Decentralized Coded Caching in Wireless Networks: Trade-off between Storage and Latency

0.54 MB·

by  Antonious M. Girgis

Decentralized Coded Caching in Wireless
Networks: Trade-off between Storage and Latency
Antonious M. Girgis∗, Ozgur Ercetin†, Mohammed Nafie∗‡, and Tamer ElBatt∗‡
∗Wireless Intelligent Networks Center (WINC), Nile University, Cairo, Egypt
† Faculty of Engineering and Natural Sciences, Sabanci University, Istanbul, Turkey.
‡ Electronics and Communications Dept., Faculty of Engineering, Cairo University, Giza, Egypt
Email: {[email protected], [email protected], mnafi[email protected], [email protected]}
Abstract—This paper studies the decentralized coded caching interference networks [3]–[7]. It is shown in [5] that caches
for a Fog Radio Access Network (F-RAN), whereby two edge- at the transmitters can improve the sum degrees of freedom
7 nodes (ENs) connected to a cloud server via fronthaul links (DoF) of the interference channel by allowing cooperation
1 with limited capacity are serving the requests of Kr users. between transmitters for interference mitigation. In [3], the
0 We consider all ENs and users are equipped with caches. A
2 decentralized content placement is proposed to independently authorsstudiedthebenefitsofcachingatboththetransmitters
store contents at each network node during the off-peak hours. and receivers in the interference channel while focusing on
n
Afterthat,wedesignacodeddeliveryschemeinordertodeliver the case of three receivers. The extension for an arbitrary
a
the user demands during the peak-hours under the objective of
J numberofreceiverswasstudiedin[4].Theauthorsin[6],[7]
minimizing the normalized delivery time (NDT), which refers to
5 the worst case delivery latency. An information-theoretic lower proposedacentralizedcodedcachingschemeforaFogRadio
2 boundontheminimumNDTisderivedforarbitrarynumberof AccessNetwork(F-RAN)withcachesequippedatedge-nodes
ENs and users. We evaluate numerically the performance of the (ENs) to minimize a novel metric named normalized delivery

] decentralized scheme. Additionally, we prove the approximate time (NDT) which measures the worst case delivery latency
T
optimality of the decentralized scheme for a special case when
withrespecttoaninterference-freebaselinesysteminthehigh
I the caches are only available at the ENs.
. signal-to-noise ratio (SNR) regime.
s
c Incontrasttopriorworks[3]–[7]thatdiscussedthecentral-
I. INTRODUCTION
[
ized content placement, we study the decentralized caching
Caching the most popular contents close to user terminals
2 problem for an F-RAN architecture with caches equipped
v is a promising solution to deal with the tremendous growth at both ENs and users. We first formulate the problem for
3 of the mobile traffic in the fifth generation (5G) wireless
arbitrary number of ENs and users. Then, we propose the
7 networks.Acachingsystemcomprisesoftwoseparatephases.
decentralized content placement in which each node stores
6
The first phase is the placement phase that occurs during the
a fraction of each file randomly and independently from
6
off-peak hours when the resources are under-utilized. During
0 other nodes. Based on the decentralized content placement,
. thisphase,thenetworknodesfilltheircacheswiththepopular we design a coded delivery scheme that exploits the network
1
contents. The second phase is the delivery phase that occurs
0 topology to minimize the NDT for the two ENs and arbitrary
during the peak hours when the network is congested, and
7 number of users case. We show that the performance of
1 hence, the caches at the network nodes can be exploited to the decentralized scheme is within a constant factor of the
: partly serve the user requests. In [1], Maddah-Ali and Niesen
v information-theoreticoptimumforthecaseinwhichonlyENs
have proposed a novel centralized coded caching scheme for
i have caches. The performance of the decentralized scheme is
X
an error-free broadcast channel. The authors showed that the
evaluated numerically for the general scenario.
r networkcongestioncanbesignificantlyreducedbyjointlyde-
a
signingtheplacementanddeliverytogeneratecodedmulticast II. SYSTEMMODEL
transmissions in the delivery phase. However, the centralized
We consider a Fog Radio Access Network (F-RAN) as
schemein[1]reliesontheknowledgeofthenumberofactive
depicted in Fig 1 comprising a cloud server that has a library
usersinthedeliveryphasetodesigntheplacementphase.This ofN files,W (cid:44){W ,···,W },eachofsizeF bits.Asetof
1 N
limits the applicability of the centralized scheme, since the K edge-nodes,EN ,···,EN ,isreadytoservetherequests
placementandthedeliveryphasesmighttakeplaceatdifferent oftK usersthrough1aK ×KKtGaussianwirelessinterference
r t r
times.Moreover,theusersinthewirelessnetworksarecharac- channel with channel coefficient h ∈ C between EN and
ki i
terizedbytheirmobilityinwhichthenumberofusersvariesin the kth user. The channels H (cid:44) [h ], i ∈ [K ], k ∈ [K ],
ki t r
the network over time. To cope with this problem, the work
are independent and identically distributed according to a
in [1] is extended in [2] to present the decentralized coded
continuous distribution. The cloud is connected to each EN
caching. Recently, the concept of coded caching is studied in via a fronthaul link of capacity C bits per symbol, where
F
the symbol refers to a channel use of the downlink wireless
This work was supported in part by the European Unions Horizon 2020
channel.EachEN ,i∈[K ],isequippedwithacachememory
researchandinnovationprogrammeundertheMarieSkodowska-Curiegrant i t
agreementNo690893. Vi of size MtF bits, while each user, k ∈ [Kr], is equipped
µwit=h Ma c/aNcheanmdeµmo=ryMZi/NofassiztheeMnorrFmalbizitesd. cWaceheresfiezretaot  rlogP
t t r r C F
eacThheENsysatnedmaotpeearcahteussienr,twreosppehcatsivees,lyn,awmheelyrepµlatc,eµmre∈nt[0p,h1a]s.e1 V1E:N1tNF ZU1s:err1NF
Cloud
anddeliveryphase.Theplacementphasetakesplacewhenthe
traffic load is low, and hence, the network resources can be C
utilized to fill the cache memories of each network node as a FrlogP
functionofthecontentlibraryW withoutanypriorknowledge
LibraryofNfiles
ofthenumberofactiveusersinthenextphase,thefutureuser V2E:N2tNF ZU2s:err2NF
demands, and channel coefficients. In the delivery phase, the
Fig. 1: A Fog Radio Access Network (F-RAN) with K = 2
kthuserrequestsafileW amongtheN filesavailableatthe t
cloud, where we considerdkd = [d ,···,d ] ∈ [N]Kr as the edge-nodes and Kr =2 users.
