Applying DCOP to User Association Problem in Heterogeneous Networks with Markov Chain Based Algorithm

0.61 MB·

by  Peibo Duan

Applying DCOP to User Association Problem in
Heterogeneous Networks with Markov Chain Based
Algorithm
Peibo Duan∗, Guoqiang Mao∗†‡§, Changsheng Zhang¶ and Bin Zhang¶
∗School of Computing and Communications
University of Technology, Sydney, Australia
†Data61 Australia
§Beijing Unriversity of Posts and Telecommunications, Beijing, China
7 §School of Electronic Information Communications, Huazhong Unriversity of Science Technology, Wuhan, China
1
¶School of Computer Application
0
2 Northeastern University
Bin Zhang is the correspondence author
n
a
J

5
Abstract—Multi-agent systems (MAS) is able to characterize DCOP model [7] 3) the application of DCOP in modeling

] the behavior of individual agent and the interaction between environmental systems, such as sensor networks [8], [9], [10],
A agents.Thus,itmotivatesustoleveragethedistributedconstraint [11], disaster evacuation [12], traffic control [13], [14] and
optimization problem (DCOP), a framework of modeling MAS,
M resource allocation [15], [16], [3]. In this paper, we take
to solve the user association problem in heterogeneous networks
. (HetNets). Two issues we have to consider when we take DCOP more attention to the application of DCOP. More precisely,
s into the application of HetNet including: (i) How to set up an we leverage DCOP to solve user association problem in the
c
effectivemodelbyDCOPtakingaccountofthenegtiveimpactof downlink of multi-tier heterogeneous networks with the aim
[
theincrementofusersonthemodelingprocess(ii)Whichkindof to assign mobile users to different base stations in different
1 algorithmsismoresuitabletobalancethetimeconsumptionand
tiers while satisfying the QoS constraint on the rate required
v thequalityofsoltuion.Aimingtoovercometheseissues,wefirstly
9 come up with an ECAV-η (Each Connection As Variable) model by each user.
8 in which a parameter η with an adequate assignment (η=3 in is generally regarded as a resource allocation problem [17],
2 thispaper)isabletocontrolthescaleofthemodel.Afterthat,a [18],[19],[20]inwhichtheresourceisdefinedbytheresource
1 Markovchain(MC)basedalgorithmisproposedonthebasisof blocks(RBs).Inthiscase,themoreRBsallocatedtoauser,the
0 log-sum-expfunction.Experimentalresultsshowthatthesolution
largerrateachievedbytheuser.Themethodstosolvetheuser
. obtained by DCOP framework is better than the one obtained
1 bytheMax-SINRalgorithm.ComparingwiththeLagrangedual associationproblemaredividedintocentralizedcontrolledand
0 decomposition based method (LDD), the solution performance distributed controlled. With regard to the centralized way, a
7
has been improved since there is no need to transform original centralentityissetuptocollectinformation,andthenusedto
1
problem into a satisfied one. In addition, it is also apparent that
decide which particular BS is to serve which user according
:
v the DCOP based method has better robustness than LDD when to the collected information. A classical representation of
i the number of users increases but the available resource at base
X stations are limited. centralized method is Max-SINR [21]. Distributed controlled
methodsattractconsiderableattentioninlastdecadesincethey
r
a I. INTRODUCTION donotrequireacentralentityandallowBSsanduserstomake
of agent and the cooperation between agents in Multi-agent autonomous user association decisions by themselves through
System (MAS), a framework, named distributed constraint the interaction between BSs and users. Among all available
optimization problem (DCOP) in terms of constraints that methods, the methods based on Lagrange dual decomposation
are known and enforced by distinct agents comes into being (LDD) [20] and game theory [22] have better performance.
