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2504 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 10, OCTOBER 2012

Scheduling in Mobile Ad Hoc Networks WithTopology and Channel-State Uncertainty

Lei Ying, Member, IEEE, and Sanjay Shakkottai, Senior Member, IEEE

Abstract—We study throughput-optimal scheduling in mobilead hoc networks with time-varying (fading) channels. Traditionalback-pressure algorithms (based on the work by Tassiulas andEphremides) require instantaneous network state (topology,queues-lengths, and fading channel-state) in order to makescheduling/routing decisions. However, such instantaneous net-work-wide (global) information is hard to come by in practice,especially when mobility induces a time-varying topology. Withinformation delays and a lack of global network state, differentmobile nodes have differing “views” of the network, thus in-ducing uncertainty and inconsistency across mobile nodes in theirtopology knowledge and network state information. In such asetting, we first characterize the throughput-optimal rate regionand develop a back-pressure-like scheduling algorithm, whichwe show is throughput-optimal. Then, by randomly partitioningthe geographic region spatially into disjoint and interference-freesub-areas, and sharing delayed topology and network state infor-mation only among nearby mobile nodes, we develop a localizedlow-complexity scheduling algorithm. The algorithm uses instan-taneous local information (the queue length, channel state andcurrent position at a mobile node) along with delayed networkstate information from nearby nodes (i.e., from nodes that werewithin a nearby geographic region as opposed to network-wide in-formation). The proposed algorithm is shown to be near-optimal,where the geographic distance over which delayed network-stateinformation is shared determines the provable lower bound on theachievable throughput.

Index Terms—Delayed network state information, mobile ad hocnetworks, throughput optimal scheduling.

I. INTRODUCTION

M OBILE ad hoc networking is one of the most innovativeemerging networking technologies and has broad ap-

plications in various domains (e.g., battlefield communications,search and rescue operations, and range extension for rural net-works). Mobile nodes communicate with each other using wire-less communication, where simultaneous nearby transmissions

Manuscript received August 18, 2009; revised June 02, 2010; accepted Jan-uary 20, 2012. Date of publication March 08, 2012; date of current versionSeptember 21, 2012. This work was supported in part by the National ScienceFoundation (NSF) under Grants CNS-0347400, CNS-0519535, CNS-0721380,CNS-0831756, CNS-0964391, and CNS-0953165, in part by the Defense Ad-vanced Research Projects Agency (DARPA) ITMANET program, and in part bythe Defense Threat Reduction Agency (DTRA) under Grants HDTRA1-08-1-0016 and HDTRA1-09-1-005. An earlier version of this paper [1] appeared inthe Proceedings of the IEEE INFOCOM conference, Rio de Janeiro, Brazil,April 2009, Recommended by Associate Editor I. Paschalidis.L. Ying was with the Department of Electrical and Computer Engineering,

Iowa State University Ames, IA 50010USA . He is nowwith the School of Elec-trical, Computer, and Energy Engineering, Arizona State University, Tempe, AZ85287 USA (e-mail: [email protected]).S. Shakkottai is with the Department of Electrical and Computer Engineering

and the Wireless Communications and Networking Group (WNCG), The Uni-versity of Texas at Austin, . The University of Texas at Austin, Austin, TX 78712USA (e-mail: [email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAC.2012.2190309

can cause significant interference. To develop a high-perfor-mance mobile ad hoc network, a key step is to design schedulingalgorithms that selectively activate a subset of links according tothe known network state information in order to avoid excessiveinterference as well as maximize network throughput. In thispaper, we study scheduling algorithms for mobile ad hoc net-works with time-varying (fading) channels. Most studies in lit-erature that build on the work of Tassiulas and Ephremides [2],[3]) (in the context of time-varying channels and/or topology)require all nodes in the network to have (globally shared) in-stantaneous network state (e.g., topology knowledge, queues-lengths, and fading channel-state) in order to make scheduling/routing decisions. However, in a mobile network with a time-varying topology, it does not seem reasonable to expect all nodesto have such instantaneous network-wide (global) information.Furthermore, in general, there is no central controller in mobilead hoc networks, so each mobile has to make transmission de-cisions based on the information it collects. Thus, a challengingproblem is to develop scheduling algorithms with channel andtopology uncertainty.We consider a network with sender-receiver (S-R) pairs,

where the S-R pairs move according to Markovian processes.We assume that each mobile knows its own current position andinstantaneous channel state, but it only has other mobiles’ infor-mation with delay . This information delay along with the lackof global network state induces uncertainty and inconsistencyin the topology knowledge and network state information (dueto the fact that different mobile nodes have different “views” ofthe network). Our focus of this paper is to first understand thefundamental network throughput region under the informationinconsistency and topology uncertainty, and then develop on-line scheduling algorithms that are optimal or near optimal. In[4], it has been shown that the network throughput region withcomplete network state is achievable even when queue lengthinformation is delayed. The focus of this paper is the impactof delayed channel and geographical information on networkthroughput.

