Adaptive Replication in Distributed Content Delivery Networks

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by  Mathieu Leconte

Adaptive Replication in Distributed Content
Delivery Networks
M. Leconte M. Lelarge L. Massoulie´
Technicolor - INRIA INRIA - E´cole Normale Supe´rieure Microsoft Research - INRIA Joint Center
[email protected] [email protected] [email protected]
Abstract—We address the problem of content replication in the data center is much more costly than by a small server,
large distributed content delivery networks, composed of a data thus requests are served from the small servers in priority. In
4 center assisted by many small servers with limited capabilities fact, we only send requeststo the data center if no idle server
1 and located at the edge of the network. The objective is to
storing the corresponding content is present. We assume the
0 optimize the placement of contents on the servers to offload
2 as much as possible the data center. We model the system data center is dimensionedto absorbany such remainingflow
constituted by the small servers as a loss network, each loss of requests. We restrict our attention here to contents which
n
corresponding to a request to the data center. Based on large require immediate service, which is why we do not allow
a
system / storage behavior, we obtain an asymptotic formula for
J requests to be delayed or queued. Stringent delay constraints
theoptimalreplicationofcontentsandproposeadaptiveschemes
are indeed often the case for video contents –for example
8 relatedtothoseencounteredincachenetworksbutreactinghere
to loss events, and faster algorithms generating virtual events in the context of a Video-on-Demand (VoD) service–, which

] at higher rate while keeping the same target replication. We represent already more than half of the Internet traffic.
I
N showthroughsimulationsthatouradaptiveschemesoutperform
In that context, we address the problem of determining
significantlystandard replication strategies both in terms of loss
s. rates and adaptation speed. whichcontentstostoreinthesmallservers.Thegoalistofind
c contentplacementstrategieswhichoffloadasmuchaspossible
[ I. INTRODUCTION thedatacenter.Inthisoptic,itisobviousthatpopularcontents

1 shouldbemorereplicated,butthereisstillalotoffreedomin
The amount of multimedia traffic transfered over the In-
v thereplicationstrategy.Inorderto comparedifferentpolicies,
0 ternet is steadily growing, mainly driven by video traffic. A we model the system constituted by the small servers alone
7 large fraction of this traffic is already carried by dedicated
as a loss network, where each loss corresponds to a request
7 content delivery networks (CDNs) and this trend is expected
to the data center. The relevant performance metric is then
1
to continue[1]. The largestCDNs are distributed,with a herd
. simplythe expectednumberof losses inducedby eachpolicy.
1 ofstoragepointsanddatacentersspreadovermanynetworks.
0 With the cooperationof the InternetService Providers, CDNs We make three main contributions:first, using a mean-field
4 heuristic for our system with a large number of servers with
can deploy storage points anywhere in the network. How-
1 limited coordination, we compute accurately the loss rate of
ever, these cooperations require a lot of negociations and are
:
v therefore only slowly getting established, and it also seems eachcontentbasedonitsreplication;buildingonthisanalysis,
i we derive an optimized static replication policy and show
X necessary to consider alternatives to a global cache-network
that it outperforms significantly standard replications in the
planning. An easily deployable solution is to make use of
r
a the ressources (storage, upload bandwidth) already present at literature; finally, we propose adaptive algorithms attaining
the optimal replication. Contrary to most caching schemes,
the edge of the network. Such an architecture offers a lot of
ourevictionrulesarebasedonloss statistics insteadofaccess
potential for cost reduction and quality of service upgrade,
statistics; in particular our algorithms do not know or learn
but it also comes with new design issues [2] as the operation
the popularities of the contents and will adapt automatically.
of a very distributed overlay of small caches with limited
We also propose a simple mechanism relying heavily on our
capabilities is harder than that of a more centralized system
mean-fieldanalysisthatallowsustospeedupsignificantlythe
composed of only a few large data centers.
rate of adaptation of our algorithms. At each step, extensive
Inthispaper,weconsiderahybridCDNconsistingofbotha
simulations support and illustrate our main points.
data center and an edge-assistance in the form of many small
servers located at the edge of the network; we call such a The rest of the paper is organized as follows: we review
system an edge-assisted CDN. The small servers are here to relatedworkinSectionIIanddescribemoreindetailoursys-
reducethecostofoperationoftheCDN,whilethedatacenter temmodelinSectionIII.InSectionIV,weanalyseourmodel
helpsprovidereliabilityand qualityofserviceguarantees.We under a mean-field approximation and obtain an asymptotic
stressthefactthatthesmallservershavealimitedstorageand expression for the optimal replication. Then, in Section V,
service capacity as well as limited coordination capabilities, leveraging the insights from the analysis we propose new
which makes cheap devices eligible for that role. As their adaptive schemes minimizing the average loss rate, as well
respective roles hint, we assume the service of a request by as ways to further enhance their adaptation speed.
2
II. RELATEDWORK trafficwithvarioustypesofcontents.Basedonthedifferences
in their respective popularity distributions, [7] advocates that
Ouredge-assistedCDNmodelisrelatedtoafewothermod- VoDcontentsarecachedveryclosetothenetworkedgewhile
els. A major difference though is our focus on modeling the user-generated contents, web and file sharing are only stored
basic constraints of a realistic system, in particular regarding in thenetworkcore.Ina loosesense, we also studysuch type
the limited capacity of servers (in terms of storage, service of two-layer systems, with the small servers at the edge of
and coordination) and the way requests are handled, i.e. the the network and the data center in the core, and address the
matching policy, which role is to determine which server an question of determining whether to store contents at the edge
incomingrequestshouldbedirectedto.Weindeedconsiderthe (and in what proportion)or to rely on the core to serve them.
