Adaptive CSMA under the SINR Model: Efficient Approximation Algorithms for Throughput and Utility Maximization

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by  Peruru Subrahmanya Swamy

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Adaptive CSMA under the SINR Model: Efficient
Approximation Algorithms for Throughput and
Utility Maximization
Peruru Subrahmanya Swamy, Radha Krishna Ganti, Krishna Jagannathan
Abstract—We consider a CSMA based scheduling algorithm ideabehindthesealgorithmsliesinusingareversibleMarkov
for a single-hop wireless network under a realistic SINR model chain to sample feasible schedules from a product form
for the interference. We propose two local optimization based
distribution called the Gibbs distribution [10]. Specifically,
approximation algorithms to efficiently estimate certain attempt
each linkadaptivelyadjustsits transmissionattemptrate (also
rate parameters of CSMA called fugacities. It is known that
6 adaptive CSMA can achieve throughput optimality by sampling known as its fugacity) in order to ensure sufficient average
1 feasible schedules from a Gibbs distribution, with appropriate service rate.
0 fugacities. Unfortunately, obtaining these optimal fugacities is In order to support a given feasible service rate vector
2 an NP-hard problem. Further, the existing adaptive CSMA using CSMA, the corresponding fugacities have to be com-
algorithmsuseastochasticgradientdescentbasedmethod,which
n usuallyentailsanimpracticallyslow(exponentialinthesizeofthe puted. Unfortunately, determining the appropriate fugacities
a network) convergence to the optimal fugacities. To address this corresponding to the desired service rates is an NP-hard
J
issue,wefirstproposeanalgorithmtoestimatethefugacities,that problem [7]. In [7], the optimal fugacities are computed as

2 cansupportagivensetofdesiredservicerates.Theconvergence a solution to an optimization problem (here after referred to
2 rateandthecomplexityof thisalgorithmareindependentof the
as the Gibbsian problem), using a stochastic gradient descent
network size, and depend only on the neighborhood size of a
] link. Further, we show that the proposed algorithm corresponds algorithm. Each iteration of the gradient descent requires
T exactlytoperformingthewell-knownBetheapproximationtothe estimatingtheaverageserviceratesunderthecurrentiterateof
I underlying Gibbs distribution. Then, we propose another local thefugacities,whichinturnentailswaitingfortheunderlying
.
s algorithmtoestimatetheoptimalfugacitiesunderautilitymax- Markovchain to reach steady-state. This ‘mixingtime’ of the
c imization framework, and characterize its accuracy. Numerical
[ underlying Markov chain could be very large (exponential in
results indicate that the proposed methods have a good degree

of accuracy, and achieve extremely fast convergence to near- the size of the network), depending on the network load and
1
optimal fugacities,andoftenoutperformtheconvergence rateof topology [11], [12]. Therefore, the existing adaptive CSMA
v
5 the stochastic gradient descent by a few orders of magnitude. algorithms [7] do not provide a practical way to estimate the
6 IndexTerms—Wirelessadhocnetwork,Distributedalgorithm, fugacities,althoughtheyeffectivelysupportthedesiredservice
0 CSMA, Gibbs distribution, Bethe approximation, Throughput ratesoncethe optimalfugacitiesare estimated[13]. Themain
6 optimality focusofthispaperisinproposingefficientmethodstoestimate
0
the fugacities, under a realistic SINR (signal-to-interference-
.
1 I. INTRODUCTION plus-noise ratio) model for the interference. Specifically, we
0
consider the following two scenarios under which fugacities
6 The problem of link scheduling for maximum throughput
are to be computed:
1 has been widely studied, particularly with emphasis on the
:
v maximum-weightschedulingframework,developedin[2],[3]. • Theserviceraterequirementsofthelinksareknown.The
i In spite of its throughput maximizing property, maximum- objective is to support these average service rates.
X
weight scheduling requires centralized control, and necessi- • The service rate requirements are not known, but each
r
a tates the solution of an NP-hard problem for each schedul- link has a utility functionof its average service rate. The
ing decision. Several works have attempted to modify the objective is to maximize the sum utility of the network.
maximum-weight algorithm, so as to make it more amenable
A simple conflict graph based interference model [7],
to simple, distributed implementation [4], [5], [6]. However,
[8], [9] is widely used in the wireless context due to its
these greedy algorithms do not achieve full throughput.
simplicity and tractability, although it does not adequately
Inaseriesofrecentpapers[7],[8],[9],aclassofdistributed
capture the complex nature of the wireless interference [14].
link scheduling algorithms called adaptive CSMA algorithms
More specifically, a conflict graph based interference model
havebeenproposedandprovento bethroughputoptimal,i.e.,
ignores the fact that whether or not two links can transmit
theycansupportanyachievableserviceratevector.Thecentral
simultaneously,dependson the transmissionstate of the other
links and their spatial locations. Some recent papers extend
A part of this work [1] has been presented at IEEE Annual Allerton
ConferenceonCommunication,Control,andComputing(Allerton)2015,held the adaptive CSMA framework to a more realistic interfer-
atMonticello, IL,USA ence models like the SINR model [15], [16], and Rayleigh
P.S.Swamy,R.K.GantiandK.Jagannathan arewiththeDepartment of
faded channels [17]. However, these papers also essentially
ElectricalEngineering,IITMadras,Chennai,India600036.Email:{p.swamy,
rganti, krishnaj}@ee.iitm.ac.in employ stochastic gradient descent on a Gibbsian function to
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estimate the fugacities, and hence suffer from impractically theiralgorithm.Ourlocalalgorithmswhichefficientlyestimate
slow convergence rates. thesefugacitiescanbeusedinconjunctionwiththetechniques
For the conflictgraph based interferencemodelassumed in in [13], [23] to obtain a practicalCSMA algorithmwith good
[7], approximatebut efficient methodsto compute the fugaci- throughputand delay properties.