with it. In last decade, the research effort of DCOP has been Hamidreza and Vijay [20] put forward a unified distributed
dedicatedonthefollowingthreedirections:1)thedevelopment algorithm for cell association followed by RBs distribution in
of DCOP algorithms which are able to better balance the a k-tier heterogeneous network. With aid of LDD algorithm,
computational complexity and the accuracy of solution, such the users and BSs make their respective decisions based on
as large neighborhood search method [1]; Markov Chain local information and a global QoS, expressed in terms of
MonteCarlosamplingmethod[2][3]anddistributedjunction minimumachievablelong-termrate,isachieved.However,the
tree based method [4] 2) the extension of classical DCOP constraintrelaxationandthebacktrackinalmosteachiteration
model in order to make it more flexible and effective for areneededtoavoidoverloadattheBSs.Inaddition,aswewill
practicalapplication,suchasexpectedregretDCOPmodel[5], show later, the number of out-of-service users will increase
multi-variable agent decomposition model [6] and dynamic since a user always selects a best-rate thereby taking up a
large number of RBs and leaving less for others. Nguyen and A. DCOP
Bao [22] proposed a game theory based method in which the
The definitions of DCOP have a little difference in differ-
users are modeled as players who participate in the game ent literatures [26], [4], [5] 1. In this paper, we formalize
of acquiring resources. The best solution is the one which
the DCOP as a four tuples model < A,V,D,C > where
can satisfy Nash equilibrium (NE). Loosely speaking, such
A = {a ,a ,...,a } consists of a set of agents, V =
1 2 |A|
solution is only a local optima. In addition, it is difficult to
{v ,v ,...,v } is the set of variables in which each variable
1 2 n
guarantee the quality of the solution.
v ∈ V only belongs to an agent a ∈ A. Each variable has
i
thereisnoresearchofmodelinguserassociationproblemas a finite and discrete domain where each value represents a
MAS. However, some similar works have been done focusing possible state of the variable. All the domains of different
on solving the resource management problem in the field variables consist of a domain set D ={d ,d ,...,d }, where
1 2 n
of wireless networks or cognitive radio network [23], [24], d is the domain of v . A constraint c∈C ={c ,c ,...,c }
i i 1 2 |C|
[25] by DCOP framework. These methods can not be directly is defined as a mapping from the assignments of m variables
appliedtouserassociationproblemmainlyduetothescaleof to a positive real value:
the models for these practical applications is relatively small.
For instance, Monteiro [24] formalized the channel allocation R(c):d ×d ×···d →R+ (1)
i1 i2 im
inawirelessnetworkasaDCOPwithnomorethan10agents
considered in the simulation parts. However, the amount of The purpose of a DCOP is to find a set of assignments of
users and resource included in a HetNet is always hundreds all the variables, denoted as X∗, which maximize the utility,
and thousands. In this case, a good modeling process along namely the sum of all constraint rewards:
with a suitable DCOP algorithm is necessary. According to
(cid:88)
the in-depth analysis above, it motivates us to explore a good argmax R(c) (2)
way to solve user association problem by DCOP. The main X∗ C
contributions of this paper are as follows:
B. System Model of User Association Problem
• An ECAV (Each Connection As Variable) model is pro- Consider a k-tier HetNet where all the BSs in the same tier
posedformodelinguserassociationproblemusingDCOP havethesameconfigurations.Forexample,atwo-tiernetwork
framework. In addition, we introduce a parameter η with including a macro BS (B ) and a femto BS, (B ), is shown in
1 2
which we can control the scale (the number of variables Fig.1.ThesetofallBSsisdenotedasB ={B ,B ,...,B }
1 2 NB
and constriants) of the ECAV model. where NB is the total number of BSs. All the BSs in the
• A DCOP algorithm based on Markov chain (MC) is kth tier transmit with the same power Pk. The total number
proposed which is able to balance the time consumption of users is denoted by NU and the set of all users is U =
and the quality of the solution. {U ,U ,...,U }.
1 2 NU
• The experiments are conducted which show that the With OFDMA technology in LTE-Advanced networks, the
results obtained by the proposed algorithm have supe- resource, time-frequency, is divided into blocks where each
rior accuracy compared with the Max-SINR algorithm. block is defined as a resource block (RB) including a certain
Moreover, it has better robustness than the LDD based time duration and certain bandwidth [17]. In this paper, the
algorithm when the number of users increases but the resource configured at each BS is in the format of RB so
available resource at base stations are limited. that its available RBs are decided by the bandwidth and the
scheduling interval duration allocated to that BS. We assume
The rest of this paper is organized as follows. In Section
theBSsintheHetNetsharethetotalbandwidthsuchthatboth
II, the definition of DCOP and the system model of user
intra- and inter-tier interference exist when the BSs allocate
associationproblemalongwithitsmixedintegerprogramming
RBs to the users instantaneously.
