞ TC Theexpression(10)isexact,sinceforC →∞itcanbeeas-
C ≤γE[V] λ(τ −θ)dθdτ =γE[V]TC (8) ilyprovedthatT (m)→∞w.p.1.Thisisobtainedexploiting
C
Z0 Z0
by the following properties: i) as C →∞ the conditional hit Profile λ(t) 0∞λ2(τ)dτ
probability for contents originating R ≥ 1 requests tends to Exponential ζ−1L1e−tt/L fo−rtζ≥0 R(ζ−21L1)2
phit(R)=1−1/R,andii)theprobabilityofobservingatleast Powerlaw(ζ>1) +1 fort≥0
one request for content m is Pr(R≥1)=1−eVm. Uniform L (cid:18)1 Lfort∈(cid:19)[0,2L] L(2ζ1−1)
The value of C (and consequently T ) for which p ap- 2L 2L
λpr(ot)a.chInedsepehdit,∞it,isinpsotessaidblheeatovidlyerdiveepeanbdosCuthnedpoonptuhlearditiyffpehrirteonficlee Triangular (L2tLL2−2t ffoorrtt∈∈[[L0,,L2]L] 32L
TABLEI
of the hit probability from phit,∞: EXAMPLESOFPOPULARITYPROFILESλ(t).FORALLPROFILESTHE
Corollary 3: AVERAGECONTENTLIFETIMEISSETEQUALTOL.OBSERVETHAT
p −p ≤ ∞λ(τ)dτ 0∞λ2(τ)dτ ISTHEKEYPARAMETERAPPEARINGIN(18).
hit,∞ hit,TC
ZTC R
Proof: Starting from (1), we obtain:
φ′ (x) = E[VexV] ≈ E[V]+xE[V2] for small values of x.
∞ φ′ − TC λ(τ −θ)dθ V
V 0
phit =1−Z0 λ(τ) (cid:16) R E[V] (cid:17)dτ = betFteurrtahpeprmrooxriem,awteethcaenreilmatpioronvbeetthweeerensuCltsanindTCoroasllafroyllo1wtso:
∞ φ′ − TCλ(τ −θ)dθ C
Z0 λ(τ)1− V (cid:16) R0E[V] (cid:17)dτ = Following the sameCre≈asoγnEin[Vg]TasCthe proof for (8),(o1b7-)
TC  φ′ − τλ(α)dα  serve that, for small values of T , TC λ(τ − θ)dθ ≪ 1.
λ(τ) 1− V 0 dτ+ C 0
Z0 " (cid:0) ER[V] (cid:1)# Now φV(− 0TC λ(τ −θ)dθ) can be Rapproximated with 1−
∞ φ′ − TCλ(τ −θ)dθ E[V] 0TC λ(Rτ −θ)dθ and the desired relation is obtained.
V 0 Using (17), we can now rewrite (16) as follows:
ZTC λ(τ)1− (cid:16) R E[V] (cid:17)dτ (13) R E[V2]C ∞
p ≈ λ2(τ)dτ (18)
  hit E2[V] γ
where we have operated the change of variable α = τ −θ. Z0
By observing that φ′(x) = E[VexV] ≤ E[V] for any x ≤ 0, Theexpressiongivenin(18)enlightensustothepotentially
it is possible to upper bound the right-most term of (13) as large effect the popularity profile of content has on the
follows: effectiveness of caching, when cache sizes are small. For
illustration, if we consider the popularity profiles given in
∞ φ′V − 0TC λ(τ −θ)dθ ∞ Table I, the hit probability is always inversely proportional
λ(τ) 1− dτ ≤ λ(τ)dτ
ZTC  (cid:16) R E[V] (cid:17) ZTC to the average content lifetime (L). But, by comparing the
thirdcolumn,it is clearthatthe actualvaluedependsstrongly
  (14)
on the shape of the profile.
Thanksto (12)itis also possibletoupperboundtheleft-most
term: III. NUMERICAL VALIDATION
TC φ′ − τλ(θ)dθ In this section we present: i) the results of applying
λ(τ) 1− V 0 dτ ≤ the approximation of LRU under non-stationary traffic (see
E[V]
Z0 " (cid:0) R (cid:1)# Sec.II-A), as given by Theorem 1; ii) the validation of the
∞ φ′ − τ λ(θ)dθ predictions of the model through Monte Carlo simulations of
λ(τ) 1− V 0 dτ =
E[V] a single cache.
