a graph theory based opportunistic link scheduling for wireless ad hoc networks

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 10, OCTOBER 2009 5075

A Graph Theory Based Opportunistic LinkScheduling for Wireless Ad Hoc Networks

Qing Chen, Student Member, IEEE, Qian Zhang, Senior Member, IEEE, and Zhisheng Niu, Senior Member, IEEE

Abstract—Taking advantage of the independent fading channelconditions among multiple wireless users, opportunistic transmis-sions schedule the user with the instantaneously best conditionand thus increase the spectrum utilization efficiency of wirelessnetworks. So far, most proposed opportunistic scheduling policiesfor ad hoc networks exploit local multiuser diversity, i.e., eachtransmitter selects its best receiver independently. However, dueto co-channel interference, the decisions of neighboring transmit-ters are highly correlated. Furthermore, the neighboring linkswithout a common sender also experience independent channelfading. Taking the contention relationship and the channeldiversity among links into account, we extend the concept ofmulti-user diversity to a more generalized one, by which a set ofsenders cooperatively schedule the instantaneously and globallybest out-going links, thus the spatial diversity of the channelvariation can be further exploited. In this paper, we formulate theopportunistic scheduling problem with fairness requirements intoan optimization problem and present its optimal solution, i.e., theoptimal scheduling policy. We also propose GOS, a distributedGraph theory based and Opportunistic Scheduling algorithm,which modifies IEEE 802.11 protocol to implement the optimalscheduling policy. Theoretical analysis and simulation resultsboth verify that our implementation achieves higher networkthroughput and provides better fairness support than the existingalgorithms.

Index Terms—Wireless ad hoc networks, multiuser diversity,opportunistic scheduling, proportional fairness, graph theory,maximum weighted independent set.

I. INTRODUCTION

THE opportunistic transmission is firstly studied for cel-lular networks, in which a base station serves multiple

wireless users which are experiencing independent channelfading. The authors showed in [1] that the total capacitycan be maximized by picking the user with the best channelto transmit. For wireless ad hoc networks, the OpportunisticAuto Rate (OAR) scheme is proposed in [2], by which a flowtransmits with higher data rate and more back-to-back packetswhen the channel condition is better. OAR only exploits thetime diversity of the channel variation, whereas a node mayhave packets destined to several neighboring nodes in wirelessad hoc networks. Therefore, each transmitter selecting itsinstantaneously best receiver simply exploits the multiuser di-versity as in cellular networks, which jointly leverages the timeand spatial heterogeneity of channels. Furthermore, multiple

Manuscript received March 21, 2007; revised February 16, 2008; acceptedMarch 4, 2008. The associate editor coordinating the review of this paper andapproving it for publication was V. Bhargava.

Q. Chen and Z. Niu are with the Tsinghua National Laboratory forInformation Science and Technology, Tsinghua University, Beijing, China (e-mail: [email protected]; [email protected]).

Q. Zhang is with the Department of Computer Science, Hong KongUniversity of Science and Technology (e-mail: [email protected]).

Digital Object Identifier 10.1109/TWC.2009.070311

co-channel senders exist in a wireless ad hoc network andthe neighboring links which are not originated from the samesender also experience independent channel fluctuations. Byintroducing the coordination among neighboring transmitters,such link diversity can be exploited and further improves thesystem performance.

However, to opportunistically schedule the links in 802.11based ad hoc networks, there are at least three unique chal-lenges due to the substantially different PHY and MAC char-acteristics. Firstly, due to the shared media in wireless ad hocnetworks, the co-channel interference has deep impact on thelink scheduling. Two links that contend with each other can notbe scheduled concurrently. Hence, we should find the optimalset of links those can be activated simultaneously to achievethe best network performance. Secondly, while selecting thelinks with good channel conditions, it is also important toconsider the fairness among the flows. To schedule a linkshould not be barely based on its channel quality but alsothe achieved throughput of its own and the neighboring flows.Thirdly, without the help of any infrastructure node, thescheduling policy should be executed in a distributed way.Not only the transmitter selects the on-going receiver but alsothe links without common senders should exchange necessaryinformation and adjust their own transmission patterns accord-ingly.

Recently, Opportunistic packet Scheduling and Auto Rate(OSAR) scheme [3] and Medium Access Diversity (MAD)scheme [4] are proposed to exploit multiuser diversity for802.11 based wireless ad hoc networks. By using theseschemes, one sender multicasts a channel probing message(e.g. Group RTS in MAD) before data transmissions. Eachreceiver replies the current channel condition and then thesender schedules the rate adaptive transmission to the receiverwith the best channel quality. In [5], the authors improved theOSAR and proposed a Contention-Based Prioritized Oppor-tunistic (CBPO) scheme to reduce the probing overhead, inwhich the channel conditions can be replied simultaneouslyby using Black-Burst (BB) contention method. However, theprevious algorithms do not consider the interaction amongneighboring transmitters, i.e. a sender individually makes itslocal decision to maximize its own performance. In [11],the authors introduce the cooperation among neighboringtransmitters into the opportunistic scheduling. It is shown thatcooperatively selecting the outgoing link remarkably increasesthe network throughput. Actually, Such cooperation leads to amore generalized multiuser diversity, i.e. the diversity amongmultiple links with and without a common sender.

Purely exploiting channel variations shows preference toflows with good channel conditions. For wireless ad hoc

1536-1276/09$25.00 c⃝ 2009 IEEE

5076 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 10, OCTOBER 2009

networks, many schemes [6]–[8] are presented to providefair scheduling, whereas none of them takes the time-varyingchannels into account. For 802.11 based networks with fadingchannels, to keep fairness among multiple flows, MAD [4]use a 𝑘-set round robin and the revenue based scheduling tomake sure each receiver can be served according to its QoSrequirement. However, by the above work, QoS requirementsare difficult to achieve, since no mechanism is presented tocoordinate the neighboring senders’ transmissions.

In this paper, we formulate the cooperative and opportunis-tic scheduling with fairness requirements as an optimizationproblem, in which the objective is to maximize the networkutility and the constraints are the contention restrictions. Theutility function is chosen to reveal the fairness among differentlinks. We solve the problem and present its optimal solu-tion, i.e. the optimal policy for our opportunistic schedulingproblem. In order to implement the optimal policy into theIEEE 802.11 based ad hoc networks, we also propose adistributed scheduling algorithm, which is inspired by a greedyalgorithm in graph theory to find the maximum weightedindependent set. In our scheme, the links exchange theirchannel information and determine their own transmissionpriorities by adjusting the length of a Traffic-control Inter-Frame Space (TIFS), which is a newly introduced by us andto be inserted into the consecutive transmissions of each link.Through such a priority-based link scheduling algorithm, alink with better channel condition accesses the channel withhigher probability, while the fairness requirements are takeninto consideration. The key contributions of this paper are: 1)A generalized multiuser diversity model is given for wirelessad hoc networks, while considering fairness requirements; 2)We present the optimal criteria to choose the globally optimalset of simultaneously transmitting links, which actually is aweighted maximum independent set in the context of the graphtheory; 3) A Graph theory based and Opportunistic Scheduling(GOS) is designed, which can be easily implemented intothe IEEE 802.11 based ad hoc networks. We also give thetheoretical analysis of the performance lower bound and theoverhead of our scheme. 4) In the GOS, we propose a newpriority based algorithm by introducing the TIFS, and theoptimal value of TIFS is analytically given.

The rest of this paper is organized as follows. Problemformulation is given in Section II. Section III represents theoptimal solutions. Section IV to VI describe the distributedimplementation of the optimal scheduling and its performanceanalysis. Section VII gives the numerical results. This paperis concluded in Section VIII.

