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IEEE Wireless Communications • February 201248 1536-1284/12/$25.00 © 2012 IEEE

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AC C E P T E D F R O M OP E N CALL

INTRODUCTIONWith the advance of broadband wireless trans-mission, the increasing popularity of mobiledevices, and the deployment of wireless sensornetworks, wireless networks are increasinglyused to serve real-time flows that require strictper-packet delay bounds. Such applicationsinclude VoIP, video streaming, real-time surveil-lance, and networked control. For example, astudy by Cisco [1]has predicted that wirelesstraffic will grow exponentially, and mobile videowill dominate wireless traffic in the near future,accounting for 62 percent of wireless traffic by2015. Serving real-time flows is also a key com-ponent of many cyber-physical systems. In oneexample of a centralized cyber-physical system, aserver may gather surveillance data from wire-less sensors, based on which it makes controldecisions and disseminates them to actuatorunits. Both surveillance data and control deci-

sions need to be delivered in a timely manner, orthe performance of the system may be degraded.In addition to delay bounds, such applicationsalso require some guarantees on their timely-throughput, defined as the throughput of pack-ets that are delivered on time.

Serving real-time flows in wireless networks isparticularly challenging since wireless transmis-sions are subject to shadowing, fading, and inter-ference, and thus usually unreliable. Further, thechannel qualities may differ from link to link. Adesirable solution for serving real-time flowsthus needs to explicitly take into account thestochastic and heterogeneous behavior of packetlosses due to failed transmissions.

In this article, we introduce a theory for serv-ing real-time flows over wireless links. The coreof this theory is a model that takes into accountall the aforementioned challenges, the delaybounds and timely throughput requirements ofapplications, and the unreliable and heteroge-neous nature of wireless transmission. Weaddress three important problems concerningthe service of real-time flows: admission control,packet scheduling, and utility maximization whentimely-throughput requirements are elastic. Wederive a sharp necessary and sufficient conditionthat characterizes when it is feasible to fulfill thedemands of all real-time flows, as well as a poly-nomial-time algorithm for admission control. Wealso introduce two on-line scheduling policiesthat are proven to fulfill every feasible system.For the scenario where timely-throughputrequirements are elastic, we formulate a utilitymaximization problem that aims to maximize thetotal utility over all flows. We introduce a bid-ding procedure and an online scheduling algo-rithm that together achieve the maximum totalutility.

We extend the model to incorporate variousreal world complexities. We consider the sce-nario where different applications may gener-ate traffic with different patterns and requiredifferent delay bounds. We also consider thesituation where channel quality may vary overtime, and where rate adaptation may beemployed to enhance transmission reliability.We introduce a framework for designing

I-HONG HOU AND P. R. KUMAR, TEXAS A&M UNIVERSITY

ABSTRACTWireless networks are increasingly used to

serve real-time flows. We provide an overview ofan emerging theory on real-time wireless com-munications and some of its main results. Thistheory is based on a model that jointly considersthe delay bounds and throughput requirementsof clients, as well as the unreliable and heteroge-neous nature of wireless links. This model canbe further extended in several aspects. It pro-vides solutions to three important problems,namely, admission control, packet scheduling,and utility maximization. The theory can also beextended to consider broadcast of real-timeflows, and to incorporate network coding.

REAL-TIME COMMUNICATION OVERUNRELIABLE WIRELESS LINKS:

A THEORY AND ITS APPLICATIONS

This material is based on work partially supported byUSARO under Contract Nos. W911NF-08-1-0238 and W-911-NF-0710287, NSF under Contracts CNS-1035378,CNS-0905397, CNS-1035340, and CCF-0939370, andAFOSR under Contract FA9550-09-0121. Any opinions,findings, and conclusions or recommendations expressedin this publication are those of the authors and do notnecessarily reflect the views of the above agencies.

