Throughput Optimizing Localized Link Scheduling for Multihop Wireless Networks Under Physical Interference Model

3.6 MB·English

by  Yaqin Zhou

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Throughput Optimizing Localized Link Scheduling
for Multihop Wireless Networks Under Physical
Interference Model
Yaqin Zhou, Xiang-Yang Li, Senior Member, IEEE, Min Liu, Member, IEEE, Xufei Mao, Member, IEEE,
Shaojie Tang, Member, IEEE, Zhongcheng Li, Member, IEEE
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Abstract—We study throughput-optimum localized link Thelinkschedulingproblemhasbeenstudiedwithdifferent
1
0 scheduling in wireless networks. The majority of results on optimizationobjectives, e.g., throughput-optimumscheduling,
link scheduling assume binary interference models that simplify
2 minimum length scheduling. Our study mainly focuses on
interferenceconstraintsinactualwirelesscommunication.While
maximizing throughput in multihop wireless networks. Nev-
n the physical interference model reflects the physical reality
a more precisely, the problem becomes notoriously harder under ertheless, a common primary issue here is to find a set of
J the physical interference model. There have been just a few conflict-free links with regard to the interference models.

4 existingresultsonlinkschedulingunderthephysicalinterference Interference models matter significantly in scheduling al-
2 model,andevenfeweronmorepracticaldistributedorlocalized gorithm design and analysis. It is well known that the time
scheduling. In this paper, we tackle the challenges of localized
complexity of a throughput-optimum scheduling that tries
] link scheduling posed by the complex physical interference
I constraints. By cooperating the partition and shifting strategies to find a maximum weighted independent set of links, is
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into the pick-and-compare scheme, we present a class of local- polynomial under the 1-hop interference model in switched
s. ized scheduling algorithms with provable throughput guarantee wired networks, whereas it becomes NP-hard under more
c subject to physical interference constraints. The algorithm in general interference models in wireless networks [4]. Here
[ the linear power setting is the first localized algorithm that
achieves at least a constant fraction of the optimal capacity the weight associated with every link is its corresponding
2 regionsubjecttophysicalinterferenceconstraints.Thealgorithm queue length. Motivated by providing efficient solutions for
v in the uniform power setting is the first localized algorithm the NP-hard problem, numerous low complexity scheduling
8 with a logarithmic approximation ratio to the optimal solution. algorithmswith suboptimalthroughputguarantee[5], [4], [6],
3 Our extensive simulation results demonstrate correctness and
[7], [8], [9], [10], [11], [12], [13], have been proposed in
7 performance efficiency of our algorithms.
4 literature.
. Index Terms—localized link scheduling, physical interference Despite of the numerous significant results gained for the
1
model, maximum weighted independent set of links (MWISL),
problem, most of them assume simple binary interference
0
capacity region.
3 models,e.g.,hop-based,range-based,andprotocolinterference
1 models [14]. Under this category of interference model, a
v: I. INTRODUCTION set of links are conflict-free if they are pairwise conflict-free.
i AS a fundamental problem in wireless networks, link Conflictoftransmissionsontwodistinctlinksispredetermined
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scheduling is crucial for improving networking perfor- independently of the concurrent transmissions of other links.
r mances through maximizing throughput and fairness. It has Thus, interferencerelationshipsbased on these modelscan be
a
recently regained much interest from networking research represented by conflict graphs, and we can leverage classic
community because of wide deployment of multihop wire- graph-theoreticaltools for solutions. However, in actual wire-
less networks, e.g., wireless sensor networks for monitoring less communication, interference constraints among concur-
physical or environment[1] [2] through collection of sensing rent transmissions are not local and pairwise, but global and
data [3]. Generally, link scheduling involves determination of additive. Conflict of distinct transmissions is determined by
which links should transmit at what times, what modulation the cumulativeinterferencefrom all concurrenttransmissions,
and coding schemes to use, and at what transmission power which is often depicted by the physical interference model,
levels should communication take place [4]. In addition to e.g.,the Signal-to-Interference-plus-NoiseRatio (SINR) inter-
its great significance in wireless networks, developing an ference model. But due to the simplification of binary inter-
efficient scheduling algorithm is extremely difficult due to ference models on physical reality, the corresponding results
the intrinsically complex interference among simultaneously no longer hold under the more realistic physical interference
transmitting links in the network. model. Recent results [15], [16] have shown that throughput
differssignificantlyunderthe two kindsofmodels.Moreover,
Yaqin Zhou, Min Liu, and Zhongcheng Li are with Research Center of
the global and additive nature of the physical interference
Network Technology, Institute ofComputing Technology, Chinese Academy
ofSciences, Beijing, China. modeldrivesprevioustraditionaltechniquesbased on conflict
Xiang-YangLiandShaojieTangarewithDepartmentofComputerScience, graphsinapplicableortrivial.Consequently,designingandan-
Illinois Institute ofTechnology, Chicago, IL,USA.
alyzing scheduling algorithms under the physical interference
Xufei Mao is with TNLIST, School of Software, Tsinghua University,
Beijing, China. model becomes especially challenging.
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Some recent research results [17], [14], [18], [19], [20], is also applicable. We prove that our revised algorithm
[21], [22], [23], [24] have addressed a few challenges related canachieveO(log V )fractionofcapacityregionwhere
| |
tothelinkschedulingproblemunderthephysicalinterference V is the number of nodes.
| |
model. To the best of our knowledge, however, all of these, 3) We conduct extensive simulations to evaluate our al-
withthroughputmaximizationorotheroptimizationobjectives gorithms. The simulation results demonstrate the cor-
suchasaminimumlengthschedule[14],justfocusoncentral- rectness of our algorithms and corroborate our theo-
ized implementation.Distributed or evenlocalized scheduling retical analysis. The comparisons with two centralized
under the physicalinterference model is more demandingout algorithms in terms of average throughput performance
of practical relevance. further show performance efficiency of our algorithms.
Though[25],[26],[27]considerdistributedimplementation
The remainder of the paper is organized as follows. In
of centralized algorithms, they fail to provide an effective
SectionII,wedefinetheexactsystemmodelsforourproblem.
localized scheduling algorithm with satisfactory theoretical
InSectionIII,wedescribethebasicideasofoursolutions,and
guarantee [25] [26] and require global propagation of mes-
the proposed localized scheduling algorithm with theoretical
sages. A scheduling algorithm without theoretical guarantee
performance analysis. In Section IV, we revise our solutions
may cause arbitrarilybad throughputperformance,and global
for the uniform power setting, and propose a localized algo-
propagation of messages throughout network is inefficient in
rithmwitha logarithmicapproximationratio.We evaluateour
terms of time complexity [28], especially for large-scale net-
solutions for both power settings in Section V. In Section VI,
works. This motivatesus to developlocalized link scheduling
we review related works on link scheduling. In Section VII
algorithms with provable throughput performance. Here by a
we discuss some related issues on proposed algorithms. We
localized scheduling algorithm we mean that each node only
conclude this paper in Section VIII.
needsinformationwithinconstantdistancetomakescheduling
decisions, while a distributed algorithm may inexplicitly need
informationfar away. Since just local informationis available II. MODELSANDASSUMPTIONS
for each node to collaborate on schedulings with globally
A. Network Communication Model
coupledinterferenceconstraint,it posessignificant challenges
to designing efficient localized scheduling algorithms with We modela wireless networkbya two-tuple (V,E), where
theoretical throughput guarantee. V denotes the set of nodes and E denotes the set of links.