II. PROBLEM FORMULATION

Consider an ad hoc network with 𝑁 links (𝑓𝑖, 𝑖 ∈ 𝒩 ), inwhich 𝑓𝑖 denotes the 𝑖th link. We assume that all of the links(single-hop flows) have saturated traffic. In this paper, we con-sider the system with fixed transmit power. Due to the fadingphenomenon, the channel condition of a certain link, i.e. theSINR (Signal to Interference plus Noise Ratio) sensed by thereceiver, is time-varying. Suppose that time is divided intotimeslots with fixed time width. Hence, throughput achievedby a link in one timeslot is linear proportional to its transmitdata rate. It is reasonable to assume that channel conditions do

not vary during a timeslot, since the channel coherence timetypically exceeds the duration of multiple packet transmissions[3]. As in literature [3], we also use the highest achievable datarate (𝜇𝑖(𝑡)), which is determined by the SINR, to represent the𝑖th link’s channel condition in timeslot 𝑡.

The contention relationship of links can be represented bya Contention Graph (CG) [8], in which vertexes are links andan edge exists between two vertexes if the corresponding twolinks contend with each other. Herein, two links are claimedto be contended if and only if any node of one link is inthe interference range of any node of another link. A node’sinterference range is the area in which the transmission ofany other node can interrupt its receiving. Throughout thispaper, we use vertex/link/flow, contended/edged/neighboringinterchangeably. Due to the fading phenomenon, the pathgain between any two nodes varies from time to time, whichleads to a time-varying contention relationship CG(𝑡). Weintroduce a contention indication function 𝑐(𝑖, 𝑗, 𝑡), whichequals 1 if link 𝑖 and link 𝑗 are edged in the contention graphCG(𝑡), otherwise zero. Moreover, by coloring vertices we canobtain several Independent Subsets (IS), in which the flowscan transmit simultaneously. A Maximal Independent Subset(MIS) (S𝑚(𝑡)) is an IS that is not subset of any other IS. Theset Ω(𝑡) = {S𝑚(𝑡)} denotes the MIS set.

We formulate an arbitrary scheduler as Q, in which Q(𝑡)denotes the scheduled transmitting link set in timeslot 𝑡. The𝑖 ∈Q(𝑡) means that link 𝑖 transmits at this moment. We alsointroduce an indicator function 𝐼𝑋 , which equals 1 if 𝑋 istrue, otherwise zero.

We formulate each link’s average throughput as 𝑏𝑖 =lim𝑡→∞ 𝑏𝑖(𝑡), in which 𝑏𝑖(𝑡) denotes the average throughputof the 𝑖th link until timeslot 𝑡 and it can be updated by usingan exponentially weighted low-pass filter [1], i.e.

𝑏𝑖(𝑡+ 1) =𝑇𝑐 − 1

𝑇𝑐𝑏𝑖(𝑡) +

1

𝑇𝑐𝜇𝑖(𝑡)𝐼𝑖∈𝑄(𝑡), (1)

in which 𝑇𝑐 is the average window size. We aim to maximizethe network utility by introducing the utility functions 𝑈𝑖(⋅),which are non-decreasing, concave and differentiable. Bymaximizing the sum of all users’ utilities

∑𝑖 𝑈𝑖(⋅), we can

control the tradeoff between efficiency and fairness. Differentshapes of utility functions lead to different types of fairness.For example, a family of utility functions parameterized by𝛼 ≥ 0 is proposed in [10]:

𝑈𝛼(𝑥) =

⎧⎨⎩

log 𝑥, if 𝛼 = 1𝑥1−𝛼

1− 𝛼, otherwise.

(2)

by which the proportional fairness is achieved as 𝛼 = 1 andthe max-min fairness as 𝛼 → ∞. If we set 𝛼 = 0, the problemreduces to system throughput maximization.

Thus we present the link scheduling problem with fairnessrequirements as

max𝑄

∑𝑖∈𝒩 𝑈𝛼(𝑏𝑖)

s.t. 𝑐(𝑖, 𝑗, 𝑡) = 0, ∀𝑖, 𝑗 ∈ 𝑄(𝑡), 𝑖 ∕= 𝑗.(3)

CHEN et al.: A GRAPH THEORY BASED OPPORTUNISTIC LINK SCHEDULING FOR WIRELESS AD HOC NETWORKS 5077

III. OPTIMAL CRITERIA OF SCHEDULING AND THEIR

GRAPH THEORETICAL ANALYSIS

Let us denote the optimal solution/policy of problem (3) by𝑄∗. Then we have the following lemma.

Lemma 1: The optimal selection of the opportunisticscheduling (3) in any timeslot 𝑡 is a maximal independentset, i.e. the optimal policy satisfies 𝑄∗(𝑡) ∈ Ω(𝑡).

This can be easily proved by contradiction. Then, we givethe following three propositions to present the optimal policiesfor the proportional fairness, the system throughput maximiza-tion and the 𝛼-utility maximization problem respectively.

Proposition 1: The optimal solution of the log-utility max-imization problem (𝛼 = 1 in problem (3)), i.e. the schedulingproblem with proportional fairness, is of the following form.

𝑄∗(𝑡) = 𝑆𝑚∗(𝑡)

where 𝑚∗ = argmax𝑚

∏𝑖∈𝑆𝑚(𝑡)

(1 +𝜇𝑖(𝑡)

(𝑇𝑐 − 1)𝑏𝑖(𝑡)). (4)

Proof:According to the objective of the problem (3) with

𝛼 = 1, the optimal scheduler 𝑄∗ should meet the con-dition:

∑𝑖∈𝒩 log 𝑏∗𝑖 ≥ ∑

𝑖∈𝒩 log 𝑏𝑖, in which the 𝑏∗𝑖 and𝑏𝑖 are obtained by the scheduler 𝑄∗ and an arbitraryscheduler 𝑄 respectively. Thus we have

∑𝑖∈𝒩 log 𝑏∗𝑖 (𝑡) ≥∑

𝑖∈𝒩 log 𝑏𝑖(𝑡), ∀𝑡. This can be simply proved by contradic-tion. With the help of Lemma 1, we have 𝑄∗(𝑡) = 𝑆∗

𝑚(𝑡). Theabove inequation is equivalent to the following condition: Forany specific 𝑡,

∑𝑖∈𝒩

log 𝑏∗𝑖 (𝑡+ 1) ≥∑𝑖∈𝒩

log 𝑏†𝑖 (𝑡+ 1), (5)

where 𝑏†𝑖 is obtained by a scheduler 𝑄† that satisfies 𝑄†(𝜏) =𝑆𝑚∗(𝜏), ∀𝜏 < 𝑡 and 𝑄†(𝑡) = 𝑆𝑚†(𝑡) (𝑆𝑚†(𝑡) is an arbitraryMIS). Thus, we have 𝑏†𝑖 (𝑡) = 𝑏∗𝑖 (𝑡).

