Alpha Fair Coded Caching

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by  Apostolos Destounis

Alpha Fair Coded Caching
Apostolos Destounis1, Mari Kobayashi2, Georgios Paschos 1, Asma Ghorbel2
1 France Research Center, Huawei Technologies Co. Ltd., email: fi[email protected]
2Centrale-Supe´lec, France, email: fi[email protected]
Abstract—Theperformanceofexistingcodedcachingschemesis techniques,likeopportunisticscheduling[17],[24],[18],which
sensitivetoworstchannelquality,aproblemwhichisexacerbated servetheuserwiththebestinstantaneouschannelquality.Thus,
when communicating over fading channels. In this paper we
a first challenge is to discover a way to allow coded caching
address this limitation in the following manner: in short-term,
techniquetoopportunisticallyexploitthefadingofthewireless
we allow transmissions to subsets of users with good channel
quality, avoiding users with fades, while in long-term we ensure channel.
7 fairnessacrossthedifferentusers.Ouronlineschemecombines(i) Apartfromthefastfadingconsideration,thereisalsoalong-
1 jointschedulingandpowercontrolforthebroadcastchannelwith termlimitationduetothenetworktopology.Theuserlocations
0 fading, and (ii) congestion control for ensuring the optimal long- mightvary,whichleadstoconsistentlypoorchannelqualityfor
2 term average performance. We restrict the caching operations to
the ill-positioned users. The classical coded caching scheme is
the decentralized scheme of [6], and subject to this restriction we
n designed to deliver equal video shares to all users, which leads
provethatourschemehasnear-optimaloverallperformancewith
a
respect to the convex alpha-fairness coded caching optimization. toill-positionedusersconsumingmostoftheairtimeandhence
J
By tuning the coefficient alpha, the operator can differentiate drivingtheoverallsystemperformancetolowefficiency.Inthe
6 user performance with respect to video delivery rates achievable literature,thisproblemhasbeenresolvedbytheuseoffairness
2 by coded caching. We demonstrate via simulations our scheme’s
superiority over legacy coded caching and unicast opportunistic amonguserthroughputs[24].Byallowingpoorlylocatedusers
] scheduling, which are identified as special cases of our general toreceivelessthroughputthanothers,preciousairtimeissaved
T
framework. and the overall system performance is greatly increased. Since
I
. IndexTerms—Broadcastchannel,codedcaching,fairness,Lya- the sum throughput rate and equalitarian fairness are typically
s punov optimization. the two extreme cases, past works have proposed the use of
c
[ alpha-fairness [15] which allows to select the coefficient α and
I. INTRODUCTION drive the system to any desirable tradeoff point in between
1
v A key challenge for the future wireless networks is the of the two extremes. Previously, the alpha-fair objectives have
0 increasing video traffic demand, which reached 70% of total been studied in the context of (i) multiple user activations
3 mobileIPtrafficin2015[1].Classicaldownlinksystemscannot [17], (ii) multiple antennas [19] and (iii) broadcast channels
7 meet this demand since they have limited resource blocks, and [20].However,herethefairnessproblemisfurthercomplicated
7
therefore as the number of simultaneous video transfers K by the interplay between scheduling and the coded caching
0
increases, the per-video throughput vanishes as 1/K. Recently operation.Inparticular,wewishtoshedlightintothefollowing
.
1
itwasshownthatscalableper-videothroughputcanbeachieved questions: what is the right user grouping and how we should
0
ifthecommunicationsaresynergisticallydesignedwithcaching design the codewords to achieve our fairness objective while
7
1 at the receivers. Indeed, the recent breakthrough of coded adapting to changing channel quality?
: caching [2] has inspired a rethinking of wireless downlink. To address these questions, we study the content delivery
v
Different video sub-files are cached at the receivers, and video over a realistic block-fading broadcast channel, where the
i
X requests are served by coded multicasts. channel quality varies across users and time. In this setting,
r By careful selection of sub-file caching and exploitation we design a scheme that decouples transmissions from coding.
a
of the broadcast wireless channel, the transmitted signal is In the transmission side, we select the multicast user set
simultaneouslyusefulfordecodingatuserswithdifferentvideo dynamically depending on the instantaneous channel quality
requests. Although this scheme–theoretically proved to scale and user urgency captured by queue lengths. In the coding
well–can potentially resolve the future downlink bottleneck, side, we adapt the codeword construction of [6] depending on
several limitations hinder its applicability in practical systems how fast the transmission side serves each user set. Combining
[3]. In this work, we take a closer look to the limitations that with an appropriate congestion controller, we show that this
arise from the fact that coded caching was originally designed approach yields our alpha-fair objective. More specifically, our
for a symmetric error-free shared link. approaches and contributions are summarized below:
If instead we consider a realistic model for the wireless 1) We impose a novel queueing structure which decomposes
channel, we observe that a naive application of coded caching the channel scheduling from the codeword construction.
faces a short-term limitation: since the channel qualities of the Although it is clear that the codeword construction needs
users fluctuate over time and our transmissions need to reach tobeadaptivetochannelvariation,ourschemeensuresthis
all users, the transmissions need to be designed for the worst through our backpressure that connects the user queues
channelquality.Thisisinstarkcontrastwithstandarddownlink and the codeword queues. Hence, we are able to show
that this decomposition is without loss of optimality. wireless channel.
2) We then provide an online policy consisting of (i) admis-
A. Fair file delivery
sion control of new files into the system; (ii) combination
of files to perform coded caching; (iii) scheduling and The performance metric is the time average delivery rate of
powercontrolofcodewordtransmissionstosubsetofusers filestouserk,denotedbyrk.Henceourobjectiveisexpressed
onthewirelesschannel.Weprovethatthelong-termvideo with respect to the vector of delivery ratesrrr. We are interested
delivery rate vector achieved by our scheme is a near in the fair file delivery problem:
optimal solution to the alpha-fair optimization problem
K
(cid:88)
under the specific coded caching scheme [6]. r∗ =argmax g(r ), (1)
k
3) Through numerical examples, we demonstrate the superi- r∈Λ
k=1
ority of our approach versus (a) opportunistic scheduling
where Λ denotes the set of all feasible delivery rate vectors–
with unicast transmissions and classical network caching
clarified in the following subsection–and the utility function
(storing a fraction of each video), (b) standard coded
corresponds to the alpha fair family of concave functions
caching based on transmitting-to-all.