ties have been proposed[18], [19] using a popularvariational The remainder of this paper is organized as follows. In
techniquecalledtheBetheapproximation[20],[21].However, Section II, we describe the SINR based interference model,
the solutions given in [18], [19], cannot be directly extended and review the adaptive CSMA algorithm. In Section III,
toSINRbasedinterferencemodel.Thisisbecausetheconflict we introduce the local Gibbsian problems and propose our
graph based interference model corresponds to a simple pair- algorithm for computing the fugacities for a given service
wise interaction model [20], while the SINR model involves rate requirements. Section IV provides a brief review of
higher order interactions. The presence of these higher order the Bethe approximation, as relevant to adaptive CSMA. In
interactions makes the extension non-trivial. For graphical Section V, we derive our main result which establishes the
models even with higher order interactions, there are well equivalence between the local Gibbs optimization, and the
knownalgorithmslikeBeliefpropagation(BP)[20]forsolving Bethe approximation. In Section VI, we consider two special
the Bethe approximationproblem. However, in the contextof cases of the SINR model and derive closed-form expressions
adaptive CSMA under the SINR model, they can directly be for the local Gibbsian problems. In Section VII, we propose
usedonlytoestimatetheserviceratesgiventhefugacities,but a local algorithm to solve the utility maximization problem
not the other way around. and quantify its performance. In Section VIII, we present
We start with the Gibbsian optimization problem corre- numericalresultstoconfirmthefastconvergence,andSection
spondingtotheoptimalfugacities,andapproximatethisglobal IX concludes the paper.
problem by decoupling it into local optimization problems
at each link. The local problems are identical in structure to II. MODELAND PRELIMINARIES
the global problem, and are referred to as the local Gibbisan
We consider a single-hop wireless network and model the
problems. The dimension of the local problem at a link
linksusingabipolemodel,introducedin[24],[25].Inabipole
is equal to the size of its immediate neighbourhood and
model, each transmitter is associated with a receiver on the
hence typically small, and independent of the network size.
Euclidean plane. A transmitter and its corresponding receiver
Therefore these local Gibbsian problems can be efficiently
are referred to as a link. Let N denote the set of all the links
solved in a scalable fashion. The local solutions are then
inthenetwork.Let|N|=N bethetotalnumberoflinks.Let
suitably combined to obtain an approximate solution to the
r denote the distance between the transmitter and receiver
globalproblem.Inparticular,fortwospecialcasesoftheSINR ii
of link i. For simplicity, we assume1 that a link distance r
based interference model, we obtain closed form solutions to ii
is much smaller than the distances of the transmitter and the
these local problems.
receiver from the other links. With this assumption, we can
We prove that the solution of our local Gibbs optimization
think of links as points in the Euclidean space R2. Let r
method corresponds exactly to the celebrated Bethe approx- ji
denote the distance between the links i,j. We assume a time
imation [20] to the global Gibbsian optimization problem.
slotted model.
The accuracy of the Bethe approximation has been empiri-
Interference model: We consider the standard path-loss
cally evidenced in various fields [20], [22]. Therefore, it is
model kdk−α,α > 2, where d is the distance between a
reasonable to expect a fair degree of accuracy in the context
receiver and a transmitter, and α is the path loss exponent.
of CSMA as well. In fact, numerical results indicate that
Let P denote the transmit power of link i. We assume white
in order to obtain the level of accuracy in the fugacities i
Gaussian thermal noise at all the receivers with variance w.
obtainedbyusingourproposedmethod,thestochasticgradient
Let x(t)=[x ]N denotethe schedule of the network at time
descent method [7] takes an inordinately long time, often i i=1
t. Specifically, x (t) = 1 denotes that the link i is active
runningintotensofmillionsoftimeunitsevenforfairlysmall i
(transmitting) in time slot t. If there is no ambiguity, we will
networks. Therefore, in practical terms, our algorithm will
also use x to denote x(t).
operatewithsubstantiallysmallerconvergencetime,compared
Although all the active links in the network can poten-
to the original implementation of adaptive CSMA.
tially contributeto the interference,the aggregateinterference
It is worth noting that our algorithm is robust to gradual
from the transmitters beyond a certain distance can be safely
changes in the desired service rates, as well as to changes
neglected [26], [27]. This approximation is standard in the
in the network topology, since the local solutions can be
literature [15] and this distance, referred to as the close-
efficiently re-computed for the new set of requirements. On
in radius is denoted by R . Let N := {k | r ≤ R }.
theotherhand,thestochasticgradientdescentislikelytotake I i ik I
For convenience, let link i be also included in the set N .
a very long time to converge to its new operating point. i
We refer to the links in N \ {i} as the neighbors of link
The impractically slow mixing time of the CSMA Markov i
i. The neighborhood relationship can be represented by an
chain is also known to result in poor delay performance[11],
interference graph G(V,E). V is the set of links in the
[12].Recentworkslike[13],[23]haveshownimproveddelay
network and two links share an edge if they are within a
performance by employing several parallel instances of this
Markov chain. However, the result in [13] assumes that the
1The results in this paper do not require this assumption. This is just to
required optimal fugacities can be pre-computedand given to keeptheexpressions concise.
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distance R . Then the total interference power at link i is where p (1) denotes p (x = 1). The adaptive CSMA algo-
I i i i
given by rithm can support any service rate in the rate region provided
appropriate fugacities are used [7], [15] for the underlying
I (x)= P r−α. (1)
i j ji Gibbs distribution.
{j∈Ni|Xj6=i,xj=1} If the desired service rates are known, these fugacities
Then, the SINR at link i is given by can be obtained by solving the system of equations in (4).
Equivalently, the fugacities can also be obtained by solving
P r−α
γ (x)= i ii . (2) the following Gibbsian optimization problem [7] referred to
i
Ii(x)+w as the global problem.
Receptionmodel:Weassumethat,ineachtimeslot,asingle The global Gibbsian problem:
packet of data is transmitted from each active transmitter. If
lnλ=arg maxG(r), (5)
the received SINR at the corresponding receiver exceeds a r∈RN
pre-determined threshold T, i.e., γ (x) ≥ T, the packet is
i where G(r):= s r −ln exp y r .
successfully received. k k k k
Rate region: A schedule x∈{0,1}N is said to be feasible, kX∈N (cid:16)yX∈I (cid:16)kX∈N (cid:17)(cid:17)
if all the active links in the schedule meet the required SINR Here {si}i∈N ∈Λ are the desired service rates. A distributed
constraint, i.e., γ (x)≥T, ∀i such that x =1. The set of all stochastic gradient descent algorithm was proposed in [7]
i i
the feasible schedules is denoted by I. In our scenario, since to solve (5). However, estimating the gradient of G(r) in a
each link transmits one data packet whenever it is successful, distributedmannerentailsthe underlyingMarkovchainofthe
the long-term service rate of a link is equal to the fraction CSMA algorithm to converge to steady-state, which takes an
of time the link is successful. The rate region Λ, which is impractically long time in general.
defined as the set of all the possible service rates is given In the next section, we consider this scenario where the
by the convex hull of the feasible schedules in I. Hence, links know their target service rates. We provide an efficient
Λ = { α x | α = 1, α ≥ 0, x ∈ I}. If a and scalable approximationto this problem, by proposing the
x∈I x x∈I x x
link scheduling policy can support any rate vector in the rate following local Gibbsian problems. The solutions of these
P P
region, then the scheduling policy is said to be rate-optimal. local problems are appropriately combined to estimate the
Adaptive CSMA: We briefly review the adaptive CSMA solution to the global problem.
algorithm[7],[16]. Inthis algorithm,eachlink iis associated
with a fugacity λ > 0 which defines the underlying Gibbs
i III. THELOCAL GIBBSIAN PROBLEMS
distribution. In each time slot, a randomly selected link i is
allowed to update its schedule x (t) based on the information Wenowintroducesomedefinitionsrequiredforthedescrip-
i
in the previous slot: tion of the local Gibbsian problems.