To avoid the logarithmic computation, (5) can be put as∏𝑖∈𝒩

𝑏∗𝑖 (𝑡+ 1) ≥∏𝑖∈𝒩

𝑏†𝑖 (𝑡+ 1). (6)

Since the average throughputs of links which are selectedneither by 𝑄∗ nor by 𝑄† are the same in both sides of (6), thelink set of concern is 𝑆∗

𝑚(𝑡)∪𝑆𝑚†(𝑡). Therefore, we have

∏𝑖∈𝑆𝑚∗(𝑡)

∪𝑆𝑚† (𝑡)

𝑏∗𝑖 (𝑡+ 1) ≥∏

𝑖∈𝑆𝑚∗ (𝑡)∪

𝑆𝑚† (𝑡)

𝑏†𝑖 (𝑡+ 1). (7)

Since 𝑋∪𝑌 can be replaced by either 𝑋

∪(𝑌 − 𝑋) or

𝑌∪(𝑋 − 𝑌 ), (7) can be written as

∏𝑖∈𝑆𝑚∗ (𝑡)

𝑏∗𝑖 (𝑡+ 1)∏

𝑖∈𝑆𝑚†(𝑡)−𝑆𝑚∗ (𝑡)

𝑏∗𝑖 (𝑡+ 1)

≥∏

𝑖∈𝑆𝑚† (𝑡)

𝑏†𝑖 (𝑡+ 1)∏

𝑖∈𝑆𝑚∗(𝑡)−𝑆𝑚† (𝑡)

𝑏†𝑖 (𝑡+ 1).(8)

By replacing 𝑏∗𝑖 (𝑡)’s and 𝑏†𝑖 (𝑡)’s in (8) using (1), we have∏

𝑖∈𝑆𝑚∗

(𝑇𝑐 − 1)𝑏∗𝑖 (𝑡) + 𝜇𝑖(𝑡)

𝑇𝑐

∏𝑖∈𝑆

𝑚†−𝑆𝑚∗

(𝑇𝑐 − 1)𝑏∗𝑖 (𝑡)𝑇𝑐

≥∏

𝑖∈𝑆𝑚†

(𝑇𝑐 − 1)𝑏†𝑖 (𝑡) + 𝜇𝑖(𝑡)

𝑇𝑐

∏𝑖∈𝑆𝑚∗−𝑆

𝑚†

(𝑇𝑐 − 1)𝑏†𝑖 (𝑡)𝑇𝑐

.

(9)

Since we have 𝑏†(𝑡) = 𝑏∗(𝑡), by multiplying∏𝑖∈𝑆𝑚∗ (𝑡)

∩𝑆𝑚†(𝑡)(𝑇𝑐 − 1)𝑏∗𝑖 (𝑡) to both sides, (9) can

be written as∏

𝑖∈𝑆𝑚∗ (𝑇𝑐 − 1)𝑏∗𝑖 (𝑡) + 𝜇𝑖(𝑡)∏𝑖∈𝑆𝑚∗ (𝑇𝑐 − 1)𝑏∗𝑖 (𝑡)

≥∏

𝑖∈𝑆𝑚† (𝑇𝑐 − 1)𝑏†𝑖 (𝑡) + 𝜇𝑖(𝑡)∏𝑖∈𝑆

𝑚† (𝑇𝑐 − 1)𝑏†𝑖 (𝑡).

(10)Therefore, the scheduler 𝑄∗ is the optimal solution if and

only if

𝑄∗(𝑡) = 𝑆𝑚∗(𝑡), 𝑚∗ = argmax𝑚

∏𝑖∈𝑆𝑚(𝑡)

(1+𝜇𝑖(𝑡)

(𝑇𝑐 − 1)𝑏𝑖(𝑡)).

(11)

Note: If the window size 𝑇𝑐 goes to infinite, eqn. (11) canbe reduced to

𝑄∗(𝑡) = 𝑆𝑚∗(𝑡), 𝑚∗ = argmax𝑚

∑𝑖∈𝑆𝑚(𝑡)

𝜇𝑖(𝑡)

𝑏𝑖(𝑡). (12)

since we have lim𝑥𝑖→0

∏𝑖

(1 + 𝑥𝑖) = 1 +∑𝑖

𝑥𝑖 by taking a

first-order approximation.Proposition 2: The optimal solution of the system through-

put maximization (𝛼 = 0 in problem (3)), is of the followingform.

𝑄∗(𝑡) = 𝑆𝑚∗(𝑡), where 𝑚∗ = argmax𝑚


𝜇𝑖(𝑡).

(13)Proof:

Similarly as the proof of Proposition 1, the optimal sched-uler 𝑄∗ to maximize the overall throughput of links shouldmeet the following condition: For any specific 𝑡,

∑𝑖∈𝑆𝑚∗ (𝑡)

∪𝑆𝑚†(𝑡)

𝑏∗𝑖 (𝑡+1) ≥∑

𝑖∈𝑆𝑚∗(𝑡)∪

𝑆𝑚†(𝑡)

𝑏†𝑖 (𝑡+1). (14)

where 𝑏†𝑖 is obtained by a scheduler 𝑄† that satisfies 𝑄†(𝜏) =𝑆𝑚∗(𝜏), ∀𝜏 < 𝑡 and 𝑄†(𝑡) = 𝑆𝑚†(𝑡) (𝑆𝑚†(𝑡) is an arbitraryMIS).

(14) can be written as∑

𝑖∈𝑆𝑚∗

(𝑇𝑐 − 1)𝑏∗𝑖 (𝑡) + 𝜇𝑖(𝑡)

𝑇𝑐+

∑𝑖∈𝑆


(𝑇𝑐 − 1)𝑏∗𝑖 (𝑡)𝑇𝑐

≥∑

𝑖∈𝑆𝑚†

(𝑇𝑐 − 1)𝑏†𝑖 (𝑡) + 𝜇𝑖(𝑡)

𝑇𝑐+

∑𝑖∈𝑆𝑚∗−𝑆

𝑚†

(𝑇𝑐 − 1)𝑏†𝑖 (𝑡)𝑇𝑐

.

(15)By adding


∩𝑆𝑚† (𝑡)

(𝑇𝑐−1)𝑏∗𝑖 (𝑡)𝑇𝑐

to both sides,


𝑖∈𝑆𝑚∗(𝑡)𝜇𝑖(𝑡)𝑇𝑐

≥ ∑𝑖∈𝑆

𝑚† (𝑡)𝜇𝑖(𝑡)𝑇𝑐

.Therefore, the optimal solution of the system throughput

maximization problem can be given as (13).

Proposition 3: In the case of the window size 𝑇𝑐 → ∞, theoptimal solution of 𝛼-utility maximization problem (3), is ofthe following form.

𝑄∗(𝑡) = 𝑆𝑚∗(𝑡), where 𝑚∗ = argmax𝑚


𝜇𝑖(𝑡)

(𝑏𝑖(𝑡))𝛼.

(16)Proof:


The optimal scheduler 𝑄∗ to maximize the network 𝛼-utilityof links should meet the following condition: For any specific𝑡,

∑𝑖∈𝑆𝑚∗

∪𝑆𝑚†

𝑈𝛼(𝑏∗𝑖 (𝑡+1)) ≥

∑𝑖∈𝑆𝑚∗

∪𝑆𝑚†

𝑈𝛼(𝑏†𝑖 (𝑡+1)). (17)

where 𝑏†𝑖 is obtained by a scheduler 𝑄† that satisfies 𝑄†(𝜏) =𝑆𝑚∗(𝜏), ∀𝜏 < 𝑡 and 𝑄†(𝑡) = 𝑆𝑚†(𝑡) (𝑆𝑚†(𝑡) is an arbitraryMIS).


𝑖∈𝑆𝑚∗

𝑈𝛼(𝑏∗𝑖 (𝑡+ 1)) +

∑𝑖∈𝑆


𝑈𝛼(𝑏∗𝑖 (𝑡+ 1))

≥∑

𝑖∈𝑆𝑚†

𝑈𝛼(𝑏∗𝑖 (𝑡+ 1)) +

∑𝑖∈𝑆𝑚∗−𝑆

𝑚†

𝑈𝛼(𝑏∗𝑖 (𝑡+ 1)).

(18)

By using the first order Taylor expression and 𝑇𝑐 → ∞, wehave

𝑈𝛼(𝑏∗𝑖 (𝑡+ 1)) = 𝑈𝛼(

(𝑇𝑐 − 1)𝑏∗𝑖 (𝑡) + 𝜇𝑖(𝑡)

𝑇𝑐)

= 𝑈𝛼(𝑏∗𝑖 (𝑡)) + 𝑈 ′

𝛼(𝑏∗𝑖 (𝑡))

𝜇𝑖(𝑡)

𝑇𝑐+ 𝑜(

𝜇𝑖(𝑡)

𝑇𝑐).