A. Main Contributions

The main contributions of this paper include:1) We first characterize the network throughput regionunder the information structure that each pair has itsown instantaneous channel and geographic information,but other pairs’ information with a delay of time slots.

2) We then propose a scheduling algorithmwhere each mo-bile first computes a location-based threshold functionbased on the global delayed information and statisticalinformation. The input of the location-based thresholdfunction for S-R pair is their current location, and theoutput is a nonnegative value used to compare with the

0018-9286/$31.00 © 2012 IEEE

YING AND SHAKKOTTAI: SCHEDULING IN MOBILE AD HOC NETWORKS WITH TOPOLOGY AND CHANNEL-STATE UNCERTAINTY 2505

current channel state of S-R pair . After obtaining the lo-cation-based threshold function, the sender of S-R paircomputes the threshold value by inputting the pair’s cur-rent location. The sender transmits if the current channelstate is better than the threshold value, and keeps silentotherwise. The key idea of this threshold-based sched-uling is using the delayed information, which is con-sistent across the network, to coordinate the nodes be-havior, and then making the actual transmission deci-sions based on instantaneous geographical and channelinformation. We show that the algorithm is throughput-optimal, i.e., it can stabilize the network as long as thetraffic is within the network throughput region. Eachmobile, however, needs to compute the threshold func-tion, and the computation complexity grows exponen-tially with network size.

3) Finally, we propose a localized scheduling algorithm,where we partition the geographic region spatially intodisjoint and interference-free sub-areas and delayedtopology and network state information are shared onlyamong nearby mobile nodes.We develop a near-optimalalgorithm for distributed scheduling. The algorithm usesinstantaneous local information along with delayed net-work state information from nearby nodes (i.e., fromnodes that were within a nearby geographic region asopposed to network-wide information). We show thatthis algorithm is near-optimal; more formally, we showthat traffic is supportable if issupportable under some throughput-optimal algorithm,where is the parameter depending on the spatialscale of the partition. In this localized algorithm, thelocation-based threshold function is computed basedon the delayed information of those nodes that werewithin the same sub-area. Therefore, the computationcomplexity of the algorithm is only determined by thesize of the sub-area and the corresponding mobile nodeswithin a sub-area (as opposed to all mobile nodes in thenetwork).

B. Related Work

Throughput-optimal routing/scheduling algorithm was firstproposed in [2], [3]. Assuming that all mobile or static nodeshave perfect global knowledge of the queue, channel andtopology state, throughput-optimal routing/scheduling algo-rithms have been developed for different networks [5]–[13].There has also been much work in developing distributed andlow-complexity implementations [14]–[19]. Please see [20],[21] for a survey.There have been some studies in the context of incomplete

network state information (missing/delayed channel, queueor topology state). To the best of our knowledge, the earliestwork to consider delayed queue-length information and itsimpact on stability of back-pressure algorithms is [4]. In adown-link/up-link wireless scenario that explore the trade-offbetween channel measurements and opportunistic gain, studiesinclude [22]–[31]. With independent and identically distributed(i.i.d.) channels and a static network, [32] has developedrouting/scheduling algorithms with noisy channel estimates.

Fig. 1. S-R pairs deployed in a square area with side-length .

In [33] and [34], the authors have studied throughput-optimalscheduling/routing in static ad hoc networks with delayednetwork state information (queue length and channel state).However, to the best of our knowledge, we are not aware of(near) throughput-optimal scheduling results with limited anddelayed channel/topology knowledge in a mobile context.

II. MODEL AND NOTATIONS

We consider a wireless network with sender-receiver (S-R)pairs. We use to denote the set of the S-R pairs. We name thesender of pair to be sender , and the receiver of pair to bereceiver .Without loss of generality, we assume the S-R pairsare deployed in a square area with side-length as in Fig. 1.1) Traffic Model: We assume single-hop traffic in this

paper, i.e., there is a traffic flow from sender to receiver foreach , and sender does not communicate with otherreceivers than receiver . We denote by the number ofpackets arriving at sender at time . We assume thatare bounded random variables and independent across timeand different sender-receiver pairs. In particular, we assume

for all and . In this paper, we only considersingle-hop flows. So the paper focuses on scheduling in amobile ad hoc network and does not address the routing issue.2) Mobility Model: We consider a discrete-time system. We