• If its SINR is inadequate, i.e., γi(x(t−1)) ≤ T, then Local schedule: Let xj ∈ {0,1}Nj, be the set of variables
corresponding to the transmission status of the link j and
x (t)=0.
i
its neighbors, i.e., x := {x |k∈N }. We refer to x as
• If γi(x(t − 1)) ≥ T, then link i exchanges control j k j j
the local schedule at j. Further, from (1), it can be observed
messageswithitsneighbors,tofindiftheycanmeettheir
that the SINR of a link depends only on the local schedule.
SINR requirements if link i gets activated. If any of its
Hence the SINR at a link j can be viewed as a function
neighbors cannot meet its requirement, then x (t)=0.
i
of x , i.e., γ (x) = γ (x ). Also, recall that a schedule x
• If all the neighbors can meet their SINR requirements j j j j
is said to be feasible, if all the active links in the schedule
even if link i gets activated, then x (t) = 1 with
probability λi , and x (t)=0 with proibability 1 . meet their required SINR threshold. Thus, 1(x ∈ I) can
Itcanbeshownt1h+aλtithealgoirithminducesa Markovch1a+iλnion b1e(xfa∈ctIo)riz=ed over t1h(eγl(oxca)l ≥schTe)d.uTlehevna,ri(a3b)lecsan{xbje}wNj=r1itteans
thestatespaceoftheschedules{0,1}N.Further,thestationary j j
distributionoftheMarkovchain,parametrizedbythefugacity as j:Qxj=1
vector λ=[λ ]N , is given by 1
i i=1 p(x)= λ 1(γ (x )≥T), ∀x∈{0,1}N. (6)
j j j
Z
1
p(x)= λj 1(x∈I), ∀x∈{0,1}N, (3) j:Yxj=1
Z
j:Yxj=1 Localfeasiblity: A localscheduley =[yk]k∈Nj ∈{0,1}Nj
where1(x∈I) is anindicatorofx beinga feasibleschedule, at link j is said to be feasible, if either the link j is inactive
andZ isthenormalizingconstant.Then,duetotheergodicity (i.e., yj =0), or it is active and meets the required threshold
of the Markov chain, the long-term service rate of a link i SINR i.e., (yj = 1 and 1(γj(y) ≥ T)). The set of all
denoted by si is equal to the marginal probability that link the feasible local schedules at j is denoted by Ij. It can
i is active, i.e., p (x = 1). Thus, the service rates and the be observed from (6) that p(x) assigns zero probability to a
i i
fugacities are related as follows: schedule x if any of its local schedule is infeasible.
Local service rate vector: Let s := {s }N denote the set
si =pi(1)= Z−1 λj, ∀i∈N, (4) of service rates of all the links in the netwioir=k1. Then the local
x:Xxi=1 j:Yxj=1 service rate vector at link j denoted by sj be defined as the
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set of all the service rates corresponding to the link j and its element of the local fugacity vector at link k) from the
neighbors, i.e., s :={s |k ∈N }. neighboring link k.
j k j
Local capacity region: The local capacity region at link j Computationalcomplexity:TheimplementationoftheNew-
is definedas the convexhull of the local feasible schedulesat ton’s method [28] in the second step of our algorithm is
link j given by feasible. This is because the gradient and the Hessian of
the local objective function F(r) can be analytically com-
Λ ={ α z | α =1, α ≥0, z ∈I }. (7)
j z z z j puted since the dimension of the problem is small. The
zX∈Ij zX∈Ij exact expressions for the gradient and Hessian at link j
The local Gibbsian problem at link j : can be computed using certain marginals of the distribution
ˆb (x ) = Z−1exp( x r ),∀x ∈ I , where Z is a
βj =argrm∈RaNxjF(r), (8) njoFrmojralkiz∈atiNojn,cloentsmtaPnt(.kr∈)Nrjeprkeskent thje probjability P(xj=1)
where βj := [βjk]k∈Nj, and the function F : RNj → R is under the disjtributiokn ˆbj. Similarly, for i,k ∈ Nj, lekt mik
defined as denote the probability P(x = 1,x = 1) under the same
i k
F(r):= skrk−ln exp ykrk . distributionˆbj. Then,thegradientandHessian ofthefunction
F(r) are given by
kX∈Nj (cid:16)yX∈Ij (cid:16)kX∈Nj (cid:17)(cid:17)
Observethat, thelocalproblemsarestructurallysimilar tothe [∇(F(r))] =s −m (r), k ∈N ,
k k k j
globalproblem,exceptthatI intheglobalproblemisreplaced
m (r)−m (r)m (r), i,k ∈N ,i6=k
obfytIhje, aloncdalNpriosbrleepmlacaetdlibnykNi jis. Ijnuspta|rNticu|.laTr,htehesodliumtieonnssioton [∇2(F(r))]ik =(mkik(r)−mki(r)2,ki=k. j
i
these local problemsare referred to as the local fugacities. In The computation of the gradient and the Hessian requires
particular, at each link j ∈N, there is a local fugacity vector the information about the local feasible schedules at a link.