In this paper we tackle these challenges of practical lo- Eachdirectedlinkl=(u,v) E representsacommunication
∈
calized scheduling for throughput maximization under the request from a sender u to a receiver v. Let l or uv
k k k k
realistic physical interference model with the commonly- denote the length of link l. We assume each node knows its
used linear and uniform power assignment. For the primary own location, which is necessary in the partition and shifting
challenge of decoupling the global interference constraints, process.
on observing that distance dominates the interference, we
partition the links into disjoint local link sets with a certain
B. Interference model
distance away from each other. By this way, we bound the
Under the physical interference model, a feasible schedule
interference from all the other local link sets with a constant
isdefinedasanindependentsetoflinks(ISL),eachsatisfying
so that independent scheduling inside each local link set is
possible. Then we shift the partitions to ensure every link P η uv −κ
getscheduled.Usingthisweimplementlocalizedschedulings. SINRuv =∆ uP· ·ηk wkv −κ+ξ ≥σ,
Furthermore,wenovellycombinethesetwostrategiesintothe w∈Tu w· ·k k
P
pick-and-compare scheme by which we can prove that our where ξ denotesthe ambientnoise, σ denotescertain thresh-
algorithms obtain a theoretical capacity guarantee. old, and T denotes the set of simultaneous transmitters with
u
We summarize our main contributions as follows. u. It assumes path gain η uv −κ 1, where the constant
·k k ≤
1) We proposea localized link schedulingalgorithmunder κ>2ispath-lossexponent,andη isthereferencelossfactor.
the linear power setting with provable throughputguar- We considerthe followingtwo transmissionpowersettings.
antee. We successfully decouple the global interference 1) Linearpowersetting:asenderutransmitstoareceiver
constraints forthe implementationof localized schedul- v always at the power P = c uv β P where c
u
ing, by proving that there exists a constant distance and β are both constant satisfyi·nkg ck> 0≤,0 < β < κ,
between any two disjoint local link sets to ensure inde- and P is the maximum transmission power.
pendentlink schedulinginside a locallinkset underthe 2) Uniform powersetting: alllinksalwaystransmitatthe
linear power setting. We then prove that our algorithm same power P =P.
u
can achieve a constant fraction of capacity region.
We use P and P alternatively to denote the transmitting
2) We further revise the algorithm to work under the l u
power of link l=(u,v).
uniform power setting, and give a detailed analysis on
We also assume all links having a length less than the
theachievablecapacityregion.Whileundertheuniform
maximum transmission radius κ ηP. The distance between
powersetting,weprovethatifthesizeofthescheduling σξ
setforlocallinksetsisupperboundedbyaconstant,the u and v satisfies r uv Rq, where r and R respectively
≤k k≤
solution to decouple the global interference constraints denotes the shortest link length and the longest link length.
We suppose that r and R are known by each node.
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C. Traffic models and scheduling In the rest of the paper we look for solutions to generate
localized link schedulings of approximately optimal weight,
Themaximumthroughputschedulingisoftenstudiedinthe
following models. It assumes time-slotted wireless systems, with a certain constant probability at least.
and single-hop flows with stationary stochastic packet arrival
process at an average arrival rate λ. The vector A(t) = III. THEALGORITHM IN THELINEARPOWERSETTING
l
Al(t) denotes the number of packets arriving at each link Inthissection,wefocusonthedesignoflocalizedschedul-
{ }
in time slot t. Every packet arrival process A (t) is assumed ing algorithm for the linear power setting. We firstly demon-
l
tobei.i.dovertime.We alsoassumeallpacketarrivalprocess strate the basic ideas of the algorithm design before involved
Al(t) have bounded second moments and they are bound by in the details of implementation.
A , i.e., A (t) A , l E. Let a vector 0,1 E
max l max | | A. Basic idea
≤ ∀ ∈ { }
denotea schedule S(t) at each time slot t, where S (t)=1 if
l
The basis of our idea is to create a set of disjoint local
link l is active in time slot t and S (t)=0 otherwise. Packets
l
link sets in which the scheduling can be done independently
departure transmitters of activated links at the end of time
without violating the global interference constraint. The de-
slots. Then, the queue length (it is also referred to as weight
coupling of the global interference constraint is based on
or backlog) vector Q(t) = Q(t) evolves according to the
l
{ } the fact that distance dominates the interference between two
following dynamics:
distinctlinks. Thatis, if a transmitting link is placeda certain
Q(t+1)=max 0,Q(t) S(t) +A(t+1). distance away from all the other transmitting links, the total
{ − }
interference it receives may get bounded. We prove later that
The throughput performance of link scheduling algorithms
thisassumptionholdsactually.Based onthis, thenwe employ
ismeasuredbyasetofsupportablearrivalratevectors,named
thepartitionstrategiestodividethenetworkgraphintodisjoint
capacity region or throughputcapacity. Thatis, a scheduling
local areas such that each local area is separated away by
policy is stable, if for any arrival rate vector in its capacity
a certain distance to enable independent local computation
region [12],
of schedulings inside every local area. Links lying outside
lim E[Q(t)]< .
t ∞ localareaswillkeepsilentto ensureseparationoflocalareas.
→∞
A throughput-optimalschedulingalgorithmcan achievethe As links lying outside local areas cannot remain unscheduled
optimal capacity region. It is well known that the policy all the time or it will induce network instability, we use the
of finding a maximum independent set of links to schedule shifting strategy to change partitions at every time slot to
regardingtotheunderlyinginterferencemodelsisthroughput- makesurethateverybackloggedlinkwillbescheduled.These
optimal[29].Unfortunately,theproblemof findinga MWISL locally computed scheduling link sets compose a new global
itself is NP-hard generally [30]. Thus we have to rely on schedule (t) at every time slot t.
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approximation or heuristic methods to develop suboptimal Inlightofthepick-and-comparescheme,wechooseamore
schedulingalgorithmsrunningin polynomialtime.A subopti- weightedschedule,denotedasS(t), betweena newgenerated
malschedulingpolicycanjustachieveafractionoftheoptimal schedule (t)andthelast-timescheduleS(t 1)usingQ(t).
X −
capacityregion.Thefractionallyguaranteedcapacityregionis Meanwhile, by Proposition 1, if we guarantee that
depicted by efficiency ratio γ [4]. P(S(t) Q(t) γS (t) Q(t)) δ (1)
∗
Asuboptimalschedulingpolicywithefficiencyratioγ must · ≥ · ≥
find a γ-approximation scheduling at every time slot t to for some constant γ > 0, δ > 0, the queue length vector
achieveγ timesoftheoptimalcapacityregion[31].Itremains Q(t) will eventually converge to a stable state. Then we can
difficult to achieve in a decentralized manner. The pick-and- get a constant approximation ratio scheduling policy for the
compare approach proposed in [5] enables that we just need optimal.
to find a γ-approximationscheduling with a constant positive
B. Detailed description
probability.Thebasic pick-and-compare[5] worksasfollows.