The authors providean overview of anemerging theory onreal-time wirelesscommunications andsome of its mainresults. This theory isbased on a modelthat jointly considersthe delay boundsand throughputrequirements ofclients, as well asthe unreliable andheterogeneousnature of wirelesslinks.

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IEEE Wireless Communications • February 2012 49

scheduling policies and derive tractable feasi-bility optimal policies for two different scenar-ios involving unreliable fading channels andrate adaptation. We also address the problemof maximizing the total utility when the con-tending flows are elastic.

In addition to unicast flows, we also addressthe problem of scheduling real-time flows thatare broadcasts. The major challenge for broad-casting is that due to the lack of acknowledg-ment (ACK), the sender may not haveimmediate feedback information on whether atransmission is successful. We extend our modelto address this challenge. The extended modelalso allows the optional usage of network cod-ing. We introduce a framework for designingscheduling policies for any arbitrary networkcoding scheme, and derive policies for variouscoding schemes.

The rest of the article is organized as follows.We describe the basic model for real-time wire-less communication. We establish a necessaryand sufficient condition for feasibility. We intro-duce two online scheduling policies that are fea-sibility optimal. We address the problem ofserving real-time flows when their timely-throughput requirements are elastic. We extendthe basic model to incorporate various realworld complexities. We develop a framework fordesigning scheduling policies, which is then usedto derive policies for two different scenarios. Weaddress the utility maximization problem whenrate adaptation is incorporated. We study theproblem of broadcasting real-time flows. Wethen conclude the article.

BASIC MODEL FORREAL-TIME WIRELESS COMMUNICATIONS

Consider a system with one access point (AP)and N wireless clients. Figure 1 illustrates anexample of such a system. Each client is associat-ed with one real-time flow that requires servicefrom the AP. We assume that time is slotted andslots are numbered by t ∈ {1, 2, 3, …}. Thelength of a time slot is set to be the time neededfor the AP to transmit one packet to a client,including all possible overheads. Time slots arefurther grouped into intervals, where an intervalconsists of T consecutive time slots in (kT, (k +1)T], for some k. The AP is in charge of schedul-ing transmissions in each time slot. This modelnaturally applies to downlink traffic. When bothdownlink traffic and uplink traffic are present,this model can still be applied by incorporatingserver-centric access mechanisms, such as 802.11PCF, WiMAX, and Bluetooth.

At the beginning of each interval, that is, attimes 1,T + 1, 2T + 1, …, each real-time flowgenerates one packet. We assume that packets inall real-time flows need to be delivered within adelay bound of T time slots if they are to be use-ful. In other words, packets that are generated atthe beginning of an interval are only useful ifthey are delivered no later than the end of theinterval. If a packet is not delivered before itsdelay bound, we say that the packet expires, andit is dropped from the system. By droppingexpired packets, it is guaranteed that the delay

of every delivered packet is at most T time slots.We consider unreliable and heterogeneous

wireless links. When the AP makes a transmis-sion for client n, the transmission is successfulwith probability pn. The values of pn may differfrom client to client, as the channel reliabilitiesfor different clients may be different. The APhas instant feedback information on whether atransmission is successful (e.g., by enforcingACKs).1

We measure the performance of a client byits timely-throughput, which is defined as thelong-term average number of successfully deliv-ered packets for the client per interval. Weassume that each client has an inelastic timely-throughput requirement. We denote the timely-throughput requirement of client n by qn.

In the following sections, we will discuss howto characterize whether it is feasible to fulfill therequirements of all clients in the system, andhow to design a feasibility optimal policy that ful-fills the requirements of any set of clients as longas the set is feasible. These terms are formallydefined as follows.

Definition 1: A system is said to be fulfilled bysome scheduling policy η, if, under η, the timely-throughput provided to each client n is at leastqn.

Definition 2: A system is feasible if thereexists some scheduling policy that fulfills it.

Definition 3: A scheduling policy is feasibilityoptimal if it fulfills every feasible system.