βj :=[βjk]k∈Nj. Specifically, this information is required to compute the
Local algorithm: Here, we propose a simple and dis- normalization constant Z . These computations are feasible
j
tributed algorithm to solve the local Gibbsian problems and becausetheO(2|Nj|)complexityinvolvedinthiscomputation
subsequently compute the approximate global fugacities by scales only with the size of the local neighborhood, and is
combining the local solutions using (9). These approximate independent of the total size of the network which could be
global fugacities can be directly used in the CSMA al- substantially large. In particular, in spatial networks where
gorithm instead of adapting the fugacties using a stochas- the neighborhood size does not scale with the network size,
tic gradient descent on the global problem which usually our algorithm is order optimal, while the convergerate of the
doesn’t converge in practical time scales. Each link in global stochastic gradient descent typically scales as O(2N).
the network executes the following algorithm in parallel. Further, it can be easily shown that the objective function of
the local problem, F(r) in (8) is a strictly concave function
Local Gibbsian method at link j which guarantees an extremely fast convergence rate for the
Newton’s method. Indeed, as the numerical results indicate,
Input: (sk, k ∈Nj); Output: λ˜j. thelocalalgorithmconvergesinthreetofourtimestepsunder
1) Obtain the service rates (s , k ∈ N ) from the neigh- most scenarios.
k j
bours. Next, we prove that the approximate global fugacites
2) Compute the local fugacities (βjk, k ∈ Nj) by solving {λ˜j}Nj=1 obtained using the local Gibbsian method (9) cor-
the local problem (8) using the Newton’s method [28]. respond exactly to performing the well known Bethe approx-
3) Obtainthefollowinglocalfugacities(β ,k ∈N )from imation to the global Gibbsian problem. In the next section,
kj j
the neighbours. we review the Bethe approximation technique.
4) Compute the approximate global fugacity λ˜ as
j
1−s |Nj|−1 IV. REVIEW OF THEBETHEAPPROXIMATION
λ˜j = j eβkj. (9) A. Product form distribution
s
(cid:18) j (cid:19) kY∈Nj LetS beafiniteset,andletXi,i=1,2,...,N,berandom
variables each taking values in S. The joint PMF (probability
Informationexchange:Onthewhole,thealgorithmrequires
massfunction)oftherandomvariablesissuccinctlydenotedas
only two steps of information exchange with the neighbours.
p(x), where x= {x ,x ,...,x }. Suppose that p(x) factors
1 2 N
Once in the first step, to obtain the service requirements of
into a product of M functions and is given by,
the neighbours, and again in the third step to obtain the local
fugacities computed at the neighbours. Except for these two 1 M
p(x)= f (x ), x∈SN. (10)
information exchanges, the algorithm is fully distributed and Z j j
can be executed independently at each link. Also, it may be jY=1
noted that a link requires only one element of the fugacity The function f (·) has arguments x that are some non-
j j
vectorcomputedateachneighbour,butnottheentirefugacity empty subset of {x ,x ,...,x }. Here Z is a normalization
1 2 N
vector. For example, link j obtains β (which is only one constant. Further, the productform distributions are generally
kj
5
represented using a graph called factor graph [20] that has 2) Bethe optimization (BO): The following optimization
two sets of nodes namely variable nodes and factor nodes problem is referred to as the Bethe optimization.
corresponding to the variables and the functions in (10)
argmin F (b ,b ), subject to (12)
respectively. An edge is drawn between a variable node and B f v
bf,bv
a factor node if the variable is an argument of that factor
b (x )≥0, i=1,...,N, x ∈S,
function.Fortheproductformdistributionconsideredin(10), i i i
we are interested in the marginal probabilities called variable bi(xi)=1, i=1,...,N,
marginals and factor marginals. Xxi
Variable node marginals: The marginal probability distribu- ˆbj(xj)≥0, j =1,...,M, xj ∈SNj,
tion pi(xi) corresponding to a variable node i, is obtained ˆbj(xj)=1, j =1,...,M,
by summing p(x) over the variables corresponding to all
Xxj
other variable nodes, i.e., pi(xi) = x\xip(x), xi ∈ S, ˆbj(xj)=bi(xi), j =1,...,M, i∈Nj, xi ∈S,
i=1,...,N.
P xjX\{xi}
Factor node marginals: Corresponding to each factor node,
where N is the set of indices of all the variable nodes
j = 1,...,M, the marginal probability function pˆ (x ), is j
j j associated with the factor node j.
obtainedby summingp(x) overall the variablesin x\x . Let
j Note: A feasible collection (b ,b ) of the Bethe optimiza-
N denote the number of arguments in f , i.e., N = |x |. f v
j j j j tion problem does not necessarily represent the marginals of
Then, pˆj(xj)= x\xjp(x), xj ∈SNj. any coherent joint distribution over SN, and are therefore
Let pv :={piP}Ni=1, pf :={pˆj}Mj=1 denote the collection of referred to as pseudo-marginals [20]. In spite of performing
all the variable node marginals and the factor node marginals the optimization over a relaxed set, the solution of the Bethe
corresponding to the distribution p(x) respectively. Next, we optimization (b∗,b∗) exactly coincides with the marginals of
f v
discusstheBetheapproximation[20]whichisusedtoestimate the true distribution (p∗,p∗), if the underlying factor graph
f v
these marginal probability distributions. is a tree. Further, for factor graphs with loops, the solution
leads to very good estimates of the true marginals [20], [22].
Infact,inmanyapplications,solvingfortheglobalminimizer
of (12) is not feasible because of its non-convexity.Hence, a
stationary point of the BFE satisfying the constraints in (12)
B. Bethe approximation is considered as a good estimate of the true marginals [20].
C. Bethe approximation for CSMA
1) Bethe free energy (BFE): Consider a probability dis-
tribution p(x) of the form (10) for which we are interested We observe that the stationary distribution of the CSMA
in finding the marginals. Let b(x) be some distribution on Markov chain p(x) given by (6) is in product from (10) with
SN, whose factor and variable marginals are given by bf = the factor functions {fj}Nj=1 defined as
{ˆb }M andb ={b }N .Here,p(x)isreferredtoasthetrue
j j=1 v i i=1 λ , if x =1,γ (x )≥T,
distribution, and b(x) is referred to as the trial distribution as j j j j
f (x )= 0, if x =1,γ (x )<T, (13)
itisusedtoestimatethetruedistributionp(x).Next,wedefine j j  j j j
theBetheentropy,whichisrequiredforthedefinitionofBFE. 1, if xj =0.
The Bethe entropy of a distribution b(x) is a weighted sum
of the entropies corresponding to the factor and the variable where xj is the localschedule of the link j defined earlier.