At every time slot, it generates a feasible schedule that has a Now we describe the details of our method.
constant probability of achieving the optimal capacity region. We firstgivea briefdescriptionofthepartitionandshifting
If the weight of this new solution is greater than the current strategies [11], as illustrated in Fig. 1. We partition the plane
solution, it replaces the current one. Using this approach intocellswithside lengthd=R,usinghorizontallinesx=i
achieves the optimal capacity region. The proposition below andverticallinesy =j forallintegersiandj.Averticalstrip
further extends this approach to suboptimal cases. withindexiis (x,y)i<x i+1 .Similarly,wedefinethe
{ | ≤ }
Proposition 1: ( [10]) Given any γ (0,1], suppose that horizontalstripj.Letcell(i,j)denotetheintersectionareaofa
∈
analgorithmhasaprobabilityatleastδ >0ofgeneratingan verticalstripiandahorizontalstripj.Asuper-subSquare(i,j)
independentset (t) of links with weight at least γ times the is the set of cells: cell(x,y)x [i K +a ,(i+1) K +
t
X { | ∈ ∗ ∗
weightoftheoptimal.Then,capacityγ Λ canbeachievedby a ),y [j K+b ,(j+1) K+b ) , anda sub-square(i,j)
t t t
· ∈ ∗ ∗ }
switching links to the new independent set when its weight is insideitisthesetofcells: cell(x,y)x [i K+a +M,(i+
t
{ | ∈ ∗
larger than the previous one(otherwise, previous set of links 1) K + a M),y [j K + b + M,(j + 1) K +
t t
∗ − ∈ ∗ ∗
will be kept for scheduling). The algorithm should generate b M) . The corresponding link set Y (or L ) consists
t ij ij
− }
the new scheduling S(t) from the old scheduling S(t 1) of links with both ends inside super-subSquare(i,j) (or sub-
−
and current queue length Q(t). square(i,j)). Let constant K = 2M +J, 0 a ,b < K.
t t
≤
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TABLE I: Summary of notations
J sidelengthofsub-square Lij linksetofsub-square(i,j) Yij linksetofsuper-subSquare(i,j)
K sidelengthofsuper-subSquare Xij(t) newscheduling forLij att Sij(t) scheduling forYij att
Zi locallinkset OPTi∗j(t) localoptimalMWISLforLij S∗(t) globaloptimalMWISLatt
R longestlinklength Si∗j(t) intersection ofLij andS∗(t) S(t) globalscheduling attimeslott
d sidelength ofcell kuvk linklength rS(l) relative interference lgetfromlinksetS
aS(l) affectness lgetfromlinksetS Q(t) queuelengthvector W(S) weightoflinksetS
ISl interference link lsuffered fromlinkset S Iml ax themaximuminterference lcanbear Imax theminimumofIml ax
nodethen participatesin computingthe new localscheduling,
(cid:54)(cid:88)(cid:83)(cid:72)(cid:85)(cid:16)(cid:86)(cid:88)(cid:69)(cid:54)(cid:84)(cid:88)(cid:68)(cid:85)(cid:72) (cid:54)(cid:88)(cid:69)(cid:16)(cid:86)(cid:84)(cid:88)(cid:68)(cid:85)(cid:72) (cid:54)(cid:88)(cid:83)(cid:72)(cid:85)(cid:16)(cid:86)(cid:88)(cid:69)(cid:54)(cid:84)(cid:88)(cid:68)(cid:85)(cid:72)(cid:54)(cid:88)(cid:69)(cid:16)(cid:86)(cid:84)(cid:88)(cid:68)(cid:85)(cid:72)
denoted as (t), for its sub-square(i,j) using the weight
ij
X
Q(t).LetS (t 1)bethesetoflinksfromS(t 1)(theglobal
ij
− −
solutionattime slott 1)fallingin the super-subSquare(i,j)
−
instead of sub-square(i,j). If S (t 1) Q(t) > (t)
ij ij
− · X ·
Q(t), let S (t)=S (t 1), else S (t)= (t), the global
ij ij ij ij
− X
solution is the union of S (t) from all super-subSquares.
ij
The pseudo-codes are given in Algorithm 1. Since every
nodeknowsthelocalityfromwhichitwillcollectinformation,
(cid:71) (cid:45)(cid:71) t (t+1) (cid:71) (cid:45)(cid:71) it then participates the corresponding local computation, and
Shifting
at last it sends (if it is a coordinator) or receives (if not)
(a) Partition(K,at,bt). Here the (b) Partition(K,a(t+1),bt). the results. It may require multihop propagation to collect
gray area is the local area the Change the partition through
andbroadcasttheneededinformationinsidesuper-subSquares.
link set of which participate in shifting to the right by one cell,
computingthenewschedule;the ensuring that links in the white The details are omitted here. Please refer to [27] for de-
links in the white area keep area in previous partitions have tailed implementation. The coordinatorcomputes a new local
silent. opportunity to be scheduled.
scheduling link set by enumeration in constant time 2|Lij|,
becausethesizeofinterference-freelinksforasub-square(i,j)
Fig. 1: The partition and shifting process. The partition
is bounded by a constant (we claim it in Lemma 1).
changes at different time slots by shifting vertically or hor-
izontally by a step of one cell-length.
Algorithm 1 Distributed Scheduling by node v under the
a , b are adjustable variables, referred to as the horizontal
t t linear power setting
and vertical shifting respectively. M is a constant, relating
to the distance for independent scheduling in local areas. We 1: state = White; active = No; Coordinator = No;
will formallydefine it later in Lemma 2. We define the above 2: Calculate which cell node v resides in regarding to the
current partition(K,a ,b );
process as Partition(K,a ,b ). By separately adjusting a , b t t
t t t t
by a step of single cell, we can getK2 differentpartitionsfor 3: if v is the closest node to the center of super-subSquare
then
a plane totally.
Note that herein nodes do not shift, it looks like that 4: Coordinator = Yes;
a virtually partitioned cover having the same size with the 5: end if
originaltopologyshifts verticallyorhorizontally.Initially,the 6: if Coordinator = Yes then
virtualcoverandtheoriginaltopologyofthenetworkcoincide 7: Collect Q(t) and Sij(t−1).
completely.Whenthevirtualcovershifts,nodeslocatedin the 8: Compute Xij(t) in sub-square(i,j);
edge area willcertainly getuncoveredif the distance between 9: if Sij(t−1)·Q(t)>Xij(t)·Q(t) then
the edge of topology and the nearest nodes is less than Kd. 10: Sij(t)=Sij(t−1);
To tackle this, we just need to assume a larger virtual cover, 11: else
e.g.,thedistancebetweentheedgeoftopologyandthenearest 12: Sij(t)=Xij(t);
nodes is equal or greater than Kd. 13: end if
Then at each time slot nodes cooperate to compute a 14: Broadcast RESULT(Sij(t)) in super-subSquare(i,j);
15: end if
globally feasible scheduling as follows.
