Utility Optimization in Heterogeneous Networks via CSMA-Based Algorithms

0.35 MB·English

by  Matthew Andrews

Utility Optimization in Heterogeneous Networks via
CSMA-Based Algorithms
Matthew Andrews Lisa Zhang
Alcatel-Lucent Bell Labs, Murray Hill, NJ Alcatel-Lucent Bell Labs, Murray Hill, NJ
[email protected] [email protected]
Abstract—We study algorithms for carrier and rate allocation intheinteriorofthemacrocellitself,notjustfromneighboring
in cellular systems with distributed components such as a het- macrocells.
erogeneous LTE system with macrocells and femtocells. Existing
LTE networks with small cells represent a hybrid of tradi-
3 work on LTE systems often involves centralized techniques
tional cellular networks and traditional ad-hoc networks. On
1 or requires significant signaling, and is therefore not always
0 applicableinthepresenceoffemtocells.MoredistributedCSMA- the one hand, basestations are running the full LTE protocols
2 based algorithms (carrier-sense multiple access) were developed which allows for the many interference mitigation schemes
n in the context of 802.11 systems and have been proven to be thattheseprotocolsprovide.Ontheotherhand,theplacement
utility optimal. However, the proof typically assumes a single
a of picocells and femtocells in an LTE network is likely to be
transmissionrateoneachcarrier.Further,itreliesontheCSMA
J unstructured and so the interference configurations are likely
collision detection mechanisms to know whether a transmission
3 is feasible. to resemble a typical ad-hoc configuration. As a consequence,

InthispaperwepresentaframeworkforLTEschedulingthat there is no hope for any centralized planning, which is a
]
I is based on CSMA techniques. In particular we first prove that possibility for cellular network interference mitigation tech-
N CSMA-based algorithms can be generalized to handle multiple
niques such as frequency reuse. We therefore need distributed
transmission rates in a multi-carrier setting while maintaining
. algorithms.
s utility optimality. We then show how such an algorithm can be
c implemented in a heterogeneous LTE system where the existing We are interested in scheduling algorithms for a heteroge-
[
Channel Quality Indication (CQI) mechanism is used to decide neous system that consists of a mixture of macrocells and
3 transmission feasibility. small cells. We wish to determine the channels, or carriers,
v used by each basestation as well as the transmission rate on
8 I. INTRODUCTION each channel. This should be done in order to maximize a
5
Interferencemitigationisafundamentalprobleminwireless utilityfunctionassociatedwiththesystem.Althoughanumber
2
networks.Theexactmethodforhandlinginterferencedepends of scheduling algorithms have been proposed in the LTE
4
. on the nature of the network, e.g. whether it is a centrally context, many of them require a non-trivial amount of signal-
1
controlledcellularnetworkoramoreunstructuredad-hocnet- ing among the transmitters. For example, in some algorithms
1
2 work. For cellular networks interference can be mitigated via a scheduling decision is preceded by a calculation of how
1 techniques such as power-control, frequency reuse, and fine- the decision would affect the overall system utility, e.g. by
: grained rate control based on channel-quality measurements exchangingpartialderivativeinformationbetweenneighboring
v
i together with some aspect of central planning. On the other transmitters. This is difficult to support in heterogeneous
X
hand,forad-hocnetworks,especiallythoserunningthe802.11 networks with small cells due to the complexity of setting
r protocol, interference is typically mitigated by a distributed up the necessary communication channels.
a
collision-based random access scheme, perhaps coupled with OurmainresultistoshowthatLTEschedulinginheteroge-
a fairly coarse-grained rate-adaptation procedure. neous networks can be performed using techniques developed
In this paper we are concerned with interference mitiga- in the context of 802.11 networks. These networks utilize
tion in cellular systems with distributed components such a Carrier-Sense Multiple Access (CSMA) protocol, often
as heterogeneous 4G LTE systems that include small cells. enhanced with a Request-to-Send/Clear-to-Send (RTS/CTS)
Small cells are basestations that aim to provide high data rate mechanism [11]. In this setup, transmitters sense the channel
coverageoverasmallhigh-trafficarea.Forexample,picocells before transmitting and proceed only if no conflicting trans-
areownedbyacellularproviderandplacedonpubliclocations missions are active.
such as lamp posts. Alternatively, femtocells are owned by an TheattractivenessofthisframeworkisthatCSMAschedul-
end-userwiththeaimofimprovingcoverageinaprivatehome ing algorithms can achieve optimal throughputs without any
or business. An important property of femtocells is that they explicit signaling. Coordination is implicit in the “collision”
canoperateinClosedSubscriberGroup(CSG)modeinwhich mechanism defined by the CSMA mechanism. In particular,
the basestation restricts the set of mobile terminals that can Jiang and Walrand showed in [10] that such mechanisms can
connect to it. Another effect introduced by femtocells is that be used to achieve any set of feasible throughputs. Among
interferencetomacrocelluserscannowcomefromafemtocell the sequence of papers that followed, Liu et al.[14] presented
utility-optimal algorithms for CSMA networks with a single • In Section VIII we give an overview of past work on
carrier and single transmission rate. Two subsequent papers scheduling andresource allocation incellular and802.11
discussed implementation issues associated with these algo- networks.
rithms [13] and presented an extension to the case of a single
transmission rate on multiple carriers [15]. However, to the
best of our knowledge no previous work has looked at how
such techniques can be applied in the LTE context. II. ABSTRACTMODEL
A number of issues arise if we are to use CSMA-based
scheduling algorithms for LTE. First, LTE networks utilize an
We begin by describing an abstract model that captures the
OFDM physical layer which consists of multiple transmission
notion of multiple transmission rates on multiple carriers and
rates over multiple carriers realized by multiple adjustable
variable-power scheduling. We consider a system in which a
power levels. Hence, in addition to deciding when to transmit
set of transmitters communicate to a set of receivers via a set
on each carrier as in [10], [14], [15], the scheduler now
of links L on a set of carriers C at transmission rates from
also chooses the power level used and the transmission rate.
a set R of positive numbers. Each link is associated with a
Second, LTE networks do not have an explicit carrier sense
transmitter and a receiver, where multiple links may share a
mechanism to detect conflicting transmissions. We need to
commontransmitterbuteachlinkcorrespondsone-to-onewith
build this capability via the existing Channel Quality In-
a receiver.