A NECESSARY ANDSUFFICIENT CONDITION FOR FEASIBILITY

In this section, we characterize when a system isfeasible, given the channel reliabilities, pn, andtimely-throughput requirements, qn, of clients.This solves the problem of admission control.

We first observe that the actual timely-throughput of a client n is determined by thelong-term average number of time slots theAP spends on transmitting packets for client nper interval. Lemma 1 in [4] shows that thetimely-throughput of client n is at least qn ifand only if the long-term average number oftime slots the AP spends on the client is atleast qn/pn. We hereafter define wn = qn/pnand call it the load imposed by client n. Veri-

Figure 1. An example that illustrates the system model. The right half of the fig-ure shows the topology that includes one AP and three clients. The left half ofthe figure shows the timeline of each client. We use an arrow to indicate thebeginning of an interval.

T

p1

p2

p3

AP

1

2

3

1 In this case, a slot is thetime to deliver the packetas well as receive theACK. The quantity pn canthen be regarded as theprobability that the trans-mission from the AP toclient n is successful, aswell as the ACK fromclient n to the AP.


IEEE Wireless Communications • February 201250

fying whether a system is fulfilled is now equiv-alent to verifying whether the average numberof time slots the AP spends on client n is atleast wn for all n.

Since the AP can make at most one transmis-sion in each time slot, we can immediately obtainthat a system is feasible only if ΣN

n=1 wn ≤ T. Inother words, the total loads imposed by allclients cannot exceed the number of time slotsavailable in an interval if the set of clients’requirements is to be feasible. However, whilethis condition is necessary for feasibility, it is notsufficient, because it may be impossible for theAP to utilize all the T time slots in every inter-val. For example, consider a system with twoclients and T = 3. Suppose in a certain interval,the AP transmits the two packets for the twoclients in the first two time slots, respectively,and both transmissions are successful. Now, asthe AP has delivered all the packets in the sys-tem, there are no more packets to be transmit-ted in the third time slot. Thus, the AP is forcedto be idle in the third time slot. This examplesuggests that we need to take the amount ofunavoidable idle time into account when evalu-ating feasibility.

While the amount of idle time slots may dif-fer from policy to policy, we show that it is thesame for all work-conserving policies [4, Lemma3].

Definition 4: A scheduling policy is a work-conserving policy if it schedules some packet fortransmission as long as there are any undeliv-ered packets, and only idles when all unexpiredpackets have been delivered.

It can be shown that, for every feasible sys-tem, there exists a work-conserving policy thatfulfills it [4, Lemma 4]. Thus, we can limit ourattention to work-conserving policies.

Let I be the long-term average number ofidle time slots in an interval under any work-conserving policy. Since, on average, the AP canonly make T – I transmissions in an interval, weobtain that a system is feasible only if ΣN

n=1 wn ≤T – I.

It turns out that this condition is also onlynecessary and not sufficient for feasibility. It canbe further refined. Observe that, if we removesome clients from a feasible system, resulting ina system that only consists of a subset of clientsof the original one, the resulting system mustalso be feasible. Thus, by letting IS denote thelong-term average number of idle time slots inan interval when only a subset S ⊆ {1, 2, …, N}of clients is present, we obtain an even morestringent necessary condition.

Lemma 1: A system is feasible only if Σn∈Swn≤ T – IS, for all S ⊆ {1, 2, …, N}. �

We note that the conditions ΣNn=1wn ≤ T – I

and Σn∈Swn ≤ T – IS, for all S ⊆ {1, 2, …, N} arenot equivalent. The following exxample demon-strates that the condition in Lemma 1 is indeeda stronger one.

Example 1: Consider a system with two clientsand interval length T = 3. The reliabilities forboth clients are p1 = p2 = 0.5. Client 1 requiresa timely-throughput of q1 = 0.876, while thetimely-throughput requirement of client 2 is q2= 0.45.

