Optimizing MDS Coded Caching in Wireless Networks with Device-to-Device Communication

0.32 MB·

by  Jesper Pedersen

Optimizing MDS Coded Caching in Wireless
Networks with Device-to-Device Communication
Jesper Pedersen†, Alexandre Graell i Amat†, Iryna Andriyanova‡, and Fredrik Bra¨nnstro¨m†
†Department of Signals and Systems, Chalmers University of Technology, Gothenburg, Sweden
‡ETIS Laboratory, ENSEA/University of Cergy-Pontoise/CNRS, Cergy-Pontoise, France
Abstract—We consider the caching of content in the mobile using maximum distance separable (MDS) codes reduces the
devicesinawirelessnetworkusingmaximumdistanceseparable macro BS downlink traffic.
(MDS) codes. We focus on a small cell, served by a base station
In this paper, we consider the use of MDS codes to cache
(BS), where mobile devices arrive and depart according to a
7 content in the mobile devices of a wireless cellular network.
Poisson birth-death process. Users requesting a particular file
1
download coded symbols from caching devices using device-to- The devices roam between neighboring cells according to a
0
device communication. If additional symbols are required to Poisson random process, similar to the modeling of device
2
decode the file, these are downloaded from the BS. We optimize mobility in [5]–[7], and request files from a library at random
n the MDS codes to minimize the network load under a global
times. Files are encoded using MDS codes of equal code
a average cache size constraint. We show that, for most practical
J scenarios, using optimized MDS codes results in significantly length but potentially different code rate, and coded symbols
3 lowernetworkloadcomparedtowhencachingthemostpopular are cached in a number of mobile devices. We formulate
2 files. Furthermore, our results show that caching coded symbols a mixed integer linear program (MILP) to find the MDS
of each file on all mobile devices, i.e., maximal spreading, is codes that minimize the network load, assuming a global

] optimal.
average cache size constraint (across all devices). We show
T
that optimizing the rates of the MDS codes may yield a
I
. I. INTRODUCTION significantly lower network load than when caching only the
s
c most popular files. We also show that, for the considered
Mobile data traffic is predicted to grow by 800% in the
[
scenario,cachingsymbolsofagivenfileonallmobiledevices,
next 6 years [1]. This imposes a severe strain on existing

1 wireless networks. One of the most promising methods to i.e., maximal spreading [9], is optimal.
v Notation: P(·) and E[·] denote probability and expectation,
reduce downlink traffic is to store content closer to end users,
9 respectively. A random variable X that follows the binomial
commonlyreferredtoascaching[2].Cachinghasbeenstudied
8
distribution with parameters n and p is denoted by X ∼
2 extensivelyinrecentyears.In[3],acentralserverwithaccess
6 toafilelibraryservesanumberofusers,eachwiththeability B(n,p). We use bold lowercase letters x to denote vectors.
0 to cache content, using a common channel. It is shown that, 0m and 1m are the length-m vectors with all entries 0 and 1,
. respectively.
1 exploitinglocalcachesassideinformation,transmittingcoded
0 multicasts, i.e., linear combinations of packets from different II. SYSTEMMODEL
7
files,cansignificantlyreducetheamountofdatathatisneces-
1 sary to transmit over the channel. This concept is investigated WeconsideralargeareaofAtot squareunits(sq.u.)consist-
: ing of many small cells with equal area A sq.u., each served
v in [4] for the specific application of wireless networks. The
Xi centralserverisomittedandrequestsforcontentareservedby by a BS. Mtot mobile devices are uniformly spread over the
large area, which gives that the number of devices in a given
mobile devices caching data, which broadcast interfile coded
r small cell, denoted by X, is binomially distributed as
a messages using device-to-device (D2D) communication.