(19)

where 𝑈 ′𝛼(⋅) is the differential function of 𝑈𝛼(⋅). By adding∑

𝑖∈𝑆𝑚∗∩

𝑆𝑚† 𝑈𝛼(𝑏

∗𝑖 (𝑡)) to both sides of (18), we have


𝑈 ′𝛼(𝑏

∗𝑖 (𝑡))

𝜇𝑖(𝑡)

𝑇𝑐≥

∑𝑖∈𝑆

𝑚† (𝑡)

𝑈 ′𝛼(𝑏

†𝑖 (𝑡))

𝜇𝑖(𝑡)

𝑇𝑐. (20)

The differential function of 𝛼−family utility is 𝑥−𝛼. There-fore, the nearly optimal solution of the network 𝛼-utilitymaximization problem can be given as (16).

Note: If the 𝛼 is set as 1, the nearly optimal solution is rightthe reduced proportional fairness scheduler (12). In anothercase that 𝛼 = 0, we get the same optimal scheduling policyas Proposition 2 claims.

In the real world, large 𝑇𝑐 is acceptable, thus the nearlyoptimal solution is reasonable. By defining the weight of the𝑖th link as 𝜇𝑖(𝑡)

(𝑏𝑖(𝑡))𝛼, we can see that the nearly optimal schedul-

ing policy 𝑄∗ selects a Maximum Weighted Independent Set(MWIS) in each timeslot. The MWIS is one concept in thegraph theory, which denotes an MIS that has the largest totalweight of links among all the MIS’s.

Graph theory is an important branch of mathematics inwhich a graph can be a symbolic representation of a networkand of its connectivity. In recent years, graph theory has beenwidely introduced into the research of the wireless ad hocnetworks, e.g. the media access scheme design [8], the systemcapacity [13] and the network connectivity [14] analysis. Inthe previous literature, the graph is used to represent thecontention or the connectivity. Whereas, in this paper, weconsider the fading channels and thus we suggest using theweighted graph in which each vertex is associated with aweight. In the view of the graph theory, to find the (weighted)maximal independent set is a classical and NP-hard discretemathematical problem. There have been proposed and ana-lyzed numerous approximation algorithms for this problem.We give some background knowledge as follows.

A. Maximum Weighted Independent Set in Graph Theory

Let 𝐺 be an undirected graph where each vertex 𝑣 has apositive weight 𝑤𝑣 . Let 𝑉 (𝐺) and 𝐸(𝐺) denote the vertex setand the edge set of 𝐺, respectively, as usual. Let 𝑊 (𝐺) bethe sum of the weights of all vertices: 𝑊 (𝐺) =

∑𝑣∈𝑉 𝑤𝑣 .

For a vertex set 𝑋 , let 𝑤(𝑋) denote the sum of the weightsof the vertices in 𝑋 . Let 𝑁𝐺(𝑣) denote the neighbor set ofvertex 𝑣 in 𝐺. For a vertex 𝑣, the weighted degree 𝑑𝑤(𝑣,𝐺)in 𝐺 is given as follows:

𝑑𝑤(𝑣,𝐺) =𝑤(𝑁𝐺(𝑣))

𝑤𝑣. (21)

The weighted average degree 𝑑𝑤(𝐺) of graph 𝐺 is defined asfollows:

𝑑𝑤(𝐺) =

∑𝑣∈𝑉 𝑤𝑣𝑑𝑤(𝑣,𝐺)

𝑊=

∑𝑣∈𝑉 𝑤(𝑁(𝑣))

𝑊. (22)

In graph theory, the most important greedy algorithm [12]to locate the MWIS can be written as follows. We select aminimum weighted degree vertex as a vertex in the weightedindependent set 𝐼 , and delete this vertex and all of its neigh-bors from the graph. We repeat this process for the remainingsubgraph until the subgraph becomes empty. It is proved thatsuch an algorithm attains the following lower bound

𝑤(𝐼) ≥ 𝑊

𝑑𝑤 + 1. (23)

In the next section, we will borrow the above idea from thegraph theory to design our heuristic scheduling policies.

IV. GOS: A GRAPH THEORY BASED AND OPPORTUNISTIC

SCHEDULING

By the optimal criteria, a scheduler should gather followinginstantaneous parameters for each timeslot: the contentiongraph, the links’ feasible data rate and achieved throughput.Then a set of flows in the MWIS are scheduled to transmitsimultaneously. The above procedures cannot be directly im-plemented into IEEE 802.11 based ad hoc networks due to thefollowing challenges: 1) exchanging feasible data rates all overthe network is impractical, since such a flooding consumes lotof bandwidth and some instantaneous values become outdatedafter a multi-hop transmission; 2) it is difficult to track thetime-varying contention graph which is needed in the optimalscheduling; 3) to schedule a set of links in a deterministicorder, as TDMA in cellular networks, is not trivial becauseof the distributed nature of an IEEE 802.11 based ad hocnetwork.

We design our Graph theory based and OpportunisticScheduling (GOS) policy based on the greedy algorithm tofind MIS in the graph theory, of which the core idea is thatthe link with the lowest weighted degree among its neighborstransmits first. In fact, the weighted degree denotes a kind ofcooperation among links, since a link with lower weighteddegree means that it contends with fewer neighboring links orit gets higher weight by itself. In other words, by using ourGOS, each link decides its own transmission pattern with theconsideration of its neighboring links’ information.

In our scheme, each link’s transmitter and receiver main-tains a degree table for the neighboring links and the link


Fig. 1. The time line of the Graph theory based and OpportunisticScheduling. The red blocks denote the piggyback information of the degreetables.

itself. Figure 1 shows a typical time line of GOS. Before adata transmission, a sender multicasts a RTS packet and itscandidate receivers reply with CTS packets which contain theirdegree tables. Then, the sender sends back-to-back packetson one of the links. After one sequence of transmission, i.e.RTS, CTS, DATA and ACK, the transmitter would hold foran interval before starting the next transmission. The lengthof inserted interval is set according to the weighted degree ofthis link. In detail, we describe several important parts of ourGOS in following subsections.

1) Degree Table Update: As we mentioned in Section II,two links are claimed to be neighboring if they are within eachother’s interference range. This range is variable according tothe distance between this node and its intended sender, andit is usually larger than the communication range and can beconservatively regarded as twice of it [9]. Thus in practice, weredefine the neighboring relationship as two links are withineach other’s 2-hop transmission range.

In order to create and update the degree tables, twomechanisms are introduced: channel probing and degree tableexchanging. The probing process is based on the Group RTSmechanism [3], by which the link’s condition is computedby its receiver when a RTS packet is arrived and the resultis sent back to the transmitters by CTS packets (one by oneif multiple candidate receivers exist). A receiver would alsoupdate the channel condition and its own degree table whenit receives any other packets, e.g. DATA packets, from itsintended senders. We modified the Group RTS mechanismto facilitate the degree table exchanging. In our scheme, areceiver measures the channel condition and updates its degreetable after receiving RTS from its intended transmitter. Thenthe receiver sends back its degree table by CTS. The transmit-ter also conveys its degree table in the DATA packets (after thepreamble part and coded in the basic data rate). It is meantthat during a sequence of RTS-CTS-DATA-ACK exchange,the transmitter and the receiver of a link synchronize theirdegree tables. In the meantime, a node overhearing the aboveCTS or DATA packets would also update its own degree table.Therefore, the channel information is spread out during thedata transmissions. Since each node maintains the informationof its neighboring links, the parameters are actually propagatedin a 2-hop transmission range.

2) Link Scheduling: Two phases of scheduling are proposedhere. The first one is that, after receiving CTS packet(s)destined to itself, the transmitter select a link which has

the lowest weighted degree among the out-going links. Thetransmitter sends certain number of back-to-back packets onthis link with the PAcket Concatenation (PAC) mechanism[4], by which nodes will transmit more data during epochs ofhigh-quality channels.