dication (CQI) mechanism. Lastly, each basestation in an
LTE network typically has its own local scheduler, such as We solve a scheduling problem, i.e. at each time instant
Proportional Fair, that governs the transmissions to the users we specify for every carrier the links that are transmitting
within the cell. We need a mechanism that allows the existing on that carrier together with the associated transmission rates.
local scheduler to work with the CSMA-based algorithm. More precisely, we represent a schedule on a carrier c ∈ C
The main result of this paper is a scheduling algorithm by a vector (r0,...,rL−1) ∈ RL0 where l = |L| and R0 =
for utility maximization in heterogeneous LTE networks. Our R∪{0}. Such a schedule is feasible if it can be realized by
methodology is motivated by the CSMA analysis of [14]. an appropriate power allocation so that every link (cid:96) ∈ L can
We believe our main contribution is in showing that the transmitfromitstransmittertoitsreceiveratrater(cid:96) oncarrier
analysis can be adapted for the case of multiple per-carrier csimultaneously.Aschedulingalgorithmdescribesaschedule
transmissionratesandpowers,and(perhapsmoreimportantly) for every carrier c∈C at every time instant. Note that a link
describing how the algorithms can be implemented using the is allowed to simultaneously transmit on multiple carriers.
CQI mechanism present in LTE. Thisabstractmodelcapturesinterferenceandpowerassign-
We begin with a more abstract version of the algorithm in ments by the notion of a feasibility region, which consists
which details of the interference are abstracted away into a of all valid schedules. In the next two sections we describe
feasibility region for the transmissions. Many of the existing our basic algorithmic framework in this abstract model for
CSMA algorithms implicitly work with this notion. We ini- which we do not concern ourselves with how the system
tially assume an “oracle” that informs a transmitter whether knows whether a schedule is feasible but simply assume an
a potential transmission would disrupt existing transmissions. oracle that indicates whether a potential transmission leads
The later sections of the paper discuss how to realize such to feasibility. As mentioned before, for 802.11 networks this
a scheme in a heterogeneous LTE system in practice. We can be approximately realized by CSMA techniques coupled
structure the paper as follows. with RTS/CTS messages. In later sections we describe how
the oracle can be realized in an LTE heterogeneous network.
• In Section II we present an abstract model of a multi-
Weconsidertheproblemofsystemutilitymaximization.In
carrier system that allows for multiple transmission rates
particular let γ indicate the transmission rate on link (cid:96) on
oneachcarrier.Thismodelassumesthateachtransmitter (cid:96),c
carrierc.ForagivenconcaveutilityfunctionU(·)wewishto
knows whether a potential transmission is feasible.
maximize the aggregate utility over all links, i.e. to maximize
• In Sections III-IV we adapt the Liu et al. single-carrier (cid:80) (cid:80)
U( γ ). Note that for each link the utility function
single-rate utility maximization algorithm for CSMA to (cid:96) c (cid:96),c
is applied to the total transmission rate on the link over all
the abstract framework of Section II to address the
carriers. This coupling between the carriers implies that we
general case of multiple transmission rates on multiple
cannot simply treat each carrier as an isolated system.
carriers.
• In Sections V and VI we give a concrete model for het- A formal version of the optimization problem is given
erogeneous LTE networks with small cells, in which we below. Consider a schedule m ∈ N where N is the set
c c
address power level and interference directly instead of of feasible schedules on carrier c. If π ∈ [0,1] indicates
m
viathenotionofafeasibilityregionfortransmission.We the fraction of time that m occurs and r indicates the
(cid:96),m
also discuss practical issues such as CQI-based collision transmission rate on (cid:96) under schedule m, then γ can be
(cid:96),c
detection and incorporating a local scheduler. viewed as a weighted sum of r where π serves as the
(cid:96),m m
• In Section VII we present simulation results. weight. Throughout the paper, notations such as (cid:126)γ indicate a
Algorithm III.1 RANDACC((cid:104)(cid:96),r,c(cid:105), λ, µ, T) Algorithm III.2 MMUO
• (cid:96): link for each time frame f do
• r : rate for each link (cid:96), carrier c and rate r do
• c: carrier RANDACC((cid:104)(cid:96),r,c(cid:105), λ(cid:96),r,c[f],µ(cid:96),r,c[f], f)
• λ: channel access rate for each (cid:96) do
• µ: expected transmission duration update service received S(cid:96)[f]
• T : time frame update virtual queue q(cid:96)[f +1] according to (2)
for each (cid:96), c and r do
t← beginning of T updateλ(cid:96),r,c[f+1]andµ(cid:96),r,c[f+1]accordingto(3)
while t∈T do
x← randomly drawn from Exp(λ)
t←t+x
The algorithm MMUO (multi-carrier multi-rate utility op-
if (cid:104)(cid:96),r,c(cid:105) leads to config. in Nc then timization) approximates a solution to (1) as follows (see
x← randomly drawn from Exp(1/µ) AlgorithmIII.2).Timeisdividedintoframesoffixedduration.
(cid:96) transmits on c at r during (t,t+x] Duringeachframef,eachpotentialtransmission(cid:104)(cid:96),r,c(cid:105)calls
t←t+x the RANDACC routine with parameters λ(cid:96),r,c[f] and µ(cid:96),r,c[f].
At the end of each frame, link (cid:96) calculates the service
received during the frame (denoted by S [f]), and updates
(cid:96)
vector (γ(cid:96),c)(cid:96)∈L,c∈C. We wish to solve: a virtual queue size parameter (denoted by q(cid:96)[f]) as follows.
max f1((cid:126)γ)=(cid:88)U((cid:88)γ(cid:96),c) (1) q(cid:96)[f +1]=(cid:104)q(cid:96)[f]+b[f]·(cid:16)U(cid:48)−1(q(cid:96)[f]/V)−S(cid:96)[f](cid:17)(cid:105)qmax (2)
(cid:96)∈L c qmin
s.t. γ ≤ (cid:88) r π ∀(cid:96),c In the above equation, b is a step size function that satisfies
(cid:96),c (cid:96),m m
property(A1)(definedlater),q andq areboundsonthe
(cid:88) πmm∈=Nc1 ∀c. Tvihretuaplosqiutievueepsairzaem,eatnedr V[x]cqqommnainxmtroa=xlsmthien(amqcmicnuaxra,cmyaoxf(qtmheina,lxg)o)-.
m∈Nc rithm.