1 Kr
demand vector. The cloud replies to user demands by sending
fronthaul links. Moreover, we define the minimum NDT for a
amessageU =(U (t))TF ofblocklengthT toEN viathe
i i t=1 F i given tuple (µ ,µ ,r), as
fronthaul link, where each fronthaul message (U )Kt cannot t r
i 1
exceed TFCF bits due to the fronthaul capacity limitations. δ∗(µt,µr,r)(cid:44)inf{δ(µt,µr,r):δ(µt,µr,r) is feasible}
Each EN , i ∈ [K ], has an encoding function mapping the (3)
i t
cvaecchtoer cdo,natnendtstheVic,hathneneflrsoHnthtaoula mmeessssaaggee XUii,(cid:44)th(Xe id(etm))aTt=nEd1 serTvheeanNyDpTosδsi(bµlet,uµsre,rrd)ermepanredssendts∈th[Ne ]wKorrsntorcmasaelizdeedlawyittho
of block length T . EN transmits message X over the inter-
E i i respecttoaninterference-freebaselinesystemoftransmission
ference channel to Kr users under an average transmit power rate log(P) in the high SNR regime.
constraint 1 (cid:107)X (cid:107)2≤P. Each user k implements a decoding
function toTEestimiate the requested file Wˆdk from the cache depWeendpsoionnt otwutothtraatnsthmeisesniodn-tod-eelnadysl,atfernocnythafourl adnelaFy-RTAFN,
contents Zk and the received message Yk (cid:44)(Yk(t))Tt=E1. and edge delay TE. The fronthaul delay TF represents the
transmissiontimetodeliverthecloudmessagestoENs,while
Yk(t)=(cid:88)Kt hkiXi(t)+nk(t), t∈[TE] (1) the edge delay TE represents the transmission time to deliver
the messages of ENs to end users. In order to obtain the
i=1
whereY (t)∈Cisthereceivedsignalofthekthuserattime fronthaul-NDT δF, we substitute in (3) T = TF. Similarly,
k
we get the edge-NDT δ by substituting T = T in (3).
t,andn (t)∼CN (0,1)denotesthecomplexGaussiannoise E E
k
Throughoutthepaper,weconsidertwotypesoftransmissions.
at the kth user. For a given coding scheme (caching, cloud
1) Serial transmission: Fronthaul and edge transmissions
encoding, EN encoding, and user decoding functions), the
occur consecutively. The cloud first transmits its messages
transmission rate of the system is defined as K F/T, where
r {U }Kt toENsthatbegindeliveringtheirmessagesX tothe
T is the end-to-end latency. We say that a coding scheme is i 1 i
users only after receiving the fronthaul message. As a result,
feasible, if and only if, we have
theoveralllatencyisthesumofthefronthaulandedgedelays.
(cid:16) (cid:17)
max max P Wˆ (cid:54)=W →0 as F →∞ Thus, the NDT of system with serial transmission is defined
d∈[N]Krk∈[Kr] dk dk as
δ =δ +δ . (4)
S F E
which is the worst-case probability of error over all possible
2) Pipelined transmission: In [6], the authors introduced
demands d and over all users. Our goal in this paper is to
the concept of pipelined transmission, where the fronthaul
design a decentralized coding scheme to minimize the end-
to-end latency T for delivering user demands d ∈ [N]Kr in and edge transmissions take place simultaneously. In other
words, ENs do not wait for the fronthaul transmission to be
the deliveryphase. Inthe following,we define thenormalized
completed in order to start transmitting their messages to the
delivery time (NDT) first discussed in [7] as a performance
users. Accordingly, the NDT of the system for the pipelined
metric for any coding scheme.
transmission is given by
Definition 1. The normalized delivery time (NDT) for any
feasible coding scheme with a given normalized cache size δP =max{δF,δE} (5)
µ , µ , and fronthaul capacity C =rlog(P), is defined as
t r F which is the maximum between the fronthaul-NDT and the
max E (T) edge-NDT. In the following, we characterize the trade-off
H
δ(µt,µr,r)=Pl→im∞Fl→im∞sup Fd/log(P) (2) between the normalized cache size µt, µr and the NDT of
the system for the decentralized caching scheme with both
where E denotes the expectation with respect to the channel serial transmission δ and pipelined transmission δ .