most basic matchingpolicy:an idle server storing the content Concerning P2P VoD models, content pre-fetching strate-
is chosen uniformly at random; and no queueing of requests,
gies have been investigated in many papers (e.g. [3], [8],
nore-directionduringserviceandnosplittingareallowed.The [9],[10]),whereoptimalreplicationstrategiesand/oradaptive
reason for modeling such stringent constraints and refusing schemeswerederived.Theyworkunderamaximummatching
simplifyingassumptionsinthisdomainisthatthisshouldlead
assumptionandmostofthemassumeinfinitedivisibilityofthe
to fundamental intuitions and qualitative results that should requests,i.e. thata requestcan be servedin parallelby allthe
apply to most practical systems. serversstoringthecorrespondingcontent;wewanttotakeinto
Our edge-assisted CDN model is very similar to the dis- account more realistic constraints in that respect.
tributed server system model of [3], [4]; we merely use a Onecanalsoexplorethedirectionofoptimizingdistributed
different name to emphasize the presence in the core of systems with a special care for the geographical aspects as
the network of a data center, which makes it clear what in [11], [12], [13]. These papers solve content placement and
performance metric to consider, while otherwise availability matchingproblemsbetweenmanyregions,whilenotmodeling
of contents for example may become a relevant metric. The in a detailed way what happens inside a region.
edge-assistedCDNmodelisalsorelatedtopeer-to-peer(P2P) Finally, most of the work in these related domains fo-
VoD systems, which however have to deal with the behavior cuses on different performance metrics: in hierarchical cache
of peers, and to cache networks, where caches are deployed networks, [14] addresses the problem of joint dimensioning
in a potentially complex overlay structure in which the flow of caches and content placement in order to reduce overall
ofrequestsenteringacacheisthe“miss”flowofanotherone. bandwidth consumed; [15] optimizes replication in order to
Previous work on P2P VoD typically does not model all the minimize search time and devises elegant adaptive strategies
fundamentalconstraintswementioned,andthecachenetworks towards that end. Surprisingly, for various alternative objec-
literaturetendstocompletelyobliteratethefactserversbecome tives and network models, the proportional (to popularity)
unavailable while they are serving requests and to focus on replicationexhibitsgoodproperties:itminimizesthedownload
alternativeperformancemetricssuchassearchtimeornetwork time in a fully connected network with infinite divisibility
distance to a copy of requested contents. of requests and convex server efficiency [16], as well as
Indistributedservernetworks,amaximummatchingpolicy the average distance to the closest cache holding a copy of
(with re-packing of the requests at any time but no splitting the requested content in a random network with exponential
or queueing) has been considered in [3], [4]. In this setting, expansion [17]; and recall that [3] also advocated using this
in [3] it was shown that replicating contents proportionallyto replication policy. Therefore, due to its ubiquity in the litera-
their popularity leads to full efficiency in large system and ture, proportional replication is the natural and most relevant
largestoragelimits, andin [4] itwas furtherprovedthatthere schemeto whichwe shouldcompareanyproposedpolicy;we
isactuallyawiderangeofpolicieshavingsuchanasymptotic will obtain substantial improvementsover it in our model.
property and an exact characterisation of the speed at which
efficiencydecreasesforagivenpolicywasobtained.However, III. EDGE-ASSISTEDCDN MODEL
thatformulafortheefficiencyistoocomplicatedtoderiveany Inthissection,wedescribeindetailouredge-assistedCDN
practical scheme from it, and anyway maximum matching at model.
any time is probably unrealistic for most practical systems. As already mentioned, the basic components of an edge-
Finding efficient content replications in cache networks is assistedCDNareadatacenterandalargenumbermofsmall
a very active area of research. The optimal replication is not servers. The CDN offers access to a catalog of n contents,
yet understood, however adaptive strategies are studied; they which all have the same size for simplicity (otherwise, they
createreplicasforcontentswhenincomingrequestsarriveand could be fragmented into constant-size segments). The data
evict contents based on either random, least frequently used center stores the entire catalog of contents and can serve all
(LFU) or least recently used (LRU) rules, or a variation of the requests directed towards it, whereas each small server
those. There is an extensive literature on the subject and we can only store a fixed number d of contents and can provide
only mention the most relevant papers to the present work. serviceforatmostonerequestatatime.We canrepresentthe
An approximationfor LRU cache performanceis proposed in informationofwhichserverstoreswhichcontentinabipartite
[5] and further supported in [6]; it allows [7] to study cache graph G = (S ∪C,E), where S is the set of servers, C the
networkswithatwo-levelshierarchyofcachesunderamixof set of contents, and there is an edge in E between a content
3
c and a server s if s stores a copy of c; an edge therefore seems reasonable to pay a particular attention to the asymp-
indicatesthat a server is able to serve requestsfor the content totics of the performance of the system as n,m → ∞. To
with which it is linked. Given numbers of replicas (D ) keepthingssimple,we can assumethatthe numberofservers
c c∈C
for the contents, we assume that the graph G is a uniform m andthe numberof contentsn grow to infinity together,i.e.
randombipartitegraphwithfixeddegreesdontheserversside n = βm for some fixed β. In addition, as storage is cheap
andfixeddegreesequence(D ) onthecontentsside. This nowadayscomparedto access bandwidth, it also makes sense
c c∈C
models the lack of coordination between servers and would to focus on a regime with d large (but still small compared
result from them doing independent caching choices. We do to n). In these regimes, the inefficiency will tend to 0 as the
notallowrequeststobequeued,delayedorre-allocatedduring system size increases under most reasonable replications (as
service,andasaresultaserverwhichbeginsservingarequest was shown in [4] under the maximum matching assumption).
will then remain unavailable until it completes its service. However,asthecostofoperationisonlytiedtolosses,further
Splitting of requestsis not considered either, in the sense that optimizing an already small inefficency is still important and
only one server is providing service for a particular request. can lead to order improvements on the overall system cost.