We can view this as an extension of the formulation of [15] The values of λ [f] and µ [f] stay unchanged during
(cid:96),r,c (cid:96),r,c
thatallowsformultiratetransmissions(andimplicitlyvariable eachframef,andareupdatedtoλ [f+1]andµ [f+1]
(cid:96),r,c (cid:96),r,c
transmission powers). at the end of frame f, so that
In reality the sets L and N can change over time due to
c
λ [f +1]·µ [f +1]=exp(r·q [f +1]). (3)
mobility. However, as in common in the literature we assume (cid:96),r,c (cid:96),r,c (cid:96)
that this happens on a slow enough timescale that it makes
Asweshallseein(4)theperformanceof RANDACC depends
sensetosolvetheutilitymaximizationproblemforthecurrent
on the product of λ and µ. The choice of this product is
network configuration.
explained in the proof of Theorem 4.
III. MULTI-CARRIER,MULTI-RATESCHEDULING IV. ANALYSIS
ALGORITHM
WenowshowthattheMMUOalgorithmleadstoanoptimal
We first describe a routine RANDACC (in Figure Algo- solutionto(1).Theproofbuildsuponthefollowingoptimality
rithm III.1), a continuous-time random access algorithm that properties of RANDACC that were shown in [10] for the
determines when a transmission will take place. We use special case of one carrier and 0/1 transmission rates.
(cid:104)(cid:96),r,c(cid:105) to denote the transmission on link (cid:96) ∈ L on carrier
c ∈ C at rate r ∈ R. Each transmission (cid:104)(cid:96),r,c(cid:105) is associated A. RANDACC for 0/1 Rates
with two parameters, λ(cid:96),r,c the channel access rate and µ(cid:96),r,c Suppose each link (cid:96) ∈ L calls RANDACC with parameters
which represents the expected transmission duration. After an (λ ,µ ) over a sufficiently long time frame. Let (cid:126)λ=(λ )
(cid:96) (cid:96) (cid:96) (cid:96)∈L
exponentially distributed waiting period with mean 1/λ(cid:96),r,c, and let µ(cid:126) = (µ(cid:96))(cid:96)∈L. Let m(cid:126)λ,µ(cid:126)(t) be the schedule at time t
RANDACC checks whether (cid:104)(cid:96),r,c(cid:105) leads to a valid schedule under the RANDACC routine. Recall r(cid:96),m is the transmission
in Nc at that time instant. If yes, the transmission starts rate on link (cid:96) under schedule m. In this case r(cid:96),m ∈{0,1}.
immediately and lasts for an exponentially distributed time
periodwithmeanµ .Notethataninvalidscheduleincludes Lemma 1 ([12]). The sequence of schedules m(cid:126)λ,µ(cid:126)(t) for t≥
(cid:96),r,c
the situation in which (cid:96) conflicts with itself, namely (cid:96) already 0 form a continuous time reversible Markov Chain with the
transmits on c, or (cid:96) conflicts with another link on c, both of following stationary distribution.
owpheircahteasriencaacpotunrteinduobuysNmca.nNneort,etwaolsolintkhsatmsaikneceaRscAhNedDuAliCnCg πm(cid:126)λc,µ(cid:126)c = (cid:80)Π(cid:96)Π:r(cid:96),m=1λ(cid:96)·λµ(cid:96)·µ ∀m,
decision simultaneously with zero probability and therefore m(cid:48) (cid:96):r(cid:96),m(cid:48)=1 (cid:96) (cid:96)
they make conflicting decisions with zero probability. where Π (·)=1.
(cid:96)∈∅
Further, the resulting link throughput γ(cid:126)λ,µ(cid:126) = where (cid:126)q∗ and(cid:126)γ∗ are such that ((cid:126)γ∗,(cid:126)πq(cid:126)∗) is the solution to the
(cid:96)
(cid:80)mπm(cid:126)λc,µ(cid:126)cr(cid:96),m is optimal for every link (cid:96) ∈ L in the following convex optimization problem over (cid:126)γ and (cid:126)π.
following sense. We say that a link throughput vector (cid:126)γ is
max f ((cid:126)γ,(cid:126)π)=
feasible if there exist π such that γ <(cid:80) r π . 2
m (cid:96) m∈Nc l,m m V (cid:88)U((cid:88)γ )−(cid:88) (cid:88) π logπ (7)
(cid:96),c m m
Lemma 2 ([10]). For any feasible link throughput vector (cid:126)γ,
there exists (cid:126)λ and µ(cid:126) such that (cid:96)∈L (cid:88)c c m∈Nc
s.t. γ ≤ r π ∀(cid:96),c
(cid:96),c (cid:96),m m
γ(cid:96) ≤γ(cid:96)(cid:126)λ,µ(cid:126). m∈Nc
(cid:88)
B. MMUO πm =1 ∀c.
Formultipletransmissionratesonmultiplecarriers,thegen- m∈Nc
eralization from Lemma 1 to Corollary 3 is straight-forward. Further, if (cid:126)γ† is the optimal solution to (1), then
This is because RANDACC that runs on one carrier does not
|f ((cid:126)γ∗)−f ((cid:126)γ†)|≤|C|log|∪ N |/V (8)
interferewiththatonadifferentcarrier.Further,eachrate-link 1 1 c c
pair can be treated as a distinct link. For a particular carrier
c, let vectors (cid:126)λ =(λ ) and µ(cid:126) =(µ ) .
c (cid:96),r,c (cid:96)∈L,r∈R c (cid:96),r,c (cid:96)∈L,r∈R
The assumptions of Theorem 4 are:
Recall N is the set of feasible schedules on carrier c.
c (A1) (cid:80)∞ b[f]=∞ and (cid:80)∞ b2[f]<∞.