Z0 " (cid:0) R (cid:1)# The results presented in this section relate the size of a
∞ φ′ (−Λ(τ)) 1−φ (−1)
1− λ(τ) V E[V] dτ =1− E[VV] (15) cache (C) to the hit probability (phit) and look at relation
Z0 between these two variables, when varying: i) the average
whichcorrespondsto p .Bycombining(14)with(15),we content lifetime (L), ii) the average content request volume
hit,∞
get the assert. (E[V]),iii) thedistributionofcontentrequestvolumes,iv)the
shape of the popularity time profile.
D. Small-cache regime
Foralltheresultsthatfollow,wesetthecontentarrivalrate
Under this regime, we get the following hit probability: (γ)to10kcontentsperdayandthecontentrequestvolume(V)
Corollary 4: For very small cache sizes, we can approxi- isassumedtobedistributedaccordingtoaParetodistribution:
mate the hit probability as: f (v) = βVβ /v1+β for v ≥ V . The choice of a Pareto
V min min
distribution is justified by two factors. First, several recent
E[V2] ∞
p ≈ T λ2(τ)dτ (16) measurement studies have shown that the Zipf law is a very
hit E[V] C
Z0 goodapproximationof the empiricaldistributionof long term
Proof: The expression in (16) is obtained from (1) by content (video) request volumes [10], [11]. Second, a Zipf-
assuming TC λ(τ−θ)dθ ≈λ(τ)T ≪1, andapproximating like distribution of content request volumes with parameter
0 C
R
1 1
sim - E[V] = 1.5
0.9 mod - E[V] = 1.5
0.1 0.8 sim - E[V] = 3
mod - E[V] = 3
0.7
sim - E[V] = 10
0.01 0.6 mod - E[V] = 10
Phit msoimd -- LL == 11 Phit 0.5
0.001 sim - L = 10 0.4
mod - L = 10
0.3
sim - L = 50
0.0001 mod - L = 50 0.2
sim - L = 300 0.1
mod - L = 300
1e-05 0
100 1000 10000 100000 1e+06 100 1000 10000 100000 1e+06
Cache Size Cache Size
Fig.1. Cachehitprobabilityunderexponentialpopularityprofile,fordifferent Fig.3. CachehitprobabilityunderexponentialpopularityprofilewithL=10
values oftheaverage contentlifetime L(expressedindays). daysfordifferent values oftheaveragecontent volume E[V].
1 1
0.5
0.1 0.2
0.1
0.01 0.05
Phit msoimd -- LL == 11 Phit 0.02 msoimd -- ββ == 11..55
0.001 sim - L = 10 0.01 sim - β = 2
mod - L = 10 0.005 mod - β = 2
sim - L = 50 sim - β = 3
0.0001 mod - L = 50 0.002 mod - β = 3
sim - L = 300 sim - β = 4
mod - L = 300 0.001 mod - β = 4
1e-05
100 1000 10000 100000 1e+06 100 1000 10000 100000 1e+06
Cache Size Cache Size
Fig.2. Cachehitprobabilityunderpowerlawpopularityprofilewithζ=3, Fig.4. Cachehitprobabilityunderexponentialpopularityprofilefordifferent
fordifferent values oftheaverage contentlifetime L(expressedindays). requestvolumedistributionsandaveragecontentrequestvolumeE[V]=1.5.
α=1/(β−1) is obtainedwhena largenumberofindividual Theresultshererevealthatthevolumeofhitsaccumulatedby
content request volumes are independently generated accord- contenthasimpactexclusivelyforlargecachesizes, i.e.when
ing to Pareto distribution. C ≥ 200k objects, and as predicted by (10), the cache hit
With regard to the different popularity profiles that content probabilityincreaseswith the volumeof contentrequests. For
may display, we consider exponential and power law profiles moderatecachesizes(C ≤10k),theeffectofthehitsbecomes
as given in Table I. Finally, the parameter ζ is used to model negligible as predicted by (18). Indeed, all curves correspond
the different time-dependent popularity profiles (popularity to the same value of E[V2].