assume that the pairs move at the beginning of each time slot,and stay still within a time slot. The mobility of each pair isMarkovian on a discrete square-lattice over the square region,i.e., the next location of a mobile is determined by its currentlocation, and does not depend on the other history information,and the next location is on a (possibly fine resolution) grid. Fur-thermore, we assume that the mobility processes are stationaryand ergodic (we abuse notation in that the square area actuallymeans the discrete-lattice over the square area). We denote by

the location of the sender at time slot , the locationof receiver at time slot , and the set of all positions on thesquare-lattice.3) Channel Model: We assume that a time-varying wireless

channel between each S-R. We denote by the channel ca-pacity of pair at time , which is the maximum number ofpackets that can be reliably transmitted over link at time .In this paper, is also referred as the channel state of linkat time .We assume the channel states are drawn from a finite set

such that . Thechannel state distribution of link at time only depends on its


state at time and the locations of the sender and receiverof link at time ,1, i.e.,

We also assume the channels are independent across S-R pairs.Note that for S-R pair , forms a Markovchain. These Markov chains are independent across S-R pairs,and the transition probabilities of these Markov chains are as-sumed to be known by all the mobiles in the network.4) Information Set for Sender : We assume that at time, sender has the location and channel information of pairat time , i.e., and delayed loca-tion and channel information of all pairs in the network, i.e.,

and for all ,where is backlog size at sender , is the delay for channeland location information, and is the delay for queue-lengthinformation. We assume that .2 Note that in practicalsystems, the delays depend on node distances and may be timevarying, so sender may have heterogenous delays from othernodes. We assume a homogeneous delay in this paper so that theproblem is tractable, and leave the heterogeneous delay case forour future research.5) Scheduling-Decision Vector: We define a vector to

be the scheduling-decision vector at time such thatif the sender transmits at time and otherwise. Notethat is a function of the information available to sender .6) Location-Based Threshold Scheduling: We study a class

of scheduling policies which we denote as location-basedthreshold scheduling policies. A location-based thresholdscheduling policy is defined by a real-valued function ,where the inputs are locations and the output is a real number.The form of the function is determined by the delayed channeland queue length information available at the sender, and isindependent of current locations and channel state (please seeSection III-C for a more precise description). After computing

(note that the form of varies over time sincethe delayed information changes over time), if S-D pair is atlocation and the channel state , thensender will transmit; and sender keeps silent otherwise. Wefurther define . We note that in thispaper we use lower-cases to denote the realizations of randomvariables, e.g., is a realization of .7) Interference Model: We assume a geographic-based col-

lision model [35] in this paper. If two links interfere with eachother, simultaneous transmissions on the two links will lead to acollision and no information (packet) can get through. Considerpair , the transmission of pair will interfere with the transmis-sion of pair if

1This channel model holds when the channel state is determined solely by thelocations of the sender and receiver.2To propagate delayed information in the network, a mobile can periodically

broadcast its location and channel state information to other mobiles in the net-work. Later, in the low-complexity algorithm, the information only needs to bebroadcast to nearby nodes of the mobile.

where is the Euclidean distance, and is a pro-tocol specified guard-zone to prevent interference. A transmis-sion from sender to receiver is successful if no sender whointerferes with pair is transmitting at the same time.8) Queue Management: We assume that each sender main-

tains a queue. Recall that the length of the queue at sender attime is denoted by . The evolution of the queue can bedescribed as

where we assume the packets arrive at the beginning of eachtime slot, and then the queue is served.

III. THROUGHPUT-OPTIMAL SCHEDULING ALGORITHM WITHTOPOLOGY UNCERTAINTY

In this section, we first characterize the network throughputregion under channel and topology uncertainty.

A. Optimal Throughput Region

It is easy to see that the transmission rates of the pairs at timeare determined by the following three parameters: 1) channelcondition , 2) network topology, which is defined by themobiles’ positions , and 3) the scheduling decision

.Now assume that , and. Under the collision-model defined in Section II, a maximum

link rate can be achieved over link if there is no other ac-tive pairs interfering with pair ; otherwise, the link rate is zero.Mathematically, we can define a link-rate vectorsuch that

if , and for any pair such that; and

otherwise.Given the delayed information

we define such that

where denotes the convex hull over all associated withdifferent

and is the location-based threshold function.Note that here we restrict the location-based threshold functions


to be functions determined by because of theinformation available at the senders.Note that is the set of supportable rates as-

suming that the delayed information is andsender transmits when the location of pair is atand channel state is larger than . Notethat is independent of because both and

are assumed to be stationary random variables.Finally, we define

where is the stationary distribution of.