In wireless networks, mobile devices that cache content X ∼B(M ,A/A ). (1)
tot tot
might move out of range. The impact of data loss when
The expected number of devices initially available in a small
mobile devices leave the network is considered in [5]–[7].
cell, denoted by M, is
Instantaneousrepairofsingledevicedeparturesisinvestigated
in [5] and scheduled repair of lost content due to multiple M =E[X]=M ·A/A . (2)
tot tot
devices leaving the network is treated in [6] and [7]. The
Users wish to download files from a library of N files that is
papers[5]–[7]considerasinglefileanditisshownthattheuse
alwaysavailableattheBSs.Weassumethatallfileshaveequal
of erasure correcting codes help to reduce the amount of data
size and that the file popularity follows the Zipf distribution
thatisdownloadedfromthebasestation(BS).In[8],multiple
[10], i.e., the popularity of file i is given by
files are cached in a number of small BSs from which mobile
devices download content. It is shown that caching content 1/iσ
p = , i=1,...,N, σ ≥0, (3)
i (cid:80)N 1/(cid:96)σ
(cid:96)=1
This work was partially funded by the Swedish Research Council under
grant2016-04253. where σ is the skewness parameter of the distribution.
Content allocation: Each file i that is to be cached is definedasthedownlinkrateandtheequivalentnumberoffiles
partitioned into k packets and encoded into n symbols using downloaded from caching devices (per t.u.) is defined as the
i
an (n,k ) MDS code [11]. The n coded symbols are cached D2Dcommunicationrate.Weassumethatthecommunication
i
in n mobile devices (possibly different for each file) in the is error free and incurs zero delay.
area, chosen uniformly at random. Hence, for each file i, the
n devices caching the file store a fraction α = 1/k of the III. NETWORKSTATISTICSANALYSIS
i i
file. Thus, In this section, we derive the probability to have a number
(cid:26)1 (cid:27) of caching devices available at the time of a request. For later
α ∈{0}∪ :k =1,...,n (cid:44)A, i=1,...,N, (4)
i i use,wedefineJ astheeventthattherearej cachingdevices
k j
i
available at the time of a request for file i. We also define C
where αi = 0 implies that file i is not cached. We define and C¯ as the events that the request comes from a caching
the vector α = (α ,...,α ) and refer to it as the content
1 N device and comes from a noncaching device, respectively. We
allocation.
have the following proposition.
The number of caching devices in a given small cell,
denoted by Y, is binomially distributed as Proposition 1. The probability of having j mobile devices
caching a coded symbol of file i at the time of a request for
Y ∼B(n,A/A ), (5)
tot file i is
and the expected number of caching devices in a given small q = M −m+j · mje−m, j ≥0. (8)
j
cell, denoted by m, is M j!
m=E[Y]=n·A/Atot. (6) Proof: The probability qj is given by
Arrival-departure process: Mobile devices roam in and out qj =P(Jj)=P(Jj|C¯)P(C¯)+P(Jj|C)P(C). (9)
of a small cell. We assume that the arrival of mobile devices
Since the number of caching devices and the number of
caching a coded symbol of a given file and the arrival of
noncaching devices are Poisson distributed with rates m and
devices not caching a symbol of the file are two indepen-
M −m, respectively, the expected fraction of mobile devices
dent arrival processes with interarrival times exponentially
cachingacodedsymboloffileiism/M.Arequestisequally
independent, identically distributed (i.i.d.) with rates λ and
1 likely to come from any mobile device and we have that
λ , respectively, per time unit (t.u.−1). Therefore, the overall
2
arrival rate is λ = λ +λ . Each device has an exponential M −m
1 2 P(C¯)= , (10)
i.i.d. lifetime with rate µ t.u.−1 before departing the small M
cell. Hence, the number of devices in the cell follows a
and
Poisson birth-death process with rate λ/µ [12]. We assume m
P(C)= . (11)
λ = mµ and λ = (M − m)µ and, hence, the expected M
1 2
numberofdevicesandtheexpectednumberofcachingdevices
Using (7),
in the small cell is M and m, respectively. This process can P(J |C¯)=π . (12)
be characterized by an M/M/∞ queueing model where the j j
instantaneous probability of having j caching devices in the If there is a request for file i from a caching device (event C)
cell is [12] thereisatleastonecachingdeviceinthecell.Theprobability
(λ /µ)j mj that there are j caching devices in the cell is the probability
πj = 1 e−λ1/µ = e−m, j ≥0. (7) that j−1 additional caching devices are available, i.e.,
j! j!