obtained by choosing:
A. Related work (cid:40)(d+x)1−α,α(cid:54)=1
Since coded caching was first proposed [2] and its potential g(x)= 1−α (2)
log(1+x/d),α=1
wasrecognizedbythecommunity,substantialeffortshavebeen
devoted to quantify the gain in realistic scenarios, including for some arbitrarily small d>0 (used to extend the domain
decentralized placement [6], non-uniform popularities [5], [7], of the functions to x = 0). Tuning the value of α changes
and device-to-device (D2D) networks [4]. A number of recent the shape of the utility function and consequently drives the
works replace the original perfect shared link with wireless system performance r∗ to different points: (i) α = 0 yields
channels [9], [10], [11]. Commonly in the works with wireless max sum delivery rate, (ii) α → ∞ yields max-min delivery
channels, the performance of coded caching is limited by the rate [15], (iii) α = 1 yields proportionally fair delivery rate
user in the worst channel condition because the wireless multi- [16].Choosingα∈(0,1)leadstoatradeoffbetweenmaxsum
castcapacityisdeterminedbytheworstuser[13,Chapter7.2]. and proportionally fair delivery rates.
This limitation of coded caching has been recently highlighted The optimization (1) is designed to allow us tweak the
in[11],whilesimilarconclusionsandsomedirectionsaregiven performance of the system; we highlight its importance by an
in [9], [10]. Our work is the first to addresses this aspect by example. Suppose that for a 2-user system Λ is given by the
jointly designing the transmissions over the broadcast channel convex set shown on figure 1.
and scheduling appropriate subsets of users. Different boundary points are obtained as solutions to (1).
Most past works deal with offline caching in the sense that If we choose α = 0, the system is operated at the point that
both cache placement and delivery phases are performed once maximizes the sum r +r . The choice α → ∞ leads to the
1 2
and do not capture the random and asynchronous nature of maximum r such that r = r = r, while α = 1 maximizes
1 2
video traffic. The papers [8], [12] addressed partly the online thesumoflogarithms.TheoperationpointAisobtainedwhen
nature by studying cache eviction strategies, and delay aspects. we always broadcast to all users at the weakest user rate and
In this paper, we explore a different online aspect. Requests use [2] for coded caching transmissions. Note that this results
for video files arrive in an online fashion, and transmissions in a significant loss of efficiency due to the variations of the
are scheduled over time-varying wireless channels. fading channel, and consequently A lies in the interior of Λ.
Online transmission scheduling over wireless channels has We may infer that the point α → ∞ is obtained by avoiding
beenextensivelystudiedinthecontextofopportunisticschedul- transmissions to users with instantaneous poor channel quality
ing[17]andnetworkutilitymaximization[14].Priorworksem- but still balancing their throughputs in the long run.
phasizetwofundamentalaspects:(a)thebalancingofuserrates
B. Transmission model
according to fairness and efficiency considerations, and (b) the
opportunistic exploitation of the time-varying fading channels. ToanalyzethesetoffeasibleratevectorsΛweneedtozoom
Related to our work are the studies of wireless downlink with in the detailed model of transmissions.
broadcast degraded channels; [21] gives a maxweight-type of Caching model. There are N equally popular files
policy and [22] provides a throughput optimal policy based on W ,...,W , each F bits long. The files are available to the
1 N
a fluid limit analysis. Our work is the first to our knowledge base station. User k is equipped with cache memory Z of
k
that studies coded caching in this setting. The new element MF bits, where M ∈[0,N]. Caching placement is performed
in our study is the joint consideration of user scheduling with during off-peak hour, and the goal is to fill the caches up to
codeword construction for the coded caching delivery phase. the memory constraint with selected bits. To this end, we need
to select K caching functions φ : FNF → FMF which map
II. SYSTEMMODELANDPROBLEMFORMULATION the files W ,...,W into the cakche c2ontents 2
1 N
Westudyawirelessdownlinkconsistingofabasestationand
Z (cid:44)φ (W ,...,W ), ∀k =1,...,K.
K users. The users are interested in downloading files over the k k 1 N
set of users with good instantaneous channel qualities, or less
information to a large set that includes users with poor quality.
Decoding. At slot t, each user k observes the local cache
contents Z and the sequence of channel outputs so far
k
y (τ), τ = 1,...,t and employs a decoding function ξ to
k k
determine the decoded files. Let D (t) denote the number of
k
files decoded by user k after t slots. The decoding function ξ
k
is a mapping
Fig.1. Illustrationofthefeasibilityregionanddifferentperformanceoperating ξk :CTslott×CKt×FF2M ×{1,..,N}Kt →FF2Dk(t).
pointsforK =2users.PointAcorrespondstoanaiveadaptationof[2]on
ourchannelmodel,whiletherestpointsaresolutionstoourfairfiledelivery The decoded files of user k at slot t are given by
problem. ξ (yTslott,Z ,hhht,dddt), and depend on the channel outputs and
k k k
states up to t, the local cache contents, and the requested files
The caching functions can be used to cache a few entire files, of all users up to t. A file is incorrectly decoded if it does not
or a small fraction from each file, or even coded combinations belongs to the set ofrequested files. The number of incorrectly
ofsubfiles[2],[8].Itisimportanttonotethatthecachingfunc- decodedfilesarethengivenby|∪ {ξ (t)}\dt|andthenumber
t k k
tions are selected once, without knowledge of future requests, of correctly decoded files at time t is:
and are fixed throughout our system operation.1
C (t)=D (t)−|∪ {ξ (t)}\dt|
Downlink channel model. We consider a standard block- k k t k k
fading broadcast channel, such that the channel state remains Definition 1 (Feasible rate). A rate vectorrrr =(r ,...,r ) is
1 K
constantoveraslotofTslot channelusesandchangesfromone saidtobefeasiblerrr ∈Λifthereexistfunctions([φk],[ft],[ξk])
slot to another in an i.i.d. manner. The channel output of user such that:
k in any channel use of slot t is given by
C (t)
yyy (t)=(cid:112)h (t)xxx(t)+ν (t), (3) rk =limsup kt ,
k k k t→∞
where the channel input xxx ∈ CTslot is subject to the power where the rate is measured in file/slot.
constraint E[(cid:107)xxx(cid:107)2] ≤ PTslot; νk(t) ∼ NC(0,ITslot) are In contrast to past works which study the performance of
additive white Gaussian noises with covariance matrix identity one-shot coded caching [2], [6], [8], our rate metric measures
of size T , assumed independent of each other; {h (t)∈C} the ability of the system to continuously deliver files to users.
slot k
are channel fading coefficients ∼ β2exp(1) independently
k C. Code-constrained rate region
distributed across time and users, with β denoting the path-
k
loss parameter of user k. Finding the optimal policy is very complex. In this paper,
Encoding and transmissions. The transmissions aim to we restrict the problem to specific class of policies given by
contribute information towards the delivery of a specific vector the following mild assumptions:
of file requests ddd(t), where dk(t) ∈ {1,...,N} denotes the Definition 2 (Admissible class policies ΠCC). The admissible
index of the requested file by user k in slot t. Here N is the
policies have the following characteristics:
video library size, typically in the order of 10K. The requests
1) Thecachingplacementanddeliveryfollowthedecentral-
aregeneratedrandomly,andwheneverafileisdeliveredtouser
ized scheme [6].
k, the next request of this user will be for another randomly
2) The users request distinct files, i.e., the ids of the re-
selected file.
quested files of any two users are different.