With these constraints, at any time, the subset of edges of G
overwhichsomeserviceisbeingprovidedforma generalized IV. ANAPPROXIMATION FORTHELOSS RATEOF A
matchingofthegraphG(theanalysisin[4],[18]reliesheavily CONTENT
on this). In this section, we propose an approximationto understand
Thecontentsofthecatalogmayhavedifferentpopularities, in a precise mannerthe relation between any fixed replication
leading to different requests arrival rates λ . We let λ be ofcontentsandthelossratesinthesystem.Thisanalyticalstep
c
the average request rate, i.e. λ = 1 λ . According to hasmanyadvantages:itallowsustoformallydemonstratethat
n c c
the independent reference model (IRM), the requests for the to optimize the system one needsto make the loss rates equal
P
variouscontentsarriveas independentPoisson processes with forallthecontents;asaconsequenceweobtainanexplicitex-
ratesλ . Theservice times ofall the requestsare independent pressionfortheoptimalreplication(SubsectionIV-B);finally,
c
Exponentialrandomvariableswithmean1,whichisconsistent in Subsection V-B, we will leverageour analytical expression
with all the servers having the same service capacity (upload for the full distribution of the number of available replicas
bandwidth) and all the contents having the same size. We let of a content to propose a mechanism enhancing the speed of
ρ denote the expected load of the system, i.e. ρ = nλ. The adaptivealgorithms.We validateour approximationand show
m
distribution of popularities of the contents is left arbitrary, that our optimized replication strategy largely outperforms
although in practice Zipf distributions are often encountered proportionalreplication through extensive simulations.
(see [7] for a page-long survey of many studies) and it
A. One-Dimensional Markov Chain Approximation for the
seemsthereforemandatorytoevaluatethereplicationschemes
Number of Available Replicas
proposed under a Zipf popularity distribution.
When a request arrives, we first look for an idle server For a given fixed constitution of the caches (i.e. a fixed
storing the corresponding content; if none is found then the graph G), the system as described in the previous section is
requestis sentto the data centerto be served ata highercost. Markovian,the minimumstate space indicatingwhichservers
As in the end all the requests are served without delay, be are busy (it is not necessarily to remember which content
it by the data center or by a small server, the performance they serve). We want to understand how the loss rate γ for
c
of the system is mainly described by the cost associated a particular content c relates to the graph G, but using too
with the service. This cost is mostly tied to the number of detailed information about G, such as the exact constitution
requestsserved by the data center, thereforethe most relevant of the caches containing c, would have the drawback that
performancemetrichereisthefractionoftheloadleft-overto it would not lead to a simple analysis when considering
the data center. Then,it makessense to view the requeststhat adaptive replication policies (as the graph G would then keep
the small servers cannot handle as “lost”. In fact, the system changing). Therefore, we need to obtain a simple enough but
consisting of the small servers alone with fixed caches is a still accurate approximate model tying γ to G.
c
lossnetworkinthesense of[19].Thisimpliesthatthesystem Theexpectedlossrateγ ofcontentcisequaltoitsrequests
c
corresponds to a reversible stochastic process, with a product arrival rate λ multiplied by the steady state probability that
c
form stationary distribution π. However, the exact expression c has no replicas available. Let D be the total number of
c
of that stationary distribution is too complex to be exploited replicas of content c and Z be its number of available
c
in our case (as opposed to [3], [4], [18] where the maximum replicas, i.e. those stored on a currently idle server. We thus
matching assumption yields a much simpler expression). We want to compute π(Z = 0) to get access to γ . However,
c c
call γ the rate at which requests for content c are lost, and γ the Markovchain describingthe system is too complicatedto
c
the average loss rate among contents, i.e. γ = 1 γ . The be able to say much on its steady state. In order to simplify
n c c
main goal is to make γ as small as possible. We refer to the the system, one can remark that in a large such system the
P
fractionoflost requestsγ/λasthe inefficiencyof the system. state of any fixed number of servers (i.e. their current caches
Finally, as in many real cases CDNs are quite large, with andwhethertheyarebusyoridle)areonlyweaklydependent,
often thousandsof serversand similar numbersof contents, it andsimilarly the numberof availablereplicasZ of anyfixed
c
4
numberofcontentsareonlyweaklydependent.Therefore,itis This yields the following steady-state probability:
naturalto approximatethe system by decouplingthe different
D +λ /θ
contents and the different servers (similar phenomenon are π(Zc =z)=π(Zc =Dc)θeDffc−z cD −cz eff , (1)
explained rigorously in [20]). In other words, this amounts to (cid:18) c (cid:19)
where the binomial coefficient does not really have a combi-
forgettingthe exact constitution of the caches; as a result, the
natorialinterpretationasoneofitsargumentsisnotaninteger,
correlationbetweencontentswhich are storedtogetherislost.
butshouldonlybeunderstoodas k+x = 1 k (i+x),
Then,theevolutionofZc becomesaonedimensionalMarkov for k,l∈N and x∈R . l l! i=k−l+1
chain,independentofthevaluesofZc′ forothercontents,and + (cid:0) (cid:1) Q
We now have an approximation for the full distribution of
we can try to compute its steady-state distribution.
thenumberZ ofavailablereplicasofcontentc, whichyields
Foranyz <D ,therateoftransitionfromZ =ztoz+1is c
c c
an approximationforthe lossrate γ =λ π(Z =0). We can
alwaysD −z:itistherateatwhichoneoftheD −zoccupied c c c
c c also compute the mode z of this distribution: z ≈ Dc−λc,
serversstoringccompletesitscurrentservice;wedonotneed c c 1+θeff
which can be used as an approximation for the expectation
todistinguishwhetheraserverisactuallyservingarequestfor
E[Z ] (in fact, simulations show the two are nearly indistin-
c or foranothercontentc′ as we assume the service times are c b b
guishable). We further simplify the expression obtained by
independent and identically distributed across contents. For
providing a good approximation for the normalization factor
any z > 0, the transitions from Z = z to z −1 are more
c
π(Z =D ) in Equation (1):
complicated. They happen in the two following cases: c c
• either a request arrives for content c and as Zc = z > 0 Dc D +λ /θ −1
it is assigned an idle servers storing c; π(Zc =Dc)= θexff c xc eff .