Note that this model predicts a nonzero probability that the P(J |C)= mj−1 e−m = jπj, (13)
number of mobile devices in a small cell is larger than M , j (j−1)! m
tot
which violates our initial assumption on the total number of
using (7). Using (10)–(13) with (7) in (9), we get (8).
devices in the area. However, we consider scenarios where
M (cid:28)M , such that this probability is negligibly small.
tot The amount of data that is downloaded from the BS and
Data download: Mobile devices request files at random
fromthemobiledevicesisgivenbythefollowingproposition.
times with the time between requests exponentially i.i.d. with
rateωt.u.−1.Hence,theexpectedtotalrequestrateinthesmall
cell is Mω t.u.−1. A device request file i with probability p , Proposition 2. Provided that there are j mobile devices
i
given by (3). Due to the MDS property, k coded symbols caching a coded symbol of file i in the small cell, then,
i
are sufficient to decode the file [11]. The user requesting for a given request for file i, the fraction of the file that is
content downloads as many coded symbols as possible (up to downloaded from the BS is
ki) from available caching devices and, if additional symbols (cid:40)
are required, these are retrieved from the BS. The equivalent γ(BS) = 1−jαi, if 0≤j <1/αi . (14)
number of files that are downloaded from the BS per t.u. is ij 0, if j ≥1/α
i
h(α)=Mω(cid:88)N (cid:88)∞ p max(cid:26)jα (cid:16)(1−2θ)q −(1−θ)πj(cid:17)+θq ,(1−θ)(cid:18)q −α jπj(cid:19)(cid:27) (23)
i i j j j i
M M
i=1j=0
The fraction of file i that is downloaded from caching devices Therefore, we consider 0.5 ≤ θ ≤ 1, where θ ≥ 0.5 stems
iftherequestcomesfromamobiledevicenotcachingacoded from the fact that the bottleneck is in the downlink.
symbol of file i is We enforce a global average cache size constraint, denoted
(cid:40) by β,1 where
γ(D2D,C¯) = jαi, if 0≤j <1/αi (15) (cid:88)N
ij 1, if j ≥1/αi αi ≤β. (21)
i=1
and if the request comes from a device caching a symbol of
This implies an average cache size constraint per device β¯ =
file i d
βn/M . The optimization problem can be formulated as
tot

(j−1)α , if 1≤j <1/α
 i i minimize h(α) (22a)
γi(jD2D,C) = 1−αi, if j ≥1/αi . (16) αi∈A
0, otherwise (cid:88)N
subject to α ≤β (22b)
i
Proof: We require ki coded symbols to decode file i. If i=1
at the time of a request for file i there are j < ki = 1/αi Wedenotebyα∗ theoptimalcontentallocationresultingfrom
devices caching a symbol of file i in the small cell, k −j (22).
i
symbols are retrieved from the BS. If j ≥k , no symbols are The objective function (22a) can be rewritten as shown
i
downloaded from the BS. Hence, the fraction of file i that is in (23) (at the top of the page) using (14) and (17)–(20).