Ateachtimeslot,thebasestationobservesthechannelstate
hhh(t) = (h (t),...,h (t)) and the request vector up to t, dddt, Sincewerestrictouractionspace,thedeliveryratefeasibility
1 K
constructs a transmit symbol using the encoding function f : region, ΛCC, of the class of policies ΠCC is smaller than the
t
{1,..,N}Kt×CK →CTslot. one for the original problem Λ. However, these restrictions
allow us to come up with a concrete solution approach. Note
xxx(t)=f (cid:0)dddt,hhh(t)(cid:1),
t thattheoptimalcacheandtransmissiondesignpolicyisalready
a very hard problem even in the simple case of broadcast
Finally, it transmits a codeword xxx(t) for the T channel
slot
transmissions with a fixed common rate, and the method in
uses over the fading broadcast channel in slot t . The encoding
[2], [6] are practical approaches with good performance. In
function may be chosen at each slot to contribute information
addition, looking at demand IDs when combining files would
to a selected subset of users J(t) ⊆ {1,...,K}. This allows
be very complex and, because of the big library sizes, is
several possibilities, e.g. to send more information to a small
not expected to bring substantial gains (it is improbable that
two users will make request for the same file in close time
1A reasonable extension is to enable infrequent updates of the caching
placementphase. instances).
III. OFFLINECODEDCACHING
Inthissectionwebrieflyreviewdecentralizedcodedcaching,
first proposed in [6], and used by all admissible policies ΠCC.
We set m = M the normalized memory size. Under the
N
memory constraint of MF bits, each user k independently
cachesasubsetofmF bitsoffilei,chosenuniformlyatrandom
for i = 1,...,N. By letting W denote the sub-file of W
i|J i
storedexclusivelyinthecachememoriesoftheusersetJ,the
cache memory Z of user k after decentralized placement is
k
given by
Z ={W : ∀J ⊆[K],∀J (cid:51)k,∀i=1,...,N}. (4)
k i|J
The size of each sub-file measured in bits is given by Fig.2. DecentralizedcodedcachingforK=3
|Wi|J|=m|J|(1−m)K−|J| (5) given in its cache. Similarly user 2 decodes B1. The same
approachholdsforcodewordB ⊕C tousers{2,3}and
as F → ∞. The above completely determine the caching 3 2
codeword A ⊕C to users {1,3}
functions. 3 1
Oncetherequestsofallusersarerevealed,theofflinescheme • A23 ⊕ B13 ⊕ C12 is intended users 1,2,3. User 1 can
decode A by combining the received codeword with
proceeds to the delivery of the requested files (delivery phase). 23
{B ,C }giveninitscache.Thesameapproachisused
Assuming that user k requests file k, i.e. d = k, the server 13 12
k
for user 2, 3 to decode B , C respectively.
generates and conveys the following codeword simultaneously 13 12
useful to the subset of users J:
IV. BROADCASTINGPRIVATEANDCOMMONMESSAGES
V =⊕ W , (6)
J k∈J k|J\{k} Inthissection,weaddressthequestiononhowthetransmit-
where ⊕ denotes the bit-wise XOR operation. The main idea ter shall convey private and multiple common messages, each
here is to create a codeword useful to a subset of users by intendedtoasubsetofusers,whileopportunisticallyexploiting
exploiting the receiver side information established during the theunderlyingwirelesschannel.Westartbyremarkingthatthe
placement phase. It is worth noticing that the coded delivery channel in (3) for a given channel realization hhh corresponds
with XORs significantly reduces the number of transmissions. to the Gaussian degraded broadcast channel. Without loss of
Compared to uncoded delivery, where the sub-files are sent generality, let us assume h1 ≥···≥hK so that the following
sequentially and the number of transmissions are equal to Markov chain holds.
|J|×|Wk|J\{k}|, the coded delivery requires the transmission X ↔Y1 ↔···↔YK.
of |W |, yielding a reduction of a factor |J|. In a
k|J\{k}
practical case of N >K, it has been proved that decentralized The capacity region of the degraded broadcast channel for
K private messages and a common message is well-known
coded caching achieves the total number of transmissions,
[13]. In this section, we consider a more general setup where
measured in the number of files, given by [6]
the transmitter wishes to convey 2K−1 mutually independent
T (K,m)= 1 (1−m)(cid:110)1−(1−m)K(cid:111). (7) messages, denoted by {MJ}, where MJ denotes the message
tot m intended to the users in subset J ⊆ {1,...,K}. Each user k
On the other hand, in uncoded delivery, the number of trans- must decode all messages {M } for J (cid:51) k. By letting R
J J
missions is given by K(1 − m) since it exploits only local denote the multicast rate of the message M , we say that the
J
cachinggainateachuser.ForasystemwithK =30usersand rate-tuple RRR ∈ R2K−1 is achievable if there exists encoding
+
normalized memory of m = 1/3, the minimum transmissions and decoding functions which ensure the reliability and the
required by uncoded delivery is 20 and that of decentralized rate condition. The capacity region is defined as the supremum
coded caching is 2, yielding a gain of factor 10. of the achievable rate-tuple, where the rate is measured in
In order to further illustrate the placement and delivery of bit/channel use.
decentralizedcodedcaching,weprovideanthree-userexample.
