On Optimal Geographical Caching in Heterogeneous Cellular Networks

0.89 MB·

by  Berksan Serbetci

On Optimal Geographical Caching in
Heterogeneous Cellular Networks
Berksan Serbetci Jasper Goseling
Stochastic Operations Research Stochastic Operations Research
University of Twente, The Netherlands University of Twente, The Netherlands
[email protected] [email protected]
6
1 Abstract—In this work, we investigate optimal geographical over time depending on the demand will serve as helpers to
0 caching in heterogeneous cellular networks. Different types of the overall system and decrease the backhaul traffic.
2 base stations (BSs) have different cache capacities. The content
Recently, there has been growing interest in caching in
libraryiscontainingfileswithpopularitiesfollowingaprobability
t cellular networks. In [4] Shanmugam et al. focus on the
c distribution. The performance metric is the total hit probability,
O which is the probability that a user at an arbitrary location in content placement problem and analyze which files should be
the plane will find the content that it requires in one of the BSs cached by which helpers for the given network topology and

8 that it is covered by. file popularity distribution by minimizing the expected total
1 We consider the problem of optimally placing content in all file delay. In [5] Poularakis et al. provide an approximation
BSsjointly.Asthisproblemisnotconvex,weprovideaheuristic
algorithm for the problem of minimizing the user content
] scheme by finding the optimal placement policies for different
T types of BSs using the information coming from the other types. requests routed to macrocell base stations with constrained
I Wedemonstratethattheseindividualoptimizationproblemsare cache storage and bandwidth capacities. In [6] Błaszczyszyn
.
s convex and we provide an analytical solution. As an illustration, et al. revisit the optimal content placement in cellular caches
c we find the optimal placement policy of the small base stations
by assuming a known distribution of the coverage number
[
(SBSs) depending on the placement policy of the macro base
and provide the optimal probabilistic placement policy which
stations (MBSs). We show how the hit probability evolves as the
2
deploymentdensityoftheSBSsvaries.Weshowthattheheuristic guaranteesmaximaltotalhitprobability.In[7]Maddah-Ali et
v
of placing the most popular content in the MBSs is almost al. developed an information-theoretic lower bound for the
2
2 optimalafterdeployingtheSBSswithoptimalplacementpolicies. caching system for local and global caching gains. In [8]
3 Also, for the SBSs no such heuristic can be used; the optimal Ioannidisetal.proposeanovelmechanismfordeterminingthe
placement is significantly better than storing the most popular
7 cachingpolicyofeachmobileuserthatmaximizethesystem’s
content. Finally, we show that solving the individual problems
0
to find the optimal placement policies for different types of BSs social welfare. In [9] Poularakis et al. consider the content
.
1 iteratively, namely repeatedly updating the placement policies, storage problem of encoded versions of the content files. In
0 does not improve the performance. [10]Bastugetal.couplethecachingproblemwiththephysical
6
layer. In [11] Altman et al. compare the expected cost of
1 I. INTRODUCTION
obtainingthecompletedataunderuncodedandcodeddataal-
:
v In recent years, there is extreme growth in data traffic locationstrategiesforcachingproblem.Cacheplacementwith
i
X over cellular networks. The growth rate of the demand is the help of stochastic geometry and optimizing the allocation
expected to increase in the upcoming years such that current ofstoragecapacityamongfilesinordertominimizethecache
r
a network infrastructures [1] will not be able to support this missprobabilityproblemispresentedbyAvrachenkovetal.in
data traffic [2]. In order to tackle this problem, an obvious [12]. A combined caching scheme where part of the available
approachistoincreasethenumberofbasestations.Thesebase cache space is reserved for caching the most popular content
stations require a high-speed backhaul to make this system in every small base station, while the remaining is used for
work properly and it is costly to connect every base station cooperatively caching different partitions of the less popular
to the core network in real life. A solution to this problem is contentindifferentsmallbasestations,asameanstoincrease
to reduce backhaul traffic by reserving some storage capacity local content diversity is proposed by Chen et al. in [13]. In
at both macro base stations (MBSs) and small base stations [14]Dehghanetal.associatewitheachcontentautility,which
(SBSs)andusetheseascaches[3].Inthisway,partofthedata is a function of the corresponding content hit probabilities
is stored at the wireless edge and the backhaul is used only to and propose utility-driven caching, where they formulate an
refresh this stored data. Data replacement will depend on the optimizationproblemtomaximizethesumofutilitiesoverall
users’ demand distribution over time. Since this distribution contents.