downloadedfromtheBSisgivenby(14).Followingthesame It is clear that the objective function (23) is a sum of
argument, it is trivial to obtain (15) and (16). piecewise linear functions of α . This allows us to rewrite
i
The expected downlink rate for a content allocation α, the optimization problem in a way that is tractable, using the
denoted by f(α), is given by epigraph formulation [13]. Using (23) and introducing a new
optimization variable t ∈ R, the optimization problem (22)
N ∞ ij
f(α)=Mω(cid:88)(cid:88)p γ(BS)q , (17) can be written equivalently as
i ij j
i=1j=0 (cid:88)N (cid:88)∞
minimize t (24a)
since the amount of data that is downloaded from the BS is ij
the same, irrespective of whether the request comes from a αtiij∈∈AR i=1j=0
caching or a noncaching device. The expected D2D commu- N
(cid:88)
nication rate, denoted by g(α), is given by subject to αi ≤β (24b)
i=1
g(α)=Mω(cid:88)N (cid:88)∞ p (cid:16)γ(D2D,C¯)P(J ,C¯)+γ(D2D,C)P(J ,C)(cid:17), tij +pij(cid:16)(2θ−1)qj +(1−θ)πj(cid:17)αi ≥θpiqj
i ij j ij j M
i=1j=0 (24c)
(18)
jπ
where, using (10)–(13), (15), and (16), t +(1−θ)p jα ≥(1−θ)p q (24d)
ij i i i j
M
γ(D2D,C¯)P(J ,C¯)+γ(D2D,C)P(J ,C) where the constraints (24c) and (24d) arise from the first and
ij j ij j
(cid:40) second term in the max function in (23). Note that we drop
jα P(J ,C¯)+(j−1)α P(J ,C), if 0≤j <1/α
= i j i j i the Mω term in (24a) since it is irrelevant to the solution of
P(Jj,C¯)+(1−αi)P(Jj,C), if j ≥1/αi the optimization problem.
(cid:40)jα (q − πj), if 0≤j <1/α The αi can only take on the discrete values given by (4).
= i j M i . (19) To handle this, we introduce the binary optimization variable
q −α jπj, if j ≥1/α
j i M i b ∈{0,1} and let
i(cid:96)
IV. MINIMIZINGTHEWEIGHTEDCOMMUNICATIONRATE 1 1
α =b ·0+b +b +...+b , ∀i, (25)
We consider the optimization of the content allocation α i i0 i1n i2n−1 in
such that the weighted communication rate such that
n
(cid:88)
h(α)(cid:44)θf(α)+(1−θ)g(α) (20) bi(cid:96) =1, ∀i. (26)
(cid:96)=0
is minimized. Minimizing the expected downlink rate cor-
responds to θ = 1. However, it might be desirable to also 1In practice, a strict cache size constraint per device would be desirable.
Unfortunately, this leads to a complicated optimization problem. Therefore,
limit the D2D communication rate for various reasons, such
to simplify the analysis, we enforce a global average cache size constraint
asdevicebatteryconstraintsandinterferencebetweendevices. instead.
20 40
α α
∗ ∗
αpop αpop
18 αprop αprop
30
16
) )
α α 20
( (
f h
14
10
12
10 0
0.05 0.25 0.5 0.75 1 1 10 20 30 40 50
n/M M
tot
Figure1. Thedownlinkrateversusn/Mtotusingdifferentcontentallocations Figure2. TheweighedcommunicationrateversusM usingdifferentcontent
forM =20andθ=1.Allmarkerscorrespondtosimulateddownlinkrate. allocations for n/Mtot = 1 and θ = 0.75. The markers correspond to
simulateddata.
Finally, we formulate an MILP that is equivalent to (22)
comparisonpurposes.Fortheresults,weconsiderafilelibrary
and (24), minimizing the objective function (24a) under the
with N = 100 files and Zipf parameter σ = 0.7. We assume
constraints (24b)–(24d), (25), and (26),
departure rate µ = 1 t.u.−1 and request rate ω = 1 t.u.−1.
(cid:88)N (cid:88)∞ Note that the optimal content allocation does not depend on
minimize t (27)
ij ω. We also assume that the average cache size constraint per
bαi(cid:96)i∈,t{ij0∈,1R} i=1j=0 device is β¯d =1 file. Monte Carlo simulation results are also
N included in the figures.