Theorem 1. The capacity region Γ(hhh) of a K-user degraded
Example 1. For the case of K = 3 users in Fig.2, let us Gaussian broadcast channel with fading gains h ≥···≥h
1 K
assume that user 1, 2, 3, requests file A,B,C, respectively. and 2K −1 independent messages {M } is given by
J
After the placement phase, a given file A will be partitioned
into 8 subfiles. R1 ≤log(1+h1α1P) (8)
Codewords to be sent are the following (cid:88) R ≤log 1+hk(cid:80)kj=1αjP k =2,...,K
• A∅, B∅ and C∅ to user 1, 2 and 3 respectively. J 1+h (cid:80)k−1α P
• A2⊕B1 isintendedtousers{1,2}.Oncereceived,user1 J⊆{1,...,k}:k∈J k j=1 j
(9)
decodes A by combining the received codeword with B
2 1
for non-negative variables {α } such that (cid:80)K α ≤1. at each z and the resulting power allocation for user k is
k k=1 k
(cid:26) (cid:27)
Proof: Please refer to Appendix A for the proof.
α∗ = z :[maxu (z)] =u (z) /P (13)
The achievability builds on superposition coding at the k j j + k
transmitterandsuccessiveinterferencecancellationatreceivers.
with λ satisfying
For K =3, the transmit signal is simply given by
(cid:34) (cid:35)
θ˜ 1
x=x +x +x +x +x +x P = max k − . (14)
1 2 3 12 13 123 k λ hk
+
where {x } are mutually independent Gaussian distributed
J V. PROPOSEDONLINEDELIVERYSCHEME
random variables satisfying the power constraint and x de-
J
notes the signal corresponding to the message M intended Thissectionpresentsfirstthequeueddeliverynetworkandits
J
to the subset J ⊆ {1,2,3}. User 3 (the weakest user) feasiblerateregionofarrivalrates,thendescribestheproposed
decodesM˜ ={M ,M ,M ,M }bytreatingalltheother control policy.
3 3 13 23 123
messages as noise. User 2 decodes first the messages M˜ and
3 A. Solution plan
then jointly decodes M˜ = {M ,M }. Finally, user 1 (the
2 2 12
strongest user) successively decodes M˜ ,M˜ and, finally, M . At each time slot t, the controller admits ak(t) files to be
3 2 1
delivered to user k, and hence a (t) is a control variable.2 As
Later in our online coded caching scheme we will need the k
our model dictates, the succession of requested files for user k
capacity region Γ(hhh), and more specifically, we will need to
is determined uniformly at random.
characterize its boundary. To this end,
Queueingmodel.Thebasestationorganizestheinformation
it suffices to consider the weighted sum rate maximization:
into the following types of queues:
(cid:88)
max θJrJ. (10) 1) User queues to store admitted files, one for each user.
rrr∈Γ(hhh)
J:J⊆{1,...,K} The buffer size of queue k is denoted by Sk(t) and
expressed in number of files.
We first simplify the problem using the following theorem.
2) Codeword queues to store codewords to be multicast.
Theorem 2. The weighted sum rate maximization with 2K−1 There is one codeword queue for each subset of users
variables in (10) reduces to a simpler problem with K vari- J ⊆ {1,...,K}. The size of codeword queue J is
ables, given by denoted by Q (t) and expressed in bits.
J
f(α)=k(cid:88)K=1θ˜klog11++hhkk(cid:80)(cid:80)kjkj==−111ααjjPP. (11) Adinecqtihdueeesuefhoinormgwpmoofalinc(yyakfiπ(lte)ps)etrvofoararimdambsliettsh,ien(tfoioi)ltlhotehweiunnsgeirtoqpdueeercauitdeioessnSs:hko((twi))
to combine together files from different user queues to be
where θ˜ denotes the largest weight for user k
k encoded into the form of multiple codewords which represent
θ˜ = max θ . the required broadcast transmissions for the reception of this
k K
K:k∈K⊆{1,...,k} file–these codewords are stored in the appropriate codeword
queues Q (t), (iii) and last it decides the encoding function
Proof: The proof builds on the simple structure of the ca- J
f . (ii) and (iii) are further clarified in the next section.
pacityregion.Wefirstremarkthatforagivenpowerallocation t
of other users, user k sees 2k−1 messages {W } for all J Definition 3 (Stability). A queue S(t) is said to be (strongly)
J
suchthatk ∈J ⊆{1,...,k}withtheequalchannelgain.For stable if
T−1
a given set of {αj}kj=−11, the capacity region of these messages limsup 1 (cid:88)E[S(t)]<∞.
is a simple hyperplane characterized by 2k−1 vertices C eee for T→∞ T
k i t=0
i = 1,...,2k−1, where C is the sum rate of user k in the
k A queueing system is said to be stable if all its queues are
RHS of (9) and eee is a vector with one for the i-th entry and
i stable. Moreover, the stability region of a system is the set of
zero for the others. Therefore, the weighted sum rate seen is
all arrival rates such that the system is stable.
maximized for user k by selecting the vertex corresponding to
the largest weight, denoted by θ˜. This holds for any k. The above definition implies that the average delay of each
Weprovideanefficientalgorithmtosolvethispoweralloca- job in the queue is finite.
tionproblemasaspecialcaseoftheparallelGaussianbroadcast In our problem, if we develop a policy that keeps user
channelstudiedin[23,Theorem3.2].Following[23],wedefine queues SSS(t) stable, then all admitted files will, at some point,
the rate utility function for user k given by be combined into codewords. If in addition codeword queues
QQQ(t) are stable, then all generated codewords will reach their
u (z)= θ˜k −λ (12) destinations, meaning that all receivers will be able to decode
k 1/h +z the admitted files that they requested.
k
where λ is a Lagrangian multiplier. The optimal solution
2Wenotethatrandomfilearrivalscanbedirectlycapturedwiththeaddition
correspondstoselectingtheuserwiththemaximumrateutility ofanextraqueue[14],whichweavoidtosimplifyexposition.
Lemma 3. The region of all feasible delivery rates Λ is the where b =m|I|(1−m)|J|−|I|−1.