is varying slowly, the stored data can be refreshed at off-peak The main contribution of this paper is to find optimal
times.Inthisway,cachescontainingpopularcontentchanging placement strategies that maximize total hit probability in
heterogeneous cellular networks. Our focus is on hetero-
geneous cellular networks in which an operator wants to
jointly optimize the cached content in macro base stations
(MBSs) and small base stations (SBSs) with different storage
capacities. This problem is not convex. Therefore, we provide
a heuristic scheme and optimize only one type of cache (e.g.
SBS) with the information coming from other types of caches
(e.g. MBS and other SBSs with different cache capacities) at
each iteration step.
We show that whether MBSs use the optimal deployment
Fig. 1. A realization of the probabilistic placement policy (J = 6 and
strategy or store “the most popular content”, has no impact K=3).Arandomnumberispicked(0.68inthiscase)andtheverticalline
on the total hit probability after deploying the SBSs with intersectswiththeoptimalb(i)values.Fromthisfigure,weconcludethatthe
contentsubset{c1,c3,c5}willbecached.
optimal content placement policies. We show that it is crucial
tooptimizethecontentplacementstrategyoftheSBSsinorder
to maximize the overall performance. We show that heuristic memoryintervalsofunitlength.Thenb(i) valuesfillthecache
j
policieslikestoringthepopularcontentthatisnotyetavailable
memory sequentially and continue filling the next slot if not
in the MBSs result in significant performance penalties.
enoughspaceisavailableinthememoryslotthatithasstarted
In Section II we start the paper with model and problem
filling in as in the end completely covering the K memory
i
definition. In Section III we present the optimal placement
intervals. Then, for any type−i cache, a random number from
strategy problem and give required tools to solve it. In
the interval [0,1] is picked and the intersecting K files are
i
Section IV we provide an iterative heuristic scheme for the
cached. An example is shown in Figure 1.
joint optimization problem. In Section V we continue with
performanceevaluationoftheoptimalplacementstrategiesfor III. CONTENTPLACEMENTPROBLEM-INDIVIDUAL
differentprobabilisticdeploymentscenarios.InSectionVIwe OPTIMIZATION
conclude the paper with discussions. A. Formulation of the problem
II. MODELANDPROBLEMDEFINITION We start this section with defining our performance met-
ric and the formulation of the optimization problem. The
Throughout the paper we will be interested in two types
performance metric is the total hit probability which is the
of base stations, namely MBSs and SBSs. However, at this
probability that a user will find the content that he requires
section we will give a more general formulation as it is
in one of the caches that he is covered by. We assume that
possibletohaveSBSswithdifferentstoragecapacitiesinsome
the placement strategy for the probability distribution over J
topologies.Weconsideraheterogeneouscellularnetworkwith
files through all L−1 types is fixed and known and we will
L-different types of base stations in the plane. These base
solve the optimization problem for only one type. In this sec-
stations are distributed according to a homogeneous spatial
tion, without loss of generality, we consider the optimization
process [16]. All base stations have a coverage radius r. Let
of type−1. For notational convenience, superscript c in the
N , i=1,...,L, denote the number of base stations of type
i notation b(i)c indicates that the placement strategy for type−i
i that are covering a user at an arbitrary location in the plane. j
Furthermore, let p(i) := P[N = n ] and p := P[N = n], is known and constant. Then, the total hit probability is given
ni i i n by
where N =(N ,...,N ) and n=(n ,...,n ).
1 L 1 L
Caches store files from a content library C = J ∞
{c1,c2,...,cJ},whereanelementcj isafilewithnormalized f(cid:16)b(1)(cid:17)=1−(cid:88)aj (cid:88)(1−b(j1))n1p(n11)q(j,n1), (2)
size 1. The probability that file cj is requested is denoted as j=1 n1=0
aj. Without loss of generality, a1 ≥a2 ≥···≥aJ. where
Thecachesoftype−ihavecachememorysizeK ≥1,i=
i
1,...,L. We use the probabilistic content placement policy q(j,n1)=P(non type−1 caches miss the file cj)
of [6]. We denote the probability that the content c is stored ∞ ∞ L
at a type−i cache as j = (cid:88) ··· (cid:88) (cid:89)(1−b(jl)c)nlP(NL2 =nL2|N1 =n1),
b(i) :=P(c stored in type−i cache), (1) n2=0 nL=0l=2
j j where nL =(n ,...,n ) and NL =(N ,...,N ) are (L−
2 2 L 2 2 L
and the placement strategy b(i) = (b(i),...,b(i)) as a J-tuple 1)-tuplesrepresentingthecoveragevectorsofnontype−1base
1 J
for any type−i cache. The content is independently placed in stations.