formulationarebrieflyintroduced.InSectionIII,weillustrate
Assuming the channel state information is available at the
the ECAV-η model. After that, a MC based algorithm is
BSs, the SINR experienced by user U , served by B in the
j i
designed in section IV. We explore the performance of the
kth tier is given by
DCOP framework by comparing with the Max-SINR and
LDD methods in Section V. Finally, Section VI draws the P g
SINR = k ij (3)
conclusion. ij (cid:80) P g +BN
Bl∈B/{Bi} k ij 0
In (3), g is the channel power gain between U and B ,
ij j i
B/{B } represents all the BSs in B except B , B is the
i i
II. PRELIMINARY bandwidthandN isnoisepowerspectraldensity.Thechannel
0
power gain includes the effect of both path loss and fading.
Path loss is assumed to be static and its effect is captured in
This section expounds the DCOP framework and system
modelofuserassociationproblemalongwithitsmixedinteger
1[26] formalized the DCOP as a three tuples model. [4] adopted a four-
programming formulation. tuplesmodelwhile[5]usedafivetuplesmodel
(a) AsimpleinstanceofHetNet (b) ECAVmodel
Fig.1. AsimpleinstanceofHetNetwithECAVmodeling
the average value of the channel power gain, while the fading commonly used formulation as follows
is assumed to follow the exponential distribution.
From the above, the efficiency of user U powered by BS (cid:88)(cid:88)
j maximize F = x r (8a)
ij ij
B , denoted as e , is calculated as
i ij i∈Bj∈U
(cid:88)
s.t. x r (cid:62)γ,∀U ∈U (8b)
ij ij j
eij =log2(1+SINRij) (4) i∈B
(cid:88)
x n (cid:54)N ,∀B ∈B (8c)
ij ij i i
GiventhebandwidthB,timedurationT andthescheduling j∈U
intervalΓ configuredateachRB,weattaintheunitrateatUj (cid:88)x (cid:54)1,∀U ∈U (8d)
ij j
upon one RB as follows
i∈B
n ∈{0,1,...,N },∀B ∈B,∀U ∈U (8e)
ij i i j
BTe
uij = Γ ij (5) xij ∈{0,1},∀Bi ∈B,∀Uj ∈U (8f)
The first constraint ensures the rate QoS requirement from
On the basis of formula (5), the rate received at U with users. Constraint (8c) indicates that the amount of RBs con-
j
n RBs provided by B in the kth tier is sumed at the same BS is no more than the total RBs N
ij i i
configurated at the BS. Constraint (8d) guarantees one user
associated with a unique BS. Constraint (8e) guarantees the
r =n u (6)
ij ij ij number of RBs a BS allocates to a user falls within the range
from zero and N . The last constraint (8f) guarantees the
i
Associated with each user is a quality-of-service (QoS) connection between a user and a BS has two states denoted
constraint. This is expressed as the minimum total rate the by a binary variable. The objective function (8a) refers to the
user should receive. Denoting the rate requiremnt of the jth sum of rate rather than a function acted on the rate such as
user by γj, the minimum number of RBs required to satisfy U(xijrij) (e.g. U(x)=log(1+x)) in some references. Gen-
γj is calculated by: erally, two phases are needed to gain the solution including:
1) transforming original problem into a satisfied one through
nij =(cid:100)γj (cid:101) (7) relaxing Constraint (8e) by nij ∈ {0,nimjin}; 2) the left RBs
min u ineachBSwillbeallocatedtousersinordertomaximizethe
ij
objective function.
in which (cid:100)·(cid:101) is a ceiling function.