J,I
same as the stability region of the system (i.e. the set of In order determine our proposed policy, namely the set of
all demand arrival rates for which there exists a policy that decisions {aaa(t),σ(t),µ(t)} at each slot t, we first characterize
stabilizes the queueing system). the feasible region Λ as a set of arrival rates aaa. We let π
h
Let a =limsup1(cid:80)t−1E[a (t)], denote the time average denote the probability that the channel state at slot t is hhh∈H
k t→∞ t t=0 k where H is the set of all possible channel states. We let Γ(hhh)
number of admitted files for user k. Lemma 3 implies the
denotethecapacityregionforafixedchannelstateh.Thenwe
following Corollary.
have the following
Corollary 4. Solving (1) is equivalent to finding a policy π Theorem 5 (Feasibility region ΛCC). A demand rate vector
such that (cid:88)K is feasible, i.e. aaa¯ ∈ ΛCC, if and only if there exist µ ∈
aπ =argmax g (a ) (15) (cid:80)
k k π Γ(h), σ¯ ∈[0,σ ],∀I ⊆{1,...,K} such that:
h∈H h I max
k=1
(cid:88)
s.t. the system is stable. σ¯ ≥a¯ ,∀k =1,...,K (17)
J k
J:k∈J
Proof: See Appendix B (cid:88)
T µ ≥ b σ¯ ,∀I ∈2K. (18)
slot I J,I J
B. Feasible Region J:I⊆J
Contrary to the offline coded caching in [6], we propose an Constraint (17) says that the service rate at which admitted
onlinedeliveryschemeconsistingofthefollowingthreeblocks. demands are combined to form codewords is greater than
Each block is operated at each slot. the arrival rate, while (18) implies that the long-term average
transmission rate µ for the subset I of users should be higher
1) Admission control: At the beginning of each slot, the I
than the rate at which bits of generated codewords for this
controller decides how many requests for each user,
group arrive. In terms of the queueing system defined, these
a (t) should be pulled into the system from the infinite
k
constraints impose that the service rates of each queue should
reservoir.
be greater than their arrival rates, thus rendering them stable.
2) Routing: The cumulative accepted files for user k are
Theorem 5 implies that the set of feasible average delivery
storedintheadmitteddemandqueuewhosesizeisgiven
rates is a convex set.
by S (t) for k = 1,...,K. The server decides the
k
combinations of files to perform coded caching. The C. Admission Control and Routing
decision at slot t for a subset of users J ⊆ {1,..,K},
In order to perform the utility maximization (15), we need
denoted by σ (t) ∈ {0,1,...,σ }, refers to the
J max to introduce one more set of queues. These queues are virtual,
number of combined requests for this subset of users. It
in the sense that they do not hold actual file demands or bits,
is worth noticing that offline coded caching lets σ =1
J but are merely counters to drive the control policy. Each user
for J = {1,...,K} and zero for all the other subsets.
k is associated with a queue U (t) which evolves as follows:
The size of the queue S evolves as: k
k
U (t+1)=[U (t)−a (t)]++γ (t) (19)
(cid:34) (cid:35)+ k k k k
(cid:88)
S (t+1)= S (t)− σ (t) +a (t) (16) whereγ (t)representsthearrivalprocesstothevirtualqueue
k k J k k
J:k∈J and is given by
If σJ(t) > 0, the server creates codewords by applying γk(t)=arg max [Vgk(x)−Uk(t)x] (20)
offline coded caching explained in Section [] for this 0≤x≤γk,max
subset of users as a function of the cache contents In the above, V > 0 is a parameter that controls the utility-
{Z :j ∈J}. delay tradeoff achieved by the algorithm (see Theorem 6).
j
3) Scheduling: The codewords intended to the subset J of The general intuition here is as follows: Observe that the
users are stored in codeword queue whose size is given number a (t) of admitted demands is the service rate for the
k
by Q (t) for I ⊆ {1,...,K}. Given the instantaneous virtual queues U (t). The control algorithm actually seeks to
I k
channelrealizationhhh(t)andthequeuestate{Q (t)},the optimizethetimeaverageofthevirtualarrivalsγ (t).However,
I k
server performs scheduling and rate allocation. Namely, since U (t) is stable, its service rate, which is the actual
k
at slot t, it determines the number µ (t) of bits per admission rate, will be greater than the rate of the virtual
I
channel use to be transmitted for the users in subset I. arrivals, therefore giving the same optimizer. Stability of all
By letting b denote the number of bits generated for other queues will guarantee that admitted files will be actually
J,I
codeword queue I ⊆ J when offline coded caching is delivered to the users.
performed to the users in J, codeword queue I evolves We present our on-off policy for admission control and
as routing. For every user k, admission control chooses a (t)
k
Q (t+1)=[Q (t)−T µ (t)]++ (cid:88) b σ (t) demands given by
I I slot I J,I J
a (t)=γ 1{U (t)≥S (t)} (21)
J:I⊆J k k,max k k
addition, data from queues Q (t),Q (t) are transmitted.
{2} {2.3}
VI. PERFORMANCEANALYSIS
In thi section, we present the main result of the paper,
that our proposed online algorithm leads to close to optimal
performance for all policies in the class ΠCC:
Theorem 6. Let r¯π the mean time-average delivery rate for
k
user k achieved by the proposed policy. Then
K K (cid:18) (cid:19)
(cid:88) (cid:88) 1
g (r¯π)≥ max g (r¯ )−O
k k rr¯r∈ΛCC k k V
k=1 k=1
T−1
limsup 1 (cid:88)E(cid:110)Qˆ(t)(cid:111)=O(V),
T
T→∞
t=0
where Qˆ(t) is the sum of all queue lengths at the beginning of
time slot t, thus a measure of the mean delay of file delivery.
Fig.3. Anexampleofthequeueingmodelforasystemwith3users.Dashed
lines represent wireless transmissions, solid circles files to be combined and
The above theorem states that, by tuning the constant V,
solidarrowscodewordsgenerated.
the utility resulting from our online policy can be arbitrarily
close to the optimal one, where there is a tradeoff between the
For every subset J ⊆ {1,...,K}, routing combines σ (t)
J
guaranteedoptimalitygapO(1/V)andtheupperboundonthe
demands of users in J given by
total buffer length O(V).