the cache memories according to the same distribution for the We define the optimization problem to find the optimal
same type of caches. The placement procedure is as follows. placement strategy maximizing the total hit probability for a
The memory of a type−i cache is divided into K continuous type−1 cache as follows:
i
Problem 1. where φ(ν¯) is the solution over b(1) of
j
(cid:16) (cid:17)
max f b(1) ∞
a (cid:88) n (1−b(1))n1−1p(1)q(j,n )=ν¯, (13)
s.t. b(11)+···+b(J1) =K1, b(j1) ∈[0,1], ∀j. (3) jn1=0 1 j n1 1
B. Solution of the optimization problem andν¯canbeobtainedastheuniquesolutiontotheadditional
constraint
In this section, we will analyze the structure of the opti-
¯b(1)+···+¯b(1) =K . (14)
mization problem. 1 J 1
Proof: From (9), (10) and (11), we have
Lemma 1. Problem 1 is a convex optimization problem.
(cid:34) ∞ (cid:35)
Proof: The objective function is separable with respect ω¯ =¯b(1) a (cid:88) n (1−b(1))n1−1p(1)q(j,n )−ν¯ , (15)
to(w.r.t.)b(1),...,b(1) (Replace(cid:80)J a =1andrewritethe j j j 1 j n1 1
objective fu1nction (2J).). 1−f(cid:0)b(1)(cid:1)j=i1s injcreasing and convex n1=0
(cid:16) (cid:17) which, when insterted into (10), gives
in b(1), ∀j. Hence, it is a convex function of b(1),...,b(1) .
j 1 J (cid:34) ∞ (cid:35)
We already showed that 1−f(cid:0)b(1)(cid:1) is convex by Lemma ¯b(j1)(cid:16)¯b(j1)−1(cid:17) aj (cid:88) n1(1−b(j1))n1−1p(n11)q(j,n1) =ν¯.
1 and the constraint set is linear as given in (3). Thus, the n1=0 (16)
Karush-Kuhn-Tucker(KKT)conditionsprovidenecessaryand From (16), we see that 0<¯b(1) <1 only if
j
sufficient conditions for optimality. The Lagrangian function
∞
corresponding to Problem 1 becomes ν¯=a (cid:88) n (1−b(1))n1−1p(1)q(j,n ).
j 1 j n1 1
J ∞
L(cid:16)b(1),ν,λ,ω(cid:17)=(cid:88)a (cid:88)(1−b(1))n1p(1)q(j,n ) n1=0
j j n1 1 Since we know that 0≤b(i) ≤1, this implies that
j=1 n1=0 j
 
+ν(cid:88)J b(j1)−K1−(cid:88)J λjb(j1)+(cid:88)J ωj(cid:16)b(j1)−1(cid:17), ν¯∈(cid:34)ajp(11)q(j,1), aj (cid:88)∞ n1p(n11)q(j,n1)(cid:35).
j=1 j=1 j=1 n1=0
(4)
If ν¯<a p(1)q(j,1), we have
j 1
where b(1), λ, ω ∈RJ and ν ∈R.
+ ∞
Let b¯(1), λ¯, ω¯ and ν¯ be primal and dual optimal. The KKT ω¯ =λ¯ +a (cid:88) n (1−b(1))n1−1p(1)q(j,n )−ν¯>0.
j j j 1 j n1 1
conditions for Problem 1 state that
n1=0
(cid:88)J ¯b(1) =K , (5) Thus, from (10), we have ¯b(j1) = 1. Similarly, if ν¯ >
j 1 a (cid:80)∞ n p(1)q(j,n ), we have
j=1 j n1=0 1 n1 1
0≤¯b(1) ≤1, ∀j =1,...,J, (6) ∞
jλ¯j ≥0, ∀j =1,...,J, (7) λ¯j =ω¯j +ν¯−aj (cid:88) n1(1−b(j1))n1−1p(n11)q(j,n1)>0.
n1=0
ω¯ ≥0, ∀j =1,...,J, (8)
j
Hence, from (9), we have ¯b(1) =0.