Given traffic , we say the traffic is within the networkthroughput region (or the traffic is supportable) if there exists ascheduling algorithm, under which there exists such that

for all .We can see that is the convex hull of the sup-

portable rates under all possible location-based threshold sched-uling given the delayed information is

.Theorem 1: Traffic is supportable only if .Proof: We first define function ,

and denote by the set of all . Note that the inputs ofare the current channel state of link , the locations of senderand receiver , and the output is a threshold value. If the delayednetwork state is fixed to be , due to the Markovianproperties of channel states and locations, the set of achievableservice rates is

which is a convex hull over different and

Therefore, traffic load is supportable only if it is in the fol-lowing region:

To complete the theorem, we next show that

It can be easily verified that

because , i.e., a location-based thresholdscheduling is a function in .Next we will show

by proving that

holds for any . We first consider the case such that the fol-lowing inequality always holds

(1)

In this case suppose that is not a location-based thresholdfunction, which implies that there exit channel states andsuch that and . Wethen construct a location-based threshold function based onsuch that if and

It can be shown that

because fixing the mobiles’ positions, the impact of a S-R pair’stransmissions on other pairs is determined by the probability theS-R pair transmits but is independent of in which channel statesthe S-R pair transmits. This inequality yields that

In case condition (1) does not hold. For example, for someand , we have

We can replace channel state with two virtual states andsuch that and

Then the same argument can be applied to prove that

The theorem therefore holds.


Fig. 2. Mobile ad hoc network example.

Fig. 3. Markovian mobility.

B. Illustrative Example

Now we use a simple example to illustrate andthroughput-region . Consider a simple example with two S-Rpairs (S-R-1 and S-R-2) as in Fig. 2, where S-R-1 is not mo-bile and located at , and S-R-2 moves betweenlocations and , respectively. As-sume that link capacities are unity for both of the links, i.e.,

(in other words, there is no channel fading).However, the two pairs interfere with each other when S-R-2is at location . The mobility of S-R-2 follows aMarkov process as shown in Fig. 3.Note that when S-R-2 is at position , the two

pairs cannot be active (i.e., successfully transmit packets) si-multaneously. Thus, we have

and

Next we assume that the information delay .Since both channels are time-invariant and the position ofS-R-1 is fixed, the information S-R-1 has is completelydetermined by the delayed information from S-R-2. Itis easy to verify that given

, is as shown in

Fig. 4(a), and given ,is as in shown Fig. 4(b).

We note that Fig. 4(a) is constructed based on the factthat S-R-2 has probability 2/3 to stay at the current location

, and has probability 1/3 to move to location

Fig. 4. Throughput regions under different delayed information. (a) Case 1.(b) Case 2.

. Since the link capacities are constant, the sched-uling decisions are on-off decisions based on the location ofthe link. Since S-R-1’s location is fixed, its scheduling decisionis either always on or always off. Consider the case that S-R-1is always off. The optimal decision of S-R-2 is to be on all thetime. Then, any such that and is supportable.On the other hand, if S-R-1 is on, then the optimal decision ofS-R-2 is to be on at location and off at location

. Then, any such that andis supportable. Fig. 4(a) is the convex hull of all supportablerates. Fig. 4(b) is constructed similarlyNote that

so the throughput region for is as shown in Fig. 5.The throughput region without any information delay is alsoshown in Fig. 5 using the dotted line as a comparison. We cansee that the throughput region shrinks when we have delays intopology/channel knowledge.

C. Throughput-Optimal Scheduling Algorithm

In this section, we propose a throughput-optimal schedulingalgorithm which stabilizes the network for within the net-work throughput region.1) Threshold-Based Scheduling: Given the delayed informa-

tion and

which is available at all senders, the senders make transmissiondecisions according to the following two steps:


Fig. 5. Throughput regions for (solid-line) and (dotted-line).

Step 1) Each sender computes a set of threshold functions(for all ) that solves the following

optimization problem:

(2)

We note that the inputs of these threshold func-tions are the current locations and channel states,but the computation of the threshold functionsthemselves does not depend on current locationsand channel states. We further note that the value

of

equals to one if no pair that interferes with pairtransmits at time given the location-based

threshold function , and equals to zero otherwise.So

is the expected transmission rate pair can achieve.The maximization problem (2) is to find the loca-tion-based threshold functions that maximize thesum of the queue-weighted expected transmissionrates

We also note that the instantaneous channel statesand locations are random variables

in this calculation. The calculation of (2) only in-volves the delayed information and the statistical in-formation, and does not require the instantaneousinformation even though link knows its instanta-neous information. Given the transition probabilitiesof the Markov chains associated with the channelsand the locations, the expectation in (2) can be com-puted and the optimization problem can be solved.