16: if state = White then
Time slot t = 0: Every node first decides in which
cell it resides by a partition using (a ,b ) = (0,0); then it 17: if receive message RESULT(Sij(t)) then
participates in the process of comput0ing0a local scheduling 18: if v ∈Sij(t) then
19: state = Red; active = Yes;
S (0) for the sub-square(i,j) it belongs to. Let the solution
ij 20: else
S(0) of time slot 0 be the union of the local solutions S (0)
ij 21: state = Black; active = No;
for all sub-squares.
22: end if
Timeslott 1:Everynodedecidesinwhichcellitresides
≥ 23: end if
by a partition starting from (a ,b ). The shifting strategy for
t t 24: end if
a and b works as follows. We let a =t mod K; and b =
t t t t
(b + 1) mod K if a = 0, or it keeps unchanged. Each
t t
5
C. Theoretical analysis and proof Lemma 3: ( [21]) There is a polynomial-timeprotocolthat
takesa p-signalset andrefinesintoa p-signalset, for p >p,
We will then prove that our algorithm is correct and has a ′ ′
increasing the numberof slots by a factor of at most 4(p′)2.
constant approximation ratio to the optimal capacity region. p
Given a network (V,E), supposing Zi is a set of disjoint That is, if a p′-signal set of links can be scheduled simul-
local link sets inside for scheduling, w∪here Z E and Z taneously in a single slot, then we can use some algorithm to
i i
Zj =φ if i=j, for any link l Zi, if l is acti∈vated, then ∩ partitiona p-signalintoatmost4(p′)2 slotssuchthateachset
6 ∈ p
Il =Iiln+Iolut oafpl-isnikgsnianlesvetercyanslobteisreafipn′e-dsiginntaolsaettm.Tohsits4l(emp′m)2apim-spiglineaslthseatt
where Il denotes cumulative interference from all other p ′
throughapolynomial-timealgorithm,e.g.,afirst-fitalgorithm.
activatedlinksinthenetwork,Il denotesthetotalinterference
in Next in Lemma 4 we present a constant approximation
from simultaneously transmitting links inside Z and Il
i out relationship between each locally computed scheduling link
denotes the total interference from transmissions outside.
set and its counterpart of the global optimal scheduling set.
Therefore, we can do independent scheduling inside Z
owuitthsioduetZco,nisfidIelratigoentsobfoIuolnudtefdrobmy acocnocnusrtraenntt, it.rea.n,smissionsi tioLnermatmioato4:thTehweewigehigthotfothfeXiinjt(etr)sehcatsioancsoentsbtayntthaeplporcoaxlilminak-
i out
set L and the global optimal MWISL S (t).
Il (1 ε) Il , Il ε I , 0<ε<1, l Z . ij ∗
in ≤ − · max out≤ · max ∈ i Proof:Thoughthe total interferencethateverylink l can
lWinkesleitnIEmaxcadnentoolteerathnetmduarxinimguamsuinctceersfsefruelnctreatnhsamtitshseiolonn,gaensdt twoelersahtoewatthaatsiutcdcoeesssfulilttlterainnsflmuiesnsicoenobnelcoocmalesco(m1p−utaεt)iIomlnaoxf,
Il representthemaximuminterferencethatanactivatedlink optimal link scheduling set for L .
max ij
l can tolerant during a successful transmission. NormallyanyISLisa1-signalset.Nevertheless,inorderto
We will give a formal statement that Il of each activated keep independence of sub-squares, the locally computed ISL
out
link l is indeed bounded by εImax in Lemma 2. However, to shouldbe a p-signalset, where p is biggerthan 1. Thatis, for
prove Lemma 2, we shall claim the following Lemma 1 in the affectness of a normal ISL it holds that:
advance.
Lemma 1: In the linear power setting, the number of
σ Il
interference-free links for a local link set Zi inside a square aS(l)=cl rl∗(l) max 1, (2)
with a size length JR is bounded by a constant. Let OPTi ·l∗ S ≤ 1−σξ/Pl · Pl ≤
refer to the optimal MWISL for Zi. That is, X∈
whereas the affectness of a locally computed ISL for sub-
(√2JR)κ 1 ξ rβ−κ square must satisfy that,
|OPTi|≤ (1 ε) (cid:20)σ − ·cη (cid:21)+1,
−
σ (1 ε) Il
locWalePorlpeottoimf|:OaTPlhMTisiW|puIbrSodoLefnfioosrteaZvaain,ilatuhbpelpeneiwrnbeAohupanpvdeen,odfixthAe.size of the aSij(t)(l)=cl·l∗∈XSij(t)rl∗(l)≤ 1−· σξ−/Pl · mPalx ≤1−(ε3.)
Lemma 2: In a given network (V,E) under the physical Therefore, by Lemma 3, any p-signal set can be refined
interferencemodelinthelinearpowersetting,iftheEuclidean into at most 4(p′)2 p-signal sets, where p >p. So a normal
distance between any two disjoint local link sets is at least p ′ ′
M R, then activated links in each local link set suffer a MWISL can be refined into 4 1 -Signal link sets at
bou×nded cumulative interference from all other activated link most.Sincethe 1 -Signall(i1n−kε)s2et(r1e−tuεr)nedbyenumeration
sets, i.e., for each activated link l in local link set Zi, ismostweighted(,1s−oε)thelocallycomputedlinkschedulingsets
Iolut≤ε·Imax, 0<ε<1, Xij(t) has a weight
1 (1 ε)2
where M is a constant, satisfying M 2πcηRβ−κ·|OPTi|ub κ W( ij(t)) − W(OPTij(t)). (4)
≥(cid:20) (κ−2)εImax (cid:21). X ≥ 4
Proof: This proof is available in Appendix B. LetS (t)=S (t) L denotethe intersectionby L and
i∗j ∗ ∩ ij ij
To analyze the theoretical performance of our method, we S (t), where S (t) is the globaloptimal MWISL at time slot
∗ ∗
first review the following definitions. t. Let W(S) denote the summed weight of links in a link set
Definition 2: (affectness [21]) The relative interference of S. It is obvious that
link l on l is the increase caused by l in the inverse 4
∗ ∗
coofntvheeniSenINceR, daetfiln,enram(le)ly=rl∗0(.l)Le=t cIll∗=/(Plηkuvkσ−κ). For W(Si∗j(t))≤W(OPTij(t))≤ (1−ε)2W(Xij(t)). (5)
be a constant that indlicates the extentl to w1h−icσhξ/t(hPelηakumvbki−eκn)t Thus it proves the lemma.
Theorem 1: S(t) = S (t) computed by our algorithm is
ij
noise approaches the required signal at receiver t . The ∪
v an independentlink set underthe physicalinterferencemodel
affectness of link l caused by a set S of links, is the sum
inthelinearpowersetting.Theweightof S(t), i.e.,W(S(t)),
of relative interference of the links in S on l, scaled by c , or
l isaconstantapproximationoftheweightoftheglobaloptimal
aS(l)=cl·l∗P∈Srl∗(l). MWISL with probability of at least 1/K2.