In order to achieve the priority based scheduling among thelinks without common senders, we propose the second phaseof scheduling in which a longer interval is inserted into twoconsecutive sequence of data transmissions for a lower prioritytransmitter. Here we call the inserted interval as the Traffic-control InterFrame Space (TIFS) (see Fig. 1), which is upto one or several packets’ transmission time. In other words,after receiving an ACK packet, a transmitter would not try tosend another RTS for several packets’ transmission time. Inaddition, if a delayed node finds that its degree has turned tobe the lowest by overhearing packets, it would reset the TIFSto 0 at once and try to send its RTS.

The optimal length of TIFS is the duration from now tillthe transmitter to be scheduled again. We give the closed-form express of the optimal TIFS in Section V. However, theoptimal value depends on several factors of the network, suchas the move pattern of the nodes, contention graph and fairnessrequirements. In order to adaptively set TIFS, we imitate theIEEE 802.11 Contention Window (CW) updating algorithm inwhich a transmitter doubles its CW size if a collision occurs:

TIFS =

⎧⎨⎩

0, if 𝑠 = 1TIFS𝑚𝑖𝑛, if TIFS = 0 and 𝑠 > 1min(TIFS × 𝑠,TIFS𝑚𝑎𝑥), otherwise,

(24)

in which the 𝑠 denotes one transmitter’s degree order amongall the transmitters in its neighboring links. The 𝑠 = 1 meansthat such a transmitter has the lowest degree. The exponentialincrease (multiply by the factor 𝑠) leads to quick convergenceto the optimal value, whereas the TIFS is reset to zero as soonas the weighted degree turns to be the smallest in the degreetable.

To evaluate the proposed distributed algorithm, two discus-sions should be given as follows: 1) The network performancedepends on the setting of the TIFSs, thus what’s the optimalsetting of TIFS should be discussed; 2) By using the optimalTIFS, how is the network utility that obtained by our schemein the worst cases and how is the overhead of our distributedalgorithm? The answers will be given in the following twosections.

V. THE OPTIMAL LENGTH OF TRAFFIC CONTROL INTER

FRAME SPACE

In our heuristic scheduling algorithm, a TIFS is insertedbetween the consecutive data transmissions for an unscheduledlink. Shorter TIFS leads to frequently access to channel evenif its channel condition has not turn be good enough. However,longer TIFS leads to longer delay while it may miss the rightchance to access the channel. Therefore, the optimal lengthshould be the expect time that an unscheduled sender turns tobe scheduled from now.

To compute the expect time, we model the fading channelas a Finite-State Markov channel (FSMC) as in literature [15],in which each state means a achievable data rate 𝜇𝑖. For eachlink, the transition probability can be computed by giving the


Fig. 2. A two-dimension Markov Chain for a two-link transmission scenario.

nodes’ mobile speed, the average signal to noise ratio and thesampling rate (see detail in literature [15]). Here we assumethat the channel conditions of different links are independent.Therefore, a 𝑁 -link fading channel can be formulated as a𝑁 -dimension discrete Markov chain (see Fig. 2 for example).Each state is identified by 𝑠𝑡𝑘(𝑡) = (𝑤1, 𝑤2, ⋅ ⋅ ⋅ , 𝑤𝑁 ), 𝑘 ∈ 𝒦and 𝒦 = {1, 2, ⋅ ⋅ ⋅ ,𝐾𝑁}, in which the 𝐾 denotes the numberof possible data rates (including zero) and 𝑤𝑖 denotes theweight of the 𝑖th link associated with a certain data rate,e.g. 𝑤𝑖 = 𝜇𝑖 for system maximization and 𝑤𝑖 = 𝜇𝑖/𝑏𝑖for proportional fairness. By using the optimal schedulingin each state, the links in the MWIS should be scheduled,i.e. 𝑄∗(𝑠𝑡𝑘) = 𝑆𝑚∗ ,𝑚∗ = argmax𝑚{∑𝑖∈𝑆𝑚

𝑤𝑖}. We canclassify the states into overlapped groups associated with eachlink 𝑆𝑇𝑖 = {𝑘∣𝑖 ∈ 𝑄∗(𝑠𝑡𝑘)}, ∀𝑖 ∈ 𝒩 . In other words, 𝑆𝑇𝑖

contains all the states in which the 𝑖th link would be scheduled.We assume that each link knows the starting state 𝑠𝑡𝑘0 , i.e.,the weights of all other links. Thus the optimal TIFS shouldbe the expect first arrival time, i.e. the expect rounds (denotedby 𝑚𝑘𝑖) that starting from state 𝑠𝑡𝑘, the state of the Markovchain firstly jumps into the state group 𝑆𝑇𝑖.

By taking other states as the transition states after one-stepjump, we have 𝑚𝑘𝑖 =

∑𝑗∈𝒦−𝑆𝑇𝑖

(1 + 𝑚𝑗𝑖)𝑝𝑘𝑗 +∑

𝑗∈𝑆𝑇𝑖

𝑝𝑘𝑗 ,

in which 𝑝𝑘𝑗 denotes the one-step transit probability fromstate 𝑠𝑡𝑘 to 𝑠𝑡𝑗 . The 𝑝𝑘𝑗’s can be computed by multiplyingeach link’s transit probability, since the transitions of linksare assumed to be independent. The above equation can berewritten as (1− 𝑝𝑘𝑘)𝑚𝑘𝑖 −

∑𝑗 ∕=𝑘,𝑗∈𝒦−𝑆𝑇𝑖

𝑝𝑘𝑗𝑚𝑗𝑖 = 1.

For different 𝑘, 𝑘 ∈ 𝒦−𝑆𝑇𝑖, the above equation holds. Wehave an equation array with 𝐽 = 𝐾𝑁 − ∣𝑆𝑇𝑖∣ equations and𝐽 unknown values (𝑚𝑖’s). Therefore, we have

(𝐼𝑖 − 𝑃𝑖)−→𝑚𝑖 =

−→𝑐 𝑖 (25)

Here 𝐼𝑖 is a 𝐽-dimension identity matrix, 𝑃𝑖 is the one-steptransit probability matrix among the states {𝑠𝑡𝑘∣𝑘 ∈ 𝒦−𝑆𝑇𝑖}.Meanwhile −→𝑚𝑖 = {𝑚𝑘𝑖∣𝑘 ∈ 𝒦 − 𝑆𝑇𝑖} and −→𝑐 𝑖 is a 𝐽-rowvector of all 1’s. Let 𝑍𝑖 = {𝑧(𝑖)𝑘𝑗 } denote (𝐼𝑖 − 𝑃𝑖)

−1. Thus−→𝑚𝑖 = 𝑍𝑖−→𝑐 𝑖.

In conclusion, the optimal TIFS for the 𝑖th link in state 𝑘

should be

𝑡𝑘𝑖 = 𝑚𝑘𝑖𝑇0 =

𝐽∑𝑗=1

𝑧(𝑖)𝑘𝑗 ⋅ 𝑇0, (26)

where 𝑇0 denotes the width of one timeslot.

VI. THE PERFORMANCE ANALYSIS OF OUR GOS

A. Performance Lower Bound of the Optimal Scheduling

As we mentioned in the Section III, the centralized andgreedy algorithm to locate MWIS obtains a lower boundas equation (23). In this paper, we propose a distributedway to located the MWIS. Distributed and centralized im-plementations differ in terms of the choice of vertex to beselected in each stage. By using our distributed and heuristicscheduling GOS with the optimal TIFS setting and the PerfectInformation (PI), more than one link can be selected at thesame time, whereas each chosen link is associated with thelowest weighted degree among its neighbors. The perfectinformation means that each link knows the right weighteddegrees of its neighboring links.