By the definition of local feasibility, p(x) assigns zero
marginal distributions of b(x) and is given by H (b ,b )=
B f v probability to a schedule x, if it is locally infeasible at any
oPfMjt=h1eHˆvja(rˆbiajb)l−e PnodNi=e1(idiin−th1e)Hfia(cbtio)r, wgrhaeprhe,dainids tHˆhej(dˆbje)gre=e lsiantkis.fyHepˆnjc(ye,)t=he0f,act∀oyr n∈/odIej,m∀ajrg∈inNals. pf ={pˆj}Nj=1 have to
− y∈SNj ˆbj(y)lnˆbj(y), Hi(bi) = − y∈Sbi(y)lnbi(y) BFE for CSMA: We shall consider the product form
denote the entropies of the factor marginal ˆb (x ), and the representationoftheCSMA stationarydistributionp(x)in(6)
P P j j
variable marginal b (x ) respectively. andcomputetheBFE.Asdefinedin(11),theBFEforagiven
i i
trial distributionb(x) has two termsU (b ,b ), H (b ,b ).
B f v B f v
Definition 1. Consider a true distribution p(x) of the form The following lemma characterizesthe first term U (b ,b ).
B f v
(10). Let b(x) be a trial distribution. Then, the Bethe free
Lemma 1. If the factor marginals of a trial distribution b ,
energy F (b) is defined as f
B
assign zero probabilities to all the infeasible local schedules,
i.e.,
FB(b)=FB(bf,bv)=UB(bf,bv)−HB(bf,bv), (11) ˆbj(y)=0, ∀y ∈/ Ij, ∀j ∈N, (14)
then U (b ,b ) reduces to U (b ,b ) = U (b ) =
B f v B f v B v
− N b (1)lnλ . Moreover, if b does not satisfy (14),
where U (b ,b )=− M ˆb(x )lnf (x ). i=1 i i f
B f v j=1 xj j j j UB(bf,bv)=∞.
P
P P
6
Proof: Proof follows from definition of factors in (13). the variable marginals {b∗} :
i i∈Nj
Using Lemma 1 and (11), the BFE for any (bf,bv) that ˆb∗j =argmaxHˆj(ˆbj), subject to, (19)
satisfies (14) is given by ˆbj
ˆb (x )=b∗(x ), ∀i∈N , x =1, (20)
j j i i j i
FB(bf,bv)=FB {ˆbj}Nj=1,{bi}Ni=1 , (15) xjX\{xi}
N (cid:16) (cid:17) ˆbj(y)=0, ∀y ∈/ Ij. (21)
= −b (1)lnλ −Hˆ (ˆb )+(d −1)H (b ),
i i i i i i i
Proof: Proof is provided in Appendix X-A.
i=1
X Further,it canbe shown[29]thatthere isa uniquesolution
where di is degree of variable node i in the factor graph. For to the maximum entropy problem (19). In other words, the
(bf,bv) that does not satisfy (14), FB(bf,bv)=∞. variable marginals {b∗i}i∈Nj uniquely characterize the factor
marginal ˆb∗. This unique characterization is stated in the
j
following Lemma.
V. LOCAL GIBBS OPTIMIZATIONGIVES THEBETHE
APPROXIMATED FUGACITIES Lemma 3. The optimalsolution of (19) will have the follow-
ing form, parametrized by vj :=[vjk]k∈Nj ∈R|Nj|.
We are now in a position to state a key result of this paper,
whichassertsthattheapproximatedglobalfugacitiesobtained ˆb∗(x )= 1 exp x v , ∀x ∈I , (22)
by solving the local Gibbsian problems correspond exactly j j Zj k jk j j
to the Bethe approximated fugacities for the global problem. (cid:16)kX∈Nj (cid:17)
Specifically, we claim that {λ˜}Ni=1 obtained through (9), are where Zj is a normalization constant, and vj is the solution
theuniquesetoffugacitiesforwhich,thedesiredservicerates to the following optimization problem defined by the variable
{si}Ni=1 areobtainedasastationarypointofthecorresponding marginals {b∗i}i∈Nj:
Bethe free energy. The precise statement is as follows:
v =arg max F(r), (23)
j
Theorem 1. Let {λ˜}N be the approximate fugacities esti- r∈RNj
i=1
matedin(9).Thenthesearetheuniquefugacitiesforwhichthe
where F(r):= b∗(1)r −ln exp y r .
marginalsgiven in (17) and (18) are obtainedas a stationary k k k k
point of the corresponding Bethe free energy given in (16). k∈Nj (cid:16)y∈Ij (cid:16)k∈Nj (cid:17)(cid:17)
P P P
Proof: Proof follows by considering the dual problem of
N (19),byassociatingdualvariableswiththeconstraintsin(20).
FB(bf,bv)= −bi(1)lnλ˜i−Hˆi(ˆbi)+(di−1)Hi(bi), It is a standard proof which can be found in [29].
Xi=1 Notethattheoptimizationproblem(23)correspondsexactly
(16)
to the local Gibbsian optimization problem (8). Hence in the
b∗(1)=s , b∗(0)=1−s , (17) light of Lemma 3, it is straight forward to verify that the
j j j j
1 variable and factor marginals defined in (17), (18) satisfy the
ˆb∗(x )= exp x β , ∀x ∈I , (18)
j j Z k jk j j condition laid in Lemma 2. Next, in Lemma 4, we derive the
j
(cid:16)kX∈Nj (cid:17) second condition required for stationary point of FB(bf,bv).
where βj is the local fugacity vector at link j obtained from Lemma 4. Let (b∗f,b∗v) be a stationary point of the BFE
(8), and Zj is the corresponding normalization constant. correspondingto the distributionp(x) in (6). Foreachi∈N,
let v be the vector which characterizes the factor marginal
Proof: First, we establish that (b ,b ) is a stationary i
f v ˆb∗.Thentheglobalfugacitieswhichdefinep(x)are relatedto
point of the Bethe free energy if and only if it satisfies the i
the factor and variables marginals (b∗,b∗) as follows:
conditions given in Lemmas 2 and 4. To that end, we prove f v
thesetwoLemmasseparately,andinAppendixX-B,weargue
1−b∗(1) di−1
that these two Lemmas together constitute the necessary and λ = i evji, ∀i∈N. (24)
i b∗(1)
sufficient condition. (cid:18) i (cid:19) jY∈Ni
Next, we prove that the factor and variable marginals in
Proof: Proof is provided in Appendix X-B.
(17), (18) satisfy the conditionsspecified in Lemmas 2 and 4.
Further, due to the definition of the approximated global
The following lemma states that a gradient zero point of
fugacities (9), the variable and factor marginals defined in
F (b ,b ) should have a maximal entropy property, subject
B f v
(17), (18) immediately satisfy the condition laid in Lemma
to constraints defined in (12), and (14).