Now, we have:

I{1} = I{2} = 2p1 + (1 – p1)p1 = 1.25,

I ≡ I{1,2} = p1p2 = 0.25.

If we evaluate the condition for the set of allclients {1, 2}, we have w1 + w2 = 2.66 < 2.75 =T – I{1, 2}. However, if we evaluate the conditionfor the subset of S = {1}, we find w1 = 1.76 >1.75 = T – I{1}. This indicates that the set ofclients is not feasible. Thus, merely evaluatingthe condition for the set of all clients is notenough. �

It turns out that the more stringent necessarycondition in Lemma 1 is actually both necessaryand sufficient.

Theorem 1 ([4], Theorem 4): A system is fea-sible if and only if Σn∈S wn ≤ T – IS, for all S ⊆{1, 2, …, N}. �

While the above theorem provides a sharpcharacterization of feasibility, it involves testingthe condition Σn∈S wn ≤ T – IS for all subsets S⊆ 1, 2, …, N. Sine there are 2N such subsets, thisappears to require exponentially many tests.However, it turns out that this condition can begreatly simplified.

Theorem 2 ([4], Theorem 5): Sort all clientsso that q1 ≥ q2 ≥ · · · ≥ qN. Let Sm := {1, 2, …,m} be the subset that consists of clients 1through m. A system is then feasible if and onlyif Σm

n=1 wn ≤ T – ISm, for all m ∈ {1, 2, …, N}. �This theorem reduces the number of tests for

checking feasibility to N. Each such test can beperformed efficiently by using the Fast FourierTransform, using which [4] has developed a poly-nomial time algorithm based on this theorem.

FEASIBILITY OPTIMAL SCHEDULING POLICIES

In this section, we introduce two scheduling poli-cies that are feasibility optimal (i.e., they fulfillevery feasible systems). Both policies are whatwe call the largest debt first scheduling policies.When employing a largest debt first schedulingpolicy, the AP calculates the debt it owes to eachclient at the beginning of each interval. The APthen prioritizes all clients according to the debtsowed to them, such that clients with larger debtsget higher priority. In each time slot of the inter-val, the AP transmits the packet for the clientwith the largest debt among those whose packetsare not delivered yet.

Figure 2 shows an example of schedulingusing the largest debt first scheduling policy. Inthis example, we assume that at the beginning ofsome interval, client 1 has the largest debt, client3 has a medium debt, and client 2 has the small-est debt. The AP first transmits the packet forclient 1, and keeps retransmitting the packet incase the previous transmission fails. The packetfor client 3 is only transmitted after the packetfor client 1 is successfully delivered. Finally, thepacket for client 2 is only transmitted after bothpackets for clients 1 and 3 are successfully deliv-ered.

wq

p

wq

p

11

1

22

1 76

20 9

= =

= =

. ,

. ,

When employing alargest debt firstscheduling policy, theAP calculates thedebt it owes to eachclient at the begin-ning of each interval.The AP then priori-tizes all clientsaccording to thedebts owed to them,such that clients withlarger debts get high-er priority.



The two scheduling policies differ in theirdefinitions of debt. The first kind of debt, thetime-based debt, is derived from the concept ofload wn defined earlier.

Definition 5: Let un(k) be the number of timeslots that the AP spends transmitting the packetfor client n in the kth interval. The time-baseddebt of client n at the beginning of the (k + 1)th

interval is defined as rn(1)(k + 1) := kwn – Σk

j=1un(j). The largest debt first policy that appliesthe time-based debt is called the largest time-based debt first scheduling policy.

The second kind of debt, the weighted-deliverydebt, is derived directly from the timely-through-put requirement qn of a client.

Definition 6: Let dn(k) be the indicator func-tion of the event that the AP delivers a packetfor client n in the kth interval. The weighted-delivery debt of client n at the beginning of the(k + 1)th interval is defined as

The largest debt first policy that prioritizesaccording to the weighted-delivery debt is calledthe largest weighted-delivery debt first schedulingpolicy.