Let 𝑣𝑖’s (𝑖 ∈ ℐ) denote the vertices selected from thecontention graph. Then we give the following lemma whichhelps to derive the lower bound of our algorithm.

Lemma 2: In our proposed GOS with the PI, the verticesin the CG can be divided into groups 𝐺′

𝑖’s which satisfy thefollowing requirements:

∙ Non-overlapped: 𝐺′𝑖

∩𝐺′

𝑗 = Φ, ∀𝑖, 𝑗 ∈ ℐ and 𝑖 ∕= 𝑗.∙ Full Division:

∪𝑖∈ℐ 𝐺

′𝑖 = 𝑉𝐶𝐺, where the 𝑉𝐶𝐺 denotes

the set of all the vertices in the CG.∙ Core Existence: 𝑣𝑖 should be within and be the core of

one and only one group, i.e. 𝐺′𝑖. The core means that

𝑣𝑖 should be associated with the lowest weighted degreeamong the group 𝐺′

𝑖.

Proof: The lemma is proved if we can give a process ofthe grouping. We present the following procedures.

Firstly, we set the groups 𝐺′𝑖 = {𝑣𝑖}, in which only a core

vertex exists in each group.Nextly, we group the other non-core vertices. For an arbi-

trary non-core vertex 𝑣𝑎, it must have at least one neighboringvertex, otherwise it should be a core. Three situations shouldbe considered: 1) If only one core exists in 𝑣𝑎’s neighborhood,𝑣𝑎 joins the group of this core; 2) If multiple cores are around𝑣𝑎, 𝑣𝑎 can randomly select to join one and only one groupfrom the groups associated with the neighboring cores; 3) Inthe cases that no core exists in 𝑣𝑎’s neighborhood, we can findat least one vertex, denoted by 𝑣𝑏, is the neighbor of 𝑣𝑎 andsatisfies 𝑤𝑣𝑏 < 𝑤𝑣𝑎 . Such a 𝑣𝑏 must exist, otherwise 𝑣𝑎 wouldbe a core. If 𝑣𝑏 have already been grouped, 𝑣𝑎 can join thesame group as 𝑣𝑏. Otherwise, 𝑣𝑏 can be treated as 𝑣𝑎 to finda group. If a group can be found for 𝑣𝑏, 𝑣𝑎 would join thesame group as 𝑣𝑏. Otherwise, a 𝑣𝑐, which is not a core andsatisfies 𝑤𝑣𝑐 < 𝑤𝑣𝑏 , can be found. 𝑣𝑐 is absolutely not the𝑣𝑎, since we have 𝑤𝑣𝑐 < 𝑤𝑣𝑏 < 𝑤𝑣𝑎 . Iteratively, a chain ofnon-core vertices can be found if 𝑣𝑎 still can not be grouped.Whereas, the number of vertices in CG is finite. Thus the chainwould definitely be terminated by reaching a grouped vertex,e.g. the vertex has a neighboring core. Then all the vertices in


this chain can be grouped into the group associated with thisgrouped vertex.

With the help of Lemma 2, we can give the performancelower bound of our algorithm as follows.

Proposition 4: The distributed and heuristic schedulingGOS with the PI obtains the following performance lowerbound, i.e., produces the independent set satisfying the in-equality

𝑤(𝐼𝐶) =∑

𝑣𝑖∈𝐼𝐶

𝑤𝑣𝑖 ≥𝑊

𝑑𝑤 + 1, (27)

where 𝑊 is the overall weight, 𝑑𝑤 is the weighted averagedegree and 𝐼𝐶 = {𝑣𝑖∣𝑖 ∈ ℐ} denotes the link set selected byour GOS.

Proof: We can divide the contention graph into non-overlapped, full divided and core existed groups according tothe Lemma 2, i.e., 𝐺′

𝑖, 𝑖 ∈ ℐ. We first argue the lower boundof 𝑑𝑤𝑊 as follows.

𝑑𝑤𝑊 =∑

𝑣∈𝑉 (𝐺)

𝑑𝑤(𝑣,𝐺) ≥∑𝑖

∑𝑣∈𝐺′

𝑖

𝑤𝑣𝑑𝑤(𝑣,𝐺′𝑖)

≥∑𝑖

∑𝑣∈𝐺′

𝑖

𝑤𝑣𝑑𝑤(𝑣𝑖, 𝐺′𝑖) =

∑𝑖

𝑤(𝐺′𝑖)𝑑𝑤(𝑣𝑖, 𝐺

′𝑖).

(28)Adding 𝑊 =

∑𝑖 𝑤(𝐺

′𝑖) to both side of the above inequal-

ity, we have

(𝑑𝑤 + 1)𝑊 ≥∑𝑖

𝑤(𝐺′𝑖)

2

𝑤𝑣𝑖

(29)

Finally we apply Proposition 1 in literature [12] with 𝑎𝑖 =𝑤𝑣𝑖 , 𝑏𝑖 = 𝑤(𝐺′

𝑖) . The inequality

(𝑑𝑤 + 1)𝑊 ≥ 𝑊 2

𝑤(𝐼𝐶)(30)

holds, which implies this proposition.The rate adaptive IEEE 802.11 MAC protocols, e.g. OAR,

exploit the time heterogeneity of each link but do not take thegeneralized multiuser diversity into account. The links withdifferent data rates supported have nearly the same probabilityto access the channel. In the case of the perfect carrier sensing,i.e. no collision happens, the link set selected by 802.11 MACis a random independent set. We can compute the averageperformance of the rate adaptive 802.11 MAC by

𝑤(𝐼𝑀 ) = 𝐸{∑𝑣∈𝐼𝑀

𝑤𝑣} = 𝑊 ⋅𝐸{ 𝑤𝑣𝑖

𝑤(𝐺′𝑖)}

= 𝑊 ⋅𝐸{ 𝑤𝑣𝑖

𝑤𝑣𝑖 +𝑁𝐺′(𝑤𝑣𝑖 )} =

𝑊

𝑑𝑤 + 1,

(31)

in which 𝐸 denotes the expect value.Proposition 1 to 3 give the optimal and centralized schedul-

ing policies, by which a set of links in the MWIS are scheduledto transmit in a timeslot. For practical implementation, wepropose the distributed and asynchronous way to approachthe optimal policy: A link with the lowest weighted degreeamong its neighbors has the highest priority to transmit. Thedistributed algorithm may not achieve the optimal performanceas the optimal policies due to the heuristic nature, whereasProposition 4 provides its performance lower bound. Further-more, the equation (31) shows that the average performance

TABLE IAVERAGE TRANSMISSION AND CARRIER SENSING RANGES

Rates (Mbps) 11.0 5.5 2.0 1.0 CS

Range (m) 399 531 669 796 1783

of the rate adaptive 802.11 MAC without collisions equals thelower bound of our proposed algorithm with PI.

B. Overhead Analysis of our Distributed Algorithm

Comparing with other opportunistic scheduling schemes,e.g. OSAR, our distributed algorithm GOS does not induceany more control packets, although we need to carry a bitmore information on the CTS and DATA packets. However,the number of packet makes difference of network throughputrather than the packet length. The extension part of the CTSand DATA packets are the degree table whose length is3+ (8+4+11+1)×𝑛 bits, in which the first 3 bits indicatethe number (𝑛) of entities in this table, and the number islimited to be no greater than 8. The (8 + 4+ 11+ 1) denotesthe information bits of one entity which is composed of onebyte for link identifier (e.g. combining the last four bits of thetransmitter’s and the receiver’s MAC address), four data ratebits (at most 16 levels), 11-bit for the weighted degree and oneimmediate neighbor indicator bit. In order to be decoded byall of the neighboring nodes, the extension parts are encodedin the basic data rate, 1Mbps. In a typical scenario, in whicha sender receives 1.5 CTS packets on average and the intervalof a data packet is the transmission time of a 1000-byte datapacket with 1Mbps data rate, the overhead induced by ourscheme is

Overhead =𝑇Added bits

𝑇Added bits + 𝑇DIFS + 4𝑇SIFS + 𝑇RTS + 1.5𝑇CTS + 𝑇ACK + 𝑇DATA

=(3 + 24× 8)× (1 + 1.5)

(3 + 24× 8)× 2.5 + 1202 + (192 + 8× (34 + 1000))

= 4.8%

In this case, our new scheme introduces lower than 5%bandwidth overhead, but leads to much higher overall through-put gain as it will be seen in the next section.