S P
matrix H, and r measures the multiplexing gain of the
III. DECENTRALIZEDCODEDCACHING
1Notations: we use ⊕ to denote the bitwise XOR operation. [0,1] refers
Inthissection,weintroducethedecentralizedcodedcaching
to the real numbers between 0 and 1. [M] denotes the set of integers
{1,···,M},and[a:b]denotesthesetofintegers{a,···,b}fora≤b. for the 2×Kr F-RAN caching problem. First, we present the
decentralizedcontentplacement.Then,wedesignthedelivery interference channel, where the system becomes a Multiple-
scheme to serve any user demands in the delivery phase. Input-Single-Output (MISO) broadcast channel with K = 2
t
transmit antennas and K receivers. In general, the sum DoF
r
A. Decentralized Content Placement for a broadcast channel with K transmit antennas and K
t r
In the placement phase each user k ∈ [Kr] independently receivers is min{Kt,Kr} [8]. In our case DoF = 2. Thus,
stores µ F bits from each file W ∈W uniformly at random. the edge transmission latency to deliver these fragments in
r j
In a similar way, each EN , i ∈ [K ], independently stores the high SNR regime is T(1) = Kr|Wdk,φ,φ|F. By using
i t E 2log(P)
µ F bits from each file W ∈ W uniformly at random. the soft-transfer mode on fronthaul links as proposed in [9],
t j
Therefore, the total number of bits stored at each user, and the cloud implements ZF-beamforming and quantizes the
at each EN is µ NF bits and µ NF bits, respectively. Thus, encoded signal to be transmitted to each EN. Therefore, the
r t
thecachememorysizeconstraintisnotviolatedateachnode. fronthaultransmissionlatencyequalstotheedgetransmission
Moreover,eachbitofafilehasprobabilityµ tobecachedata latency multiplied by the ratio between the transmission rates
r
specific user and probability µ to be cached at a specific EN. log(P)/rlog(P), yielding a fronthaul transmission latency
t
Accordingly, each file Wj ∈ W is partitioned into fragments TF(1) = Kr2|Wrldokg,(φP,φ)|F. Thus, fronthaul-NDT and edge-NDT
Wj,St,Sr for St ⊂ [Kt] a subset of ENs, and Sr ⊂ [Kr] a for this stage are given by
subset of users, in which the fragment W denotes the
bitsoffileWj storedexclusivelyinthecachj,eStm,Sermoryofeach δF(1) = K2rr |Wdk,φ,φ |≈ K2rr (1−µt)2(1−µr)Kr
EN i ∈ S and each user k ∈ Sr, but these bits do not exist (7)
t K K
in the caches of other nodes. For |St |=t, |Sr |=i, each file δE(1) = 2r |Wdk,φ,φ |≈ 2r (1−µt)2(1−µr)Kr
Wj is split into (cid:80)Kt=t0(cid:80)Ki=r0(cid:0)Ktt(cid:1)(cid:0)Kir(cid:1)=2Kt+Kr fragments. In our example, the first user receives fragment A and the
0,0
Example 1. Consider an F-RAN with Kt =2 ENs, Kr =2 second user receives fragment B0,0 at the end of this stage.
users,andN =2filesdenotedbyW (cid:44){A,B}.Accordingto 2) Delivery of fragments Wdk,φ,Sr, | Sr |≥ 1: These
the decentralized content placement, we can represent the file fragmentsareavailableatleastatoneuserandatnoneofENs.
(cid:16)
A as follows2 A = A ,A ,A ,A ,A ,A ,A For each subset Sr ⊂[Kr] of cardinality |Sr |=i∈[2:Kr],
0,0 0,1 0,2 0,12 1,0 1,1 1,2
,A1,12,A2,0,A2,1,A2,2,A2,12,A12,0,A12,1,A12,2,A12,12(cid:17) tehaechclEoNud, wsehnedrse aWmdku,φlt,iScra\s{tkm} ersesfaegrset⊕okt∈heSrbWitdsko,φf,Sfirl\e{kW} dtok
such that A0,0 consists of bits of file A that are not stored at requested by user k which are available exclusively at users
any node, and A2,12 consists of bits of file A in the cache Sr except user k. Then ENs deliver the fronthaul message to
mBe.mBoyrystorfonagll lnaowdeosfexlacregpet EnuNm1.beTrhs,etshaemesizfeolloofwsfrafogrmfienlet Kk˜ ∈r uSse,rtsh.eSkinthceusthereckatnhreucsoervehraiststhneeedbeitds bWitsdk˜W,φ,Sr\{k˜} for
Wj,St,Sr is approximately equal to from rthe multicast transmission. There are (cid:0)Kr(cid:1)dsku,φb,sSert\s{ko}f
i
|Wj,St,Sr | ≈ µ|tSt|(1−µt)Kt−|St|µ|rSr|(1−µr)Kr−|Sr|(6F) cRa(r2d)in=al(cid:88)iKtyr (cid:32)i,Ksor(cid:33)th(e1r−atµe o)2fµthi−is1(tr1a−nsµmi)sKsiro−ni+i1sFgiven by
i t r r
with probability approaching one for large file size F. In our i=2
example, |A1,2 |≈µt(1−µt)µr(1−µr)F. = (1−µt)2(1µr)(cid:104)(cid:88)Kr (cid:32)Kr(cid:33)−(1−µ )Kr (8)
µ i r
B. Coded Delivery r i=0
(cid:105)
Consider any demand vector d ∈ [N]Kr. In Exam- −Krµr(1−µr)Kr−1
ple 1, let the first user request file A and the sec- (1−µ )2(1−µ )(cid:104) (cid:105)
= t r 1−(1−µ )Kr −K µ (1−µ )Kr−1 F
ond user request file B. Therefore, the first user needs µ r r r r
r
fragments (A0,0,A0,2,A1,0,A1,2,A2,0,A2,2,A12,0,A12,2) to where the expected size of ⊕ W with |S |=i
decode file A, and the second user needs fragments is(1−µ )2µi−1(1−µ )Kr−ki∈+S1rF.dAk,sφ,aSrre\{suk}lt,fronthaurland
(B0,0,B0,1,B1,0,B1,1,B2,0,B2,1,B12,0,B12,1)todecodefile edge trantsmisrsion delayrs are T(2) = R(2) , T(2) = R(2) ,
B. In the following, we explain the delivery scheme to send F rlog(P) E log(P)
respectively. Moreover, the fronthaul-NDT and edge-NDT are
these fragments. The proposed delivery scheme is divided
given by
into five stages. In each stage, the cloud and ENs exploit
R(2) R(2)
the wireless channel to deliver a group of fragments of the δ(2) = , δ(2) = (9)
F rF E F
requested files depending on the cache contents at the nodes.