Corollary 3. The schedule sequence m(cid:126)λc,µ(cid:126)c(t) for t≥0 is a (A2) If p(cid:126)fo=0∈(cid:60)L is a solutionft=o0
continuous time reversible Markov Chain with the following +
(cid:88) (cid:88)
stationary distribution. U(cid:48)−1(p /V)− r πp(cid:126) =0 ∀(cid:96)∈L
(cid:96) (cid:96),m m
πm(cid:126)λc,µ(cid:126)c = (cid:80) Π(cid:96):r(cid:96)Π,m>0λ(cid:96),r(cid:96)λ,m,c·µ(cid:96),·r(cid:96)µ,m,c ∀m∈Nc (4) then q ≤po ≤qc mf∈oNrcall (cid:96)∈L.
n∈Nc (cid:96):r(cid:96),n>0 (cid:96),r(cid:96),n,c (cid:96),r(cid:96),n,c min (cid:96) max
The above expression shows that the stationary distribution The parameters b[·] will only be used in the analysis, not in
only depends on the product of the parameters λ and µ. thealgorithmitself.Inaddition,theparametersqmin andqmax
Whenever MMUO invokes RANDACC, it does so with λ and areunderourcontrol.Henceforanyprobleminstancewecan
µ parameters that are set according to (3). For any vector of make sure that Assumptions (A1) and (A2) hold.
virtual queues (cid:126)q = (q ) , we denote by πq(cid:126) the resulting Proof: There are several steps in the proof. In the first
(cid:96) (cid:96)∈L
distribution on Nc from the RANDACC routine. From (4) we two steps we follow the framework of [14] to show that
have, the dynamics of MMUO can be captured by a system of
(cid:16) (cid:17) differential equations. In the third step we must deviate from
(cid:80)
exp r ·q
πq(cid:126) = (cid:96):r(cid:96),m>0 (cid:96),m (cid:96) ∀m∈N (5) the approach of [14] in order to handle multiple transmission
m (cid:80) (cid:16)(cid:80) (cid:17) c rates on multiple carriers.
exp r ·q
m(cid:48)∈Nc (cid:96):r(cid:96),m(cid:48)>0 (cid:96),m(cid:48) (cid:96) We begin by replacing the discrete time frames of MMUO
The resulting link throughput is therefore, with a more convenient continuous interpolation. For notation
(cid:88) weusesquarebrackets[·]indexedwithintegersf fordiscrete
γq(cid:126) = πq(cid:126) r ∀(cid:96),c (6)
(cid:96),c m (cid:96),m sequencesdefinedonframesf,andweuseroundbrackets(·)
m∈Nc indexedwithrealnumberstforacontinuousscaledversionof
For utility optimization, our goal is to show that the virtual time. For the discrete time sequence of virtual queue vectors
queuesunderMMUOconvergetoavector(cid:126)q∗sothattheabove (cid:126)q[f] for integral frames f = 1,2,..., we define as follows
link throughput under (cid:126)q∗ maximizes the utility as defined its continuous interpolation (cid:126)q(t) for all real positive numbers
in (1). Note that for this problem we cannot treat each carrier t. We also define a continuous version S (t) for the discrete
(cid:96)
in isolation since the throughput of a link is aggregated service sequence S [f]. For f =1,2,..., let t =(cid:80)f b[i].
(cid:96) f i=1
over all carriers. Note also that the optimization problem
For t∈[t ,t ), let
f f+1
(7) in Theorem 4 differs from that of (1) in its objective
t −t t−t
function, but shares the same constraints. The motivation of q (t) = q [f]· n+1 +q [f +1]· n
(cid:96) (cid:96) t −t (cid:96) t −t
this reformulation is to obtain a more useful set of KKT n+1 n n+1 n
conditions. Further, Theorem 4 will state that the optimal S(cid:96)(t) = S(cid:96)[f]
values of the two objective functions can be arbitrarily close.
(In other words the continuous time process t is created from
The main result of this section is captured in the following
the discrete time frames scaled by the intervals [t ,t ).)
f f+1
theorem. It relies on two standard but technical assumptions
The following lemma is shown in [14] and says that the
(A1) and (A2) which we detail below.
continuous sequence (cid:126)q(t) converges to the solution of the
Theorem4. Underassumptions(A1)and(A2),foranyinitial system of stochastic differential equations (9) which can be
condition (cid:126)q[0], MMUO converges in the following sense. viewed as a continuous version of (2). The equations are
stochastic due to the term S (t) which is the result of the
lim (cid:126)q[f]=(cid:126)q∗ (cid:96)
stochastic process of MMUO.
f←∞
Lemma5([14]). Letp(cid:126)∗bethesolutiontothefollowingsystem Hence the KKT conditions for (7) are:
of differential equations with variable p(cid:126)=(p)(cid:96)∈L. VU(cid:48)((cid:88)γ )=ν ∀(cid:96),c (11)
(cid:96),c (cid:96),c
(cid:104) (cid:105) c
p˙ = U(cid:48)−1(p /V)−S (t) ·1 , (9) (cid:88)
(cid:96) (cid:96) (cid:96) p(cid:96)∈[qmin,qmax] −1−logπm+ r(cid:96),mν(cid:96),c−ηc =0 ∀m∈Nc,∀c(12)
(cid:96)
(cid:88)
where1p(cid:96)∈[qmin,qmax] isanindicatorvariableforwhetherp(cid:96) is γ(cid:96),c ≤ r(cid:96),mπm ∀(cid:96),c (13)
intherangeof[q ,q ].Fixanytimeinstantτ.Ifp(cid:126)∗(τ)=
min max m∈Nc
(cid:126)q(τ), then lim sup (cid:107)p(cid:126)∗(t)−(cid:126)q(t)(cid:107)=0. (cid:88)
τ→∞ t∈[τ,τ+T] ν ×(γ − r π )=0 ∀(cid:96),c (14)
(cid:96),c (cid:96),c (cid:96),m m
The second step of the proof says that any fixed point m∈Nc
of the stochastic system (9) will also be a fixed point of ν(cid:96),c ≥0 ∀(cid:96),c (15)
(cid:88)
an associated deterministic system of equations. Note that π −1=0 ∀c (16)
m
the RANDACC routine may not converge to the stationary m∈Nc
distribution of (4), or equivalently (5), with the given λ and
Inequalities (11) and (12) state the gradient of the Lagrangian
µ parameters within each frame before the parameters are
L((cid:126)γ,(cid:126)π : (cid:126)ν,(cid:126)η) is zero with respect to the variables (cid:126)γ and
updated to their respective new values for the next frame.