III. FORMULATIONWITHDCOP
In this section, we expound and illustrate the ECAV model
C. Mixed Integer Programming Formulation along with its modified version ECAV-η.
A. ECAV Model
The formulations of user association problem by mixed
linear programming are similar in a series of papers (see Before giving the formulation based on DCOP, we firstly
the survey literature [19]). in this paper, we present a more introduce the definition of candidate BS:
Definition1. wedeclareB ,i∈BisacandidateBSofU ,j ∈ thesame,denotedas{B ,B },whilethecandidateBSsofU
i j 1 2 3
U if the rate at U is above the threshold γ with nij RBs and U are respectively {B } and {B }. We assume the total
j min 4 1 2
provided by B . Simultaneously, nij should be less than the RBsconfiguratedatB andB is8and10.Forsimplicity,we
i min 1 2
total number of RBs (N ) configurated at B . assumetherateofeachuserservedbyoneRBprovidedbyB
i i 1
is0.8bit/s.And1bit/sofeachuserisservedbyB .Then,the
After confirming the set of candidate BSs of U ,j ∈ U, 2
j ECAVmodelisshowninFig.1(b).Therearetwoagentsnamed
denoted by, CBj, Uj sends messages to its candidate BSs so A and A . The variables in A are V1,V2 and V3 where Vj
thateachB ,i∈NBgetsknowledgeofitspossibleconnected 1 2 1 1 1 1 i
i refers to a connection between user U and B . Similarly, the
users. We define each possible connection between U and its j i
j variablesinA areV1,V2andV4.Assumingthethresholdrate
candidate BS B as a variable, denoted by Vj. In this case, 2 2 2 2
i i is 3 bit/s, we can calculate that at least (cid:100) 3 (cid:101)=4 RBs needed
all the variables are divided into NB groups according to the 0.8
for the users served by B , thus the domain of each variable
1
potentialconnectionbetweenusersanddifferentBSs.Thedo-
in A is {0,4,...,8}. Also, the domain of each variable in
mainofeachvariableVj,denotedbyDj ={0,nij ,...,N }, 1
i i min i A is {0,3,...,10}. The black lines in each agent are two
where Vj = 0 if no RB is allocated to U , otherwise, 2
i j 3-nry intra-constraints, thus C = {C1 ,C2 }. The
Vj (cid:62) nij . We define each group as an agent. Thus, an intra intra intra
i min red lines connecting two agents are two intra-constraints, thus
n-ary constraint exists among n variables (intra-constraint) to
C ={C1 ,C2 }.WeuseC1 andC1 toillustrate
guaranteethatthereisnooverloadatB .Notethatausermay inter inter inter intra inter
i how the reward of constraint works in different conditions.
have more than one candidate BS, there are constraints (inter-
Considering C1 , the reward is −∞ when all the variables
constraints) connecting the variables affiliated to different intra
associated with C1 have the same assignment 4. Thus the
agents on account of the assumption that a unique connection intra
total number of RBs consumed by three users is 12 which is
exists between a user and a BS. Generally speaking, the
more than 8 RBs configurated at B . Otherwise, the reward
1
utility (objective) function in the DCOP model is the sum
is 0.8 × 4 × 3 = 9.6 (bit/s) calculated according to (6).
of constraint rewards which reflects the degree of constraint
Considering C1 , the reward is −∞ when the assignment
violations. We define the reward R(c) of inter- and intra- inter
of V1 is 3 and the assignment of V1 is 4 because it means
constraints in the ECAV model as follows. For ∀c∈C 2 1
inter U will connect with more than one BSs (B and B ), which
2 1 2
violates the assumption of unique connection. Otherwise, the
(cid:40)
−∞, ∃Vj1,Vj2 ∈ψ(c),Val(Vj1/j2)>(09a) reward is 0 (9b)). If there is no constraint violated, the final
R(c)= i i i
utility calculated by the objective function is the total rate in
0, Otherwise (9b)
the whole HetNet (constraint (10b)).
For ∀c∈C
intra
B. ECAV-η Model
R(c)=  −∞, (cid:80)Vij∈ψcVal(Vij)>(1N0ia) ageTnhtes asncadlecoonfstarnainEtCs,AiVs remlaotdeedl,toretfheerrninugmtboerthoefnuusemrsb,eBr Sosf
(cid:88)
 Vji∈ψ(c)rij, otherwise (10b) and the candidate BSs hold at each user. However, some
candidate BSs of the user can be ignored because these
In constraint (9a), ψ(c) is the subset of variables connected BSs are able to satisfy the requirement of the user but with
by constraint c. Val(Vij) represents the assignment of Vij. A massive RBs consumed. It can be illustrated by the number
reward(weuse−∞inthispaper)isassignedtotheconstraints of RBs a BS allocate to a user is inversely proportional to the
if there at least two variables are non-zero at the same time geographicaldistancebetweenthem.Inthisway,weintroduce
(unique connection between a user and a BS). Otherwise, the a parameter η with which we limit the number of candidate
reward is equal to zero. In constraint (10a), the reward is −∞ BSs of each user is no more than η. The following algorithms
once there is a overload at the BS. Otherwise, the reward is present the selection of top η candidate BSs (denoted by CˆB)
the sum of the rates achieved at users. and the modeling process of ECAV-η.