Inourexample,thetransmissioninthisstageis(A ⊕B ),
1)DeliveryoffragmentsW :SincetheseK fragments 0,2 0,1
are not available at any nodesd,kt,hφe,φcloud sends {Wr }Kr so that the first user can recover fragment A0,2 from the re-
dk,φ,φ k=1 ceivedsignal andcachedfragment B .Similarly, thesecond
to each EN via the fronthaul links. Hence, each EN can 0,1
user can recover fragment B from the received signal and
perform cooperative zero-forcing (ZF) beamforming on the 0,1
cached fragment A .
0,2
3) Delivery of fragments W , | S |≥ 1: In this
2We write A0,1 as a shorthand for A{φ},{1}, and A12,1 as a shorthand dk,St,Sr r
forA{1,2},{1} stage, each subset St ⊂ [Kt] of ENs, | St |∈ {1,2},
10
9
20 10 8 δdec
15 8 7 δdec in [2]
dec*δδ/SS10 dec*δδ/PP46 δµ()r 56
4
5
0 2 0 3
0 0µ.r5 1 1 0.5µt 0 µ0.r5 1 1 0.5µt 12
0
0 0.2 0.4 0.6 0.8 1
(a) Serial transmission (b) Pipelined transmission µ
r
Fig. 2:The gap betweenthe decentralizedNDT in Theorem1 Fig. 3: Comparison between the decentralized scheme in (22)
and the lower bound in Theorem 2 for r =1 and Kr =100. and the decentralized scheme [2] for Kr =10.
tcraacnhsemcitosntaenmtsulttoicaasstusbisgentaSl ⊕⊂k∈[SKrW]dokf,Suts,Serr\s{fko}rf|roSm|≥the2ir, escdhgeemteran[1sm0]i.ssIinonoulartecnacsye itsheTs(u5)m=D2oKFr|iWsdKk2,rK{+1r}1,φ.|FThluesa,ditnhge
r r r E 2Kr log(P)
so that each user k ∈ Sr can recover the needed fragment to the fronthaul-NDT and edge-NDT Kr+1
W from the received signal and its cache contents.
Thduks,,φ,tShre\{rka}te of this transmission is given by δF(5,a) =0, δE(5,a) =(Kr+1)µt(1−µt)(1−µr)Kr. (13)
b) Another achievable transmission scheme for this stage is
=R(3(cid:0))1=−(cid:88)t(=211−(cid:88)iK=µr2t(cid:32))22t(cid:1)(cid:33)(1(cid:32)−Kiµr(cid:33)r)µ(cid:104)tt1(−1−(1µ−t)µ2−)tKµrir−1(1−µr)Kr−i+(110F) tTahnhadutsf,trhabegomtchelonEutNsds(cid:8)tWhraandvkse,m{22iK}ts,φr(cid:9)ffrK1raagrgmmtoeennEttsNs1(cid:8)anWvdiadtkht,he{e1s}yf,φrso(cid:9)tenK1mthrabuteolcloiEnmNkes2s.
µ r a broadcast channel with DoF min{K ,K } by applying ZF
r t r
−K µ (1−µ )Kr−1(cid:105)F. beamformingontheinterferencechannel.Hencethefronthaul-
r r r
NDT and edge-NDT are given by
As a result, the edge transmission latency is T(3) = R(3) , K K
E log(P) δ(5,b) = r |W |= rµ (1−µ )(1−µ )Kr
yielding the the fronthaul-NDT and edge-NDT F r dk,{1},φ r t t r
δF(3) =0, δE(3) = RF(3). (11) δE(5,b) =Kr |Wdk,{1},φ |=Krµt(1−µt)(1−µr)Kr(.14)
The following theorem gives the NDT of the proposed de-
In our example, EN first transmits a message (A ⊕B ),
1 1,2 1,1 centralized scheme for the serial and pipelined transmissions.
then, EN transmits a message A ⊕B . Finally, both of
2 2,2 2,1
ENs cooperatively transmit a message A ⊕B . Hence, Theorem1. Forthe2×K F-RANwithdecentralizedcaching
12,2 12,1 r
the first user obtains fragments A ,A ,A , and the placement, each EN having a normalized cache size µ ∈
1,2 2,2 12,2 t
second user obtains fragments B ,B ,B at the end of [0,1], each user having a normalized cache size µ ∈ [0,1],
1,1 2,1 12,1 r
this stage. and the fronthaul gain r > 0, the proposed coded delivery
4) Delivery of fragments W :Each of the two ENs scheme achieves NDT for serial transmission
dk,{1,2},φ
has these K fragments. Hence, each EN can implement ZF (cid:40)
r δ(a)+δ(a) 0<r ≤K
beamforming on the interference channel to transmit these δdec (cid:44) F E r (15)
fragmentswithDoF =2.Thus,theedgetransmissionlatency S δF(b)+δE(b) r >Kr
is T(4) = Kr|Wdk,{1,2},φ|F, yielding the fronthaul-NDT and and achieves NDT for pipelined transmission
E 2log(P)
edge-NDT (cid:110) (cid:110) (cid:111) (cid:110) (cid:111)(cid:111)
δdec =min max δ(a),δ(a) ,max δ(b),δ(b) (16)
K K P F E F E
δ(4) =0, δ(4) = r |W |≈ rµ2(1−µ )Kr.