λ¯ ¯b(1) =0, ∀j =1,...,J, (9) j
ω¯j(cid:16)¯b(j1)−j 1j(cid:17)=0, ∀j =1,...,J, (10) solFviinnagllyJ, esqinucaetio(cid:80)nsJj=o1fb((j113))issataisfdyeincgrea(s1i4n)ggfivuenctthioenuniniquνe,
solution ν¯.
∞
−a (cid:88) n (1−b(1))n1−1p(1)q(j,n )+ν¯−λ¯ +ω¯ =0, IV. JOINTOPTIMIZATION
j 1 j n1 1 j j
n1=0 Finding the optimal placement strategy for all types of
(11) caches jointly is an interesting problem. However, this opti-
∀j =1,...,J. mizationproblemisnotconvex.Hence,weprovideaheuristic
algorithm to see how the overall system performance is im-
Theorem 1. The optimal placement strategy for Problem 1 is
proved.Theprocedureisasfollows.Ateachiterationstepwe
¯b(1) = 10,, iiff νν¯¯<>aajp(cid:80)(11)∞q(j,1n)p(1)q(j,n ), (12) fithnadt tthheeoppltaimceamlesntrtatsetgraytefogrieassfpoerciofitchetyrpteypoefscaocfhecaacshseusmianrge
j  φ(ν¯), otherwisje, n1=0 1 n1 1 known and fixed. Then we continue with the same procedure
forthenexttypeandwecontinueiteratingoverdifferenttypes.
V. PERFORMANCEEVALUATION Optimal Placement Strategy for SBSs [(cid:50)(cid:51)(cid:55)(cid:18)(cid:50)(cid:51)(cid:55)]
2
In this section, we will specify different network coverage
(2)
b
modelsandshowtheperformancesoftheplacementstrategies. Others 1
In both models we use Zipf distribution for the file populari- y
ties.Theprobabilitythatauserwillaskforcontentc isequal eg1.5 b(2) + b(2)
j at 1 2
to a = j−γ , (17) nt str b1(2) + b2(2) + b3(2)
j (cid:80)Jj=1j−γ me 1
e
where γ >0 is the Zipf parameter. ac b(2)
pl b(2) 3
A. Poisson Point Process (PPP) Model mal 0.5 2
The caches follow a two-dimensional (2D) spatial homo- pti b(2)
O
geneous Poisson process with type−i caches independently 1
distributedintheplanewithdensityλi >0,i=1,...,L.The 0
number of type−i caches within radius r follows a Poisson 0 2 4 6 8 10
distribution with parameter t =λ πr2, i.e., (cid:1)(cid:1) /(cid:1)
i i 2 1
p(nii) =P(ni type−i caches within radius r) Fig.2. Optimalplacementstrategy¯b(2)forSBSswithdifferentdeployment
= tniie−ti. (18) densitiesλ2 [OPT/OPT](PPP).
n !
i
Optimal Placement Strategy for SBSs [(cid:51)(cid:50)(cid:51)(cid:18)(cid:50)(cid:51)(cid:55)]
The user is covered by n type−i caches and distribution
i 2
of the different type of caches is independent of each other.
(2)
Therefore, the total coverage distribution probability mass b
1
fduisntcrtiibountiopnns will be the product of individual probability egy1.5 Others b(2) + b(2)
at 1 2
pn =p(n11)p(n22)...p(nLL). (19) str b(2) + b(2) + b(2)
nt 1 2 3
Consider the case of two types of caches in the plane. e
m 1
Type−1 caches represent MBSs and type−2 caches represent
e
c
SBSs with K1 and K2-slot cache memories. The content a
library size is J = 100. We set K1 = 1 and K2 = 2. We al pl b3(2)
set the Zipf parameter γ =1 and taking aj according to (17). m0.5 b (2)
Also we set λ1 = 0.5 and r = 1. We will consider various pti 2
O
values for the deployment densities of SBSs, namely λ2. Our b (2)
1
aimistocomparetheoptimalplacementstrategywithvarious
0
heuristics.Therefore,wewillcomparevariousscenariosinthe 0 2 4 6 8 10
remainder of this subsection. (cid:1)(cid:1) /(cid:1)
2 1
For the [OPT/OPT] scenario first we find the opti-
mal solution for MBSs assuming that there are no SBSs Fig.3. Optimalplacementstrategy¯b(2)forSBSswithdifferentdeployment
in the plane. Solving this optimization problem gives densitiesλ2 [POP/OPT](PPP).