Definition 3: (p-signal set [21]) We define a p-signal set to Proof: The proof consists of two phases. We first prove
beonewheretheaffectnessofanylinkisatmost1/p.Clearly, that S(t)= S (t) is an independentset. We nextderivethe
ij
∪
any ISL is a 1-signal set. approximation bound S(t) achieves.
6
Thedistancebetweenanydistinct (t )isnolessthan2M
mn ∗
X
l3 cells. Thenwe considerthe distancebetweena disjointsubset
of Φpq (t ) and a (t ). Since (t ) locates in
l4 Partition pqi′j′ i′j′ ∗ Xmn ∗ Xmn ∗
at time slot t* subS-sqSuare(m,n), which is M cells away from the border of
l1 l2 super-subSquare(m,n),thedistance betweena disjointsubset
of Φpq (t ) and a (t ) is still no less thanM cells.
Partition pqi′j′ i′j′ ∗ Xmn ∗
at time slot
t*-1 CoSmpSrehensively,S(t∗) consistsof disjointsubsets which are
separated by at least M cells.
Note that a disjoint subset of S(t ) does not equalize to a
∗
Sij(t∗) since a Si′j′(t∗ 1) may be reserved completely in
−
Fig. 2: Partitions at time slot t∗−1 and time slot t∗ differentsuper-subSquaresattime slot t∗. Here we denotethe
disjoint subset by ψ (t ), and S(t )= ψ (t ).
Phase I: We rely on induction to infer that at every time i ∗ ∗ i ∗
By Lemma2, foreachlinkl ψ (t ), where ψ (t ) comes
slot S(t) is a union of disjoint activated local link sets that ∈ i ∗S i ∗
are separated by at least M cells from each other. Then we from the former part of equation (9), we have ISl(t∗) ≤ Imlax.
Meanwhile,foreachl ψ (t ),wherel ψ (t )comesfrom
have that S(t) is an independent link set under the physical ∈ i ∗ ∈ i ∗
interferencemodelbyLemma2.Thefollowingarethedetails. thelaterpartof(9),itholdsthat ISl(t∗) ≤Imlax. Thenwehave
For any link l S(t), assuming l S (t), the total inter- Il Il , l S(t ), (10)
∈ ∈ ij S(t∗) ≤ max ∀ ∈ ∗
ferencelsuffersfromalltheothersimultaneouslytransmitting
indicating that S(t ) is an independent set.
links in S(t) is denoted by Il . ∗
S(t) Next we consider situations at time slot t +1. Similarly,
Attimeslot0,everylocalactivatedlinksetS (0)= (0) ∗
is kept 2M cells away from each other, sioj S(0)Xiisj an S(t∗+1) composes of disjoint subsets separated by no less
than M cells. Using the same techniqueas at time slot t , we
independent set by Lemma 2. ∗
can get that S(t +1) is still an independent set.
At time slot 1, either S (0) or (1) is chosen to be a ∗
ij Xij By induction we can infer that S(t) is a union of disjoint
part of S(1). For those super-subSquares whose S (1) =
ij activated subsets separated by M cells at least, thus it is an
Si′j′(0) ∩ Yij(0), their distance is kept at least 2M cells independentset. Hereinwefinish thefirstphraseoftheproof.
away. For those super-subSquares whose S (1) = (1),
ij Xij Phase II: We derivetheapproximationratiobythepigeon-
their distance is also kept at least 2M cells away. And the
hole principle, Lemma 4, and Proposition 1.
distance between the two kinds of link set is at least 2M 1
− Let D(t) denote the link set of the removed strips at
cells away. So S(1) is an independent set.
Partition(K,a ,b ). D (t) represents a subset of S (t), links
At some time slot t , t > 1, for some super-subSquares, t t ∗ ∗
∗ ∗ ofwhichfallinsideD(t),i.e.,D (t)=D(t) S (t).Recalling
S (t ) consists of disjoint subsets from several different ∗ ∗
ij ∗ ∩
Si′j′(t∗ 1) which fall into Yij(t∗), i.e., that there are K2 different partitions for a plane totally. If
− we tried all these partitions in a single slot, each cell(i,j)
Sij(t∗)=S(t∗−1)∩Yij(t∗)= {Si′j′(t∗−1)∩Yij(t∗)}. (6) would appear in the “removed” strips at most 2KM times.
i[′j′ Thus the weight of all D (t) in the K2 partitions should be
∗
For instance, as illustrated in Fig. 2, link sets {l1,l2} 2KMW(S∗(t)), i.e.,
and l ,l respectively get scheduled in two differ-
{ 3 4} K2 1
ent super-subSquares (i.e., super-subSquare(1,2) and super- −
subSquare(2,2)) at time slot t 1, the links l ,l ,l ,l W(Di∗(t))≤2KMW(S∗(t)). (11)
∗ − { 1 2 3 4} i=0
are then get scheduled in the same super-subSquare(1,2) at X
Since actually we only experience one partition at time
time slot t .
∗
slot t, by the pigeonhole principle the corresponding
We then let
Partition(K,a ,b ) has a probability at least 1/K2 to be an
t t
Φiij′j′(t∗)=Si′j′(t∗−1)∩Yij(t∗) (7) optimalpartition,theweightofwhoseremovedlinksinD∗(t)
is no greater than that of other partitions. Instantly we have
for brevity. Each Si′j′(t∗ 1) is kept at least M cells away
from each other, so is eac−h Φij . the following with probability of 1/K2 at least,
i′j′
Clearly, S(t ) can be divided into two separated subsets,
∗
boynenfeowrSlmy(etc∗do)=mbypustSoemd(eXt∗si)ju(bts∗e)t,sio.ef.,S(t∗ −1), the other form(e8d) W(Di∗(t))≤ K12 ·KXi2=−01W(Di∗(t))≤ 2KMW(S∗(t)), (12)
ij
[ij and then,
2M
W( S∗(t))=W(S∗(t) D∗(t)) (1 )W(S∗(t)). (13)
=[pq i[′j′Φpi′qj′(t∗)[ m[n, Xmn(t∗) (9) ∪ ij \ ≥ − K
mn6=pq . FollowingLemma 4,the followingholdswith a probability
    of 1/K2,
Since Φpq (t )isasubsetofS(t 1),itiscomposed
pqi′j′ i′j′ ∗ ∗− (1 2M)W(S∗(t)) 4 W( (t)). (14)
by disjoiSnt sSubsets with a mutualdistance of M cells at least. − K ≤ (1 ε)2 ∪Xij
−
7
So by Proposition 1 we get that the approximation ratio for constant. Given a set of links and weights associated with the
the optimal is 4 links, the algorithm works as follows:
(1−2·MK)(1−ε)2. Phase I: Remove every link whose associated weight is at
most wmax where w denotes the maximum weight among
D. Time and communication complexity n max
all links and n is the number of all given links. Let w
min
We then give a brief analysis on time complexity and
denote the minimum weight from the remaining links.
communication complexity.