E2[V]
shapes). Fig. 4 reports the cache hit probability for different values
Figs.1and2reportthehitprobabilityfordifferentvaluesof of the parameter β, associated to the distribution of content
content lifetime, with respect to using; i) an exponential (see request volumes. From the figure we see that the shape of
Fig. 1) and ii) power law with parameter ζ = 3 (see Fig. 2) requests volumes has a significant impact on the cache hit
popularity profile. In both cases, we set V =1 and β =3, probability.As expected, by decreasing β (i,e., increasing the
min
as a consequence,we obtain E[V]=1.5 requestspercontent. correspondent parameter α for the associated Zipf law), the
The first point to observe is that the model estimates agree cache performance is improved. In particular we observe that
strongly with the simulation results for the hit probability in thecachingperformancebecomemuchmoresensitiveto β as
all the cases. Second, as predicted by the model, the average β getssmallerthan2(i.e.,asthecorrespondingZipfparameter
content lifetime (L) deeply impacts the cache performance. α increases above 1). However, the impact of β (i.e., α) on
Indeed, for a given cache size (C), a given content’s hit caching performancedoes not appearin our scenario to be as
probability increases significantly as its lifetime is reduced. strong as it does in the classical stationary case, where a sort
In particular, for moderate cache sizes, the hit probability of“phasetransition”isobservedasαcrossesthevalue1[10].
is roughly inverse proportional to the content lifetime, as Finally,Fig. 5 reportsthe cache hitprobabilityfordifferent
predicted by Corollary 4. content popularity profiles (i.e, varying ζ) - while keeping
Fig.3reportsthecachehitprobabilityfordifferentvaluesof the average lifetime as constant at L = 10 days. From the
theaveragecontentrequestvolumeE[V].Allplotsrefertothe figure, we see that the content popularity profile appears to
samevalueofβ =3anddifferentvaluesofV =E[V]β−1. haveamoderateimpactonthecachingperformance(forsmall
min β
1 1
0.5 0.5
0.2 0.2
0.1 0.1 sim - S
1
0.05 0.05 mod - S1
Phit 0.02 msoimd -- ζζ == 22..22 Phit 0.02 msoimd -- SS22
0.01 sim - ζ = 2.5 0.01 sim - S3
0.005 mod - ζ = 2.5 0.005 mod - S3
sim - ζ = 3 sim - S4
0.002 mod - ζ = 3 0.002 mod - S4
0.001 msoimd -- ζζ == 44 0.001 msoimd -- SS55
100 1000 10000 100000 1e+06 100 1000 10000 100000 1e+06
Cache Size Cache Size
Fig.5. Cachehitprobabilityunderpowerlawpopularityprofilefordifferent Fig.6. Cachehitprobabilityunderexponentialpopularityprofilefordifferent
values ofζ andthesameaveragecontent lifetime L=10days. classesconfigurations.
Popularity class L [days] S1 S2 S3 S4 S5
caches) as long as the average content lifetime L is kept 1 1 40% 30% 20% 10% 5%
constant.Fortheextremecase wherethesize cache C =100, 2 10 10% 20% 30% 40% 45%
p varies from 0.0032 and 0.001 as ζ is decreased from 4 to 3 50 10% 20% 30% 40% 45%
hit
4 300 40% 30% 20% 10% 5%
2.2 (see the simulation curves).
TABLEII
IV. EXTENSION TOAMULTI-CLASSSCENARIO AVERAGECONTENTLIFETIMELANDFRACTIONSOFCONTENTSINEACH
CLASSFORSETUPSFROMS1TOS5.
In this section, we consider a more realistic scenario in The proof for Theorem 2 (not reported here for the sake of
which contents can be partitioned into K classes, such that brevity) follows the same lines of Theorem 1.
eachclassisassociatedwithadifferentpopularityprofileλk(t) In order to evaluate the multi-class scenario, we partition
andadifferentrequestvolumedistributionVk,for1≤k ≤K. contentsinto K =4 classes, each characterisedby a different
Thisgeneralisationofourtrafficmodelisneededinorderto popularity profile and lifetime. For simplicity, here, we only
capture the variability of the popularity profiles exhibited by considered 5 different configurations as specified in Table II.
realcontents.Indeedthepopularityprofiledependsheavilyon Fig.6reportsthecachehitprobabilityforeachsetup.First,
thenatureofcontents,forexample,thepopularityevolutionof from the figure, we see that the model predictions of the
music videosis typically significantly differentto videos con- cache hit probabilityfordifferentcache sizes align accurately
taining sport highlights. However, recent experimental stud- with the results of the simulation. Second, Fig. 6 shows
ies [7] have shown that the popularity evolution of different that the heterogeneity of contents weakly impacts the cache
contents can be clustered to relatively few groups exhibiting performance,andthatthecachehitprobabilityincreaseswhen
similar temporal popularity profiles. the average content lifetime (L) of each setup decreases, i.e.