It is easy to find that a variable in the ECAV model Algorithm 1 is the pseudo code for determining CˆB. It
with non-zero assignment covers constraint (8b) and (8e) in is executed by each user distributely. More precisely, a user
the mixed integer programming formulation. Moreover, intra estimates its total candidate BSs CB by the procedure from
and inter-constraints respectively cover constraint (8c) and line 5 to 9. Based on 4 to 7, the candidate BSs of a user
constraint(8d). Therefore, The global optimal solution X∗ is ordered according to the unit number of RBs consumed at
obtained from the ECAV model is consistent with the one such user served by different BSs (from line 22 to 28). The
obtained from the mixed integer programming formulation, timeconsumptionofAlgorithm1mainlyconsistsoftwoparts.
denoted as X 2. One is the dermination of CB with time complexity O(NB).
To better understand the modeling process, we recall the The other is the ordering operation with time complexity
instance in Fig.1 where the candidate BSs of U1 and U2 are O(NB2). As a result, the total time expended of Algorithm 1
is O(NB+NB2). With CˆB, we present the pseudo code in
2WesayX∗isconsistentwithX whenthetotalratecalculatedbyobjective
relation to the building of ECAV-η model.
function (2) and (8a) is equal. This is because there may be no more than
oneoptimalsolution. As for Algorithm 2, it firstly sets up the agents distributely
(line 6). It takes O(1). After that, each user determines
Algorithm 1 CBˆ of user U ,j ∈U based on η variables, domains as well as inter-constraints from line 8 to
j j
Input: The information of HetNet (B, U, γ, η) 14. This is also carried out in parallel with O(1). Finally, the
Output: The set of candidate BS CˆB based on η intra-constraintsareconstructedbyeachagentwithO(1)(line
Initialize: CB ←φ, CBˆ ←φ 16 to 17). The total time complexity is O(3).
j j
procedure GETALLCANDIDATEBS
IV. MARKOVCHAINBASEDALGORITHM
for i∈B,j ∈U do
if r ≥γ then DCOP, to some degree, is a combinatorial optimization
ij
CB (cid:83){B } problem in which the variables select a set of values to
j i
end if maximizetheobjectivefunctionwithoutorwiththeminimum
end for constraintviolation.WeuseS todenotethesetofallpossible
end procedure combination of assignments of variables. Also, we call each
procedure GETPARTIALCANDIDATEBS elements∈Sasacandidatesolution.ConsideringanECAV-
BubbleSort (CB ) (cid:46) sorting by SINR η model in which the four tuples are as follows:
j
if |CBj|>η then • A={A1,A2,...,ANB}
for n from 1 to η do • V ={Vij| a connection between Uj and Bi}
CBˆnj ←CBnj (cid:46) get η candidate BSs • D ={Dij|Dij ={0,Nmijin}}
end for • C =Cinter∪Cintra
else We are able to rewrite the model in the following way:
CBˆ ←CB
j j
end if
(cid:88)
end procedure max Val(Vij) (11a)
s∈S
procedure BUBBLESORT(CBj) i∈|B|,j∈|U|
for m from 1 to |CBj| do s.t.(cid:64)Vj1,Vj2 ∈ψ(c),Val(Vj1/j2)>0 (11b)
i i i
forinf SfrIoNmR|C(BCBj|nt)o>mS+IN1Rdo(CBn−1) then (cid:88) rij >Nmijin (11c)
j j
exchane CBnj and CBnj−1 Vij∈ψ(c)
end if After that, a convex log-sum-exp approximation of (11a) can