• orarequestarrivesforanothercontentc′ anditisserved Xx=0 (cid:18) (cid:19)!
by a server which also stores content c. Tothatend,weusethefactthatweaimatsmalllossrates,and
Thefirsteventoccursatrateλ andthesecondoneatexpected thus most of the time at small probabilities of unavailability.
c
rate Therefore, the bulk of the mass of the distribution of Zc
|{s∈S : s is idle and c,c′ ∈s}| should be away from 0, which means that the terms for x
λc′E |{s∈S : s is idle and c′ ∈s}| , close to Dc should be fairly small in the previousexpression.
c′6=c (cid:20) (cid:21) We thus extend artificially the range of the summation in this
X
wherec∈sindicatesthatthecontentc isstoredontheserver expression and approximate π(Z =D ) as follows:
c c
s. At any time and for anyc′, |{s∈S : s is idle and c′ ∈s}|
is equal to Zc′. The term |{s ∈ S : s is idle and c,c′ ∈ s}| π(Z =D )−1 ≈ Dc+⌊λc/θeff⌋θx Dc+λc/θeff
is equal in expectation to the number of idle servers storing c c eff x
x=0 (cid:18) (cid:19)
c′ (i.e. Zc′) times the probabilitythat such a server also stores ≈ (1+Xθeff)Dc+λc/θeff.
c. As we forgetaboutthe correlationsin the caching,this last
probability is approximated as the probability to pick one of We obtain the following approximation for π(Zc =0):
tch′ewdhe−n1wreemdiasipnaintcghmatemraonrdyomslotthseinZacnavidalielabselervreerpslisctoasrinogf π(Zc =0)≈ 1+θefθf Dc(1+θeff)−θλecff Dc+Dλc/θeff .
contentcbetweenalltheremainingslotsinidleservers.Thus, (cid:18) eff(cid:19) (cid:18) c (cid:19)
Finally, as we are interested in large systems and large
Z (d−1)
E[|{s∈S : s is idle and c,c′ ∈s}|]≈ c′ Z , storage/replication,wecanleveragethefollowingasymptotic
c
(1−ρ )md
eff behavior:
where ρeff = ρ(1 − γ/λ) is the average load effectively k+x ≈ ex(Hk−CEuler),
absorbed by the system, so that the total number of memory k Γ(x+1)
(cid:18) (cid:19)
slotsontheidleserversis(1−ρeff)md.We alsoneglectedthe for k ∈ N large enough and where Γ is the gamma function:
Zc′ memory slots occupied by c′ in these idle servers when Γ(1+x) = 0+∞txe−tdt; Hk is the k-th harmonic number:
computing the total number of idle slots (1 − ρeff)md. We Hk = ki=1R1i; and CEuler is the Euler-Mascheroni constant:
obtain the following approximation for the expected rate at C =lim (H −lnk)≈0.57721.For large number of
Euler k→∞ k
which the second event occurs: replicasPD , we thus obtain the approximation:
c
d−1 ρ d−1
Zcc′6=cλc′(1−ρeff)md ≈Zc1−efρfeff d =Zcθeff, π(Zc =0)≈C(λc) 1+ θ1 −DcDcθλecff,
X (cid:18) eff(cid:19)
cwohnetreentwcecnoemglpeacrteeddttohethreataegλgrcegaattwehaircrihvarleqrauteestosfarreriqvueesftosr, whereweletC(λc)= e−CEλulcerλc/θeff andthetermDcθλecff is
awnidthwreealseotnθaebffly=hi1g−ρheρffeefffdfe−dc1t.ivNeoloteadthρatefft,hecoirnrteesrpeostnidnsgtroegliamrgee, an approximationof eθλ(e1cff+HθDeffc).θeTffhΓe(1+apθλpecffr)oximationfor π(Zc =
valuesofθeff,asρeff →1impliesθeff →∞.TheMarkovchain 0) immediately yields an approximation for the loss rate γc:
obtained satisfies the local balance equations: for z <Dc, 1 −Dc λc
γ =λ π(Z =0)≈λ C(λ ) 1+ Dθeff. (2)
(Dc−z)π(Zc =z)=(λc+(z+1)θeff)π(Zc =z+1), c c c c c (cid:18) θeff(cid:19) c
5
Properties class1 class2 class3
Note that the expression of θ involves the average effective
eff numberofcontents ni 200 400 400
load ρeff, which itself depends on the average loss rate γ. We popularity λi 9 3 1
thushaveafixedpointequationinγ (justasin[5],[6]),which numberofreplicas Di 200 67 23
wecaneasilysolvenumerically.Indeed,theoutputvalueγ of simulationdata/approximation class1 class2 class3
ourapproximationisadecreasingfunctionoftheinputvalueγ nb.ofavailable replicas E[Zi] 21.7/21.6 7.28/7.25 2.51/2.50
103×lossrates γi 0/10−5 3.31/2.36 79.4/76.3
used in ρ , which implies simple iterative methods converge
eff simulationdata/approximation Zipf,α=0.8 Zipf,α=1.2
exponentially fast to a fixed-point value for γ. proportional: 103×inefficiency 5.69/5.46 11.8/11.6
B. Approximation-Based Optimized Replication TABLE I: Properties of the content classes, and numerical
results on the accuracy of the approximation.
In this subsection, we exploit the approximation obtained
in Equation (2) to understand which replication strategy min-
imizes the total loss rate in the system. In other words, we
approximately solve the optimization problem: C. Accuracy of the Approximation and Performance of the
Optimized Replication
D ≤md
min γ s.t. c c (3)
PDc ∈N, ∀c Intheprocessofapproximatingthelossrateγc ofacontent
c, we performed many approximations based on asymptotic
Note that the approximation from Equation (2) is consistent
behaviors for large systems and large storage / replication.
with the intuition that γ is a decreasing, convex function of
Dc. Indeed, letting x =c 1+θeθffeffeθeff(Dλcc+1) ≤ 1 since we have IEtquisatitohner(e2fo)riesnnoetcteososafrayrotoff,cwhehcickhwwheedthoerintthheisfsoercmtiuolnavoiaf
Dc+1≥ λc in the regime of interest, we compute simulation. The systems we simulate are of quite reasonable
θeffln(cid:16)1+θ1eff(cid:17) size (with a few hundred contents and servers). However,
γ (D +1)−γ (D )=∆γ (D )=−γ (D )(1−x)≤0, the accuracy of the approximation should only improve as
c c c c c c c c
∆γ (D −1)=−γ (D ) 1 −1 ≤0. the system grows. We use two scenarios for the popularity
c c c c x
distribution of the contents: a class model (similar to that
The loss rate γ is a convex function of D as(cid:0)shown(cid:1)by
c c of [4]), which allows us to compare between many contents
1 with similar characteristics, and a Zipf popularity model,
∆γ (D )−∆γ (D −1)≥γ (D ) x+ −2 ≥0.