It has been shown that both largest debt firstpolicies fulfill every feasible systems, and arehence feasibility optimal.

Theorem 3 ([4, Theorems 2 and 3]): Both thelargest time-based debt first policy and thelargest weighted-delivery debt first policy arefeasibility optimal. �

SIMULATION RESULTSWe now present simulation results when applyingthe two largest debt first policies for VoIP traffic.We follow the G.711 codec, which is an ITU-Tstandard for audio compression, and the IEEE802.11b to decide parameters for the simulation.Important parameters are summarized in Table 1.

We assume that there are two groups, A andB, of clients. The timely-throughput requirementof each client in group A is 0.99 packets perinterval, while that of each client in group B is0.8 packets per interval. The channel reliabilityof the nth client in each group is (60 + n) per-cent. From the admission control result devel-oped earlier, it follows that a system with 6group A clients and 5 group B clients is feasible,while a system with 6 group A clients and 6group B clients is not.

We compare the two largest debt first poli-cies against two standard techniques. One is theIEEE 802.11 DCF mechanism, and the other isa centralized policy where the AP assigns priori-ties to clients randomly at the beginning of eachinterval. The two policies are referred to as DCFand Random, respectively. We use the total time-ly-throughput deficit, defined as Σn(qn – actualtimely-throughput of client n)+, to evaluate theperformance of a policy. A system is fulfilled bya policy if, under that policy, the total timely-throughput deficit converges to zero.

The simulation results for the feasible set isshown in Fig. 3. It can be seen that both largestdebt first policies fulfill the feasible set, as thetotal timely-throughput deficits converge to zero

over time under these two policies. On the otherhand, both the DCF policy and the Random pol-icy fail to fulfill this feasible set. This resultshows these two policies are not adequate forserving real-time flows.

The simulation result for the infeasible set isshown in Fig. 4. In this setting, the total time-lythroughput deficits remain strictly positive andbounded away from zero under all four policies.This illustrates that it is indeed infeasible to ful-fill this system, and that the conditions for feasi-bility are sharp. Moreover, the two largest debtfirst policies achieve smaller total timely-through-put deficits than the other two policies. This sug-gests that the largest debt first policies havebetter performance even for infeasible systems.

UTILITY MAXIMIZATION FORELASTIC TRAFFIC

In previous sections, we have supposed that thetimely-throughput requirement of each client, qn,is specified by the client and inelastic. In thissection, we discuss scenarios where the timely-throughput requirements are elastic.

PROBLEM FORMULATIONSuppose now that the value of qn is tunable bythe AP, for each n. Suppose also that each clientn receives some utility based on the timely-

r kkq d j

pnn j

kn

n

( )( ) :( )

.2 11+ =− =∑

Figure 2. An example illustrates the largest debt first scheduling policy. The sizeof the money bag indicates the amount of debt that the AP owes to the client.We use F to denote a failed transmission, and S to denote a successful one.

F

p1

p2

p3

AP

1

2

3F S

F S

Table 1. Simulation setup.

Bit rate 64 kb/s

Packetization interval 20 ms

Payload size per packet 160 bytes

Time slot length 610 ps

Transmission data rate 11 Mb/s



throughput that is provided. The relationbetween the utility derived and the timely-throughput provided is characterized by eachclient’s utility function, Un(qn). The utility func-tion, Un(·), is assumed to be concave and infinite-ly differentiable. Different clients may havedifferent utility functions. In the sequel, we use[xn] to denote the vector containing (x1, x2, …,xN).