IsnecoFounrdeuxsaemrprlEeec,etihveesfi2frrsatgumseedrnkr,t{eB1c,e2i}v,φesaftrathg2emeenntdtAo1f2t,h0i,sranstda(g1the2.e) δδFF((ba))==((11−−µµttrr))µ2µ2(r(r11−−µµrr))(cid:104)(cid:104)11−−((11−−µµrr))KKr(cid:16)r − K2r(cid:17)µr(cid:105)(1−µr)Kr−1(cid:105)
12,0 −Krµ (1−µ )Kr−1 1−3µt
wweh5e)rpeDrotephloeivseEerNytwsoocfhfromaoegstmehoethdnestsmfWoertdhkto,rSadtn,tφshm,a|titStmitnign|=imth1iez:seIesntfthrhaeigsomsvteeanrgatesll,, δE(a) = (1−µrµr)(cid:104)−1(cid:0)−2K(r1r−−µµr()1Kr−r µ )(cid:1)µ 1(−1µ−t µ )Kr−1(cid:105)
2 t t r r
NDT as we will elaborate in Theorem 1. (cid:104) (cid:105)
δ(b) = (1−µr) 1−(1−µ )Kr − Krµ (1−µ )Kr−1
a) Each EN has a dedicated fragment to each user. Hence, the E µr r 2 r r
(17)
systemcanbetreatedas”K ×K X-channel”,wherethesum
t r
DoF is KtKr by applying the interference alignment (IA) Proof: See Appendix A
Kt+Kr−1
Lower Bound: To evaluate the performance of the de- For µ =0: The network becomes an F-RAN with caches
r
centralized scheme, we derive a generic lower bound on the at ENs only. In [6], the authors provide a centralized coded
optimum NDT for the K ×K F-RAN caching problem for caching scheme for this network within a constant factor 2
t r
serial and pipelined transmissions. from the lower bound. From Theorem 1, the decentralized
scheme for µ →0 achieves
Theorem 2. For an F-RAN with fronthaul capacity C = r
F
 (cid:104) (cid:105)
rKlrogu(sPer)s,,KeatchENwsi,theaacnhowrmithalaizendorcmacahliezesdizceaµchr,etshiezeoµptti,maunmd δdec = K2r (cid:20)(1−rµt)2 +1(cid:21)+µt(1−µt) 0<r ≤Kr(23)
NDT δS∗ for serial transmission is lower bounded by S  K2r (1−rµ2t) +1 r >Kr
where δLB = δF +δE isδthS∗e≥soδlLuBtion of the following lin(e1a8r) δPdec = KK22rrr (cid:0)1−µ2t(cid:1) rr2≥≤rr1 <r1 (24)
programming  K2rr (1−µt)2 0<r <r2
min δF +δE where r2 =(1−µt)2/ (1+2µt(1−µt)/Kr), r1 =(cid:0)1−µ2t(cid:1).
subject to sδ +(K −s)rδ ≥f(s) s∈M (19)
E t F Theorem 3. For pipelined transmission, the decentralized
δ ≥0, δ ≥(1−µ ) (20) scheme (24) is optimum for µ ∈ [0,1], r ≥ (cid:0)1−µ2(cid:1) and
F E r t t
µ ∈[0,1/2),0<r.Letδ∗ betheoptimalcentralizedcaching
For pipelined transmission, the optimum NDT δ∗ is lower t S
bounded by P scheme for serial transmi√ssion. We have δδSd∗ec ≤ 3 for µt ∈
(cid:2) (cid:3) S
[0,1], r ≥1 and µ ∈ 0, 2−1 , 0<r.
t
(cid:26) (cid:26) (cid:27) (cid:27)
f(s)
δ∗ ≥max max ,(1−µ ) Proof: The proof is presented in Appendix C
P s∈[0:min{Kt,Kr}] s+(Kt−s)r r Theorem 3 shows that the proposed decentralized scheme
(21)
In (19) and (21), M (cid:44) [0:min{K ,K }], and f(s) = is approximately optimal for pipelined and serial transmis-
t r
Kr(1−sµr)−(Kr−s)[(Kr−s)µr+(Kt−s)µt]. sions. (cid:2)H√owever,(cid:3)the optimality is not maintained in case
µ ∈ 2−1,1 , 0 < r < 1 for serial transmission and
t
Proof: The proof of Theorem 2 uses the cut-set lower µ ∈ [1/2,1], 0 < r < (cid:0)1−µ2(cid:1) for pipelined transmission.
t t
bound on any achievable NDT. The proof can be found in
The reason is that when the fronthaul gain is low and the
Appendix B
aggregate size of ENs’ caches is high enough to store all the
IV. PERFORMANCEANALYSIS
library files, the best strategy is to store different contents in
In this section, we evaluate the performance of the decen-
ENs so that the fronthaul link would not be used. However,
tralized scheme introduced in Section III. We first evaluate
this strategy is not valid in the decentralized scheme due to
numerically the performance of the decentralized scheme for
the random content placement.
the general case. Then, we discuss two special cases.