4. Hence, the conditions specified by Lemmas 2 and 4 are
Lemma 2. Let (b∗,b∗) be a stationary point of the BFE satisfied by (b∗,b∗) defined in (17), (18) which provesthat it
f v f v
corresponding to the distribution p(x) in (6). Then, for each isastationarypointoftheBFE(16).Further,theuniquenessof
j ∈ N, this factor marginal ˆb∗ is related to its corre- thefugacitiesfollowsfromthefactthatthemaximumentropy
j
sponding variable marginals {b∗} , through the following problem (19) has a unique solution, and (24) in Lemma 4.
i i∈Nj
constrained entropy maximization problem, parameterized by This concludes the proof of Theorem 1.
7
VI. SPECIAL CASES VII. UTILITY MAXIMIZATION
Inthissection,weconsidertheutilitymaximizationproblem
In this section, we describe two special cases of the SINR
and provide an approximationalgorithm to solve the problem
interference model called the Conflict graph model [9] and
in a distributed manner. The problem is defined as follows.
the conservative SINR model considered in [15]. For these
Supposeeachlinkiinthenetworkisassociatedwithaconcave
models,wederivesimpleclosedformexpressionsforthelocal
utility function of its service rate U : [0,1] → R . Our
fugacities, which are otherwise to be obtained by solving the i +
objective is to find the service rates that maximize the system
optimization problem (8).
wide utility, i.e.,
max U (y ), (27)
i i
A. Conflict graph model y∈Λ
i
X
In the conflict graph model [9], two links cannot transmit and subsequently compute the global fugacities that corre-
simultaneously if one link is within the interference range of spond to these optimal service rates.
the other link. Then Ii, the set of local feasible schedules at In [7], an iterativealgorithmto update the globalfugacities
a link i, has a simple structure. Specifically, the set Ii has is proposed.However,each iterationof the algorithmrequires
only one feasible local schedule with link i being active (i.e., an underlying slowly mixing Markov chain to reach steady
the local schedule with just link i being active and all the state. Henceit suffersfromimpracticallyslow convergenceto
neighbors inactive). In all other feasible local schedules link the optimal fugacities. To address this issue, we propose an
i is inactive. By taking advantage of this simple structure of iterativealgorithmwhichupdatesthelocalfugacitiesinsteadof
Ii, the local fugacities are derived as follows. directly updating the global fugacities. These local fugacitiy
updates are computationally simple and do not require any
Theorem2. Fortheconflictgraphmodel,thelocalfugacities
Markov chain to convergence.These local fugacities are then
i.e., solution of the local Gibbsian problem (8) at a link i has
used to obtain the approximate global fugacities using (9)
a closed form expression given by
proposed in Section III.
s (1−s )di−2 (1−s −s )−1, if j =i, Our local algorithm can be explained as follows. Each
i i i k
eβij = k∈Ni\{i} link in the network executes this algorithm in parallel. The
s (1−s −s )−1,Q if j ∈N \{i}. algorithm involves solving an one dimensional optimization
 j i j i
problem(28)relatedtotheoriginaloptimizationproblem(27).
(25)
 Here θ > 0, is a parameter of the algorithm that can be
Proof: Proof is provided in Appendix X-C tuned.Thesolutionof thisoptimizationproblemsj(t) is used
in the subsequent steps of the algorithm to update the local
fugacities (29) and the global fugacities. We will later show
that the update equation (29) is inspired by a subgradient
B. Conservative SINR model
descentalgorithmforarelatedoptimizationproblem.Theterm
In the conservative SINR model considered in [15], the α(t) corresponds to the step-size of the subgradient descent
neighboringlinks of a link i are rankedin an ascendingorder algorithm.Anystandardstep-sizesthatsatisfytheconvergence
based on the interferencethey incur to link i. Then,the set of criteria for a subgradientdescent method can be used [28]. A
neighboring links N is partitioned into two disjoint sets NA typical example is α(t)= 1.
i i t
and NB. The set NA is obtained by adding the neighboring
i i
links one by one, starting from the link incurring the lowest Local algorithm for utility maximization at link j
interference to the highest, till the aggregate interference of
all the links in NiA is small enough to maintain the required 1) At t=0, initialize βjk(t)=0, ∀k ∈Nj.
SINR threshold T at link i. All the neighboring links which
are not in NiA are said to be intolerable and are included in Computing global fugacities:
the set NB. Letda, db be the sizes of the sets. Thenthe local
i i i
fugacities can be related to the service rates as follows.
2) Obtain the following local fugacities (β (t), k∈N )
kj j
from the neighbours.
Theorem 3. For the conservative SINR model, the local
3) Compute s (t) as
fugacities have the following closed form expressions: j
si(1−si)dbi−1 (1−si−sk)−1, if j =i, sj(t)=argqm∈[a0,x1]θUj(q)−q βkj(t). (28)
k∈NB kX∈Nj
eβij =ssjj((11−−ssij)−−1s,j)−1Q,i iiff jj ∈∈NNiiBA., 4) lCoocmalpfuutgeacthiteiesap(pβrkojx(itm),akte∈glNobja)lanfudgsacj(itty) uλ˜sji(ntg) (f9r)o.m
 (26) Updating the local fugacities:
Proof: Proof is provided in Appendix X-D.
8
5) Consider the distribution
ˆb (x ;t)=Z−1exp x β (t) ,∀x ∈I , max θ Uj(yj)+ H(ˆbj), subject to (31)
j j j k jk j j y,{ˆbj} j j
(cid:16)kX∈Nj (cid:17) ˆb (xX)≥0, ∀j,xX∈I ; ˆb (x )=1,∀j;
j j j j j j
and for each k ∈ Nj, let mjk(t) := xj:xk=1ˆbj(xj;t) xXj∈Ij
represent the marginal probability corresponding to xk y ∈[0,1]N; y = ˆb (x ), ∀j =1...N,k ∈N .
P k j j j
under this distribution.
6) For each k ∈ N , obtain s (t) computed at the neigh- xjX:xk=1
j k
Here, the constraint set is specified by explicitly expanding
bour k, and update the local fugacity at j using
the definition of Λ involved in description of Λ . It can be
j B
β (t+1)=β (t)+α(t)(s (t)−m (t)). (29) observedthatfora largeθ, this problemclosely approximates
jk jk k jk
theoriginalproblem(27)astheentropyisboundedfromabove
and below. The advantage of defining this new optimization
Information exchange: The required information exchange problem is that it is amenable to a distributed solution [7]. In
and the complexity of the algorithm in an iteration, depend particular,our localalgorithmsolvesthis problemas stated in
onlyonthesizeofthelocalneighbourhood,andisindependent the following Lemma.
of the total size of the network.