¯b(1) = (0.7136,0.2723,0.0141,0,...,0). The optimal place-
ment strategy results in storing the first three most popular
files into MBSs with given probabilities and the resulting hit For the [POP/OPT] scenario we consider a heuristic for the
probability is f(cid:0)¯b(1)(cid:1) = 0.1649. Then we set b(1)c = ¯b(1), placement in MBSs: The most popular content c1 is stored at
takeitasaninput,addSBSsintotheplaneandfindtheoptimal MBSs with probability (w.p.) 1, i.e., ¯b(1) =(1,0,...,0). The
placement strategy ¯b(2) for SBSs. We observe that using the resulting hit probability is f(cid:0)¯b(1)(cid:1) = 0.1527. It is important
iterative procedure from Section IV, repeatedly updating ¯b(1) to note that the hit probability under this heuristic policy is
and¯b(2), does not improve the hit probability. From Figure 2, not significantly different from the optimal hit probability.
afterdeployingSBSsweseethatwhenthedeploymentdensity Next we add SBSs. We set b(1)c = ¯b(1), take it as an
of SBSs is low compared to MBSs, only the first three most input,addSBSsintotheplaneandfindtheoptimalplacement
popularfilesarestoredinSBSs.Asthedeploymentdensityof strategy ¯b(2) for SBSs. From Figure 3, after deploying SBSs
SBSsincreases,weseethatprobabilityofstoringlesspopular we see that when the deployment density of SBSs is low
files in the caches increases. compared to MBSs, only c and c are stored in SBSs with
2 3
Total hit probability vs. Coverage Ratio (cid:1) /(cid:1) Optimal Placement Strategy for SBSs [(cid:50)(cid:51)(cid:55)(cid:18)(cid:50)(cid:51)(cid:55)]
2 1
0.6 2
(2)
b
y 0.5 gy 1
bilit ate1.5 Others b1(2) + b2(2)
r
a 0.4 st (2) (2) (2)
b b + b + b
t
o n 1 2 3
r e
p 0.3 m 1
t e
hi [OPT/OPT] ac
al 0.2 [POP/OPT] pl
ot [OPT/H1] mal0.5
T 0.1 [POP/H1] pti b(2) b(2)
[POP/H2] O b(2) 2 3
0 0 1
0 20 40 60 80 100
0 2 4 6 8 10
(cid:1)(cid:1) /(cid:1) MM
2 1
Fig. 5. Optimal placement strategy¯b(2) for SBSs for different M values
Fig. 4. The total hit probability evaluation for different SBS deployment [OPT/OPT](M-or-None).
densities(PPP).
a MBS, then it doesn’t receive any service from other caches
probability 1 in order to increase the chance of the user to get
either at all. Therefore, we have
these files. As the deployment density of SBSs increases, we
see that probability of storing c in SBSs slightly increases. (cid:16) (cid:17) (cid:26) 0 if max{nL}>0,
1 P NL =nL|N =0 = 2
However, we see that since we cannot get c and c from any 2 2 1 1 if max{nL}=0,
2 3 2
of the MBSs, probabilities of deploying c and c are always
2 3 and
higher than c . Probability of storing other files also increases
1
(cid:40)
as the λ2 increases. P (cid:16)NL =nL|N =n (cid:17)= 0 if (cid:80)Ll=2nl (cid:54)=(M −n1)+,
From Figure 4, we see that the difference between the hit 2 2 1 1 1 if (cid:80)L n =(M −n )+,
l=2 l 1
probabilityunderthe[OPT/OPT]andthe[POP/OPT]vanishes
as the density of SBSs increases, i.e. we can use a heuristic when n >0.