Fig. 2 shows the gap between the achievable decentralized V. CONCLUSION
NDT in Theorem 1 and the lower bound in Theorem 2 for
In this paper, we have studied the problem of decentralized
serial and pipelined transmissions with fronthaul gain r = 1
caching in Fog radio Access Networks. For two edge-nodes
and a number of users K = 100. It is observed that the
r and arbitrary number of users, a coded delivery scheme has
maximum gap is about 20 for serial transmission and 10 for
been developed to minimize the latency for delivering user
pipelined transmission. Moreover, we observe that the gap
demands. We have shown that even with the decentralized
decreases rapidly as µ → 1, however, there is only a subtle
r placement,theproposeddeliveryschemeachievesasignificant
decrease as µ increases.
t performance as compared with the derived lower bound.
Forµ =1or/andr →∞:Inthiscase,eachENhasallthe
t
libraryfiles,andhence,thesystembecomesaMISObroadcast
APPENDIXA
channel with two transmit antennas. From Theorem 1, the
PROOFOFTHEOREM1
decentralized scheme achieves NDT as
InSubsectionIII-B,weproposedtwodeliveryschemesthat
(cid:20) (cid:21) have the same first fourth stages of transmission and differ
(1−µ ) K
δdec(µr)= µ r 1−(1−µr)Kr − 2rµr(1−µr)Kr−1 only at the last stage. In order to obtain the fronthaul-NDT
r
(22) andedge-NDTpresentedin(17)foreachscheme,wesumthe
Comparingwiththeperformanceofthebroadcastchannelwith over all fronthaul-NDT and edge-NDT of the five stages:
asingletransmitantennain[2],NDTin(22)isminimumwith
term K2r (1−µr)Kr,sinceKr fragmentscachedatnoneofthe δ(a) =(cid:88)4 δ(i)+δ(a,5)
users are transmitted with DoF =2 in our case as compared F F F
to DoF = 1 in [2]. In Fig 3, we see that this reduction is i=1 (25)
4
significant when µr is low; however, the two systems achieve δ(a) =(cid:88)δ(i)+δ(a,5)
E E E
the same performance when µ approaches to one.
r i=1
The same analysis holds for the second delivery scheme (b). where (a) follows from the fact that the files are indepen-
For serial transmission, the NDT of the system is the sum dent. Step (b) follows from the chain rule. Step (c) follows
of the fronthaul-NDT and edge-NDT (see (4)), so that ENs from Fano’s inequality, where
choose the delivery scheme that minimizes the summation of
(cid:0) (cid:1)
the fronthaul-NDT and edge-NDT. It is obvious that the first H W[1:Kr]|W[Kr+1:N],Y[1:s],U[1:Kt−s],V[1:Kt−s],Z[1:Kr] ≤εFF
scheme(a)haslowerNDTthanthesecondscheme(b)asr <
since the users are able to decode their requested files from
K . Thus, the NDT for serial transmission is given by (15).
r their cache contents and the transmitted messages of ENs. ε
F
For pipelined transmission, the NDT of the system is the
is a function of F that vanishes as F →∞. Step (d) follows
maximum between fronthaul-NDT and edge-NDT (see (5)).
from the data processing inequality, where X are func-
Therefore, we first obtain the performance of each delivery [1:Kt]
tionsofW ,andthefactthatthetransmittedandreceived
scheme by taking the maximum between fronthaul-NDT and [1:Kr]
signals are independent on the unrequested files. Step (e)
edge-NDT, then, we choose the delivery scheme that has the
follows from the capacity bound of K ×s broadcast channel
t
minimum NDT between them as in (16).
over T time slots in the high SNR regime, where s≤K , in
E (cid:16) (cid:17) t
APPENDIXB addition to H Z[1:Kr],Y[1:s] |W[1:N] = 0. Step (f) follows
PROOFOFTHEOREM2 from the fact that conditioning reduces entropy, and the fact
ThelowerboundissimilartotheonepresentedinProposi- thatcachesof(Kt−s)ENshave(Kt−s)(Kr−s)µtF bits
tion1in[7]whichisbasedonthecut-setargument.Consider of (Kr−s) files. Step (g) follows from the capacity bound
the user demands d, where each user k ∈ [Kr] requests a on (Kt−s) fronthaul links over TF times slots. The second