end for be made by:
end for
end procedure
(cid:88) 1 (cid:88) (cid:88)
max Val(Vj)≈ log( exp(β Val(Vj))
s∈S i β i
i∈|B|,j∈|U| s∈S i∈|B|,j∈|U|
(12)
where β is a positive constant. We then estimate the gap be-
Algorithm 2 ECAV-η
tween log-sum-exp approximation and (11a) by the following
Initialize:
proposition in [27]:
{A,V,D,R}←φ (cid:46) elements in DCOP model
R←φ (cid:46) utility upon each constraint Proposition2. Givenapositiveconstantβ andnnonnegative
procedure SETECAV values y1,y2,...,yn, we have
for i∈B do
(cid:83)
A {A }
i
n
end for 1 (cid:88)
max y ≤ log( exp(βy ))
forV∀j(cid:83)∈{VUj,}m∈CBˆj do i=1,2,...,n i β i=1 i (13)
i 1
DDij(cid:83)←{D{j0},nimjin} ≤i=m1,2a,.x..,nyi+ βlogn
i
C(cid:83){Cj } In addition, the objective function (11a) has the same optimal
inter
R(cid:83){R(Cj )} (cid:46) based on (9b), (9a) value with the following transformation:
inter
end for (cid:88) (cid:88)
max p Val (Vj)
forCi∈(cid:83){ACido } ps(cid:62)0s∈S si∈|B|,j∈|U| s i (14)
R(cid:83){Rin(tCria )} (cid:46) based on (10a), (10b) s.t.(cid:88)p =1
intra s
end for s∈S
end procedure in which (cid:80) Val (Vj) is the reward with a
i∈|B|,j∈|U| s i
candidate solution s. For simplicity, we use g =
β
1log((cid:80) exp(β(cid:80) Val(Vj)). Hence, on the ba- BSs with their transmission powers respectively 46, 35, and
β s∈S i∈|B|,j∈|U| i
sis of formulations (12) and (13), the estimation of (11a) can 20 dBm. The macro BS is fixed at the center of the square,
be employed by evaluating g in the following way: and the other BSs are randomly distributed. The path loss
β
between the macro (pico) BSs and the users is defined as
max(cid:88)p (cid:88) Val (Vj)− 1 (cid:88)p logp L(d) = 34+40log10(d), while the pass loss between femto
ps(cid:62)0s∈S si∈|B|,j∈|U| s i β s∈S s s (15) rBeSpsreasenndtsustehres EisucLli(dde)an=d3i7sta+nc3e0lboegt1w0(ede)n. tThheeBpSasramanedterthde
(cid:88)
s.t. p =1 users in meters. The noise power refers to the thermal noise
s
s∈S at room temperature with a bandwidth of 180kHz and equals
Assumings∗ andλ∗ aretheprimalanddualoptimalpoints to-111.45dBm.Onesecondschedulingintervalisconsidered.
Without special illustration, 200 RBs are configured at macro
with zero duality gap. By solving the Karush-Kuhn-Tucker
BS, as well as 100 and 50 RBs are configured at each pico
(KKT)conditions[27],wecanobtainthefollowingequations:
and femto BS. In addition, all the results are the mean of 10
instances.
(cid:88) 1 1
Val (Vj)− logp − +λ=0,∀s∈S
s i β s∗ β B. Experimental Results
i∈|B|,j∈|U|
(16a) We firstly discuss the impact of different assignments of η
(cid:88) on the performance of ECAV model from the point of view
p =1 (16b)
s∗ of the runtime and the quality of solution. More precisely,
s∈S we generate different number of users ranging from 20 to
λ≥0 (16c)
100 with the step interval of 10. The time consumed by the
Then we can get the solution of p as follows: MC based algorithm is displayed in Fig.2. It is clear to see
s∗
that more time is needed when the number of users increases.
exp(β(cid:80) Val (Vj)) Also, the growth of runtime is depended on the value of η.