Step 2) Sender transmits with rate if its current loca-tion is and

The following theorem shows that the algorithmproposed above is throughput optimal. The detailedproof is provided in Appendix A.

Theorem 2: Assume a traffic such thatand for all and , the time-average of theexpected total queue-length are bounded under the threshold-scheduling algorithm, in other words, there exists a constantsuch that the following inequality holds for all :

Remark: While this maximization problem (2) is similar tothat in the back-pressure algorithm [2], [3], the transmission rate

is computed based on the delayed information and sta-tistical information instead of the instantaneous channel state in-formation. In our simulations, we will show that directly usingthe instantaneous channel state information will lead to a sig-nificant performance loss with the presence of topology andchannel-state uncertainty because the instantaneous informationis not globally shared among all mobiles in the network.Remark: Since the delayed information and ,

and channel state distributions are available at all mobiles, eachmobile can solve the optimization problem (2), and makes theirtransmission decisions based on its current location, instanta-neous channel state, and the threshold value. The optimizationproblem (2) involves network-wide delay information, so thecomplexity is at the scale as the network size. Note that even, the problem of finding the optimal schedule is an NP-hardproblem in general, which implies that the complexity growsexponentially with network size. In the next section, we willdevelop a low-complexity (where the decision depends only on“local” delayed information) and near optimal implementation.

IV. LOW-COMPLEXITY AND NEAR-OPTIMAL IMPLEMENTATION

In this section, we propose a scheduling algorithm whose in-formation and computation complexity is independent of thenetwork size. The idea is to partition the geographic region spa-tially into disjoint and interference-free sub-areas and only sharedelayed topology and network state information among nearbymobile nodes. Then the computation complexity is determinedby the size of the sub-area. We note that this partition idea hasbeen successfully used in literature to develop low complexityapproximations for NP-hard problem such as low complexityscheduling algorithms [36], [37].In this section, we have additional assumptions as follows:• The distance a mobile can move at the beginning of a timeslot is no more than .

• The distance between a sender and a receiver is upper andlower bounded. Denote by and the lower andupper bounds, respectively.


Fig. 6. Square is partitioned into vertical and horizontal stripes.

Fig. 7. The 1st horizontal-stripe group and the 1st vertical-stripe group.

• Each mobile is equipped with a GPS or appropriate tech-nology (e.g., cell tower based triangulation), so themobileshave knowledge of their geographic locations.

We partition the square area into horizontal stripes and ver-tical stripes as in Fig. 6, where the length of the short side of astripe is and

(3)

Without the loss of generality, we assume that is a positiveinteger (recall that is the side-length of the square area). Sowe have horizontal stripes and vertical stripes. Thehorizontal stripes are indexed with and the vertical stripesare indexed with for .Next we choose such that and are both

positive integers, and then group the stripes as follows:• the horizontal-stripe group consists of horizontalstripes ;

• the vertical-stripe group consists of vertical stripes.

Choosing , Fig. 7 shows the 1st horizontal-stripe groupand 1st vertical-stripe group.

Fig. 8. Active sub-area and its neighboring inactive area are denoted by( is ignored without causing confusion).

Based on the partition and grouping proposed above, we firstintroduce a dynamic partition algorithm:1) Dynamic Partition: We assume that the mobiles share a

common random variable such that at each time slot, uni-formly takes an integer value from 0 to When

, the unit square is partitioned by the stripes belongingto groups (horizontal and vertical) and all senders in thestripes belonging to the groups keep silent. The square areahence is divided into two types of sub-areas: i) inactive area:the groups; and ii) active area: the area outside the inactivearea. The active area consists of disjoint active sub-areas. Forexample, there are nine active sub-areas (indexed from 1 to 9)in Fig. 7.Since is a common random variable shared by all the mo-

biles, the partition, which is random, is known by all the mobilesso that the mobiles can determine their current type (active orinactive) and the area they are located. We consider schedulingschemes such that senders (transmitters) within an inactive-areakeep silent (i.e., do not transmit), which guarantees that trans-missions within different active sub-areas will not interfere witheach other (this follows because the inactive area is “wider” thanthe radio range ).We let denote the index of an active sub-area given par-

tition , indicate the area including active sub-area andneighboring inactive area as shown in Fig. 8. Note that is asquare with side-length if we ignore the edge effect.Furthermore, let denote the set of senders in the cell .We observe that the mobiles in an active sub-area at time

must be in the active sub-area or neighboring inactive stripesat time . This follows directly from (3), and the fact thata mobile can move a distance no more than per time slot.So we have

We also note that since there is no transmission in the inac-tive-area, the transmissions in different active sub-areas do notinterfere.In this section, we further assume that each sender has the

delayed information andfor all such that

i.e., sender has the delayed information from pair who was inthe disk centered at and with a radius at time .