c c c c c c which is deemed more realistic. We evaluate the accuracy
x
(cid:18) (cid:19) of the approximation using proportional replication (other
As a consequence, the optimization problem (3) is approxi-
replications were also simulated and yield similar results). To
mately convex, and we thus obtain an approximate solution
compute the approximate expressions, we solve numerically
by making the first derivatives of the loss rates ∆γ equal.
c the fixed point equation in γ. We simulate systems under a
Keeping only the dominant orders in D , we have
c reasonably high load of 0.9. As often, it is mainly interesting
γ λ to know how the system behaves when the load is high, as
∆γc(Dc)=− c 1− c . (4) otherwiseitsperformanceisalmostalwaysgood.However,as
1+θ D
eff (cid:18) c(cid:19) requestsarriveandareservedstochastically,iftheaverageload
In the first order, equalizing the first derivatives in (4) using weretoocloseto1thentheinstantaneousloadwouldexceed1
theexpressionin(2)leadstoequalizingthenumberofreplicas quite often, which would automatically induce massive losses
foreverycontent,i.e.settingDc =D+o(D))whereD = mnd and mask the influence of the replication. In fact, it is easy
is the average number of replicas of contents. Going after the to see that we need to work with systems with a number of
second order term, we get servers m large compared to ρ in order to mitigate this
(1−ρ)2
effect.
lnD
D =D+(λ −λ) +o(lnD). (5) The setup with the class model is the following: there
c c
θeffln 1+ θ1eff are n = 1000 contents divided into K = 3 classes; the
(cid:16) (cid:17) characteristics of the classes are given in the first part of
Wethereforeobtainthattheasymptoticallyoptimalreplication
Table I. The popularities in Table I together with n = 1000
is uniform with an adjustment due to popularity of the order
andρ=0.9resultinm=3800servers.Eachservercanstore
of the logof the averagenumberofreplicas. Thisagreeswith
d=20 contents, which is 5% of the catalog of contents. We
the results in [4] and goesbeyond.Finally,inserting backthis
let the system run for 104 units of time, i.e. contents of class
expression for D into Equation (2) yields
c 3 should have received close to 104 requests each.
−D Figure 1 and the second part of Table I show the results
1 λ (1+o(1))
γc = 1+ Dθeff , (6) for the class model: the left part of Figure 1 shows the
θ
(cid:18) eff(cid:19) distributions of the numbers of available replicas for all the
which shows that the average loss rate γ under optimized contentsagainsttheapproximationfromEquation(1)foreach
replication behaves as the loss rate of an imaginary average class; the right part shows the loss rates for all the contents
content (one with popularity λ and replication D). against the approximation from Equation (2) for each class.
6
stationary probability000000...012...123555 sacccilllpmaaapsssursssol a123xtiimona tdioanta stationary loss rate00000....0000.12468 sacccilllpmaaapsssursssol a123xtiimona tdioanta of nb. of available replicas123456 naZZbpiipp.pff ar01ov..xa82imilaabtlieo nreplicas stationary loss rates00000...012..12555 laZZopiispppsff r 01ora..x82tiemsation
g
00 nu5mb1e0r o1f 5ava20ilab2l5e r3e0plic3a5s 00 200inde4x0 0of c6o0n0ten8t00 1000 lo00 1 2 3 4 5 6 00 50 100 150 200
log of the rank of the content popularity rank of the content
Fig. 1: Class model: distribution of number of available
Fig. 2:Zipf model:expectednumberof available replicasand
replicas and loss rates Vs approximation.
loss rates Vs approximation.
x 10−3
Although it is not apparent at first sight, the left part of 8
Figure1actuallydisplaysaplotforeachofthe1000contents,
s
but the graphs for contents of the same class overlap almost ate6
perfectly, which supports our approximation hypothesis that s r
s
o
thestationarydistributionofthenumberofavailablereplicasis y l4
ar
notverydependenton the specific serverson which a content n
o
iscachedorontheothercontentswithwhichitiscached.This stati2
Zipf 0.8
behavior is also apparent on the right part of Figure 1, as the
Zipf 1.2
loss rates for the contents of the same class are quite similar. 0
0 50 100 150 200
From Figure 1 and Table I, it appearsthat the approximations popularity rank of the content
fromEquations(1)and(2)arequiteaccurate,withforexample Fig. 3: Zipf model: loss rates under optimized replication.
a relative errorof around5%in the inefficiencyofthe system
(9.68×10−3 Vs 9.20×10−3). We consider such an accuracy
is quite good given that some of the approximationsdone are
Equation (1); the right part shows the loss rates for all the
based on a large storage / replication asymptotic, while the
contents against the approximationfrom Equation (2). Again,
simulationsetupis with a storagecapacityofd=20contents
theresultsfrombothFigure2andTableIconfirmtheaccuracy
onlyandcontentsofclass3(responsibleforthebulkoflosses)
of the approximations.
have only 23 replicas each.