1
placement policy for MBSs as finding the optimal placement Consider the case of two types of caches in the plane.
strategyforSBSscompensatetheperformancepenaltycaused Type−1 caches represent MBSs and type−2 caches represent
by ill-adjusted placement of MBSs. However, as we will SBSs with K and K -slot cache memories. The content
1 2
see next, no straightforward heuristic seems to exist for the library size is J = 100. We set K = 1 and K = 2. Also,
1 2
placement policy in SBSs. we set the Zipf parameter γ = 1 and taking a according to
j
The first heuristic is to use is to store the two most popular (17). Also we set λ =0.5 and r =1.
1
files in SBSs, denoted by [OPT/H1] and [POP/H1] for an For the [OPT/OPT] scenario, optimal placement strategy
optimal and a heuristic policy in MBS, respectively. The for MBSs is¯b(1) =(0.7136,0.2723,0.0141,0,...,0) and the
second heuristic is to store the second and third most popular resulting hit probability is f(cid:0)¯b(1)(cid:1) = 0.1649. Then we set
files in SBSs [POP/H2]. From Figure 4 it is clear that these b(1)c =¯b(1),takeitasaninputandfindtheoptimalplacement
heuristic policies achieve significantly lower hit probability strategy¯b(2) fordifferentM values.Weobservethatusingthe
than the optimal policy. iterative procedure from Section IV, repeatedly updating ¯b(1)
and ¯b(2), does not improve the hit probability. From Figure
B. M-or-none deployment model
5, we see that the probability of storing less popular files
Type−1 caches represent MBSs and follow a two- increases as M increases.
dimensional (2D) spatial homogeneous Poisson process with For the [POP/OPT] scenario, we have ¯b(1) = (1,0,...,0)
density λ >0, and number of MBSs within radius r follows and the resulting hit probability is f(cid:0)¯b(1)(cid:1) = 0.1527. Then
1
a Poisson distribution satisfying (18) with i=1. we set b(1)c = ¯b(1), take it as an input, add SBSs into the
We assume that if a user is covered by at least one MBS plane and find the optimal placement strategy ¯b(1) for SBSs.
(type−1 cache), then it should be covered by M caches in From Figure 6, as c is stored in MBSs w.p. 1 and SBSs are
1
total.Asaresult,networkoperatorsserveuserswithproviding present in the system only when n >0, c is never stored at
1 1
them M caches as long as they are connected to one of the SBSs. We see that probability of storing c and c decreases
2 3
macro base stations. If a user is not in the coverage area of andprobabilityofstoringotherfilesincreasesasM increases.
Optimal Placement Strategy for SBSs [(cid:51)(cid:50)(cid:51)(cid:18)(cid:50)(cid:51)(cid:55)] use the optimal deployment strategy. It is shown that heuristic
2
policies for SBSs like storing the popular content that is not
(2)
1.8 b yet available in the MBS results in significant performance
y 1
g
e 1.6 (2) (2) penalties. To conclude, using the optimal deployment strategy
at b1 + b2 for the SBSs is crucial and it ensures the overall network
str 1.4 Others (2) (2) (2) to have the greatest possible total hit probability independent
nt 1.2 b1 + b2 + b3 of the deployment policy of MBSs. Finally, we have shown
e
m 1 that solving the individual problem to find optimal placement
e
c strategy for different types of base stations iteratively, namely
a 0.8
pl repeatedly updating the placement strategies of the different
al 0.6 types, does not improve the hit probability.
m
0.4
pti b(2) b(2) REFERENCES
O 0.2 3
2 [1] “CiscoVNIGlobalMobileDataTrafficForecast,2015-2020”,SanJose,
0 CA,USA,Feb.2016.
0 10 20 30 40 50 60 70 80 90 100
[2] J.Andrews,H.Claussen,M.Dohler,S.Rangan,M.Reed,“Femtocells:
MM
Past,PresentandFuture”,IEEEJournalonSelectedAreasinCommu-
nications,vol.30,no.3,pp.497-508,April2012.
Fig. 6. Optimal placement strategy¯b(2) for SBSs for different M values [3] N. Golrezaei, A. F. Molisch, A. G. Dimakis, G. Caire, “Femtocaching
[POP/OPT](M-or-None). and Device-to-Device Collaboration: A New Architecture for Wireless
Video Distribution”, IEEE Communications Magazine, vol. 51, no. 4,
Total hit probability vs. M pp.142-149,April2013.