PhaseII:Partitiontheremaininglinksintologwmax groups
We define the time complexity as the time unitsrequiredby wmin
such that the links of the i th group G have weights within
the runningofAlgorithm1 insideeachlocalareain the worst − i
[2iw ,2i+1w ]. For each group G of links, it finds an
case[28].Itincludestimeunitsforlocalinformationcollection min min i
independent set of links among it by adopting the method in
and local computation time at the center node. At every time
[32]. Totally it will get logwmax ISLs, one for each group.
slot, the coordinators shall collect queue information and last wmin
Thenitchoosesthe onewith themaximumweightamongthe
schedulingstatusofalllinksinsidethesuper-subSquare.After
logwmax ISLs as the final solution.
computation,thecoordinatorsbroadcasttheschedulingresults wmin
TheremoveoperationofPhaseIsimplyexcludesthoselinks
throughout the super-subSquares. It may require multihop
with smaller weight so that we can get a smaller number of
propagation to collect and broadcast the needed information
groups when partition is done according to weight. In that
inside super-subSquares. We implement it as follows.
way, it is more likely to get a better approximation ratio to
To avoid collision, the coordinator can first compute a tree
the optimal in practical, though the theoretical guarantee is
that determinesthe sequenceof transmissions at each node in
not explicitly improved. We also notice that the resulted link
the super-subSquare based on topology information. As the
schedulingset is an ISL by the methodin [32]. With the help
root node it broadcasts the results to its children (or child).
ofthe followinglemma presentedin [32], we will getthatthe
The children acquire the orderof transmissions and relay this
the resulted link set is indeed with limited number of links.
message as the specified order. The message is relayed until
Lemma 5: ([32])Consideralink l=(u,v)andasetN of
it reaches every node. Then the required information (or the
nodes other than u whose distance from u is at most ρ uv .
schedulingresults)canbecollected(orbroadcast)accordingto
k k
If link l succeeds in the presence of the interference from N,
the sequence.We supposethat the the parentnode can packet
then
the children’s messages into its own packet for collection. In
(ρ+1)κ uv
theworstcaseeachnodehastotransmitonebyoneatdifferent N 1 (k k)κ
mini slot, then it causes O(n ) time complexitywhere n is | |≤ σ − R
ij ij (cid:20) (cid:21)
the number of nodes inside super-subSquare(i,j). Herein we
The corollary below asserts a upper bound of the number
just give a basic scheme, the time complexity may be further
of successfully transmitting links in a local link set with size
reduced with the help of other techniques.
JR JR when utilizing Algorithm 1 in [24].
After all required information gathered, the local compu- ×
Corollary 1: Thecardinalityoftheresultedlinkscheduling
tation complexity at the center node is O(2|Lij|), which can
set by Algorithm 1 in [24] is upper bounded, i.e.,
be ignored comparing to the time for information collection.
ThusthetotaltimecomplexityisO(n ).The communication
ij (√2JR +1)κ r
complexity is the total number of basic messages transmitted (t) r 1 ( )κ
ij
|X |≤ σ − R
during each scheduling in the worst case [28]. The commu- (cid:20) (cid:21)
nication complexity is O(V ), and the number of messages
| | Proof:In the methodpresentedin [32], it addsfirstly the
transmitted in each local area is O(n ).
ij shortest link among all the candidates to the scheduling set.
Andthedistancebetweenanypairofnodeswillbenogreater
IV. THE ALGORITHM IN THEUNIFORM POWERSETTING than √2JR. Therefore, let r be the shortest link length and
We now extend the framework to the uniform power set- then ρ= √2JR, we derive the upper bound.
r
ting. Instead of enumeration, we compute a MWISL of the
We let (t) denote a upper bound of the size of the
ij ub
candidate links for each sub-square by adopting the method |X |
local computed ISL for Z by Algorithm 1 in [24]. Then we
i
proposed in [24]. The reader may wonder why we do not
present the lemma below to show that in the uniform power
use enumerationdirectly as we do in the linearpowersetting.
setting, if any two disjoint local link sets of a given network
The difficulty lies behind the fact that the cardinality of the
areseparatedbyacertainconstantdistanceaway,independent
optimal MWISL in each sub-square is no longer bounded by
scheduling inside each local link set without consideration of
a constant in the uniform power setting, which drives M no
interference outside is possible.
longer bounded by a constant and simple enumeration time
Lemma 6: In a given network (V,E) under the physical
consuming.Except for the difference,the generalstructure of
interference model in the uniform power setting, if the Eu-
thealgorithmisthesamewiththatofthelinearpowersetting.
clidean distance between any two disjoint local link sets is
The following we will give a detailed analysis, and derive a
at least M R , then activated links in each local set suffer
logarithmic bound on throughput capacity. ×
negligiblecumulativeinterferencefromallotheractivatedlink
We first describe the main idea of Algorithm 1 in [24] for
sets, i.e., for each activated link l in local link set Z ,
i
computationofMWISLinsideeachsub-square,andshowthat
the cardinality of the resulted link set is upper bounded by a Il ε I , 0<ε<1,
out ≤ · max
8
whereI isthemaximuminterferencethatthelongestlinks Since µ=log V , it is clear that Algorithm 1 can achieve
max
| |
in E can tolerant during a successful transmission, M is a O( 1 )oftheoptimalcapacityregionintheuniformpower
log V
constant, satisfying M 2πηP·|Xij(t)|ub κ1 setting| b|y Proposition 1.
≥(cid:20)(κ−2)εImaxRκ (cid:21) . The time complexity and communicationcomplexity is the
Proof: The proof is available in Appendix C. same as the corresponding of the linear power assignment.
In light of Lemma 6, the partition strategy to enable
distributedimplementationwillremaineffectiveintheuniform V. PERFORMANCE EVALUATION
power setting.
Wedosimulationexperimentstoverifythecorrectness,and
Next, we reveal the relationship between the newly com-
evaluate the throughput performance of our proposed algo-
putedschedulingsetforsub-square(i,j)andthecorresponding
rithms. The general setting of our experiments is as follows.
S (t). It will help us to derive an approximation ratio of the
i∗j We consider a network with 500 nodes, half of which as
global solution generated by our algorithm later.
sendersrandomlylocatedonaplanewithsize200 200units,
Lemma 7: Theweightofthenewlycomputedlinkschedul- ×
the other half as receivers positioned uniformly at random
ing set inside each sub-square(i,j), i.e., W( (t)), has a
approximationratioof 4µ totheweightoftXheijintersection inside disks of radius R around each of the senders. Packets
(1 ε)2 arrive at each link independently in a Poisson process 1 with
set by the corresponding−local link set L and the global
ij the same average arrival rate λ. Initially, we assign each link
optimal MWISL S (t).
∗ k packets where k is randomly chosen from [100,300]. The
Proof:Thenewlycomputedlocalschedulingsetforeach
path loss exponent is set to be 3.
sub-square should be a (1 ε)-signal set, so its weight is
We conduct three series of experiments to fully evaluate
at least (1−ε)2 times the corr−esponding1-signalset. Since the our algorithms. In the first series, we verify the correctness
4
solutionforMWISLwithuniformpowerassignmentproposed ofouralgorithms,thatis, whethereachactivatedlinkreceives
in [24] finds an ISL with affectness no greater than 1, it is boundedinterferencefromallotherlinksoutsidethelocal link
obvious that set it belongs to. The other two series of experiments focus
4µ on the evaluationof average throughputperformancein terms
W(S (t)) W(OPT (t)) W( (t)), (15)
i∗j ≤ ij ≤ (1 ε)2 Xij of total backlog (the total number of unscheduled packets).