(cid:126)π. Inequalities (13), (14) and (15) state that the first set of
HencetheserviceS (t)intheabovedifferentialequationsisa
(cid:96) constraints of (7) hold and has zero duality gap. The last
stochastic quantity. In other words, when MMUO invokes the
equality (16) states the second set of constraints of (7) hold.
wRoAuNldDAcoCnCverroguetitnoe,itisfleoancgh-tefrrammaevferaisgesu(cid:80)ffici(cid:80)ently lonrg, S·(cid:96)π(tq(cid:126)) Weintroduceanewvariablep(cid:126)=(p)(cid:96)∈L andsettheprimary
c m∈Nc (cid:96),m m variables via
and we could replace the stochastic term S (t) in (9) by its
(cid:96)
long term average. This would result in the following system π =πp(cid:126) as in (5) ∀m
m m
(10). Unfortunately, in reality, the frame f may not be long
enough. However, the following result still allows us to say and the dual variables via
that systems (9) and (10) are closely related, regardless of the
ν = p ∀(cid:96),c
convergenceofRANDACC.Asweshallsee,(10)alsoprovides (cid:96),c (cid:96) (cid:32) (cid:33)
a connection to solving (7) via the KKT conditions. (cid:88) (cid:88)
η = log exp( r q ) −1 ∀c.
c (cid:96),m (cid:96)
Lemma 6 ([14]). Every limit point of system (9) is almost m∈Nc (cid:96)
always a fixed point of the following system. We can see that the KKT conditions (12), (15) and (16) are
easily satisfied due to the definition of πp(cid:126) in (5) and as long
m
p˙(cid:96) =(cid:104)U(cid:48)−1(p(cid:96)/V)−(cid:88) (cid:88) r(cid:96),m·πmp(cid:126) (cid:105)·1p(cid:96)∈[qmin,qmax], asWp(cid:126)e∈n(cid:60)exL+t.aim to satisfy the remaining KKT conditions (11),
c m∈Nc
(13)and(14)viathesubgradientmethod.From(11),(14)and
∀(cid:96)∈L, (10)
the fact that ν is set to p , we wish to have,
(cid:96),c (cid:96)
where πmp(cid:126) is defined in (5). p(cid:96)×(U(cid:48)−1(p(cid:96)/V)−(cid:88) (cid:88) r(cid:96),mπmp(cid:126) )=0. ∀(cid:96)
In the third step of the proof we show that the system c m∈Nc
(10) leads to a solution of the optimization problem (7). (It If this does not hold (and hence we are not yet at optimality),
effectively solves the problem via the subgradient method). the standard subgradient method updates the p(cid:96) according to
For this step we must deviate from the analysis of [14] in the following system of differential equations.
order to handle the multi-carrier and multi-rate aspects of (7). (cid:88) (cid:88)
p˙ =U(cid:48)−1(p /V)− r πp(cid:126) . (17)
In particular, the Lagrangian of (7) is given by: (cid:96) (cid:96) (cid:96),m m
c m∈Nc
Due to the convexity of the problem (7) the system (17) will
L((cid:126)γ,(cid:126)π :(cid:126)ν,(cid:126)η)
(cid:32) (cid:33) eventually converge to a fixed point, p(cid:126)∗. Note that the system
= (cid:88) V ·U((cid:88)γ(cid:96),c)−(cid:88)ν(cid:96),cγ(cid:96),c (17) is identical to (10) as long as p(cid:96) ∈ [qmin,qmax] for all
(cid:96) ∈ L. Due to assumption (A2) and the definition of ν ,
(cid:96)∈L c c (cid:96),c
(cid:32) (cid:33) the solution to the dual of (7) without the constraint p ∈
(cid:96)
(cid:88) (cid:88) (cid:88) (cid:88)
+ ν(cid:96),c r(cid:96),mπm− πmlogπm [qmin,qmax] falls in the range of [qmin,qmax] and is therefore
c (cid:96)∈L m∈Nc m∈Nc equivalent to the fixed point p(cid:126)∗ of the system (17).
(cid:32) (cid:33) With p(cid:126)∗ chosen, KKT condition (11) is satisfied. We now
(cid:88) (cid:88)
− ηc πm−1 . set
c m∈Nc γ = (cid:88) πp(cid:126)∗r ∀(cid:96),c,
(cid:96),c m (cid:96),m
.
m∈Nc
which satisfy the remaining conditions of (13) and (14). of feasibility regions. For basestation i, let U be the set of
i
Finally, since associated users. To abuse notation, we also use U to denote
i
the set of links that are incident to i as there is a one-to-one
f ((cid:126)γ∗,(cid:126)π∗) ≥ f ((cid:126)γ†,(cid:126)π†)
2 2 correspondence between the users and links.
f ((cid:126)γ∗) ≤ f ((cid:126)γ†)
1 1 The maximum transmit power p for basestation i is given
i
and the entropy (cid:80)mπmlogπm ≤ log|∪c Nc|, the proof of and fixed. The scheduling problem is how to distribute pi
Theorem 4 is complete. among the resource blocks c ∈ C and among the users
We conclude by briefly summarizing in what sense we in Ui. Let pi,c(t) be the power allocation of pi on re-
have shown that MMUO is optimal. We have shown that source block c at time t; let pi,c,j(t) be the allocation
(cid:80)
an appropriate continuous interpolation of the virtual queue o(cid:26)f pi,c(t) on user j ∈ Ui. Note cpi,c(t) ≤ pi, and
dynamics approaches in the limit a vector p(cid:126)∗ that defines an pi,c,j(t) = pi,c(t) for one j ∈Ui . That is, p is
p (t) = 0 for j(cid:48) (cid:54)=j i,c
optimal dual solution of (7) (via the KKT conditions). Via i,c,j(cid:48)
allocated entirely to one chosen user j ∈U .