VII. SIMULATION RESULT

Our simulation experiments are conducted by ns-2 (ver-sion 2.29). For system maximization problem, we compareGOS with OAR, OSAR, and the optimal scheduling. More-over, three schemes with proportional fairness requirements:GOS prop, OSAR prop and Optimal prop are compared. Byusing the optimal scheduling policies (Optimal and Opti-mal prop), there is no overhead of the information exchangingand the flows are scheduled in a collision-free way. In allschemes, the data packet size is set to 1000 bytes, of whichthe available transmit rates of data packets are set to 1Mbps,2Mbps, 5.5Mbps and 11Mbps according to IEEE 802.11bstandard. The number of packets in one back-to-back trans-mission is set according to the data rate selected: 1 for 1Mbps,2 for 2Mbps, 5 for 5.5Mbps and 11 for 11Mbps. All thecontrol packets (GRTS, CTS and ACK) are transmitted with


1 2 3 4 5 64

4.5

5

5.5

6

6.5

7

The method of setting TIFS

Net

wor

k T

hrou

ghpu

t (M

bps)

TIFS = 10ms

100ms1000ms

AaptiveTIFS

opt

Optimal

Fig. 3. Network throughput obtained by different methods of setting theTIFS. The first three methods set all the inserted TIFS as constant values(10ms, 100ms or 1000ms), the fourth is the adaptive adjusting algorithm(eqn. (24)), the fifth is the optimal TIFS setting (eqn. (26)) and the last is theoptimal scheduling without collisions and overhead.

the basic data rate, 1Mbps. The values of receiver sensitivitiesfor different data rates are chosen based on the settings ofORiNOCO 802.11b card1. Thus, the average transmission andcarrier sensing ranges can be computed by the two-ray groundreflection model and the result is shown in Table I.

In our simulation, the Rician fading channel model weuse is the same as the one used in literature [2], [3]. Toevaluate the performance of OSAR with proportional fairnessrequirements, we modify OSAR prop’s scheduling criteria as:Each sender selects the link with maximal 𝜇𝑖/𝑏𝑖 among itsown outgoing links.

We use the fixed route policy and report the end-to-endeffective throughput. We set the 𝑇𝐼𝐹𝑆𝑚𝑖𝑛 = 5ms and𝑇𝐼𝐹𝑆𝑚𝑎𝑥 = 500ms in the adaptive TIFS adjusting algorithm(eqn. (24)). By using different methods of setting the TIFS,we give the network throughput in Fig. 3 for a simple scenariowhere two 450-meter links are parallel with a distance of50 meters. Fig. 3 shows that the adaptive algorithm achieveshigher network throughput than the first three static settingsand its performance is nearly the same as the optimal TIFSsetting method obtains. In the following simulations, we usethe adaptive TIFS adjusting algorithm, since the optimal TIFScan not be easily computed as described in Section V.

A. Two-Transmitter Scenario

Firstly, we simulate a two-transmitter scenario with fiveCBR flows as illustrated in Fig. 4. In this scenario, eachtransmitter has multiple candidate receivers. The distancebetween any sender and its receiver of each flow is 500m.Meanwhile, the distance between two transmitters is 1800m,which is larger than the average carrier sensing range.

As Fig. 5(a) depicts, under the system maximization objec-tive, GOS prefers to transmit on link 1, 4 and 5, since theymaximize the spatial reuse of channel. Comparing with OSAR,GOS reduces throughput of link 2 and 3, but improves link

1For 802.11b, we use the specifications for the ORiNOCO 11b Client PCCard which can be found at http://www.proxim.com/.

Fig. 4. A two-transmitter scenario and its contention graph. There are fourmaximal independent subsets: Ω = {(𝑓2), (𝑓3), (𝑓1, 𝑓4), (𝑓1, 𝑓5)}.

1, 4 and 5 by almost 100%. Totally, the network throughputof GOS is 35% higher than that of OSAR. Furthermore, ourGOS achieves about 90% of the network throughput achievedby the optimal scheduling.

To evaluate our scheduling policies to achieve propor-tional fairness, Fig. 5(b) shows the each link’s normalizedthroughput, i.e. 𝑏𝑖/𝑏∗𝑖 , in which 𝑏∗𝑖 is the throughput obtainedby the Optimal prop. Our policy GOS prop achieves wellfairness among links and its network throughput is about 70%of the optimal one. Let FI denote the proportional fairness

indexes, which is given by FI =(∑

𝑖∈𝒩 𝑏𝑖/𝑏∗𝑖 )

2

𝑁∑

𝑖∈𝒩 (𝑏𝑖/𝑏∗𝑖 )2. The FI of

the Optimal prop, GOS prop and OSAR prop are 1, 0.9993and 0.9497 respectively. It shows that comparing with theOSAR prop, our GOS prop obtains higher throughput andhigher fairness index.

B. Random Topologies

In this experiment, we simulate random topologies, inwhich the transmitters are uniformly distributed in a squarearea. More importantly, each transmitter has one or tworeceivers which are also in this square area. Firstly, we setthe side length of square as 800m, and the distance betweeneach transmitter and its intended receiver is fixed at 450m.Therefore, under this setting, the links can hear from eachother. Figure 6(a) shows the network throughput versus thenumber of links in the square, where each transmitter isassociated with only one receiver. It is shown that our GOSgets nearly the same throughput as the Optimal schedulinggets, and it outperforms the OSAR up to 50% when thenumber of links equals 8. As the number of links increases,the Optimal and GOS get higher network throughput since thedegree of diversity increases. However, without exploiting thediversity among links which do not have a common sender, thethroughput achieved by OSAR drops as the number of linksincrease due to higher collision probability. The OSAR andOAR achieve the same throughput, since no multi-receiverdiversity can be exploited here. In Fig. 6(b), when eachsender is associated with two receivers, the OSAR gets higherthroughput than OAR which does not exploit multiuser diver-sity. However, our GOS still obtains the highest performancedue to the coordination among different senders. In Fig. 8,we also present the throughput obtained by the policies withproportional fairness requirement. In this 800×800 area, thesix links are symmetrical, thus the throughput of each linkobtained by each policy is almost equal. Our GOS obtains25% higher throughput than the OSAR.

Secondly, we set the side length of the square area as 2km.Each transmitter has one or two candidate receivers which


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are uniformly distributed in a round area with a radius of𝐷𝑚𝑎𝑥 from their intended transmitters. By setting 𝐷𝑚𝑎𝑥 =800𝑚, the network throughput of different schemes versus thenumber of links is shown by Fig. 7. Similarly as before, ourGOS performs much better than OSAR with a 30% throughputgain. In addition, we simulate an topology with 3 sendersand 6 receivers and the FI of the Optimal prop, GOS propand OSAR prop are 1, 0.9972 and 0.8816 respectively, whichreveals that our policy provides much better fairness supportthan the OSAR does.