Wenowturntoevaluatingtheperformanceoftheoptimized
We now turn to the Zipf popularity model. In this model, replication from Equation (5). Figure 3 shows the resulting
thecontentsarerankedfromthe mostpopulartotheleast;for loss rates under the optimized replication, with both Zipf
a given exponent parameter α, the popularity of the content exponents. This figure is to be compared with the right part
of rank i is given by λi = i−αj−αλ. We use two different of Figure 2, which showed the same results for proportional
valuesfortheZipfexponentPα,j≤0n.8and1.2,asproposedin[7]. replication.Itisclearthattheoptimizedreplicationsucceedsat
Theexponent0.8ismeanttorepresentcorrectlythepopularity reducingtheoverallinefficiencyγ/λcomparedtoproportional
distribution for web, file sharing and user generated contents, replication (from 5.7×10−3 to 5.1×10−4 for α=0.8, and
whiletheexponent1.2ismorefitforvideo-on-demand,which from 1.2×10−2 to 1.6×10−6 for α=1.2). Note that in the
has a more accentuated popularity distribution. We simulate case of Zipf exponent α = 1.2 the popularity distribution is
networks of n = 200 contents and m ≈ 2000 servers of more“cacheable”asitismoreaccentuated,andtheoptimized
storage capacity d = 10 under a load ρ = 0.9. This yields replication achieves extremely small loss rates. However, the
an average content popularity λ = ρm ≈ 9. Note that under loss rates for all the contents are not trully equalized, as
n
proportional replication the numbers of replicas of the most popularcontentsarestilltoomuchfavored(ascanbeseenfor
popularcontentsareactuallylargerthanthenumberofservers Zipf exponent α=0.8). This may be because the expression
mforα=1.2;wethusenforcethatnocontentisreplicatedin for optimized replication is asymptotic in the system size and
more than 95% of the servers. As expected and confirmed by storagecapacity.Hence,thereisstillroomforadditionalfinite-
the simulations, this is anyway benefical to the system. Each size corrections to the optimized replication.
setup is simulated for at least 104 units of time. As an outcome of this section, we conclude that the ap-
We show the results for the Zipf modelwith both exponent proximationsproposedareaccurate,evenatreasonablesystem
values in Figure 2 and in the third part of Table I: the left size. In addition, the optimized scheme largely outperforms
part of Figure 2 shows the expected number of available proportionalreplication.Inthenextsection,wederiveadaptive
replicas for all the contents against the approximation from schemes equalizing the loss rates of all the contents and
7
achieving similar performances as the optimized replication. equalizethe loss rates. Thiswouldbe achievedfor exampleif
we picked a content for eviction uniformly at random among
V. DESIGNING ADAPTIVEREPLICATION SCHEMES all contents. However, the contents eligible for eviction are
Inapracticalsystem,wewouldwanttoadaptthereplication only those which are available (although some systems may
in an online fashion as much as possible, as it provides allowmodifyingthecachesofbusyservers,asservingrequests
more reactivity to variations in the popularity of contents. consumes upload bandwidth while updating caches requires
Such variationsare naturally expected for reasons such as the only download bandwidth). Therefore, the most natural and
introductionofnewcontentsinthecatalogoralossofinterest practicalevictionruleistopickanavailablecontentuniformly
for old contents. Thus, blindly enforcing the replication of at random (hereafter, we refer to this policy simply as the
Equation (5) is not always desirable in practice. Instead, we RANDOM policy). Then, the eviction rate for content c is
can extract general principles from the analysis performed in given by η ∝ π(Z > 0) = 1− γc. So, at equilibrium, we
Section IV to guide the design of adaptive algorithms. can expectcto have c γc = Cste, ∀λcc ∈ C. In a large system,
1−γc
Inthis section,we firstshow howbasic adaptiverulesfrom with large storage / repλlcication and loss rates tending to zero,
cachenetworkscanbetranslatedintoourcontextbasedonthe thedifferencewithareplicationtrullyequalizingthelossrates
insightsfromSectionIVtoyieldadaptivealgorithmsminimiz- is negligeable. If we are confronted to a system with a large
ingtheoveralllossrate.Then,weshowhowthemoredetailed numberofveryunpopularcontentsthough,wecancompensate
information contained in Equation (1) allows the design of for this effect at the cost of maintaining additional counters
fasterschemesattainingthesametargetreplication.Finally,we for the number of evictions of each content.
validate the path followed in this paper by evaluating through Once the rule for creatingreplicasif fixed,we immediately
simulations the adaptive schemes proposed. obtain a family of adaptive algorithms by modifying the
evictionrulesfromthecachenetworkcontextaswedidabove
A. Simple Adaptive Algorithms for Minimizing Losses
for the RANDOM policy. Instead of focusing on “usage” and
The analysis in Section IV-B shows that the problem of
the incoming requests process as in cache networks with the
minimizing the average loss rate γ is approximatelya convex LFU and LRU policies, we react here to the loss process.
problem, and that therefore one should aim at equalizing
This yields the LFL (least frequently lost) and LRL (least
the derivatives ∆γc(Dc) of the stationary loss rates of all recently lost) policies. RANDOM, LRL, and LFL are only
the contents. In addition, Equation (4) points out that these
three variantsof a generic adaptive algorithmwhich performs
derivatives are proportional to −γc in the limit of large the following actions at every loss for content c:
replication / storage, and thus equalizingthe loss rates should
1) create an empty slot on an available server, using the
provideanapproximatesolutionfortheoptimizationproblem.
eviction rule (RANDOM / LRL / LFL);
An immediate remark at this point is that it is unnecessary to
2) add a copy of the content c into the empty slot.
store contents with very low popularity λ if the loss rate of
c
the other contents is already larger than λ . Thethreeevictionrulesconsideredhererequireaveryelemen-
c
An adaptive replication mechanism is characterized by two tary centralized coordinating entity. For the RANDOM rule,
rules: a rule for creating new replicas and another one for this coordinator simply checks which contents are available,
evicting contentsto make space for the new replicas. In order picks one uniformly at random, say c′, and then chooses a
to figure out how to set these rules, we analyze the system random idle server s storing c′. The server s then picks a
in the fluid limit regime, with a separation of timescales such random memory slot and clears it, to store instead a copy of
that the dynamics of the system with fixed replication have c. In the case of LRL, the coordinator needs in addition to
enough time to converge to their steady-state between every maintain an ordered list of the least recently lost content (we
modification of the replication. Achieving this separation of call such a list an LRL list). Whenever a loss occurs for c,
timescales in practice would require slowing down enough the coordinator picks the least recently lost available content
the adaptation mechanism, which reduces the capacity of the c′ based on the LRL list (possibly restricting to contents less
system to react to changes. Therefore, we keep in mind such recently lost than c) and then updates the position of c in
a separation of timescales as a justification for our theoretical the LRL list. It then picks a random idle server s storing c′,
analysis but we do not slow down our adaptive algorithms. which proceeds as for RANDOM. Finally, for LFL, the coor-
When trying to equalize the loss rates, it is natural to use dinator would need to maintain estimates of the loss rates of
the loss events to trigger the creation of new replicas. Then, eachcontent(bywhatevermeans,e.g.exponentiallyweighted
new replicas for content c are created at rate γ , and we movingaverages);when a losshappens,the coordinatorpicks
c
let η be the rate at which replicas of c are deleted from the available content c′ with the smallest loss rate estimate
c
the system. In the fluid limit regime under the separation of and then proceeds as the other two rules. This last rule is
timescale assumption, the number of replicas of c evolves more complicated as it involves a timescale adjustement for
according to D˙ = γ − η , where all the quantities refer theestimation ofthe lossrates, thereforewe willfocusonthe
c c c
to expectations in the steady-state of the system with fixed first two options in this paper. Note that the LFL policy does
replication. At equilibrium, we have D˙ = 0 and γ = η not suffer from the drawback that the eviction rate is biased
c c c
for all the contents c, thus we need η = Cste for all c to towardspopularcontentsduetotheirsmallunavailability,and
c
8
this effect is also attenuated under the LRL policy. We point get fast adaptation).