different file W . To obtain the constraint 19, let us consider term in 26 can be upper bounded as follows
dk
s ∈ {0,···,min{K ,K }} users. From the the received
t r (cid:0) (cid:1)(a) (cid:0) (cid:1)
H Z |W ,Y = H Z |W ,Y
signals of s users which are functions of the transmitted [1:Kr] [Kr+1:N] [1:s] [1:s] [Kr+1:N] [1:s]
(cid:0) (cid:1)
messages X , the cache contents of (K −s) ENs, and +H Z |W ,Y ,Z
[1:Kt] t [s+1:Kr] [Kr+1:N] [1:s] [1:s]
the fronthaul messages received by the (Kt−s) ENs, we (b) (cid:0) (cid:1) (cid:0) (cid:1)
can reconstruct the transmitted messages X[1:Kt] at high SNR ≤ H Z[1:s]|W[Kr+1:N] +H Z[s+1:Kr]|W[1:s]∪[Kr+1:N]
regime by solving Kt equations in Kt unknowns. Therefore, (=c)sK µ F +(K −s)2µ F
r r r r
we can argue that the Kr user demands can be decodable (27)
by observing the received signals of s users Y[1:s], the cache where (a) follows from the chain rule. Step (b) follows from
contentsof(Kt−s)ENsV[1:Kr−s],thefronthaulmessagesof that conditioning reduces entropy, and s users are able to
(Kt−s) ENs U[1:Kt−s], and the cache contents of Kr users decode files W[1:s] from their cache contents and the received
Z[1:Kr]. signals. Step (c) follows from that caches of s users have
K µ F bits of K files, and caches of (K −s) users have
(cid:0) (cid:1)(a) (cid:0) (cid:1) r r r r
K F =H W = H W |W
r [1:Kr] [1:Kr] [Kr+1:N] (K −s)µ F bits of (K −s) files. Substituting from 27
r r r
(b) (cid:0) (cid:1)
= I W ;Y ,Z ,V ,U |W into 26, we get
[1:Kr] [1:s] [1:Kr] [1:Kt−s] [1:Kt−s] [Kr+1:N]
(cid:0) (cid:1)
+H W[1:Kr]|W[Kr+1:N],Y[1:s],Z[1:Kr],V[1:Kt−s],U[1:Kt−s] (sTE +r(Kt−s)TF)log(P)≥Kr(1−sµr)F
(c) (cid:0) (cid:1) (cid:104) (cid:105) (28)
≤ I W[1:Kr];Y[1:s]|W[Kr+1:N] + −(Kr−s) (Kr−s)µr+(Kt−s)µt F +εFF
(cid:0) (cid:1)
I W ;Z ,V ,U |W ,Y +ε F
[1:Kr] [1:Kr] [1:Kt−s] [1:Kt−s] [Kr+1:N] [1:s] F By taking the limit P →∞ and the limit F →∞, we reach
(≤d)I(cid:0)X ;Y (cid:1)+I(cid:0)W ;Z |W ,Y (cid:1)+ to the inequality 19 as
[1:Kt] [1:s] [1:Kr] [1:Kr] [Kr+1:N] [1:s]
(cid:0) (cid:1)
I W[1:Kr];V[1:Kt−s],U[1:Kt−s]|W[Kr+1:N],Y[1:s],Z[1:Kr] +εFF sδE +r(Kt−s)δF ≥Kr(1−sµr)
(29)
(≤e)sTElog(P)+H(cid:0)Z[1:Kr]|W[Kr+1:N],Y[1:s](cid:1)+ −(Kr−s)[(Kr−s)µr+(Kt−s)µt]
(cid:0) (cid:1)
I W[1:Kr];V[1:Kt−s],U[1:Kt−s]|W[Kr+1:N],Y[1:s],Z[1:Kr] +εFF ToobtaintheconstraintδE ≥(1−µr),considerthecut-set
(cid:0) (cid:1)
=sT log(P)+H Z |W ,Y at the first user
E [1:Kr] [Kr+1:N] [1:s]
+I(cid:0)W[1:Kr];V[1:Kt−s]|W[Kr+1:N],Y[1:s],Z[1:Kr](cid:1)+ F =H(W1)=H(cid:0)W1|W[2:N](cid:1)
(cid:0) (cid:1)
I W[1:Kr];U[1:Kt−s]|W[Kr+1:N],Y[1:s],Z[1:Kr],V[1:Kt−s] +εFF (=a)I(cid:0)W ;Y ,Z |W (cid:1)+H(cid:0)W |W ,Y ,Z (cid:1)
=sT log(P)+H(cid:0)Z |W ,Y (cid:1) 1 1 1 [2:N] 1 [2:N] 1 1
+HE(cid:0)V[1:Kt−s]|W[Kr[+1:1K:Nr]],Y[K[1:rs+],1Z:N[1]:Kr][1(cid:1):s] (≤b)I(cid:0)W1;Y1|W[2:N](cid:1)+I(cid:0)W1;Z1|W[2:N],Y1(cid:1)+εFF
(cid:0) (cid:1)
+H U[1:Kt−s]|W[Kr+1:N],Y[1:s],Z[1:Kr],V[1:Kt−s] +εFF (c) (cid:0) (cid:1)
≤ T log(P)+H Z |W ,Y +ε F
(f) (cid:0) (cid:1) E 1 [2:N] 1 F
≤ sT log(P)+H Z |W ,Y +
E [1:Kr] [Kr+1:N] [1:s] (d)
(Kt−s)(Kr−s)µtF +H(cid:0)U[1:Kt−s](cid:1)+εFF ≤ TElog(P)+µr+εFF (30)
(g) (cid:0) (cid:1)
≤ sTElog(P)+H Z[1:Kr]|W[Kr+1:N],Y[1:s] + where (a) follows from the chain rule. Step (b) follows from
(K −s)(K −s)µ F +(K −s)T rlog(P)+ε F (26) theFano’sinequality.Step(c)followsfromthecapacitybound
t r t t F F
of the broadcast channel. Step (d) follows from the fact that From (39) and (23), we have
the cache of each user has µrF bits of each file. By taking δdec (cid:0)1−µ2(cid:1)
limitsP →∞,L→∞,wegetδ ≥(1−µ ).Thiscomplete S ≤ t +1≤2 (40)
E r δ∗ r
the proof of lower bound on the NDT for serial transmission. S
Note that fronthaul messages U[Kt−s], ENs’ messages since (cid:0)1−µ2(cid:1)≤1 and r ≥1, we have (1−µ2t) ≤1.