Based on this assumption, we can easily verify that independentof the value of , for all mobiles in active sub-area at timeslot , they have the delayed information and

for all such that .Based on the assumption above, we have that i) a mobile in

active area has the delayed information of all mobiles in thesame active area, and ii) the transmissions in different activeareas do not interfere with each other. Therefore, we can decom-pose the network-wide optimization problem into those corre-sponding to individual active sub-areas, and each mobile onlyneeds to know the local delayed state information.Recall that mobiles know which tile they are located in. The

topology and channel information are exchanged among mobilenodes within distance with delay . The queue-lengthinformation is also assumed to be periodically exchanged withdelay , and we assume that , i.e., the queue-lengthinformation can be exchanged slower than the channel state in-formation.In this setting, we propose the following localized scheduling

algorithm.2) Localized Threshold-Based Scheduling: During each time

slot, the mobiles use the common random variable to generatean integer value belonging to 0 to . The squarearea is partitioned according to the dynamic partition algorithm,and the partition is known by all mobiles.Considering an active sub-area , the mobiles in the sub-

area know andfor all such that . The senders then maketransmission decisions according to the following two steps:Step 1: Sender computes a set of threshold function

that solves the following optimiza-tion problem:

(4)

where

and the threshold function needs to satisfy thatif (sender needs

to keep silent if it is in the inactive-area).Step 2: Sender transmits with rate if its current loca-

tion is and

Note that there are stripes in each partition andeach of them has size , so the overall size of inactive-areasis

Recall that is uniformly chosen from 0 to ,so the probability a mobile is located in the inactive area is

We define

Since the link capacity is upper bounded by , thethroughput-loss because of a mobile moving into inactive-areasis upper bounded by , which goes to zero when

.In the next theorem, we prove that given a traffic such

that

where is an identity vector, then the network is stochas-tically stable under the localized threshold-based scheduling.Theorem 3: Given

the time-average of the expected total queue-length is boundedunder the localized threshold-based scheduling algorithm.

Proof: To prove the theorem, the idea is to consider a vir-tual network, where when sender is in an inactive-area, thearrivals are , and sender can transmit with rate

without causing any interference to other pairs.Consider a modified localized threshold-based sched-

uling, which is the same as the original algorithm ex-cept that if (instead of

). It is easy to see that the queue-evolutionof the virtual network under the modified algorithm is the sameas the original network. Thus, we analyze the virtual networkwith arrival .Given and , let

denote the optimal solution to (2), and denote the thresholdfunction of the modified algorithm. First, for an active area ,we have

(5)


Next for an inactive area , we have

(6)

Inequalities (5) and (6) imply value of (2) under the virtualnetwork is larger than the one under the original network. Thisis the key fact and can be used to prove theorem. The detailedproof is provided in Appendix B.Remark: Note that is an upper bound on the

throughput-loss because the senders may have no packet totransmit or should keep silent in the global optimization solu-tion (2). So the actual performance loss due to the localizationcould be smaller, which will be observed in the simulation inSection V. Also the uniformness and randomness are criticalto guarantee the asymptotically small throughput loss for anymobility patterns (recall that we only require the mobility pro-cesses to be stationary and ergodic). A deterministic partitionor a nonuniform partition may result in that some mobilesare in inactive areas with a higher probability than othermobiles. These mobiles may suffer higher throughput lossesthan other mobiles, and the throughput losses hence will behighly correlated with the distribution of the traffic load. Theuniform and random partition guarantees the upper bound onthe performance loss is independent of both node mobility andtraffic load.

V. SIMULATIONS

In this section, we further study the performance of our algo-rithms using simulations.

A. Simulations in a Small-Size Network

We consider a network as shown in Fig. 9, in which there arefour sender-receiver pairs. We assume that the distance betweena sender and a receiver is 2, which is fixed. At the beginningof each time slot, S-R-1 and S-R-2 (S-R-3 and S-R-4) move asfollows:• An S-R pair moves to its left with probability 1/3 (2/3)or right with probability 2/3 (1/3) if the pair is not at theboundary of the network.

• If the S-R pair is at the left boundary, the S-R pair movesaway from the boundary with probability 2/3 (1/3), or staysat the current position with probability 1/3 (2/3).

• If the S-R pair is at the right boundary, the S-R pair movesaway from the boundary with probability 1/3 (2/3), or staysat the current position with probability 2/3 (1/3).

Fig. 9. Network used in simulations.