i∈|B|,j∈|U| s i
ps∗ = (cid:80) exp(β(cid:80) Val (Vj)) (17) Specially,thereisanexplosivegrowthofruntimewhenweset
s∈S i∈|B|,j∈|U| s i ηfromfourtofive.Aspreviouslystated,thisiscausedbymore
On the basis of above transformation, the objective is candidate BSs considered by each user. However, the quality
to construct a MC with the state space being S and the of the solutions with different values of η is not obviously
stationary distribution being the optimal solution p when improved according the results in Table I. For instance, the
s∗
MC converges. In this way, the assignments of variables will average rate achieved at each user is only improved no more
be time-shared according to p and the system will stay than0.1bit/swhenthenumberofusersare100withthevalues
s∗
in a better or best solution with most of the time. Another of η are 3 and 5. It is difficult to make a theoretical analysis
important thing is to design the nonnegative transition rate of the realationship between η and the quality of the solution.
q between two states s and s(cid:48). According to [28], a series We leave this research in future works.
s,s(cid:48)
ofmethodsareprovidedwhichnotonlyguaranteetheresulting From above analysis, we set η = 3 in the following
MC is irreducible, but also satisfy the balance equation: experiments in order to balance the runtime and performance
p q =p q .Inthispaper,weusethefollowingmethod: ofthesolution.Inaddition,wetesttheperformanceoftheMC
s s,s(cid:48) s(cid:48) s(cid:48),s
based algorithm comparing with its counterparts Max-SINR
(cid:88)
q =α[exp(β Val (Vj))]−1 (18) and LDD based algorithms.
s,s(cid:48) s i
i∈|B|,j∈|U|
TABLEI
The advantage of (18) is that the transition rate is indepen- THEAVERAGERATE(BIT/S)ACHIEVEDATEACHUSER
dent of the performance of s(cid:48). A distributed algorithm, named
Wait-and-Hp 3 in [28], is used to get the solution after we Users η=1 η=2 η=3 η=4 η=5
transform DCOP into a MC. However, as the existence of 50 12.47 12.82 13.10 13.22 13.59
inter- and intra- constriants in, a checking through the way of 80 8.12 8.37 8.55 8.68 8.77
messagepassingismadeinordertoavoidconstraintviolation. 100 6.11 6.37 6.73 6.81 6.83
V. EXPERIMENTALEVALUATION In Fig.3, we check the connection state between 200 users
A. Experimental Setting and BSs in different tiers. A phenomenon we can observe
from the figure is that there are more or less some users out
In this section, we test the performance of the MC based
of service even we use different kinds of algorithms. It is not
algorithmwithdifferentassginmentsofη intheECAVmodel.
only caused by the limited resource configured at each BSs,
A simulated environment including a three-tiers HetNet cre-
but also related to the positions of such kinds of users. They
ated within a 1000m×1000m square is considered. In the
arelocatedattheedgeofthesquareandhardlyservedbyany
system, there is one macro BS, 5 pico BSs and 10 femto
BS in the system. Further, more users are served by macro
3tosavespace,weadvisereaderstogetmoredetailsfromliterature[28] BS in Max-SINR algorithm because a larger SINR always
eixsts between the users and macro BS. As a result, the total
non-servedusersinMmax-SINRalgorithmsaremorethanthe
othertwoifthereisnoschemeforallocatingtheleftresource.
On the other hand, the number of non-served users in MC are
less than LDD when η = 3 since the user U will select a
j
BS B with the maximal QI in each iteration of the LDD
i ij
algorithm. In other words, the users prefer to connect with
a BS which can offer better QoS even when more resources
are consumed. Therefore, some BSs have to spend more RBs
which leads to the resource at these BSs being more easily
used up.
Fig.2. TheruntimeofECAV-η(η=1−5)withdifferentnumberofusers
intheHetNet
Fig.5. TheCDFsoftherateacheivedatusers
Fig.3. Theconnectionbetween usersand BSsaccordingto theallocation
schemeobtainedthroughdifferentalgorithms
Fig. 6. The total rate against the number of RBs held at the macro base
station
InFig.4,weproduceastatisticofthenumberofnon-served
users when we change the total number of users configured
in the HetNet. The average number of non-served users for
each algorithm along with the standard deviation is presented
inthefigure.ComparedwithFig.3,amoreclearresultsimply
that more than 60 (at worst, around 70) non-served users in
the Max-SINR algorithm. The LDD based algorithm comes
the second with approximate 20 users. The best resutls are
Fig.4. Non-servedusersintheHetNetwithadifferentnumberofusers
obtained by the MC algorithm with no more than 20 users
even the total users in the HetNet is 240.
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