The goal of the AP is to choose a feasiblevector [qn] so as to maximize the total utility ofall clients. Since Theorem 1 already has estab-lished a necessary and sufficient condition forfeasibility, we can formulate the problem of serv-ing elastic traffic as the following SYSTEM opti-mization problem:

SYSTEM:

(1)

(2)

over 0 ≤ qn ≤ 1, ∀n. (3)

PROBLEM DECOMPOSITION AND ABIDDING PROCEDURE

Since the utility functions of clients are all con-cave, the above optimization problem is indeed aconvex one, and standard techniques for solvingconvex optimization problems can be employed todetermine the optimal solution that maximizes thetotal utility of the system. However, this probleminvolves 2N feasibility constraints, as in Eq. 2,resulting in a huge computation overhead for stan-dard techniques. Further, in practice, the AP maynot even know the utility functions of all clients.To alleviate these challenges, we can employ adecomposition technique introduced in [10], anddecompose the SYSTEM problem into two sub-problems. The total utility in the system can bemaximized by jointly solving these two subprob-lems. In the sequel, we will establish online algo-rithms that solve these two subproblems.

The decomposition is best described by a bid-ding procedure. Suppose that, at the beginningof each interval, each client n pays the AP a cer-tain amount of money, ρn. The AP then choosesa feasible vector [qn] to achieve weighted pro-portional fairness among clients, where theweight of client n is ρn. It is known that weightedproportional fairness can be achieved by maxi-mizing ΣN

n=1 ρn log qn. Thus, the AP aims tosolve the following problem:

ACCESS POINT:

(4)

(5)

over 0 ≤ qn ≤ 1, ∀n. (6)

On the other hand, after receiving a timely-throughput of qn, each client n estimates itsprice per unit of timely-throughput as ψn :=ρn/qn. Client n assumes a linear relation betweenpayment and timely-throughput, and aims tofind a new ρn so as to maximize its own net utili-ty, that is, the difference between its utility andthe amount of its payment. More formally, clientn aims to solve the following problem:

CLIENTn:

(7)

over 0 ≤ ρn ≤ ψn. (8)

Under this bidding procedure, the AP aims toachieve weighted proportional fairness among

Max Unn

nn

ρψ

ρ,

,,

⎛

⎝⎜⎞

⎠⎟−

s.t.q

pT I S Nn

nS

n S

≤ − ∀ ⊆ …∈∑ , { , , , }.1 2

Max ρn nn

Nqlog ,

=∑

1

s.t.q

pT I S Nn

nS

n S

≤ − ∀ ⊆ …∈∑ , { , , , }.1 2

Max U qn nn

N( ),

=∑

1

Figure 4. Performance of an infeasible set.

Time (s)10

1

0

Tota

l tim

ely

thro

ughp

ut d

efic

it

2

3

4

5

6

7

8

2 3 4 5

Weighted-deliveryTime-basedRandomDCF

Figure 3. Performance of a feasible set.

Time (s)10

1

0

Tota

l tim

ely

thro

ughp

ut d

efic

it

2

3

4

5

6

7

8

2 3 4 5

Weighted deliveryTime-basedRandomDCF

HOU LAYOUT 2/10/12 3:05 PM 2


clients, and does not need to know the exactutility functions of clients. On the other hand,each client n only needs to selfishly solve its ownCLIENTn problem, and does not need to con-sider the behavior of other clients. The total util-ity in the system is maximized when the solutionsto the two subproblems converge:

Theorem 4 ([6, Theorem 2]): There existsnon-negative vectors [qn], [ψn], and [ρn], with[ρn] = [qnψn], such that:• Given that each client n pays ρn per interval,

[qn] solves the ACCESS POINT problem.• For all n such that ψn > 0, ρn solves the

CLIENTn problem.In addition, if [qn], [ψn], and [ρn] are all vectorswith positive entries, [qn] solves the SYSTEMproblem.

AN ONLINE SCHEDULING POLICYFOR ACCESS POINT

In our bidding procedure, we require each clientn and the AP to solve the CLIENTn andACCESS POINT problems, respectively. SolvingCLIENTn is easy, as it only involves the utilityfunction of client n. On the other hand, while, incontrast to solving SYSTEM, solving ACCESSPOINT does not require knowledge of utilityfunctions, it still involves all the 2N feasibility-constraints. However, we can show the interest-ing results that there exists an on-line schedulingpolicy that not only incurs negligible computa-tion overhead, but also requires no knowledge ofchannel reliabilities, [pn].