We can formalisethe multi-classscenario by assuming that when the number of contents with fast popularity dynamics
everygeneratedcontent(m)isassociatedwitharandommark (belonging to class 1) is large with respect to the rest. In
Wm takingvaluesin {1,...,K},such thatthemarkspecifies fact, as already seen in Fig. 1, contents attracting users’
the class the content belongs to. Assuming {Wm} to be i.i.d. attention for very limited periods of time (L = 1 days) are
randomvariables,thetotalstochasticintensityattime t ofthe largelyresponsibleforimprovingthecacheperformancewhen
request process is given by: adopting the LRU strategy.
Λ(t)= VmλWm(t−τm)= VmλWm(t−τm)I{τm≤t} V. CONCLUSIONS
Xm Xm This work has proposed(and validated with a Monte Carlo
Under this assumption, we can now state the following:
simulation) a simple but highly accurate approximatedmodel
Theorem2: ConsideracacheofsizeC implementingLRU
foranLRU(LeastRecentlyUsed)cacheundernon-stationary
policy,operatingunderamulti-classnon-stationarypopularity
traffic conditions. The proposed model is flexible and can
model,withtotalstochasticintensity:Λ(t)= V λ (t−
m m Wm be easily extended to consider complex and realistic traffic
τ ). Extending Che’s approximation, the hit probability is
m scenarios.Ithastheadvantageofbeingcomputationallycheap
P
given by:
whencomparedwiththeMonteCarlosimulations-especially
p =1− K Pr{W =k} ∞λ (τ)φ′Vk − 0Tcλk(τ −θ)dθ dτ whOenurcarecshueltssizsehsowarethlaartgcea.ching performance is deeply im-
hit k=1 1 Z0 k (cid:16) R E[Vk] (cid:17) pacted by the dynamics resulting from the popularity of
X
content.In particular,we find thatwhen cache sizes are small
where T is the solution to the equation:
C
as in the case anticipated for ICN networks, the content hit
∞ K TC probability is largely sensitive to the popularity profile of
C =γ 1− Pr{W =k}φ − λ (τ −θ)dθ dτ
1 Vk k contents.
Z0 " 1 Z0 !#
X
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APPENDIXA top popular contents to the total traffic is significant. Indeed,
LRUUNDER STATIONARYPOPULARITY observethatthatH(M)=Θ(1)whenM growslarge,i.e.the
aggregate contribution of tail contents is marginal.
First we briefly resume Che’s approximation of LRU in a
Forα<1,instead,onlycachingasignificantportionofthe
classical traffic scenario in which every content m, in a finite
huge catalog we can achieve significant cache hit probability.
catalogofsizeM,presentsatime-invariantpopularityprofile.
In this case H(M) diverges as M → ∞ showing that the
More in details, we assume that requests for content m arrive
aggregate contribution of tail contents is dominant.
atthecacheaccordingtoatimehomogeneousPoissonprocess
with intensity λ . Let Λ = λ be the resulting total
m m m
arrival intensity of content requests at the cache.
P
Now,thankstoChe’sapproximation,discussedinSec.II-B,
a content m is present in the cache at time t, if and only if a
time less than T has passed since the last requestfor content
c
m, i.e., if at least a request for such content has arrived in
the interval(t−T ,t]. Since requestsarrivalsare Poisson, the
c
probabilityp (m)thatatatleastonerequesthasarrivedinthe
h
interval(t−T ,t] is givenby:p (m)=1−e−λmTc. Observe
c h
thatp (m)represents,byconstruction,also,thehitprobability
h
for content m, as immediate consequenceof PASTA property
of arrivals.
Considering a cache of size C, by construction: C =
I . When averaging both sides, we obtain:
m {mincache}
PC = E[I ]= p (m)= (1−e−λmTc).
{mincache} h
m m m
X X X
By solving the aboverelationship, we obtain T , and then the
c
average hit probability on the cache as:
p = p p (m) (19)
hit m h
m
X
A. Numerical evaluation
WetesttheChe’sapproximationinthestationarypopularity
case. We set a catalogsize equalto M =107 contents.Fig. 7