Lemma5. Thelocalutilitymaximizationalgorithmconverges
Accuracy: The accuracy of the above algorithm can be to a limit s such thatthe optimizationproblem (31) attainsits
understood by splitting the error into two parts. The first part maximum at (s,{ˆb }) for some {ˆb }.
j j
is the error involved in estimating the optimal service rates
Proof: The outline of the proof is to show that the
in (27). The second part is the error involved in estimating
local utility maximization algorithm corresponds to the dual
the global fugacities corresponding to these service rates.
subgradient method [28] for (31). The proof is provided in
We characterize the first part of the error in Theorem 4,
which states that the gap to the optimum utility is O(1). Appendix X-E.
In other words, the local algorithm provides a good estimaθte Consider (s,{ˆbj}), the solution to the maximization prob-
of the optimal service rates when θ is large. The second lem (31). We now complete the proof of Theorem 4 by
part of the error is determined by the accuracy of the Bethe showing that s satisfies (30).
approximation.Thiserroris evidencedto be reasonablysmall From the definition of ΛB, for every y ∈ ΛB there exists
in many applications [20], [22]. some{ˆbj}suchthat(y,{ˆbj})isfeasiblefortheproblem(31).
Hence,
Theorem 4. Under the local utility maximization algorithm,
the service rates s(t) = [s (t)]N converge to some s =
j j=1 θ max U (y )≤θ max U (y ) + H(ˆb )
[ssajt]iNjs=fie1s∈ [0,1]N, i.e., limt→∞s(t) = s. Further, the limit s y∈ΛBXj j j y∈ΛBXj j j  Xj j
(a)  
≤ θ U (s)+ H(ˆb ), (32)
N N log|I | j j
U (s )≥max U (y )− j j . (30) j j
Xj=1 j j y∈Λ Xj=1 j j P θ where(a)followsfromtheXfactthat(s,X{ˆb })maximizes(31).
j
Since Λ⊆Λ , we have
Proof: We begin the proof by defining Bethe approxi- B
mated capacity region as the set of service rates given by
max U (y )≤ max U (y ). (33)
j j j j
y∈Λ j y∈ΛB j
Λ :={y ∈[0,1]N |[y ] ∈Λ , ∀j ∈N}, X X
B k k∈Nj j
The fact that entropy H(ˆb )≤log|I |, along with (32), (33)
j j
where Λ is the local capacity region described in (7). This completes the proof of Theorem 4.
j
definition is motivated from the concept of pseudo-marginals
discussed in Section IV. Compared to a set of fully con- We conclude this section by comparing our utility maxi-
sistent marginals (i.e., there is a coherent joint distribution mization algorithmwith prior works. Our algorithmis similar
corresponding to the marginals), the pseudo-marginals have in spirit to the subgradient descent based algorithm (on a
to satisfy only a relaxed set of constraints. Hence it follows modified maximization problem) proposed in [7]. However,
that the actual capacity region is contained in the Bethe the key differenceis in the complexityinvolvedin computing
approximated capacity region, i.e., Λ⊆ΛB. thesubgradients.Specifically,inouralgorithm,thecomplexity
Now,wedefineanewoptimizationproblembymakingtwo of computing the subgradient is independent of the total size
changestotheoriginalutilitymaximizationproblem(27).The of the network; On the other hand, it could be exponentially
firstchangeistorelaxtheconstraintsetΛin(27)byreplacing large in the network size in [7]. Another work that is close
it with Λ . The second change is to scale the objective to our work is [19], where the authors propose an algorithm
B
function by a factor θ, and add some local entropy terms. with polynomial rate convergence. However, the algorithm is
The resulting maximization is problem given by specific to conflict graph interference model, and cannot be
9
directly extended to the SINR interference model considered independent of the network size. Moreover, as discussed in
in our work. Section III, the complexity or information exchange involved
ineachiterationoftheNewton’smethoddependonlythesize
VIII. NUMERICAL RESULTS of the local neighborhood, and independent of the network
size.Hence,theoverallperformanceofthealgorithminterms
In this section, we present simulations to evaluate the
of complexity or convergencetime is very good.
performanceofouralgorithms.Asdiscussedearlier,theBethe
approximation is exact if the underlying factor graph is a
tree. Further, a factor graph with short cycles is the most nt1.5
adversarial setting for Bethe approximation [20]. Hence, it e 30 links
would be interesting to understand the performance of our adi 20 links
algorithms for random network topologies with short cycles e gr 1 10 links
as illustrated in Figure 1. h
of t0.5
m
7
r
o
9 N
0
6 11 1 2 3 4 5 6
Iteration
4
5 3 8 1 12
14 Fig.2. Convergence ofthelocalGibbsianmethod
4
3 15 B. Utility maximization setting
10
2 Here, we consider the case of computing the fugacities for
2 6
7 theproportionalfairnessutilitysetting,i.e.,Ui(si)=logsi for
allthelinks.Weupdatethefugacitiesusingthelocalalgorithm
1
for utility maximization proposed in Section VII. In Figure
5
13 3, we plot the norm of the subgradient corresponding to the
0
0 1 2 3 4 5 6 7 8 maximization problem (31), which indicates the convergence
Fig.1. Illustrationofaninterferencegraphofafifteenlinknetworkusedfor ofthe algorithm.The convergenceisplottedfortwo networks
simulations. Eachnode represents alink in thenetwork. Anedge is present of size 15 and 20. It can be observed that the algorithm
iftwolinks areintheinterference range.
converges within 200 iterations.
Simulation setting: We generate a spatial random network
by uniformly placing the transmitter nodes on a two dimen- 1.2
sional square plane of length 8. Each transmitter is associated ent 1 15 links
di
with a receiveratadistanceof0.5 ina randomdirection.The gra0.8 20 links
resulting interference graph for a 15 link network is plotted ub
in Figure 1. The path loss exponentα is set to 3, the close-in he s0.6
radius RI is set to 2.4, and the threshold SINR is set to 15 m of t0.4
dB. The transmit power of the links is set to 1. or0.2
N
We present the simulation results for the two scenarios 0
0 50 100 150 200
considered in this paper namely: A) Service requirementsare
Iteration
given, and B) Utility maximization setting.