The scheduling policy is called the weightedtransmission (WT) policy. Let un(k) be the num-ber of time slots the AP spends transmitting thepacket for client n in the kth interval. At thebeginning of the (k + 1)th interval, the AP sortsall clients in increasing order of

In each time slot within the (k + 1)th interval,the AP transmits the packet for the client thathas the smallest

among those whose packets are yet to be deliv-ered.

It can be shown that this simple onlinescheduling policy solves the ACCESS POINTproblem.

Theorem 5 ([6, Theorem 6]): Given [ρn], thetimely-throughput [qn] resulting from the WT pol-icy is a solution to the ACCESS POINT problem.

SIMULATION RESULTSWe now present some simulation results. Asmentioned earlier, we follow the G.711 codecand IEEE 802.11b standard for setting parame-ters of the simulation. We assume that the utilityfunction of each client n is

where γn is a positive integer and αn ∈ (0, 1).We consider four different policies:• A policy that applies the WT scheduling policy

and the bidding procedure, which we call WT-Bid

• A policy that only applies the WT schedulingpolicy and does not apply the bidding proce-dure (i.e., clients do not update their bids),which we call WT-NoBid

• A random policy (Rand) that assigns prioritiesto clients randomly at the beginning of eachinterval

• A policy that assigns higher priorities to clientswith larger γn, and breaks ties randomly at thebeginning of each interval, which we call P-Rand.

The performance of each policy is evaluated byits resulting total utility in the system, Σn Un(qn).

We consider two different settings, both with30 clients. In the first setting, we set pn =(50+n)%, γn = (n mod 3)+1, and αn =0.3+0.1(n mod 5), for each client n. In the sec-ond setting, we set pn = (20 + 2n)%, γn = 1,and αn = 0.3 + 0.1(n mod 5), for each client n.

Simulation results for the two settings areshown in Fig. 5 and Fig. 6, respectively. In bothsettings, the WT-Bid policy outperforms all theother three policies.

A GENERALIZED MODEL FORUNICASTING REAL-TIME FLOWS

The model introduced in previous sectionshas assumed that each client generates onepacket at the beginning of each interval andrequires the same delay bound, the channel reli-ability pn does not change over time, and alltransmissions take exactly one time slot. In prac-tice, however, different applications may gener-ate traffic at different patterns, may requiredifferent delay bounds, channel qualities mayvary over time due to fading, and the amount oftime needed for a transmission can depend onthe transmission rate which can differ fromclient to client and change over time. In this sec-tion, we generalize the model earlier to incorpo-rate all these concerns.

We still consider a system with one AP and

U qq

n n nn

n

n

( ) ,=−

γα

α 1

jk

n

n

u j=∑ 1 ( ).

ρ

jk

n

n

u j=∑ 1 ( ).

ρ

Figure 5. Performance of the first setting.

Time (s)10

-19

Tota

l uti

lity

-21

-17

-15

-13

-11

-9

2 3 4 5

WT-BidWT-NoBidRandP-Rand



N clients, and assume that time is slotted andgrouped into intervals, where one interval con-sists of T time slots. We suppose that differentclients may generate packets at different pat-terns. At the beginning of each interval, eachclient may or may not generate one packet.The process of packet generation is assumed tobe a stationary, irreducible Markov chain withfinite states. The long-term average probabilitythat only a subset S of clients generates a pack-et in an interval is denoted by R(S). Further,different clients may require different delaybounds. We denote the delay bound of client nby τn, which is assumed to be smaller or equalto T. If client n generates a packet at the begin-ning of an interval, this packet needs to bedelivered no later than the tn

th time slot of theinterval, or the packet expires and is droppedfrom the system.