(cid:88)
subject to αi ≤β Fig. 1 shows the downlink rate in (17) versus the code
i=1 length n, normalized to the total number of mobile devices
(cid:16) π (cid:17)
tij +pij (2θ−1)qj +(1−θ) j αi ≥θpiqj in the area Mtot, for M = 20. Note that n/Mtot = 1
M
corresponds to maximal spreading [9], i.e., storing a coded
jπ
t +(1−θ)p jα ≥(1−θ)p q symbol of a given file on as many devices as possible. In this
ij i i i j
M
case, the global average cache size constraint (21) becomes
n
α −(cid:88) bi(cid:96) =0, ∀i a strict cache size constraint, i.e., β = β¯d. We recall that
i n−(cid:96)+1 n/M = m/M, using (2) and (6). We observe that the
(cid:96)=1 tot
(cid:88)n downlink rate decreases with n/Mtot for both the optimal
b =1, ∀i and proportional content allocations, i.e., maximal spreading
i(cid:96)
(cid:96)=0 [9] is optimal. This optimality is observed for the weighted
V. NUMERICALRESULTS communication rate in (20) for several values of N, σ, M,
β¯ , and θ. For the popular content allocation, the downlink
d
In Figs. 1–3 we evaluate the downlink rate (17) and the
rate increases with n/M . This is because β decreases as
tot
weighted communication rate (20) for several content allo-
n/M increases and a smaller number of complete files can
cations α. Besides the optimal content allocation α∗, we tot
becachedinthemobiledevices.Forn/M =1,thedownlink
tot
consider a popular content allocation, where each of the (cid:98)β(cid:99)
rateusingthepopularcontentallocationisalmost80%higher
most popular files is cached in n (possibly different) mobile
compared to when using the optimal content allocation.
devices (i.e., using an (n,1) repetition code). The popular
In Fig. 2, we show the weighted communication rate h(α)
content allocation is given by
in (20) versus the expected number of mobile devices in the
αpop =(1(cid:98)β(cid:99),0N−(cid:98)β(cid:99)). (28) small cell, M, for n/Mtot = 1, i.e., maximal spreading, and
θ = 0.75. The expected accumulated cache capacity of the
We also consider a proportional content allocation, devices in the small cell, Mβ¯ , grows with M but so does
d
the expected request rate Mω. An increase of Mβ¯ leads
α =βp, (29) d
prop
to a decrease of h(α), while an increase of Mω leads to
wherep=(p ,...,p ).Weremarkthattheproportionalcon- an increase of h(α). It is observed in the figure that h(α)
1 N
tent allocation typically leads to noninteger k file partitions. increases with M, i.e., the effect of an increase in weighted
i
Albeit impractical, this content allocation is interesting for communication rate due to the increased request rate domi-
23 1
α α
∗ ∗
αpop αpop
21 αprop αprop
19
)
α i 1/2
( α
h
17
1/3
1/4
15 .
.
.
13 0
0.5 0.6 0.7 0.8 0.9 1 1 5 10 15 20 25 30 35
θ i
Figure 3. The weighted communication rate using the various content Figure 4. Various content allocations as a function of file rank for the 35
allocationsasafunctionofθ forn/Mtot=1/6andM =30.Themarkers mostpopularfilesforn/Mtot=1/6,M =30,andθ=0.6.
correspondtosimulateddata.
integer linear program to minimize the weighted sum of the
nates the performance. For small M, h(α) is similar for all downlink rate and the device-to-device communication rate
considered content allocations. However, for larger M, MDS under a global average cache size constraint. We showed that
coded caching with optimized code rates yields significantly optimizedMDScodedcachingmayyieldasignificantlylower
lowerweightedcommunicationrate.Comparedtotheoptimal weighted communication rate compared to when caching
content allocation, the popular content allocation and the (uncoded) the most popular files. Furthermore, our results
proportional content allocation involve roughly a 50% and a show that caching coded symbols of a particular file on all
10% increase, respectively, in the weighted communication devices, i.e., maximal spreading, is optimal.
rate for M =30.
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