[4] K. Shanmugam, N. Golrezaei, A. G. Dimakis, A. F. Molisch, G.
Caire,“FemtoCaching:WirelessContentDeliveryThroughDistributed
0.7 Caching Helpers”, IEEE Transactions on Information Theory, vol. 59,
no.12,pp.8402-8413,December2013.
y
[5] K.Poularakis,G.Iosifidis,L.Tassiulas,“ApproximationAlgorithmsfor
bilit 0.6 MCoombimleunDicaatatioCnasc,hvionlg.6in2,Snmoa.1ll0,Cpepll.N36e6tw5-o3r6k7s”7,,IOEcEtoEbeTrra2n0s1a4c.tions on
ba 0.5 [OPT/OPT] [6] B. Błaszczyszyn, A. Giovanidis, “Optimal Geographic Caching In
o [POP/OPT] CellularNetworks”,IEEEInternationalConferenceonCommunications
r (ICC)2015,pp.3358-3363,London,UK,June2015.
p [OPT/H1]
0.4 [7] M.A.Maddah-Ali,U.Niesen“FundamentalLimitsofCaching”,IEEE
hit [POP/H1] Transactions on Information Theory, vol. 60, pp. 2856 - 2867, May
[POP/H2] 2014.
al 0.3 [8] S. Ioannidis, L. Massoulie, A. Chaintreau, “Distributed Caching over
ot Heterogeneous Mobile Networks”, SIGMETRICS 2010, pp. 311-322,
T NY,USA,June2010.
0.2
[9] K.Poularakis,L.Tassiulas,“ExploitingUserMobilityforWirelessCon-
tent Delivery”, IEEE International Symposium on Information Theory
0.1 Proceedings(ISIT)2013,pp.1017-1021,Istanbul,Turkey,2013.
0 20 40 60 80 100 [10] E.Bas¸tug˘,M.Bennis,M.Debbah,“Cache-enabledSmallCellNetworks:
Modeling and Tradeoffs”, 11th International Symposium on Wireless
MM
CommunicationsSystems,pp.649-653,2014.
[11] E. Altman, K. Avrachenkov, J. Goseling, “Distributed Storage in the
Fig. 7. The total hit probability evaluation for different M values (M-or- Plane”, NetworkingConference,IFIP2014,pp.1-9,Trondheim,Nor-
None). way,June2014.
[12] K.Avrachenkov,X.Bai,J.Goseling,“OptimizationofCachingDevices
withGeometricConstraints”,arXivpreprintarXiv:1602.03635,2016.
[13] Z.Chen, J.Lee,T.Q. S.Quek,M. Kountouris,“CooperativeCaching
From Figure 7, we see that the hit probabilities under the
andTransmissionDesigninCluster-CentricSmallCellNetworks”,arXiv
[OPT/OPT] and the [POP/OPT] are identical and increase as preprintarXiv:1601.00321,2016.
M increases, i.e. we can use a heuristic placement policy [14] M. Dehghan, L. Massoulie, D. Towsley, D. Menasche, Y. C. Tay, “A
UtilityOptimizationApproachtoNetworkCacheDesign”,Proc.IEEE
for MBSs. For heuristic SBS deployment policies, [OPT/H1],
Infocom,SanFrancisco,CA,USA,April2016,toappear.
[POP/H1] and [POP/H2] policies achieve significantly lower [15] M. Dehghan, L. Massoulie, D. Towsley, D. Menasche, Y. C. Tay, “A
hit probability than the optimal policy. UtilityOptimizationApproachtoNetworkCacheDesign”,Proc.IEEE
Infocom,SanFrancisco,CA,USA,April2016,toappear.
VI. DISCUSSIONANDCONCLUSION [16] F. Baccelli and B. Blaszczyszyn, Stochastic Geometry and Wireless
Networks,VolumeI-Theory,ser.FoundationsandTrendsinNetworking
InthispaperwehaveshownthatwhetherMBSsusetheop- Vol.3:No3-4,pp249-449. NoWPublishers,2009,vol.1.
timaldeploymentstrategyorstore“themostpopularcontent”,
hasverylimitedimpactonthetotalhitprobabilityiftheSBSs
are using the optimal deployment strategy. It is important to
optimize the content placement strategy of the SBSs and the
total hit probability is increased significantly when the SBSs