X , and received messages Y cannot exceed T time t r
[1:Kt] [1:s] p 2) For 1≤r <K : From (39) and (23), we have
slots for pipelined transmission, where T denotes end-to-end r
p
latency in case of pipelined transmission. Thus, the lower δdec (1−µ )2 2µ (1−µ )
S ≤ t +1+ t t ≤3 (41)
boundontheNDTforpipelinedtransmissioncanbeobtained δ∗ r K
S r
in a similar proof for serial transmission by substituting
TE = TF = Tp in (26). By taking limits P → ∞, L → ∞ where (1−rµt)2 ≤ 1 as (1−µt)2 ≤ 1 and r ≥ 1. In addition,
and combing with the fact that δ∗ ≥(1−µ ), we obtain the 2µt(1−µt) ≤1, since K ≥2 and µ (1−µ )≤1.
lower bound presented in (21). P r 3K)rFor 0 < r < 1, µr ∈ (cid:2)0:√2t−1(cid:3): Ftrom the inequali-
t
ties 36, 37, we have
APPENDIXC
PROOFOFTHEOREM3 K K
δ∗ ≥ r (1−2µ )+ r +µ (42)
Forpipelinedtransmission,wehavethreeinequalitiesonthe S 2r t 2 t
minimum NDT (δ∗) from Theorem 2 by substituting K =2 Using (23) and the lower bound (42), we obtain
P t
and µr =0 in (21)δ,P∗as≥foKllrow(1s2−r 2µt) (31) δδSdS∗ec ≤ K2r (cid:104)K2(rr1−(r1µt−)22+µt1)(cid:105)++Kµ2rt(+1µ−tµt) (43)
δP∗ ≥ KKr(11−+µrt)+µt (32) =1+ K2rr (1−µ2t2K2µrrt)−+µK2t2r +µt ≤2
δ∗ ≥ r (33) √
We consider two mPain reg2imes of fronthaul gain r: wcohmerpeletKe2rsr(t1hµ−e2t2Kpµ2rtrr)o−+oµfK2t2ro+fµTthe≤ore1mfo3r. µt ∈ (cid:2)0: 2−1(cid:3). This
1) For r ≥ (cid:0)1−µ2(cid:1): It can be easily seen that the
t
decentralized scheme in (24) achieves the lower bound from REFERENCES
the inequality (33). [1] M. A. Maddah-Ali and U. Niesen, “Fundamental limits of caching,”
2) For 0 < r < (cid:0)1−µ2(cid:1) and µ ∈ [0,1/2): Using IEEETrans.Inf.Theory,vol.60,no.5,pp.2856–2867,2014.
t t
[2] ——,“Decentralizedcodedcachingattainsorder-optimalmemory-rate
inequality (31), we have
tradeoff,”IEEE/ACMTrans.Netwo.,vol.23,no.4,pp.1029–1040,2015.
δPdec = K2rr (1−µt)2 = (1−µt)2 ≤1 (34) [3] Fla.teXnuc,yMin.Tcaaoc,hea-nadidKed.Lwiuir,e“leFsusnidnatmerefenrteanlctreadneeotwffobrketsw,”eeanrXsitvorpargeeprainndt
δP∗ K2rr (1−2µt) (1−2µt) arXiv:1605.00203,2016.
[4] J.Hachem,U.Niesen,andS.Diggavi,“Degreesoffreedomofcache-
for 0<r <r2 and µt ∈[0,1/2). Similarly, we have aidedwirelessinterferencenetworks,”arXivpreprintarXiv:1606.03175,
δPdec = K2rr (cid:0)1−µ2t(cid:1) = (cid:0)1−µ2t(cid:1) ≤1 (35) [5] M20.1A6..Maddah-AliandU.Niesen,“Cache-aidedinterferencechannels,”
δP∗ K2rr (1−2µt) (1−2µt) [6] Ain.IESeEnEguISpItaT,,2R0.15T,anpdpo.n8,09a–n8d13O.. Simeone, “Cloud and cache-aided
for r < r < (cid:0)1−µ2(cid:1) and µ ∈ [0,1/2). Combining wireless networks: Fundamental latency trade-offs,” arXiv preprint
2 t t arXiv:1605.01690,2016.
with the fact that δPdec ≥ 1, this proves the optimality of [7] R.TandonandO.Simeone,“Cloud-aidedwirelessnetworkswithedge
the decentralized scδhP∗eme for pipelined transmission when caching:Fundamentallatencytrade-offsinfogradioaccessnetworks,”
µt ∈[0,1], r ≥(cid:0)1−µ2t(cid:1) and µt ∈[0,1/2), 0<r. [8] Hin.IWEEeiEngIaSrItTen,,2Y0.16S,tepinpb.e2r0g2,9a–n2d0S3.3S.hamai,“Thecapacityregionofthe
From Theorem 2 by substituting with µ = 0 and K = 2 gaussianmultiple-inputmultiple-outputbroadcastchannel,”IEEETrans.
r t
Inf.Theory,vol.52,no.9,pp.3936–3964,2006.
in (19), we have s∈{0,1,2} inequality constraints for serial
[9] O. Simeone, O. Somekh, H. V. Poor, and S. Shamai, “Downlink
transmission as follows multicellprocessingwithlimited-backhaulcapacity,”EURASIPJournal
K onAdvancesinSignalProcessing,vol.2009,no.1,pp.1–10,2009.
δ ≥ r (1−2µ ) (36) [10] V.R.CadambeandS.A.Jafar,“Interferencealignmentandthedegrees
F 2r t of freedom of wireless networks,” IEEE Trans. Inf. Theory, vol. 55,
δ +rδ ≥K (1−µ )+µ (37) no.9,pp.3893–3908,2009.
E F r t t
[11] A.M.Girgis,O.Ercetin,M.Nafie,andT.ElBatt,“Decentralizedcoded
K
δ ≥ r (38) caching in wireless networks: trade-off between storage and latency,”
E 2 arXivpreprintarXiv:1701.06673,2017.
We consider three different regimes of fronthaul gain r:
1) For r ≥ K : From the inequality (38), we obtain the
r
lower bound on the minimum NDT for serial transmission
K
δ∗ ≥ r (39)
S 2