(3) the optimal virtual queue sizes determine channel access i
Powersettingsandtransmissionratesarerelatedthroughthe
parameters λ and µ for which the corresponding link
(cid:96),r,c (cid:96),r,c
channel quality information (CQI). CQI values are defined on
throughputsprovideanoptimalprimalsolutiontoproblem(7).
pairs of links and resource blocks. During every time slot t,
V. AMORECONCRETEMODEL:HETEROGENEOUSLTE the values of CQI (t) for all c∈C and (cid:96)∈U are reported
c,(cid:96) i
SYSTEM to basestation i. We assume that each basestation has perfect
In this section we present a more concrete model for the CQI reporting.
scheduling problem so that it more closely matches resource Let rc,(cid:96)(t) be the transmission rate along link (cid:96) on carrier
allocation in LTE heterogeneous networks. c during time slot t. Specifically, for link (cid:96)=ij between the
We begin with a brief system description. We consider basestation i and the associated user j, we define
downlink transmissions from a set of basestations to a set of
r (t) = w ·F(p (t)·CQI (t)) (18)
mobile users in a time-slotted system. We assume an OFDM- c,(cid:96) c i,c,j c,(cid:96)
based air interface in which the spectrum is divided into a set CQI (t) = gijc(t) (19)
of carriers called resource blocks (RBs), each of which can c,(cid:96) Nc+(cid:80)i(cid:48)(cid:54)=ipi(cid:48)cgi(cid:48)jc(t)
be scheduled separately. For example a 20MHz LTE system
In (19), g represents the path loss between i and j on
is typically divided into 100 resource blocks. In the time ijc
resource block c, and N is the background noise on c. Both
dimension a time slot corresponds to a Transmission Time c
g and N depend on c since radio propagation conditions
Interval (TTI) which has a typical duration of 1ms in an LTE ijc c
and background interference may be different on different
system.
frequencies. The product of p and CQI is commonly
We consider a heterogeneous network in which the bases- i,c,j c,(cid:96)
referred to as signal-to-interference-plus-noise ratio, SINR.
tations are divided into two classes, namely macrocells and
(We can therefore think of CQI as the SINR for a unit
femtocells.(Foreaseofdescriptionweusetheterms“macros” c,(cid:96)
power transmission.) In (18), w is the bandwidth of resource
and “femtos”. However, our discussions also apply directly c
block c and F(·) represents spectral efficiency as a function
to networks with picocells.) Macros typically have a much
of SINR. For example F(·) could be a suitably discretized
higher max transmit power than femtos, since macrocells
version of the Shannon function log(1+x). We assume that
provide wide-area coverage, whereas femtocells (which may
F(·) is such that r is always a member of a discrete set
be privately owned) provide focused coverage in one specific c,(cid:96)
R∪{0}.
location, e.g. a house or apartment. At any time instant
The primary scheduling decision is to determine the power
each mobile user associates with one basestation. Each macro
levels p . In the literature this problem is sometimes known
acceptsanassociationwithanymobileuser.Afemtohowever i,c
as inter-cell interference coordination (ICIC). The secondary
may be in “Closed Subscriber Group” mode (CSG) and only
scheduling decision is to allocate p to the user-level power
accept an association with a small subset of users. We remark i,c
p . Typically, each basestation in an LTE network has its
that femtos have two notable effects that are departures from i,c,j
ownlocalschedulerforuser-levelallocation,inwhichcasethe
traditional cellular networks. First, they may create strong
scheduling freedom is at the inter-cell level. For concreteness
interferencetoamacrocellfromwithinthecellitself,whereas
we assume the local scheduler uses the Proportional Fair (PF)
in a macro-only network interference to a cell mostly comes
algorithm. In the following section we give details on how
from outside that cell. Second, a mobile user may not be able
to allocate p and p . For user-level allocation p , we
toassociatewiththebasestationwiththestrongestsignalifthe i,c i,c,j i,c,j
consider two cases depending on whether a local scheduler
basestationisafemtoinCSGmodeandcannotassociatewith
exists.
the user. We assume that each mobile user associates with the
basestation for which the received signal is strongest, among
VI. IMPLEMENTATION
those that the user is able to associate with.
Unlike in the abstract model, we address basestation power In Section III we described the utility-optimal MMUO
allocation and interference directly instead of via the notion algorithm for the abstract model. In this section we present
MMUO-basedheuristicsfortheLTEresourceallocationprob- are set then which user is chosen by PF does not affect the
lem in heterogeneous networks. We address a number of interference experienced in other cells.)
issues. First, scheduling decisions need to be made in slotted The PF algorithm works as follows. For each link (cid:96) ∈ U
i
time rather than in continuous time as in MMUO. Second, basestation i maintains an estimate R of the recent average
(cid:96)
schedulingdecisionsareaboutsettingpowerlevelsratherthan transmit rate on link (cid:96), and allocates power p exclusively to
i,c
transmission rates as in MMUO. Third, we discuss how to the link (cid:96) that maximizes the ratio r˜ (t)/R , where r˜ (t)
c,(cid:96) (cid:96) c,(cid:96)
incorporatealocalschedulersuchasProportionalFair.Fourth, is the nominal rate if user j has power allocation p (t) =
i,c,j
perhaps most significantly, we show CQI-based methods for p (t). Again from (18), we have
i,c
feasibility detection. This replaces the feasibility oracle and (cid:0) (cid:1)
r˜ (t)=w ·F p (t)·CQI (t) for link (cid:96)=ij (21)
the CSMA collision detection mechanism. Lastly, since the c,(cid:96) c i,c,j c,(cid:96)
basestations are divided into two classes, macro and femto, After each scheduling decision R is updated for each link
(cid:96)
interference can be reduced by not having every basestation according to an exponential filter. This addresses the third
compete on every resource block. issue.
We begin with a basic heuristic that bypasses the last
B. Methods for Implementing the Feasibility Oracle
two issues. We then describe three methods through which
feasibility can be detected in practice. We conclude with a We now examine options for the only part of the algo-
modified heuristic in which macros and femtos have priority rithm that requires coordination among basestations, namely
on different sets of resource blocks. feasibilitydetection.Sinceresourceblockpowerassignmentis
typicallydoneonaslowertimescalethanindividualtimeslots,
A. Basic Heuristic weareinterestedindeterminingwhetherasetoftransmissions
will be feasible over multiple timeslots. In particular, we do
OurbasicheuristicworksverymuchinthespiritofMMUO.
not want to declare a transmission feasible if this is only true
To address the first issue regarding slotted time each frame
for a single timeslot due to fast fading.
now consists of an integral number of time slots. When the
We discuss multiple mechanisms which all use techniques
subroutine RANDACC is called with parameters λ and µ, the
that have been proposed in the standardization process for
time between transmission attempts (resp. the transmission
heterogeneousnetworks(e.g.[1]).Ourinitialmechanismsuse
period) is drawn from a geometric distribution with mean
the existing CQI channel with one extra piece of information
1/λ (resp. mean µ). One problem is that two links may make
which we call the activity indicator. We also allow for a
decisionsduringthesametimeslot.Wecansetthe1/λvalues
basestation to “overhear” a link to which it is not associated.