outthatitispossibletooperatethesysteminafullydistributed As a first step towards setting the probability p (Z ), we
c c
way (finding a random idle server first and then picking a write Equation (1) as follows:
content on it by whatever rule), but this approach is biased
z
λ +iθ
into selecting for eviction contents with a large number of π(Z =0)=π(Z =z) c eff .
c c
D −i+1
available replicas (i.e. popular contents), which will lead to a c
i=1
Y
replication with reduced efficency. It is of interest for future
This shows that, for any fixed value z with π(Z = z) ≥
c
work to find a way to unbias such a distributed mechanism.
π(Z =0), we can generate events at rate γ by subsampling
c c
B. Adapting Faster than Losses at the time of a request arrival with Zc = z with a first
probability
In the algorithms proposed in the previous subsection, we z
λ +iθ
usethelossesofthesystemastheonlytriggerforcreatingnew q (z)= c eff .
c
D −i+1
replicas.Itisconvenientastheseeventsaredirectlytiedtothe i=1 c
Y
quantity we wish to control (the loss rates), however it also
Ifonthecontraryz issuchthatπ(Z =z)<π(Z =0),then
c c
has drawbacks. Firstly, it implies that we want to generate a
the value of q (z) given above is larger than 1 as we cannot
c
newreplicaforacontentpreciselywhenwehavenoavailable
generate events at rate γ by subsampling even more unlikely
c
serverforuploadingit,andthuseitherwesendtworequestsat
events. If we generated virtual losses at rate γ as above for
c
atimetothedatacenterinsteadofone(onefortheuserwhich
eachvalueofz,thenthetotalrateofvirtuallossesforcontentc
wecouldnotserveandonetogenerateanewcopy)orwemust wouldbeλ Dc min{π(Z =z),π(Z =0)}, whichclearly
delaythecreationofthenewreplicaandtolerateaninedaquate c z=1 c c
still depends on c. We thus proceed in two additional steps
replication in-between, which also hinders the reactivity of R
towardssettingp (Z ):wefirstrestricttherangeofadmissible
c c
the system. Secondly, unless in practice we intend to slow
values of Z , for which we may generate virtual losses, by
c
down the adaptation enough,there will always be oscillations
excluding the values z such that π(Z =z)<π(Z =0). In
c c
inthereplication.IflosseshappenalmostasaPoissonprocess,
theregimeθ ≥1,thiscanbedoneinacoarsewaybyletting
eff
then only a bit of slow-down is necessary to moderate the z∗ = Dc−λc and rejecting all the values z >z∗. Indeed,
oscillations, but if they happen in batches (as it may very c θeff c
well be for the most popular contents) then we will create z∗
c D −iθ
many replicas in a row for the same contents. If in addition qc(zc∗)= Dc −ieff ≤1,
we use the LRL or LFL rules for evicting replicas, then the i=0 c
Y
samecontentswillsuffermanyevictionssuccessively,fuelling and the distribution of Z is unimodal with the mode at
c
againtheoscillationsinthereplication.Finally,itisverylikely a smaller value Dc−λc ≈ ρz∗. Now, subsampling with
thatpopularitiesareconstantlychanging.Ifthesystemisslow probability q (Z ) wθheffen a requecst arrives for content c and
c c
to react, then the replication may never be in phase with the Z ≤ z∗ would generate events at a total rate z∗γ . Thus,
c c c c
current popularities but always lagging behind. For all these it suffices to subsample again with probability minc′∈Czc∗′ to
z∗
reasons,itisimportanttofindawaytodecoupletheadaptation obtain the same rate of virtual losses for all the ccontents
mechanismsfromthelossesinthesystemtosomeextent,and (another approach would be to restrict again the range of
at least to find other ways to trigger adaptation. admissible values of z, e.g. to values around the mode z ).
c
Weproposeasolution,relyingontheanalysisinSectionIV. To sum up, our approach is to generate a virtual loss for
Alossforcontentcoccurswhen,uponarrivalofarequestfor content c at each arrival of a request for c with probability
b
c, its number of available replicas is equal to 0. In the same
min z∗
way, whenevera requestarrives, we can use the currentvalue p (Z )= c′∈C c′1(Z ≤z∗)q (Z ).