 
σ (t)=σ 1(cid:88)S (t)> (cid:88) bJ,IQ (t). (22) For proving the Theorem, we use the Lyapunov function
J max k F2 I (cid:32) K (cid:33)
  1 (cid:88) (cid:88) 1
k∈J I:I⊆J L(t)= U2(t)+S2(t)+ Q2(t)
2 k k F2 I
D. Scheduling and Transmission k=1 I∈2K
Inordertostabilizeallcodewordqueues,theschedulingand and specifically the related drift-plus-penalty quantity,
resourceallocationexplicitlysolvethefollowingweightedsum defined as: E{L(t+1)−L(t)|S(t),Q(t),U(t)} −
(cid:110) (cid:111)
ratemaximizationateachslottwheretheweightofthesubset VE (cid:80)K g(γ (t))|S(t),Q(t),U(t) . The proposed
k=1 k
J corresponds to the queue length of QJ algorithm is such that it minimizes (a bound on) this
(cid:88) quantity. The main idea is to use this fact in order to compare
µ(t)=arg max Q (t)r . (23)
J J
rrr∈Γ(hhh(t)) the evolution of the drift-plus-penalty under our policy and
J⊆{1,...,K}
two ”static” policies, that is policies that take random actions
We propose to apply the power allocation algorithm in Section (admissions,demandcombinationsandwirelesstransmissions),
IV to solve the above problem by sorting users in a decreasing drawn from a specific distribution, based only on the channel
orderofchannelgainsandtreatingQJ(t)asθJ.Inadition,we realizations (and knowledge of the channel statistics). We can
assumethatthenumberofchannelusesinonecoherenceblock prove from Theorem 4 that these policies can attain every
is large enough such that the decoding error from choosing feasible delivery rate. The first static policy is one such that it
channel codes with rate µ(t) is very small. In this case, no achieves the stability of the system for an arrival rate vector
feedback from the receivers is given. aaa(cid:48) such that aaa(cid:48) +δ ∈ ∂ΛCC. Comparing with our policy, we
deduce strong stability of all queues and the bounds on the
E. Example
queue lengths by using a Foster-Lyapunov type of criterion.
We conclude this section by providing an example of our
In order to prove near-optimality, we consider a static policy
proposedonlinedeliverynetworkforK =3usersasillustrated that admits file requests at rates aaa∗ = argmax (cid:80) g (a )
inFig.3.AtslotttheserverdecidestocombineW requested aaa k k k
1 and keeps the queues stable in a weaker sense (since the
by user 2 with W8 requested by user 2 and to process W4 re- arrival rate is now in the boundary ΛCC). By comparing the
quested by user 1 uncoded. Therefore σ (t)=σ (t)=1
{1,2} {1} drift-plus-penalty quantities and using telescopic sums and
and σ (t) = 0 otherwise. Given this codeword construction,
J Jensen’s inequality on the time average utilities, we obtain the
codeword queues have inputs as described in Table I. In
near-optimality of out proposed policy.
TABLEI
The full proof is in Appendix C.
CODEWORDQUEUESINPUTS.
Queue Input VII. NUMERICALEXAMPLES
Q W ;W
{1} 8,∅ 8|3 In this section, we compare our proposed delivery scheme
W ;W ;W ;W
4|∅ 4|{2} 4|{3} 4|{2,3} with the following two other schemes, all building on decen-
Q W ;W
{2} 1|∅ 1|{3}
Q W ⊕W ;W ⊕W tralized cache placement in (4) and (5).
{1,2} 1|{1} 8|{2} 1|{1,3} 8|{2,3}
three schemes although our proposed scheme provides a gain
compared to the two others.
VIII. CONCLUSIONS
We provided an algorithm to solve the problem of ensuring
fairness in the long term delivery rates in wireless systems
employing decentralized coded caching. Our results imply
that appropriately combining the opportunism arising from the
fading channels with the multicasting opportunities that arise
(a)Sumrate(α=0) (b)Proportionalfairutility(α=1)
from coded caching can mitigate the harmful impact of users
Fig.4. Performanceresultsvsnumberofusersforα=0andα=1
withbadchannelconditionsinstandardcodedcachingschemes
and provide significant increase in the performance of the
• Unicast opportunistic scheduling: for any request, the system.
server sends the remaining (1−m)F bits to the corre-
sponding user without combining any files. Here we only APPENDIXA
exploitthelocalcachinggain.Ineachslottheservesends PROOFOFTHEOREM1
with full power to user
log(1+h (t)P) A. Converse
k∗(t)=argmax k ,
k Tk(t)α We provide the converse proof for K = 3 and the general
where T (t) = (cid:80)1≤τ≤t−1µk(τ) is the empirical average case K >3 follows readily. Notice that the channel output of
k (t−1) user k in (3) for n channel use can be equivalently written as
rate for user k up to slot t.
• Standard coded caching: we use decentralized coded yyyk =xxx+ν˜k, (24)
caching among all K users. For the delivery, non-
opportunistic TDMA transmission is used. The server where ν˜k = √ν(hkk) ∼ NC(0,NkIn) for Nk = h1k and In
sends sequentially codewords VJ to the subset of users identity matrix of size n. Since N1 ≤ N2 ≤ N3, we set
J at the weakest user rate among J: M˜ = ∪ M the message that must be decoded by
(cid:18) (cid:19) k k∈K⊆[k] K
µ (t)=log 1+P min(h (t)) . user k (user k decodes all bits that user k(cid:48) ≥ k decodes) at
J k∈J k rate R˜ . More explicitly, M˜ = {M }, M˜ = {M ,M },
k 1 1 2 2 12
Once the server has sent codewords {VJ}∅=(cid:54) J⊆{1,..,K}, M˜3 ={M3,M13,M23,M123}. By Fano’s inequality, we have
every user is able to decode one file. Then the process is nH(M˜ ) ≤I(M˜ ;Y |M˜ ,M˜ )
repeated for all the demands.  1 1 1 2 3
nH(M˜ ) ≤I(M˜ ;Y |M˜ ) (25)
We consider the system with normalized memory of m = 2 2 2 3
0.6, power constraint P = 10dB, file size F = 103 bits and nH(M˜3) ≤I(M˜3;Y3).