We assume that two transmissions collide if the senders are atthe same location or next to each other. We further assume thelink rates are one packet per time slot for all links and all time,and the information delay . The arrivals are assumed tobe Bernoulli arrivals with rate .1) Threshold-Based Scheduling Algorithm Versus the

Mismatch Algorithm: We first compare the performanceof threshold-based scheduling algorithm with an algorithmwhere the senders treat the most recent information they haveas instantaneous information (i.e., they ignore the fact thatit is delayed) and make scheduling decisions based on theback-pressure algorithm. Note that the back-pressure algorithmis throughput optimal when global instantaneous informationis used. The algorithm is named as the mismatch schedulingalgorithm because the information the senders use is differentfrom that for which the algorithm was designed for.2) Mismatch Scheduling Algorithm: Given the delayed in-

formation and

which is available at all senders, the senders make transmissiondecisions according to the following two steps:Step 1: Sender computes decision vector that solves

the following optimization problem:

where if ; otherwise. Note that sender has its own instan-

taneous channel-state, location, and queue informa-tion, and delayed information of other pairs (whichare all however treated as instantaneous informationby sender in this algorithm).

Step 2: Sender transmits if .

We choose , where isthe arrival rate under which the network is critically loaded(we say the network is critically loaded if the average queuelengths are larger than 100). We executed the simulation for

iterations under both algorithms, and computed the averagequeue-lengths. The results are shown in Fig. 10, which includesthe average queue-lengths of S-R-1 for different values of(similar results hold for other sender-receiver pairs).


Fig. 10. Threshold-based algorithm the mismatch algorithm.

Fig. 11. Localized threshold-based algorithm localized mismatch algo-rithm.

From Fig. 10, we can see that the threshold-based sched-uling algorithm (which is throughput-optimal) has smaller av-erage queue-length. We also have that under thethreshold-based algorithm, and under the mis-match algorithm. Our simulation shows that the mismatch algo-rithm is not throughput optimal, and the threshold-based algo-rithm yields around 8% throughput improvement.3) Localized Threshold-Based Algorithm Versus Local-

ized Mismatch Algorithm: We divide the network into ver-tical-stripes with , and the localized threshold-basedalgorithm with . We compare the performance of thelocalized threshold-based scheduling algorithm with a localizedmismatch algorithm under this tiling. The localized mismatchalgorithm is an algorithm where sender only has delayedinformation of the pairs in the same subnetwork (tile). Thescheduling decision is computed similar to the mismatch algo-rithm with the delayed information limited to the subnetwork(tile) the sender is in.We choose . For each , we

executed the simulation for iterations under both algo-rithms, and computed the average queue-lengths. The resultsare shown in Fig. 11. Again, we can see that the localizedthreshold-based scheduling algorithm yields a smaller averagequeue-length. In this simulation, under the local-ized threshold-based scheduling, and under thelocalized mismatch algorithm.From the simulations above, we can see that properly

exploiting delayed information can improve the network

Fig. 12. Ring-like network with 100 positions.

throughput and reduce the backlogs. Furthermore, according toTheorem 3, the throughput loss is upper bounded by

The actual throughput loss is , which is muchsmaller than 0.4375. Note the throughput loss characterized inTheorem 3 is a worst case bound since it holds for any mobilityprocesses and traffic loads. Our simulation result shows that thelocalized threshold-based algorithm in fact may support a muchlarger throughput region than that in Theorem 3.

B. Simulations in a Medium-Size Network

In this subsection, we consider a medium-size network (aring-like network with 100 positions) as shown in Fig. 12. Re-ceivers are positioned on the outer-ring and senders are posi-tioned on the inner-ring. We again assume that the distance be-tween a sender and its corresponding receiver is fixed, and twotransmissions collide if the senders are at the same location ornext to each other. As the number of S-D pairs increases, the ringnetwork has “long-range” dependence, but with limited (ex-pected) one-hop neighbors (as the mobility model is a randomwalk). Due to computational complexity, we cannot simulatethe threshold-based scheduling algorithm and mismatch algo-rithm that require network-wide delayed information. In this setof simulations, we focus on the localized threshold-based algo-rithm and localized mismatch algorithm, where the network isdivided into cells, each containing 5 positions (i.e., and

).1) Performance Comparison With 80 S-D Pairs and Various

Traffic Loads: In this simulation, we place 80 S-D pairs on thering-like network, each pair has a flow with arrival rate . We


Fig. 13. Localized threshold-based algorithm localized mismatch algo-rithm.