c c z∗ c c c c
of Zc to estimate the loss rate of c. Indeed, Equation (1) tells c
us how to relate the probability of Zc =z to the loss rate γc, The rate γc at whichvirtuallosses are generatedfor contentc
for any z ∈N. Of course, successive samples of Zc are very isthengivenbyγc×minc′∈Czc∗′,whichisindependentofcas
correlated(notethatlossesmayalsobecorrelatedthough)and planned.Whenevera virtuallossoccurs,we canuse whatever
e
wemustbecarefulnottobetooconfidentintheestimatethey algorithmwe wanted to use in the first place with real losses;
provide. A simple way to use those estimates to improve the there is no need to distinguish between the real losses and
adaptation scheme is to generate virtual loss events, to which the virtual ones. For example, if we use the LRL policy, we
any standard adaptive scheme such as those introduced in the update the position of c in the LRL list and create a new
previous subsection may then react. To that end, whenever replica for c by evicting a least recently lost available content
a request for c arrives, we generate a virtual loss with a (from a server which does not already store c). If we choose
certain probability pc(Zc) depending on the current number to test for virtual losses at ends of service for c (which yields
of available replicas Z . The objective is to define p (Z ) so the same outcomein distribution, as the system is reversible),
c c c
that the rates (γ ) of generation of virtual losses satisfy the new replica can simply be uploaded by the server which
c c∈C
γ = Cste×γ for all c ∈ C (so that the target replication just completed a service for c. Furthermore, in practice, we
c c
still equalizes leoss rates) and Cste is as high as possible (to advocate estimating the values of zc∗ and pc(Zc) on the fly
e g
g
9
rather than learning these values for each content: θ can
eff 0.015 8
be computed from ρ , which is naturally approximated by fixed optimized fixed proportional
eff RANDOM fixed optimized
aatncchvnauuenermrreeracbsangotetitemirmonoopauvofutmeeftserbetfethrzohrvec∗reeorcmaffseun;rwirdenrsqeciplum′ne∈ceat(iCsslZntatszrucplc∗)fmyo′o;,,pbriwutewclrieahbsroieccofciahnnonbglncyutasasepennynrpetvsrcsbeoeeeudxrspv.sifmdaeForrarraystoteemettdoxoλawtmhctmhheaepbesinelynet,eotvtawtahieanneerl stationary loss rates00.0.0015 LRLRRALLN DviOrtuMa lvirtual log of nb. of replicas4567 RLRLRRAALLNN DDviOOrtuMMa lvirtual
a service ends for one of them. 0 3
0 50 100 150 200 0 1 2 3 4 5 6
In the next subsection, we evaluate the performance of the popularity rank of the content log of the rank of the content
adaptative schemes proposed and the virtual losses mecha-
nism. Fig. 4: Adaptive schemes: loss rates and replication.
C. Evaluation of the Performance through Simulations
400
Before getting to the simulation results, let us just mention 10% most popular contents
350 10% least popular contents
that the complexity of simulating the adaptive algorithms RANDOM
Ignrdoweesd,veitryisfaesatsywittoh stehee tshyasttesmimsuilzaetin(gwitthhens,ysmtemanfdord)a. eplicas235000 LRLRRALLN DviOrtuMa lvirtual
fixed duration t requires Ω(mdλt) operations. Furthermore, er of r200
the time needed for the RANDOM algorithm to converge, mb150
u
whenstartedatproportionalreplication,isroughlyoftheorder n100
of max D /γ, where γ is the average loss rate for the 50
c∈C c
limit replication, which decreases exponentially fast in d as 0
0 1 2 3 4 5 6 7 8 9 10
seenin Equation(6).Therefore,ifwe wanttocompareallthe time x 104
adaptive schemes, we cannot simulate very large networks. Fig. 5: Adaptive schemes: time evolution.
Anyway,our intentionis to show thatour schemeswork even
for networks of reasonable size.
As in Section IV-C, we used Zipf popularity distributions and secondly that this replication equalizes the loss rates, as
with exponents 0.8 and 1.2 and a class model to evaluate the intended.Inaddition,theadaptiveschemesperformevenbetter
performance of the various schemes. The results are qualita- thanthe optimizedstatic replication,which suffersfromfinite
tively identical under all these models, so we only show the network / finite storage effects, as they manage to find the
resultsforZipfpopularitywithexponentα=0.8.Wecompare right balance between popular and unpopular contents.
thevariousschemesintermsofthereplicationachievedandits We compare the adaptation speed of the various schemes
associated loss rates, as well as the speed at which the target on Figure 5, where we plot both the evolution of the average
replication is reached. We do not slow down the dynamics number of replicas of the 10% most popular contents and
of the adaptive schemes even though this necessarily induces that of the 10% least popular ones, starting from proportional
someoscillationsinthereplicationobtained.Nonetheless,this replication. As expected, the LRL schemes are faster than
setup is already sufficient to demonstrate the performance of the RANDOM ones, but more importantly this plot clearly
the adaptive schemes. It would be interesting to quantify the demonstrates the speed enhancement offered by the virtual
potential improvement if one reduces the oscillations of the loss method of Section V-B. Regarding the benefits of such
replication obtained (e.g. by quantifying the variance of the anenhancedreactioncapability,thereisaninterestingproperty
stationary distribution for the number of replicas for each whichwe didnotpointoutnorillustratewith thesimulations:
content); we leave this out for future work. Also, we did not thevirtuallossschemehasthepotentialtofollowaconstantly
accountfortheloadputonthedatacentertocreatenewcopies evolvingpopularityprofileatnocost,astherequiredcreations
of the contents; one can simply double the loss rates for the of replicas to adapt to the changing popularities can be done
adaptiveschemestocapturethiseffect.Notethatifadaptation without requesting copies of the contents to the data center.
speedisnotapriority,onecantradeitofftoalmostcancelthis
factorof2.Finally,concerningthevirtuallossmechanism,we VI. CONCLUSION
estimateallthequantitiesinvolvedonthefly,asrecommended We addressed the problem of content replication in edge-
in the previous section. assisted CDNs, with a special attention to capturing the
In Figure 4, we show results for the various adaptive mostimportantconstraintsonservercapabilitiesandmatching
schemes.Ontheleftpartofthefigure,weshowthestationary policy. Based on large system and large storage asymptotics,
loss rates of all the contents; on the right part we show we derived an accurate approximationfor the performance of
in log-log scale the stationary expectation of the numbers anygivenreplication,therebyallowingofflineoptimizationof
of replicas for each content. This plot shows firstly that the replication. In addition, levaraging the insights gathered
all the adaptive schemes converge to the same replication in the analysis, we designed adaptive schemes converging to
10
the optimal replication. Our basic adaptive algorithms react
to losses, but we proposedfurther mechanisms to move away
from losses and adapt even faster than they occur.
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