Fig.3. Convergence ofthelocalutility maximization algorithm
A. Service requirements are given
We consider symmetric service rate requirements for all
C. Bethe approximation error and comparison with SGD
the links, and execute the local Gibbsian method proposed
The accuracy of our algorithms depends on the error
in Section III. We use the Newton’s method at each link to
generated due to the Bethe approximation. We define the
obtainthe localfugacities.We repeatthe experimentforthree
approximation error e(st), as the maximum deviation from
random network topologiesof differentsizes. In Figure 2, we
the required service rate, among the links whose service
haveplottedtheconvergenceratetothelocalfugacitiesforthe
requirements are not met. In particular, for a given target
worst case link in each of the three topologies. Specifically,
service rate vector st = [st]N , e(st) = max (st − sa),
the norm of the gradientfor the local objective functionF(r) i i=1 i i
i
in(8)isplottedforeachiterationoftheNewton’smethod.As where, sa = [sa]N are the service rates supported by using
i i=1
asmallernormindicatestheclosenesstotheoptimalsolution, the approximatedfugacities{λ˜}N . We considertwo random
i=1
itcanbeusedtounderstandtheconvergencerate.FromFigure networks with sizes 15 (shown in Figure 1) and 20 links, and
2, it can be observed that the proposed algorithm converges comparetheerrorofourapproximationwiththeresidualerror
almost instantaneously, typically within 4 to 5 iterations, of the SGD algorithm [7] after running it for 109 time-slots
10
of the CSMA algorithm. In Figure 4, we plot the error by a distributed algorithm, namely the local Gibbsian method to
operating the network at different loads. We have normalized efficientlyestimate these fugacities.The convergencerate and
the errorbydividingwiththetargetservicerateforwhichthe the complexity of this algorithm depend only on the maxi-
network is being operated. It can be observed from Figure 4 mum size of local neighbourhoodand are independent of the
that for practical time-scales, our algorithm results in better network size. We proved that our approximation corresponds
accuracy than the SGD. For example, at a moderate load of exactlyto performingthe well knownBethe approximationto
0.6,theerrorofourlocalGibbsalgorithmiswithin10percent, the global Gibbsian problem. We also proposed an approx-
while the error of the SGD algorithmis about30-40percent. imation algorithm to estimate the fugacities under a utility
maximization framework,and characterizedits accuracy.Fur-
ther,weconsideredtwospecialcasesoftheSINRinterference
0.8
Local Gibbs 20 links model, and derived closed form solutions to the fugacities.
d Error0.6 SSLoGGcDDa l 21G05i blliibnnskk ss15 links Oalguorrniuthmmesriccaanlrleesaudlttsoinadgicoaotdetdheagtrteheeopfroacpcousreadcaypapnrdoxiimmpartoiovne
ze0.4 theconvergencetimebyafewordersofmagnitude,compared
mali to the existing stochastic gradient descent methods.
Nor0.2
X. APPENDIX
0 A. Proof of Lemma 2
0 0.2 0.4 0.6 0.8 1
Load If any factor marginal ˆb does not satisfy (21), then the
j
Fig.4. Normalizederrorasfunctionofthenetworkload,after109time-slots Bethe freeenergywillbe∞duetoLemma1. Hence,suchˆbj
cannotbeoptimal.Next,letusconsidertheBetheoptimization
for the BFE defined in (15):
1 argmin F (b ,b ), subject to (34)
B f v
or0.8 SLoGcDal Gibbs bf,bv
err
zed 0.6 ˆbj(xj)=bi(xi), j ∈N,i∈Nj,xi ∈{0,1}, (35)
ormali0.4 xjX\{xi}
N ˆb (x )=1, j =1...N, (36)
j j
0.2
Xxj
0100 102 104 106 108 bi(1)+bi(0)=1, i=1...N. (37)
time slot
As all the constraints of (34) are linear, they can be elimi-
Fig.5. Normalized errorasafunctionoftimeataloadof0.6 nated by suitable variable transformation to get an equivalent
unconstrained optimization problem [28].
InFigure5,weplotthenormalizederrorasafunctiontime, Letuslookatthe optimizationvariablesin (34). Ifwe con-
by using the time-averageservice rates observedfrom CSMA sider the distribution b (·), there are 2 optimization variables
i
algorithm. Here, our local Gibbs algorithm computes the ap- associated with it, namely b (0) and b (1). If we consider
i i
proximatefugacities,andusesthesestaticfugacities(i.e.,they the distribution ˆb (.), there is an optimization variable cor-
i
arenotadaptedduringthealgorithm)intheCSMAalgorithm. responding to each argument ρ ∈ {0,1}Ni, i.e., there are
The SGD algorithm starts with some initial fugacities, and 2|Ni| variables associated with it namely {ˆbi(ρ)}ρ∈{0,1}Ni.
adapts the fugacities by observing the corresponding service For convenience, we split the arguments ρ ∈ {0,1}Ni into
rates. The update rules used to simulate the SGD are taken
two sets. We use A to denote the set of arguments in which
from[30].AlthoughtheSGDalgorithmisprovablythroughput
there is at most one non-zero element, i.e., A = {ρ ∈
roepstiidmuaall,eitrrcoarnisberaothbesrerlvaergdethcaotmfpoarrperdactoticoaulrtiampep-rsocxailmesa,titohne {0,1}Ni | j∈Niρj ≤1}, and use B to denote the set of all
the other arguments in {0,1}Ni.
algorithm. This is because, in the SGD based algorithm, the P
Variabletransformation:Letusperformthe followingvari-
underlying CSMA Markov chain has to converge to a new
able transformations to eliminate the constraint equations,
steady state distribution for every update in the fugacities.
and thereby obtain an equivalent unconstrained optimization
problem:
IX. CONCLUSIONS (i) Replace the variable b (1) by a new variable y . Hence,
i i
We considered the adpative CSMA algorithm under the by expressing b (0) = 1 − y , we can eliminate the
i i
SINR interference model which is known to be throughput constraints in (37).
optimal. Under this model, we first studied the problem of (ii) Let a new variable z (which is a vector of length |B|)
i
computing the optimal attempt rates (fugacities) to support replace the variables corresponding to the arguments in
given service requirements. The existing stochastic gradient B, i.e., z :=(ˆb (ρ), ρ∈B).
i i
descent method to solve this problem suffers from impracti- (iii) If an argument ρ ∈ A has the non-zero element in jth
cally slow convergence. To address this issue, we proposed position (i.e., ρ = 1), then its corresponding variable
j