VIII. CONCLUSION

In this paper, we study the opportunistic scheduling problemfor IEEE 802.11 based wireless ad hoc networks, in which thefading effect of the channel, the contention among neighboringlinks and the utility-based fairness among links are taken intoconsideration. By exploiting the generalized multiuser diver-sity, i.e. by assigning the links with or without the commonsender in better channel conditions with higher priority toaccess the channel, the network throughput can be dynamically

increased. We formulate the utility-based scheduling problemas an optimization problem and we present the optimal so-lutions for different utility functions. In order to implementthe optimal schedulers into IEEE 802.11 based ad hoc net-works, we propose the graph theory based and opportunisticscheduling scheme, GOS, which is a distributed algorithm andimitates the greedy algorithm to locate the maximum weightedindependent set in graph theory. We give the analytic analysisof the performance bound and the overhead of our GOS, whichboth verify its efficiency. The simulation results also show thatGOS achieves more than 30% network throughput gain andprovides better fairness support than the existing work.

ACKNOWLEDGMENT

The research was support in part by the National BasicResearch Program of China (973 Program: 2007CB310607),NSFC/Hong Kong Research Grants Council Joint Re-search Scheme under Grant 6073116031, the Interna-tional S&T Cooperation Program of China (ISCP) (No.2008DFA12100), grants from RGC under the contract 622508


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Fig. 7. Network throughput versus the number of links in the random scenarios within an 2km×2km square area.

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and RPC06/07.EG05, NSFC/RGC grant N HKUST609/07,and the NSFC oversea Young Investigator grant under No.60629203.

REFERENCES

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[2] B. Sadeghi, V. Kanodia, A. Sabharwal, and E. Knightly, “OAR: anopportunistic auto-rate media access protocol for ad hoc networks,”ACM Wireless Networks, vol. 11, no. 1, pp. 39–53, Jan. 2005.

[3] J. Wang, H. Zhai, Y. Fang, and M. C. Yuang, “Opportunistic mediaaccess control and rate adaptation for wireless ad hoc networks,” inProc. IEEE ICC’04, vol. 1, pp. 154–158, June 2004.

[4] Z. Ji, Y. Yang, J. Zhou, M. Takai, and R. Bagrodia, “Exploitingmedium access diversity in rate adaptive wireless LANs,” in Proc. ACMMOBICOM’04, pp. 345–359, Sept. 2004.

[5] M. Zhao, H. Zhu, W. Shao, V. O. K. Li, and Y. Yang, “Contention-basedprioritized opportunistic medium access control in wireless LANs,” inProc. IEEE ICC’06, June 2006.

[6] H. Luo and S. Lu, “A topology-independent wireless fair queueingmodel in ad hoc networks,” IEEE J. Select. Areas Commun., vol. 23,no. 3, pp. 585–597, Mar. 2005.

[7] H. L. Chao and W. Liao, “Fair scheduling in mobile ad hoc networkswith channel errors,” IEEE Trans. Wireless Commun., vol. 4, no. 3, pp.1254–1263, May 2005.

[8] X. L. Huang and B. Bensaou, “On max-min fairness and scheduling inwireless ad hoc networks: analytical framework and implementation,”in Proc. ACM MOBIHOC’01, pp. 221–231, Oct. 2001.

[9] K. Xu, M. Gerla, and S. Bae, “How effective is the IEEE 802.11RTS/CTS handshake in ad hoc networks,” in Proc. IEEE Globecom’02,vol. 1, pp. 17–21, Nov. 2002.

[10] J. Mo and J. Walrand, “Fair end-to-end window-based congestioncontrol,” IEEE/ACM Trans. Networking, vol. 8, no. 5, pp. 556–567,Oct. 2000.

[11] Q. Zhang, Q. Chen, F. Yang, S. Shen, and Z. Niu, “Cooperative andopportunistic transmssion for wireless ad hoc networks,” IEEE Network,pp. 14–20, vol. 21, no. 1, Jan./Feb. 2007.

[12] A. Kako, T. Ono, T. Hirata, and M. M. Halldorsson, “Approximationalgorithms for the weighted independent set problem,” in Proc. 31stInternational Workshop on Graph-Theoretic Concepts in ComputerScience 2005, pp. 341–350, June 2005.

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Qing Chen (StM’05) received his B.S. degree fromthe Department of Electronic Engineering, TsinghuaUniversity, Beijing, China, in 2003. Currently, he isa Ph.D. candidate in the Department of ElectronicEngineering, Tsinghua University. From November2005 to May 2006, he was a visiting researcher atthe Department of Computer Science, Hong KongUniversity of Science and Technology, Hong Kong.His research interests include power control, MAClayer scheduling and quality of service in wirelessad hoc networks.


Qian Zhang (M’00-SM’04) received the BS, MS,and PhD degrees from Wuhan University, China, in1994, 1996, and 1999, respectively, all in computerscience. She joined the Hong Kong University ofScience and Technology in September 2005 as anAssociate Professor. Before that, she was at Mi-crosoft Research Asia, Beijing, China, from July1999, where she was the research manager of theWireless and Networking Group. She has publishedmore than 200 refereed papers in international lead-ing journals and key conferences in the areas of

wireless/Internet multimedia networking, wireless communications and net-working, and overlay networking. She is the inventor of about 30 pendingpatents. Her current research is on cognitive and cooperative networks, dy-namic spectrum access and management, as well as wireless sensor networks.She has also participated many activities in the IETF ROHC (Robust HeaderCompression) WG group for TCP/IP header compression.

Dr. Zhang is the associate editor for the IEEE TRANSACTIONS ON WIRE-LESS COMMUNICATIONS, IEEE TRANSACTIONS ON MULTIMEDIA, IEEETRANSACTIONS ON MOBILE COMPUTING, IEEE TRANSACTIONS ON VE-HICULAR TECHNOLOGIES (2004-2008), COMPUTER NETWORKS and COM-PUTER COMMUNICATIONS. She has also served as guest editor for the IEEEWIRELESS COMMUNICATIONS, IEEE JOURNAL ON SELECTED AREAS IN

COMMUNICATIONS, IEEE COMMUNICATION MAGAZINE, ACM/SPRINGER

JOURNAL OF MOBILE NETWORKS AND APPLICATIONS (MONET), andCOMPUTER NETWORKS.

Dr. Zhang received TR 100 (MIT Technology Review) world’s top younginnovator award in 2004, the Best Asia Pacific (AP) Young Researcher

Award elected by the IEEE Communication Society in 2004, and the BestPaper Award by the Multimedia Technical Committee (MMTC) of the IEEECommunications Society and Best Paper Award in QShine 2006, IEEEGlobecom 2007, and IEEE ICDCS 2008. She received the Overseas YoungInvestigator Award from the National Natural Science Foundation of China(NSFC) in 2006. Dr. Zhang is Chair of the Multimedia CommunicationTechnical Committee of the IEEE Communications Society. She is alsoa member of the Visual Signal Processing and Communication TechnicalCommittee and the Multimedia System and Application Technical Committeeof the IEEE Circuits and Systems Society.

Zhisheng Niu (M’98-SM’99) graduated fromNorthern Jiaotong University, Beijing, China, in1985, and got his M.E. and D.E. degrees from Toy-ohashi University of Technology, Toyohashi, Japan,in 1989 and 1992, respectively. He worked forFujitsu Laboratories Ltd., Kawasaki, Japan, from1992 to 1994, and currently is a professor at the De-partment of Electronic Engineering, Tsinghua Uni-versity, Beijing, China. His research interests includeteletraffic theory, mobile Internet, radio resourcemanagement of wireless networks, and cognitive

radio networks. Dr. Niu is a fellow of the IEICE, a senior member of the IEEE,director of IEEE Asia-Pacific Board, and council member of Chinese Instituteof Electronics. He was also serving as the TPC co-chairs of APCC2004 andIEEE ICC2008, general chairs of APCC2009 and WiCom2009, and postersession chair of Mobicom2009.

a graph theory based opportunistic link scheduling for wireless ad hoc networks

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