number of channel uses per slot T = 102. We divide users
slot Consider
into two classes of K/2 users each: strong users with β = 1
k
and weak users with βk =0.2. I(M˜ ;Y )=H(Y )−H(Y |M˜ ). (26)
3 3 3 3 3
We compare the three algorithms for the cases where the
objective of the system is sum rate maximization (α=0) and Since nlog(2πeN ) = H(Y |M˜ ,X) ≤ H(Y |M˜ ) ≤
3 3 3 3 3
proportionalfairness(α=1).TheresultsaredepictedinFig.4a H(Y ) ≤ nlog(2πe(P +N )), there exist 0 ≤ α ≤ 1 such
3 3 3
and 4b, respectively. that
Regarding the sum rate objective, standard coded caching
performs very poorly, indicative of the adverse effect of users H(Y3|M˜3)=nlog(2πe((1−α3)P +N3)). (27)
with bad channel quality. It is notable that our proposed
Using (26) and (27) we obtain
scheme outperforms the unicast opportunistic scheme, which
maximizes the sum rate if only private information packets are I(M˜ ;Y )
3 3
to be conveyed. The relative merit of our scheme increases as
=H(Y )−H(Y |M˜ )
thenumberofusersgrows.Thiscanbeattributedtothefactthat 3 3 3
our scheme can exploit any available multicast opportunities. ≤nlog(2πe(P +N ))−nlog(2πe((1−α )P +N ))
3 3 3
Ourresulthereimpliesthat,inrealisticwirelesssystems,coded (cid:18) N +P (cid:19)
caching can indeed provide a significant throughput increase =nlog 3 . (28)
N +(1−α )P
3 3
when an appropriate joint design of routing and opportunistic
transmission is used. Next consider
Regarding the proportional fair objective, we can see that
I(M˜ ;Y |M˜ )=H(Y |M˜ )−H(Y |M˜ ,M˜ ). (29)
the average sum utility increases with a system dimension for 2 2 3 2 3 2 2 3
Using the conditional entropy power inequality in [13] , we From (25), (28), (33) and (38), it readily follows that ∃ 0 ≤
have α ,α ,α ≤1 such that α +α +α =1 and
1 2 3 1 2 3
H(Y3|M˜3)=H(Y2+n3−n2|M˜3)  (cid:16) (cid:17)
≥nlog(22H(Y2|M˜3)/n+22H(n3−n2|M˜3)/n) H(M˜1) ≤log(cid:16)1+ αN11P , (cid:17)
(27) and (30) i=mpnlylog(22H(Y2|M˜3)/n+2πe(N3−N2)) (30) HH((MM˜˜32)) ≤≤lloogg(cid:16)11++ NN23α++2α(Pαα131PP+α2,)P(cid:17).
By replacing H(M˜ ) with (cid:80) R and N with 1 we
nlog(2πe((1−α3)P +N3)) k k∈K⊆[k] K k hk
obtain the result
≥nlog(22H(Y2|M˜3)/n+2πe(N −N )) 
3 2 R ≤log(1+h α P)
 1 (cid:16) 1 1 (cid:17)
equivalent to R +R ≤log 1+h2(α1+α2)P
H(Y2|M˜3)≤nlog(2πe((1−α3)P +N2)). (31) R32+R1132+R23+R123 ≤log(cid:16)1+h1+31(+hαh21α3+P1αP2)P(cid:17),
Since nlog(2πeN2) = H(Y2|M˜2,M˜3,X) ≤ B. Achievability
H(Y |M˜ ,M˜ ) ≤ H(Y |M˜ ), there exists α such that
2 2 3 2 3 2 Superpositioncodingachievestheupperbound.For1≤k ≤
0≤1−α2−α3 ≤1−α3 and 3, generate random sequences unk(mk), mk ∈ [1 : 2nR˜k] each
H(Y2|M˜2,M˜3)=nlog(2πe((1−α2−α3)P +N2)). (32) i.i.d. NC(0,αkP). To transmit a triple message (m1,m2,m3)
the encoder set X = un(m ) + un(m ) + un(m ). For
1 1 2 2 3 3
Using (29), (31) and (32) it follows
decoding:
I(M˜2,M˜3;Y2)=H(Y2|M˜3)−H(Y2|M˜2,M˜3) • Receiver 3 recover m3 from Y3 = un3(m3) +
(un(m )+un(m )+n ) by considering un(m ) +
≤nlog(2πe((1−α )P +N )) (33) 1 1 2 2 3 1 1
3 2 un(m ) as noise. The probability of error tends to zero
−nlog(2πe((1−α −α )P +N )) 2 2 (cid:16) (cid:17)
(cid:18) N2 +(31−α )P2 (cid:19) as n→∞ if R˜3 ≤log 1+ N3+(αα31P+α2)P .
=nlog 2 3 . (34) • Receiver 2 uses successive cancellation. First, it decodes
(1−α2−α3)P +N2 m from Y = un(m ) + (un(m )+un(m )+n )
3 2 3 3 1 1 2 2 2
Last we consider by considering un(m ) + un(m ) as noise. The prob-
1 1 2 2
ability of error tends to zero as n → ∞ if R˜ ≤
I(M˜ ;Y |M˜ ,M˜ ) (cid:16) (cid:17) 3
1=H1 (Y12|M˜32,M˜3)−H(Y1|M˜1,M˜2,M˜3) log(cid:16)1+ N2+(αα31P+α2)P(cid:17). Since N2 ≤ N3 and R˜3 ≤
=H(Y1|M˜2,M˜3)−H(Y1|M˜1,M˜2,M˜3,X) Sloegcon1d+,itNs3u+b(tααr3a1P+ctαs2o)Pffu,nt(hme l)ataenrdcroencdoviteiornunis(msat)isffiroemd.
=H(Y1|M˜2,M˜3)−H(Y1|X) Y˜2 = un(m )+(un(m3)+3n ) by treating2un(2m ) as
2 2 1 1 2 1 1
=H(Y |M˜ ,M˜ )−nlog(2πeN ) (35) noise. The probability of error tends to zero as n→∞ if
1 2 3 1
(cid:16) (cid:17)
Using the conditional entropy power inequality in [13] , we R˜2 ≤log 1+ N2α+2αP1P .
have • Receiver 1 uses successive cancellation twice. First, it de-
codesm fromY =un(m )+(un(m )+un(m )+n )
H(Y2|M˜2,M˜3) by consi3dering u1n1(m13) +3un2(m21) as1 noise2. Th2e pro1b-
=H(Y1+n2−n1|M˜2,M˜3) abil(cid:16)ity of error tends(cid:17)to zero as n → ∞ if R˜3 ≤
≥nlog(22H(Y1|M˜2,M˜3)/n+22H(n2−n1|M˜2,M˜3)/n) log(cid:16)1+ N1+(αα31P+α2)P(cid:17). Since N1 ≤ N3 and R˜3 ≤
=nlog(22H(Y1|M˜2,M˜3)/n+2πe(N2−N1)) (36) log 1+ N3+(αα31P+α2)P , the later condition is satisfied.