Fig. 14. Localized threshold-based algorithm localized mismatch algorithmwith different numbers of S-D pairs.

choose . For each , we exe-cuted the simulation for 30 000 iterations under both algorithms,and computed the average queue-lengths. The results are shownin Fig. 13. Again, we can see that the localized threshold-basedscheduling algorithm yields a smaller average queue-length. Inthis simulation, under the localized threshold-based scheduling, and under the localized mis-match algorithm.2) Performance Comparison With Different Numbers

of S-D Pairs: In this simulations, we varied the numberof S-D pairs from 20 to 80, and compare theunder the localized threshold-based algorithm and the lo-calized mismatch algorithm. The results are shown inFig. 14, which shows the performance gain, defined to be

, versus the numberof S-D pairs. We can see that the performance gain increasesas the number of S-D pairs. The reason is that when the net-work becomes denser, scheduling becomes more important forresolving collisions.

VI. CONCLUSION

In this paper, we studied throughput-optimal scheduling formobile ad hoc networks with information delays. We char-acterized the network throughput region under channel andtopology uncertainty. We also proposed a scheduling algorithmwhere the scheduling decisions are made based each mo-bile’s instantaneous information and delayed information from

local geographic regions. Future directions of this researchinclude: 1) considering multi-hop traffic flows instead of onlypeer-to-peer communications; and 2) considering physical in-terference model and design joint power control and schedulingalgorithms.

APPENDIX APROOF OF THEOREM 2

We define a Lyapunov function such that

Since is a Markov chain, given any , there exists aconstant such that

First, we have

where

(7)

and is the optimal solution to problem (2) giventhe delayed information and .Now given that , according to the definition

of , there exists a set of such that

(8)

Assuming that , we have that

(9)

(10)

(11)

(12)

(13)

(14)


where the does not appear in the condition of the con-ditional expectation because the rate a S-R pair achieves doesnot depend on the queue-lengths given (the computation ofhowever depends on the delayed queue-lengths).According to the definition of , it is easy to see that the

optimization problem (2) can be rewritten as

(15)

where the superscript means the transpose. Also we have that

Thus, we have that

which implies that

holds for all . Thus, we can conclude that

where the last inequality (a) holds due to the definition ofand

inequality (b) holds due to (8).Taking expectation at both sides of the inequality above, we

can further obtain that

which implies that

and

where the last inequality holds because .

APPENDIX BPROOF OF THEOREM 3

We consider the virtual network and let denote thequeue-lengths of the virtual network. We define a Lyapunovfunction such that

First, it can be shown that there exists , which is inde-pendent of such that

where

(16)

(17)

and is the threshold functions computed underthe modified algorithm defined in Section IV.Define to be the throughput region of the virtual network.

Given that

it can be easily verified that there exists a set ofsuch that

(18)


Assuming that , we have that

Recall that is the optimal decision policy in the originalnetwork with the complete delayed network information. Ac-cording to inequalities (5) and (6), we have that

Furthermore, the event that is in the inactive-area is inde-pendent of both and .Thus, we can conclude that

According to the definition of (18), we have that

Following the analysis in Appendix A, we can conclude thatthe sum of the aggregate queue-lengths of the virtual network isbounded.Note that the queue-lengths of the virtual network is the same

as those in the real network since the fictitious arrivalsare served by the fictitious services . Therefore the the-orem holds.

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Lei Ying (M’08) received the B.E. degree from Ts-inghua University, Beijing, China, in 2001 and theM.S. and Ph.D. degrees in electrical engineering fromthe University of Illinois at Urbana-Champaign, Ur-bana, in 2003 and 2007, respectively.During fall 2007, he was a Postdoctoral Fellow at

the University of Texas at Austin, Austin. He was anAssistant Professor at the Department of Electricaland Computer Engineering, Iowa State University,Ames, from January 2008 to August 2012. He is cur-rently an Associate Professor at the School of Elec-

trical, Computer, and Energy Engineering, Arizona State University, Tempe.His research interest is broadly in the area of information networks, includingwireless networks, mobile ad hoc networks, P2P networks, cloud computing,and social networks.Dr. Ying received a Young Investigator Award from the Defense Threat Re-

duction Agency (DTRA) in 2009, NSF CAREER Award in 2010, and is namedThe Northrop Grumman Assistant Professor (formerly the Litton Industries As-sistant Professor) at the Department of Electrical and Computer Engineering atIowa State University for 2010–2012.

Sanjay Shakkottai (M’02-SM’11) received thePh.D. degree from the Electrical Engineering Depart-ment, University of Illinois at Urbana-Champaign in2002.He is with The University of Texas at Austin,

where he is currently an Associate Professor andthe Engineering Foundation Centennial TeachingFellow in the Department of Electrical and ComputerEngineering. His current research interests includenetwork architectures, algorithms, and performanceanalysis for wireless and sensor networks.

Dr. Shakkottai received the NSF CAREER Award in 2004.

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