largeenoughsothatthisrarelyhappens.Ifthisdoeshappenwe
Our later mechanisms show how the algorithm could be
assume that both conflicting transmissions cease. A detailed
implemented if we indeed have a channel for exchanging
explanation of how rare collisions affect the performance of
informationbetweenbasestations(suchastheX2channelthat
utility-optimal CSMA was given in [14] and we can apply a
is defined in LTE). The bit-rate of such channels is typically
similar analysis to MMUO.
limitedandsowestressthatallweneedtoexchangeareshort
The output of MMUO, as described above, specifies the
messages such as the activity values. No detailed exchange of
transmission rate r (t) for every transmission (cid:104)(cid:96),r,c(cid:105) that
c,(cid:96) channel state is required.
takes place during time slot t. To obtain power settings,
Method1: Inthismethod,anactivityindicatorisreported
equation(18)providesthedirecttranslationfromtransmission
along with the CQI. Specifically, let y (t) be the binary
rates to power levels. (cid:96),r,c
activityindicatorthatissettooneifandonlyifMMUOmakes
p (t)= F−1(rc,(cid:96)(t)/wc), for link (cid:96)=ij. (20) a transmission (cid:104)(cid:96),r,c(cid:105) during time slot t. When CQIc,(cid:96)(t) is
i,c,j CQI (t) reported to basestation i for which (cid:96) ∈ Ui, y(cid:96),r,c(t) is also
c,(cid:96)
reported if it is set to 1. Each basestation i listens to all
For a given basestation i and resource block c, the feasibility CQI that it can decode, not just the CQI for links in U . If
i
oracle guarantees that one user j ∈ U has positive power i hears y = 1 for some (cid:96)(cid:48) on resource block c, then
i (cid:96)(cid:48),r(cid:48),c
allocation p (t). Let p (t)=p (t) for this user j. The every potential transmission (cid:104)(cid:96),r,c(cid:105), for (cid:96)∈U and r ∈R, is
i,c,j i,c i,c,j i
(cid:80)
transmission now takes place as long as p (t)≤p . This declared infeasible. Note that this method is similar in spirit
c i,c i
addresses the second issue. to the Clear-to-Send (CTS) mechanism for 802.11.
If it is the case that we can specify both user-level as Method2: ThismethodislessstringentthanMethod1in
well as inter-cell power allocations, we are done. However, declaringinfeasibility.Foreachactivityindicatory (t)=1
(cid:96),r,c
as discussed in Section V, in many instances we only have we define the safety margin to be the ratio between the
the freedom for specifying p since the user-level power is currently achievable transmission rate r˜ (t) and the actual
i,c c,(cid:96)
determined by a local scheduler such as the Proportional Fair rate r (t) that is used by MMUO. This achievable rate
c,(cid:96)
(PF) algorithm. In this case we run the MMUO algorithm “in can be computed from the CQI (t) values together with
c,(cid:96)
thebackground”tocomputethep valuesandthendetermine the current power levels. We assume that the safety margins
i,c
which user in U receives the transmission power p using are transmitted on the CQI channel along with the activity
i i,c
the PF algorithm. (We remark that once the power levels indicators. For some threshold υ >1 we say that the activity
indicator is safe if the safety margin is above υ, vulnerable if
the margin is between 1 and υ, and in outage if the margin is
below 1. (Note that if we are in outage then user (cid:96) could not
receive data at rate r for the current CQI values.) Method 2 is
thesameasmethod1exceptthatbasestationidoesnotrefrain
Fig.1. Networkconfiguration:1macroand2femtoseachwith2users.
from declaring a potential transmission on (cid:96)∈U on block c
i
feasible,evenifitoverhearsanactivityindicatory (t)=1
(cid:96)(cid:48),r(cid:48),c
as long as this indicator is currently safe.
potential to reduce interference if each class has priority on a
The exact value of υ could be a network-wide config-
different set of resource blocks. We now describe a heuristic
uredparameter.Alternativelyeachbasestationcouldgradually
to achieve this. In particular if basestation i is a macro we
lower a local estimate of υ until it observes links going into
bias it towards low numbered resource blocks by only letting
outage.
one of its users be active on a resource block if it also has
Method 3: This method applies probing to feasibility
active users on all lower numbered resource blocks. More
detection.WheneverabasestationineedstodecideifMMUO
formally, if (cid:96) ∈ U then (cid:104)(cid:96),r,c(cid:105) is feasible if for all c(cid:48) < c
could transmit on (cid:104)(cid:96),r,c(cid:105), for (cid:96) ∈ U , it briefly sets power i
i there exists (cid:96)(cid:48) ∈ U and r(cid:48) ∈ R such that y = 1.
level p on resource block c and observes the effects on i (cid:96)(cid:48),r(cid:48),c(cid:48)
i,c
Similarly, if basestation i is a femto we bias it towards high
other users. Here p is the power necessary to carry out
i,c
numbered resource blocks by only letting one of its users be
the transmission (cid:104)(cid:96),r,c(cid:105) and can be calculated as in (20).
active on a resource block if it also has active users on all
If basestation i overhears that any activity indicator moves
higher numbered resource blocks. More formally, if (cid:96) ∈ U
into outage then it sets p back to 0 and declares (cid:104)(cid:96),r,c(cid:105) i
i,c then (cid:104)(cid:96),r,c(cid:105) is feasible if for all c(cid:48) > c there exists (cid:96)(cid:48) ∈ U
infeasible. This method has the drawback that it could send i
and r(cid:48) ∈R such that y =1.
neighboringusersintooutageforshortperiods(andthiswould (cid:96)(cid:48),r(cid:48),c(cid:48)
need to rectified by more robust channel coding on the data
VII. SIMULATIONRESULTS
channels).However,ithastheadvantagethatbasestationigets
a much better sense of the “damage” that might be caused by We now provide an example to show how the algorithms
setting a particular power level p on resource block c. work. We consider a simple toy example since it allows us
i,c
Methods 4-6: The next three methods are essentially to compute the optimal schedule. We consider three omnidi-
the same as Methods 1-3. However, instead of basestations rectional basestations, 1 macro and 2 femtos, together with
overhearing activity indicators and their associated safety six users, two for each basestation. The two macro users are
status,eachbasestationwoulddirectlycommunicatetheirown at distance 100m (user 0) and 780m (user 1) respectively.