−
The total backlog fluctuates slightly in a region if the arrival
where µ = log V is the approximation bound of the algo-
| | rate vector lies in the achievable capacity region of a link
rithm in [24].
schedulingalgorithm.Oritincreasesdramaticallyifthearrival
Then, we will state an exact bound on the throughput
rate vector exceeds the achievable capacity region. If the
performance of our algorithm in Theorem 2.
total backlog increases unboundedly to infinity, the network
Theorem 2: The union of the local computed scheduling
will become unstable. Thus we can roughly approximate
link sets, i.e., S(t) = S (t), is feasible under the uniform
ij an average supportable arrival rate through the changes of
power setting. The weight of S(t) achieves a fraction of
S backlog.
O(log V ) times the optimal solution.
| | In the second series, we study how some related variables
Proof: We first show that the resulted scheduling link
affect the performance of the algorithms. In the third series,
set by our algorithm is an independent link set with uniform
we further study the performance efficiency of the proposed
powerateverytimeslot.Usingthesametechniqueintheproof
algorithms by comparisons with two centralized algorithms:
of Theorem 1, we can inductively conclude that each global
1) Greedy Maximal Schedule (GMS), which is observed
scheduling set S(t) is composed by disjoint link sets which
to perform throughput optimally in simulations [4]. It
are keptat least M cells away from each other. Therefore,by
greedily adds to the scheduling set the most weighted
Lemma 6, we can get that for each link l S(t), the total
∈ linkthatisindependentwith thepreviouslyaddedlinks.
interference it receives satisfies that,
2) RandomizedAlgorithms, where each time slot links are
Il (1 ε) Il +ε I Il (16) randomlypicked, and addedto currentschedulingset if
S(t) ≤ − · max · max ≤ max.
they do not conflicts with previous selected links. The
It then follows that S(t) is an independentlink set. process is repeatedly performed until no new links can
Wenextprovetheapproximationboundontheperformance
be added.
of our algorithm. Remember that a Partition(K,a ,b ) has a
t t
probability of 1/K2 to be a optimal partition. In that case, The reasons why we do not compare with distributed im-
with the help of Lemma 6 it follows that plementationof the two centralizedalgorithmsare as follows.
Undoubtedly, these two are distributed in nature according
(1 ε)2(K 2M)
W(∪Xij(t)) ≥ − 4Kµ− W(∪Si∗j(t)) (17) to some locality-sensitive approaches [28], if considering
binary interference models. And previous works on binary
(1 ε)2(K 2M)
− − W(S∗(t)) (18) interference models indeed implement them in a distributed
≥ 4Kµ
manner, i.e., [34] proposing a local greedy link scheduling
Hence, we get algorithmand[8]makingschedulingdecisionsbyprobability.
(1 ε)2(K 2M) 1 Thekeypointisthattheeffectofthiscategoryofinterference
P W(S(t)) − − S∗(t) . (19)
≥ 4Kµ ≥ K2
(cid:0) (cid:1) 1The results hold under any arrival process satisfying the strong law of
largenumbers,usingthefluidmodelapproach [33][12].
9
0.07 Inference outside 1 Affectness inside εImax. From Fig. 3(b), we can furtherconfirm that the trans-
0.06 Minfaexriemnucme otoultesridaeble 0.8 TMoatxailm auffmec atnffeescstness inside mittinglinkssuffernegligibleinterferenceoutside.For Fig. 4,
0.05
Interference000...000234 Affectness00..46 orweuectessiihvdoeewsaavttheerdaiagfvfeeerraeafngfteectoitmnuetessissdlepoetisrntitenrrafnFesrimgen.ict4tei(napg)e,rlaitnnradkntsihnmeiFtttioignt.agl4la(ibnn)kd.
0.2
0.01 Theseresultscooperatetoshowthattheoutsideinterferencea
00 1 2 3 L4ink ID5 6 7 8 9 00 1 2 3 T4ime sl5ot 6 7 8 9 transmittingreceivesisindeedboundedbyεImax,andindicate
the correctness of Algorithm 1.
(a) Interference vs.links (b) Affectness vs.links
Next we set anotherset of experimentsto study the perfor-
Fig. 3: The interference/affectness each randomly chosen transmit- manceofAlgorithm1aloneunderdifferentvaluesofvariables
ting link receives with the linear power assignment. (a) shows the thatmay impactthe averagethroughputperformanceactually.
outsideinterferenceeachlinkreceives.(b)showstheinsideaffectness The variables K and ε dominate the theoretical bound of
M
for each link, and the total affectness each receives from all other Algorithm 1. When ε is fixed, a larger K implies a bigger
M
simultaneously transmitting links. fraction of the optimal capacity region, but with a smaller
probability of 1 to achieve it. ε denotes a weighting factor
0.07 Inference outside 1 Affectness inside K2
0.06 Maximum tolerable Total affectness between the interference a activated link suffers inside and
inference ouside 0.8 Maximum affectness inside
0.05 outsidethesub-square.Abiggervalueof εindicatesasmaller
Inteference000...000234 Affectness00..46 vvaalluuee ooff MMK athsemoraeltliecralεlyl.eaTdhserteofoareb,igthgoeurgfhracutniodnearlthcaepfiacxietdy
0.2 region, the probability to achieve this region gets smaller
0.01
01 0 20 30 40 50 60 70 80 90 100 01 0 20 30 40 50 60 70 80 90 100 because of the resulted bigger M. Since the running time of
Time slot Time slot the simulationis muchshorterthanthe time forthe algorithm
(a) Averageinterferencevs.timeslot (b) Averageaffectness vs.timeslot
to achieve the theoretical value, the probability will impact
the actual average throughput in our experiments. Typically,
Fig.4:Averageinterferencepertransmittinglinkreceivesatdifferent
a small probability of 1 may cause poorer throughput
time slots with the linear power assignment. (a) shows the average K2
performance. An experiment study of the two variables are
outsideinterferencepertransmittinglinkreceives.(b)showstheaver-
illustrated in Fig. 5 and Fig. 6.
ageinsideaffectness,andtheaveragetotalaffectnesspertransmitting
In Fig. 5 we compare the total backlog by increasing the
link receives from all other simultaneously transmitting links.
feasible valuesof K where ε servesas an constant. Fig. 5(a),
M
is localized and binary. It is easy to keep feasibility of the 5(b),5(c)respectivelyshowsthecomparisonsoftotalbacklog
final global scheduling set by inactivating all other links in at time slot 1000 with increasing arrival rate when ε = 0.2,
the interference range of an activated link surely or w.h.p.. ε = 0.4, ε = 0.8. Note that the value of K must be greater
M
However, it is not so obvious and easy to guarantee under than 2, or the size of the sub-squares will be 0. It shall also
the physical interference model. It is hard to localize these be noticed that the feasible values of K are different when ε
M
approachesbecause the interference effect is globaland addi- varies,sinceεaffectsthevalueofM.Itthenexplainswhywe
tive. Simplyinactivatingall otherlinks in some neighborhood set different values for K for varying ε. All the three figures
M
of an activated link will not help much to keep the final show that the throughput performance generally improves as
schedulingsetfeasible.Consideringthesedifficulties,wefocus K increases, which coincides with the theoretical results we
M
on comparisons with the centralized visions. derive. Though the relevant probability shall become smaller
as K increases, there is no obvioussign shown in Fig. 5. (a),
M
A. Under Linear Power Setting 5.(b).WecanseeanobviousimpactinFig.5(c)where K can
M
Inthissubsectionweconductthethreeseriesofexperiments besetbiggervalues.Itshowstheaveragethroughputbecomes
for the linear power assignment. a little worse at K =7 than K =6.