Second, it subtracts off un(m ) and decodes un(m )
3 3 2 2
(32) and (36) imply by treating un(m ) as noise. The probability of error
1 1 (cid:16) (cid:17)
nlog(≥2πne(lo(1g(−22αH2(Y−1|αM˜32),PM˜3+)/Nn+2))2πe(N2−N1)) tSeinndcse tNo1z≤eroNa2sannd→R˜2∞≤iflogR˜(cid:16)21≤+lNo2gα+2αP11P+(cid:17)N, 1tαh+2eαP1lPater.
condition is satisfied. Last, it subtracts off un(m ) and
equivalent to 2 2
recover un(m ). The probability of error tends to zero as
H(Y1|M˜2,M˜3)≤nlog(2πe((1−α2−α3)P +N1)). (37) n→∞ if1R˜11≤log(cid:16)1+ αN11P(cid:17).
Let α =1−α −α . Combining the last inequality with (35)
1 2 3 APPENDIXB
we obtain
PROOFOFLEMMA3
(cid:18) (cid:19)
N +α P
I(M˜1;Y1|M˜2,M˜3)≤nlog 1 N1 1 . (38) theDseynsotteemAfko(rt)ustheer knuumpbetor osflofitlte.sAthlasto,hanvoetebetheantaddumeitttoedoutor
restriction on the class of policies ΠCC and our assumption stabilizes the system for that) a rate vector in th δ− interior of
aboutlongenoughblocklengths,therearenoerrorsindecoding ΛCC. We then have the following Lemmas:
the files, therefore the number of files correctly decoded for
Lemma 7 (Static Optimal Policy). Define a policy π∗ ∈ΠCC
user k till slot t is D (t). Since D (t)≤A (t),∀t≥0,∀k =
k k k
that in each slot where the channel states are h works as
1,..,K, if suffices to show that for every arrival rate vector
follows: (i) it pulls random user demands with mean a¯∗,
aaa¯∈ΛCC, there exists a policy in ΠCC for which the delivery k
and it gives the virtual queues arrivals with mean γ¯ = a¯∗
rate vector is rr¯r =aaa¯. k k
as well (ii) the number of combinations for subset J is a
We shall deal only with the interior of ΛCC (arrival rates at
random variable with mean σ¯∗ and uniformly bounded by
the boundaries of stability region are exceptional cases). Take J
σ ,(iii)selectsoneoutofK+1suitablydefinedratevectors
any arrival rate vector aaa¯ ∈ Int(ΛCC). From [14, Theorem max
µl ∈Γ(hhh),l=1,..,K+1withprobabilityψ .Theparameters
4.5] it follows that for any there exists a randomized demand l,hhh
above are selected such that they solve the following problem:
combination and transmission policy πRAND, the probabilities
of which depending only on the channel state realization each
K
slot, for which the system is strongly stable. In addition, max(cid:88)g (a¯∗)
k k
any arrival rate vector can be constructed via a randomized aaa¯
k=1
admission policy. Since the channels are i.i.d. random with (cid:88)
s.t. σ¯ ≥a¯∗,∀k ∈{1,..,K}
a finite state space and queues are measured in files and J k
J:k∈J
bits, the system now evolves as a discrete time Markov chain
K+1
(S(t),Q(t),H(t)), which can be checked that is aperiodic, (cid:88) b σ¯ ≥(cid:88)π (cid:88) ψ µl(hhh),∀I ∈2K
irreducible ad with a single communicating class. In that case, J,I J hhh l,hhh I
J:I⊆J hhh l=1
strong stability means that the Markov chain is ergodic with
finite mean. Then, π∗ results in the optimal delivery rate vector (when all
Further, this means that the system reaches to the set of possible policies are restricted to set Π).
states where all queues are zero infinitely often. Let T[n] be
the number of timeslots between the n−th and (n + 1)−th Lemma 8 (Static Policy for the δ− interior of ΛCC). Define
visit to this set (we make the convention that T[0] is the time a policy πδ ∈ΠCC that in each slot where the channel states
are h works as follows: (i) it pulls random user demands with
slot that this state is reached for the first time). In addition,
let Aˆk[n],Dˆk[n] be the number of demands that arrived and meana¯δk suchthat(aaa¯+δ)∈ΠCC,andgivesthevirtualqueues
randomarrivalswithmeanγ¯ ≤aaa¯+(cid:15)(cid:48) forsome(cid:15)(cid:48) >0(ii)the
were delivered in this frame, respectively. Then, since within k
numberofcombinationsforsubsetJ isarandomvariablewith
this frame the queues start and end empty, we have
meanσ¯δ anduniformlyboundedbyσ ,(iii)selectsoneout
J max
[Aˆk[n]=Dˆk[n],∀n,∀k. ofK+1suitablydefinedratevectorsµl ∈Γ(hhh),l=1,..,K+1
with probability ψδ . The parameters above are selected such
In addition since the Markov chain is ergodic, l,hhh
that:
A(t) (cid:80)N Aˆ [n]
a¯k =tl→im∞ t =Nl→im∞ (cid:80)nN=0 Tk[n] (cid:88) σ¯δJ ≥(cid:15)+a¯δk,∀k ∈{1,..,K}
n=0 J:k∈J
and K+1
D(t) (cid:80)N Dˆ [n] (cid:88) b σ¯δ ≥(cid:15)+(cid:88)π (cid:88) ψ(cid:48) µl(hhh),∀I ∈2K
r¯ = lim = lim n=0 k J,I J hhh l,hhh I
k t→∞ t N→∞ (cid:80)N T[n] J:I⊆J hhh l=1
n=0
Combining the three expressions, rr¯r = aaa¯ thus the result for some appropriate (cid:15) < δ. Then, the system under πδ has
follows. mean incoming rates of aaa¯δ and is strongly stable.
APPENDIXC The proof of the performance of our proposed policy is
PROOFOFTHEOREM6 based on applying Lyapunov optimization theory [14] with the
following as Lyapunov function
From Lemma 3 and Corollary 4, it suffices to prove that
urensduelrtinthgetoimnleinaevperoalgiceyatdhme iqsuseiounesraatreessmtroanxgimlyizsetabthlee adnedsirthede L(ZZZ)=L(S,Q,U)= 1(cid:32)(cid:88)K U2(t)+S2(t)+ (cid:88) Q2I(t)(cid:33).
utility function subject to minimum rate constraints. 2 k k F2
k=1 I∈2K
We first look at policies that take random decisions based
only on the channel realizations. Since the feasibility region Defining its drift as
ΛCC is a convex set (see Theorem ), any point in it can be
achieved by properly time-sharing over the possible control ∆L(Z)=E{L(Z(t+1))−L(Z(t))|Z(t)=Z}
decisions. We focus on two such policies, one that achieves
theoptimalutilityandanotheronthatachieves(i.e.admitsand , using the queue evolution equations and the fact that