activity indicators and safety margins to all their neighboring Each femto has two users (users 2-5) at distance 10m. The
basestations. This can be done using a channel such as the exactconfigurationtogetherwiththeusernumberingisshown
X2 channel in LTE that provides communication between in Figure 1. We assume that user 1 has to be associated
neighboring basestations. Note that this is a lightweight com- with the macro since the nearby femto is in CSG mode. The
munication since the activity indicator only has 2 possible transmitpowerofthemacrois46dBmandforthefemtositis
values and the safety status has only 3 possible values. 8dBm. The system bandwidth is 5MHz and the noise density
In particular the basestations would not be exchanging any is -165dBm/Hz. We split the system bandwidth into three
detailed channel state information. resource blocks. The pathloss is represented by a COST-231
Relationship to current LTE proposals: We now briefly Hata model. In particular the path loss at distance d meters is
discuss how the above methods could fit with mechanisms assumedtobe0.525∗d−3.523.Forsimplicityweconsidertwo
that have been proposed in LTE standards for interference instantaneous transmission rates, a “low” rate of 8bits/sec/Hz
coordination. In the document [1] on RF requirements for and a “high” rate of 16bits/sec/Hz. We consider blocks.
femtos,threeoptionsareproposedforcommunicationbetween For this configuration we aim to achieve user through-
macrosandfemtos.Thefirstisdirectover-the-aircommunica- puts that solve problem (7). We can compute the follow-
tion.Thesecondisover-the-airvia“victim”users.Thiswould ing optimal solution offline via a standard subgradient al-
correspond to the overhearing methods 1-3 proposed above gorithm. For brevity we use notation of the form 0h2(cid:96)4(cid:96)
in that the victim user broadcasts channel quality information to represent a schedule, which means user 0 is receiving
thatindicatestoaninterfererwhetheritissafetotransmit.The data at the high rate and users 2 and 4 are receiving
third option is via an existing backhaul which corresponds to data at the low rate. In the optimal solution for schedules
methods 4-6 above. m∈{0h2(cid:96)4(cid:96),0h3(cid:96)4(cid:96),0h2(cid:96)5(cid:96),0h3(cid:96)5(cid:96)} we have π =8.4%,
m
Enhancedheuristic:Resourceblockprioritization:: Note for schedules m ∈ {1h4(cid:96),1h5(cid:96)} we have π = 10.7%,
m
that throughout this section we have assumed that attempts and for schedules m ∈ {2h4h,2h5h,3h4h,3h5h} we have
by a link to access resource block c are governed by λ π =11.2%. Figure 2 shows the corresponding optimal link
(cid:96),r,c m
and µ and are performed independently across resource throughputs as computed by the subgradient algorithm.
(cid:96),r,c
blocks. However, since in heterogeneous networks we have ThebehaviorofMMUOandMMUOwithProportionalFair
two classes of basestations (macros and femtos) there is are similar. In the interest of space, we present the plot for
node has a local energy based on the interference that it both
causes and receives. Nodes then pick new states based on
their local energy. Gibbs sampler techniques have also been
used to motivate greedy algorithms for LTE resource block
selection, e.g. [2]. Another popular technique, e.g. used in
[19],[18],istosetpowerlevelsaccordingtoagradientascent
approach. In particular each transmitter adjusts power levels
so as to improve network utility in its neighborhood. Both
Fig. 2. Link throughputs under the optimal solution. Top curves for
the Gibbs sampler and the gradient ascent based methods
users/links 0, 4 and 5; middle curves for users 2 and 3; bottom curve for
user1.(Notethatsomecurvescoincide.) require information exchange on how much interference each
transmitter causes to each receiver. For the Gibbs sampler
methods interference information needs to be exchanged in
order to calculate local energy levels. For the gradient ascent
methods nodes need to exchange “partial derivative” informa-
tion to indicate how the interference they experience would
be affected by a change in a neighbor’s power levels. We
remarkthatMMUOdoesnotrequiresuchdetailedinformation
exchange. It bases its calculations on CQI messages that
are already included in LTE, augmented with the activity
indicators (and possibly safety margins).
CSMA-based Algorithms: In the classic CSMA setup all
Fig.3. LinkthroughputsunderMMUOincombinationwithPF.Topcurve
for user 0; second curve for users 4 and 5; third curve for users 2 and 3; links wish to access a single channel. Jiang and Walrand
bottomcurveforuser1. [10] showed that CSMA can achieve any set of feasible
throughputs.Sincethisresult,anumberofpapershavelooked
athowtomakechannelaccessratesdependentonlocalqueue
MMUO with PF only (in Figures 3). As we can see, the link sizesinordertokeepthesystemstable,e.g.[7],[8],[5],[16],
throughputscloselyapproximatetheoptimalratesinFigure2. [17]. As already discussed, we have based our analysis on the
In both plots, user 0 (the closer user to the macro basestation) work [14] (later extended in [15], [13]) that analyzed utility
has the highest throughput while user 1 (the far user to the maximization in a CSMA setting.
macro basestation) has the lowest throughput. Among the
femto users, users 4 and 5 have higher throughputs since they IX. CONCLUSION
create less interference to a macro user than users 2 and 3. In this paper we have presented a CSMA-based scheduling
We conclude with a brief discussion of how the algorithms algorithmforheterogenousLTEnetworkswithbothmacroand
for testing feasibility work in this context. In particular sup- smallcells.Ourmaincontributionistwofold.Mathematically,
pose that we are in configuration 1h4(cid:96) and suppose that user our algorithm handles the general multiple transmission rates
3 wants to transmit at the high rate. This is infeasible. User 3 on multiple carriers and achieves utility optimality. For the
may discover this by either a) overhearing the CQI reported practical setting, the communication among the basestations
by user 1 and realizing it is sufficiently close to the minimum utilizes the existing CQI-based technology and hence the
acceptable CQI or b) briefly probing the channel at the high additional signaling is minimal.
rate and then discovering that user can no longer support its
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