M M
We first plot the numerical results for the first series
of experiments in Fig. 3, 4. For convenience, we call the x 105 4x 105
interference/affectnesseachlinkreceivesfromallother simul- 3.5 eeepppsssiiilllooonnn===000...248 3.5 eeepppsssiiilllooonnn===000...248
3 3
tsaaufnfpeeecort-unsseulsybsS.tqrSauinnasrcmeeiittthtienbgemlolainxngikmssutoomutastosidleeor/uaintbssliieddeeinittnehtreefercfreoernrercenescpeoo/finnedsaiincdhge Total backlog12..255 Total backlog12..255
1
linkforsuccessfultransmissionisdifferent,thecorresponding 1 0.5
0.5
inside and total interference each link receives is shown as 0 0.2 0.4 0.6 0.8 1 00 0.2 0.4 0.6 0.8 1
Average arrival rate Average arrival rate
the value of affectness for better illustration. Recall that, for (a) K =3 (b) K =4
a successful transmission, the inside and total affectness shall M M
be less than 1 ε and 1 respectively. Fig. 6: Total backlog vs. average arrival rate vs. different values of
−
Inthissetofexperiments,wesetεasmallvalueof0.3.For ε at time slot 1000 in the linear power setting
Fig. 3, we randomly choose a time slot and 10 transmitting
links. As shown in Fig. 3(a), the outside interference each We briefly illustrate the impact of ε at fixed K in Fig. 6.
M
chosen link receives is much smaller than the constant bound Thoughthetheoreticallyachievablecapacityregionshall have
10
3.5x 105 3.5x 105 3.5x 105
K/M=3 K/M=3 K/M=3
3 KK//MM==34.5 3 KK//MM==34.5 3 KK//MM==45
K/M=6
Total backlog12..1255 Total backlog12..1255 Total backlog12..1255 K/M=7
0.5 0.5 0.5
00 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Average arrival rate Average arrival rate Average arrival rate
(a) ε=0.2 (b) ε=0.4 (c) ε=0.8
Fig. 5: Total backlog vs. average arrival rate vs. different values of K at time slot 1000 in the linear power setting
M
been greater with a smaller ε, it seems to show contradicted The setting of the first series of experimentsis the same as
resultsin Fig. 6(a).The apparentcontradictionlies behindthe the linear power setting. The correspondingnumerical results
probability of 1 which becomes quite small because of a are plotted in Fig. 9, 10. From the two figures it concludes
K2
much bigger M caused by a smaller ε. Fig. 6(b) shows a that the transmitting links receive much smaller interference
similar consequence caused by the crucial impact of ε both outside than the theoreticalbound.Therefore,it indicates that
on the achievablecapacityregionandthe relevantprobability. our algorithm is correctly applicable to the uniform power
We then focus on comparisons with the two centralized setting as well.
algorithms in Fig. 7, 8. We set ε = 0.8, K = 6 to conduct Then we then conduct another set of experiments to study
M
the following simulations. These simulation results indicate how K and ε affect the performance.
M
that our distributed scheduling algorithm(denoted by DS in We showtheeffectof K atdifferentvaluesof εwhereε=
M
figures)achievescomparableperformancewiththecentralized 0.2,0.4,0.8respectivelyinFig.11.Fig.11(a),11(b)and11(c)
GMS and the heuristic RA. plotthetotalbacklogchangesasincreasingaveragearrivalrate
under different values of K. The trends are a little different
M
3x 105DS 3x 104DS from those in the linear power setting. From the three figures
2.5 GRAMS 2.5 GRAMS wecanseethattheaveragethroughputperformancegetsbetter
Total backlog1.125 Total backlog1.125 aopsfaritKnl1yc2rcesaahusosinwedgsbMnKyothuoenbdvaeilogruoasrifitihnxmfleudfeoεnr.cceTohomenpduthetiecnrgereanssueinwltgss.pcIhrtoembdauablyiinlibgtyes
0.5 0.5 inside sub-squares since it selects candidate links based on
00 0.2 0.4 0.6 0.8 1 00.1 6 0.165 0.17 0.175 0.18 their distance with previous selected links. Thus a larger area
Average arrival rate Average arrival rate
implies more links get scheduled. This improvement remits
(a)Totalbacklogvs.averagearrivalrate (b) Zoominof(a)
the influence of the decreasing probability of 1
K2.
Fig.7:Totalbacklogvs.averagearrivalratevs.differentalgorithms
at time slot 1000 in the linear power setting
0.05 1
Inference outside Affectness inside
Maximum tolerable Total affectness
0.04 inference outside 0.8 Maximum affectness inside
tthharIetneoFauilrgg.oarl7gitohwrmiteshmcaosmmapavyaerreaagptephreaorxtroiimvtaaallterblaaytceksulionpgcproecrahtsaeansg.meIsatxosihmfoutwhmes Interference00..0023 Affectness00..46
averagerate nogreaterthan 0.2,and the othertwo supportno 0.01 0.2
greater than 0.3. We zoom in a subgraph of the Fig. 7(a) in 00 1 2 3 4 5 6 7 8 9 00 1 2 3 4 5 6 7 8 9
Link ID Time slot
the Fig. 7(b), where the average arrival rate is in [0.16,0.18].
(a) Interference vs.links (b) Affectness vs.links
It shows that our algorithm can keep the total backlog stable
in the region. Fig. 8 then illustrates detailed comparisons of Fig. 9: The interference/affectness each randomly chosen transmit-
achievable capacity region for the three algorithms. It shows tinglink receives with theuniform power assignment. (a) shows the
thatouralgorithmcansupportacomparabletrafficarrivalrate outsideinterferenceeachlinkreceives.(b)showstheinsideaffectness
vector with the two centralized algorithms. From the three for each link, and the total affectness each receives from all other
subfigures we can see that our proposed algorithm can keep simultaneously transmitting links.
the total backlogstable at an arrivalrate no greaterthan 0.18,
while the counterpart of the greedy algorithm and random Thesimilartrendoccurswhenεincreasesatdifferentvalues
algorithm is 0.28 and 0.26. of K We give a brief illustration in Fig. 12. Both the figures
M.
show that a larger ε generatesbetterperformanceat fixed K
M.
Since the affectness of local ISLs computed by the algorithm
B. Under Uniform Power Setting
of[24]insideeachsub-squaremaybemuchsmallerthan1 ε,
−
Now we conduct the same three series of experiments for it explains why ε has little influence on the theoretical bound
the uniform power assignment. actually. But ε has much more influence on the value of M