We consider fading channels. We assume thatchannel conditions evolve as a stationary, irre-ducible Markov chain with finite states. Whilechannel conditions can vary from interval tointerval, we assume that they are static during aninterval. We use c ∈ C to denote the channelstate in an interval, which includes the channelconditions of all clients.

We consider both systems with and withoutrate adaptation. When rate adaptation is notemployed, the AP uses the same data rate totransmit packets for all clients. Therefore, weassume that all transmissions take one timeslot. In this case, channel conditions determinechannel reliabilities. To be more specific, wesuppose that under channel state c a transmis-sion for client n will be successful with proba-bility pc,n.

On the other hand, when rate adaptation isemployed, the AP chooses a suitable data ratefor each client, so as to prevent transmissionsfrom failing. Different data rates effectivelyresult in different times needed for a transmis-sion. We model this by supposing that underchannel state c a transmission for client ntakes sc,n time slots. In this case, all transmis-sions are successfully delivered. Since theirdurations can vary according to channel condi-tion, reliability is bought at the expense ofdata rate.

A FRAMEWORK FORSCHEDULING POLICIESAND ITS APPLICATIONS

In this section, we study the problem of design-ing feasibility optimal scheduling policies underthe above generalized model. We introduce aframework for designing such policies. We alsodemonstrate the utility of this framework byderiving feasibility optimal policies for two spe-cific scenarios.

This framework starts by extending the con-cept of debt introduced earlier.

Definition 7: A variable rn(k), whose value isdetermined by the past history of the client n upto the kth period, or time slot kT, is called apseudo-debt if:• rn(0) = 0, for all n.• At the beginning of each period, rn(k) increas-

es by a constant strictly positive number zn =zn(qn), which is an increasing linear functionof qn.

• rn(k + 1) = rn(k) + zn(qn) – μn(k), where μn(k)is a non-negative and bounded random vari-able whose value is determined by the behav-ior of client n. Further, μn(k) = 0 if the APdoes not transmit any packet for client n.

• The system is fulfilled if and only if

as k → ∞, for all n and all ε > 0.It can be shown that for the basic model from

earlier, both the time-based debt, rn(1), and the

weighted-delivery debt, r n(2), are also pseudo-

debts.Example 2: The time-based debt can be

expressed as rn(1)(k + 1) = rn

(1)(k) + wn – un(k),where

and un(k) is the number of time slots that theAP spends transmitting packets for client n inthe kth interval.

On the other hand, the weighted-deliverydebt can be expressed as

where dn(k) is the indicator function that the APdelivers a packet for client n in the kth interval.

Furthermore, when the generalized model isconsidered, we can define a delivery debt, r n

(3)(k+1) := kqn – Σk

j=1 dn(j). We then have r n(3)(k +

1) = r n(3)(k) + qn – dn(k), and thus delivery debt

is also a pseudo-debt. �We can establish the following sufficient con-

dition for a scheduling policy to be feasibilityoptimal for the considered scenario.

Theorem 6 ([5, Theorem 3]): Let rn(k) be apseudo-debt, ck be the channel state of the kthinterval, and Sk be the set of clients that has gen-erated packets at the beginning of the kth inter-val. Apolicy that maximizes ΣN

n=1E{rn(k)+μn(k)|[rn(k)], ck, Sk} for each interval kis feasibility optimal,where x+ := max{x, 0}.

The above theorem provides a framework for

r k r kq

p

d k

pn nn

n

n

n

( ) ( )( ) ( )( )

,2 21+ = + −

wq

pnn

n=

Probr k

kn{ }

,<⎧⎨⎩

⎫⎬⎭→ε 1

Figure 6. Performance of the second setting.

Time (s)10

-18

Tota

l uti

lity

-19

-20

-17

-16

-15

-14

-13

2 3 4 5

WT-BidWT-NoBidRandP-Rand



designing scheduling policies: For each scenario,one should first choose a suitable pseudo-debt,and then aim to maximize

for each interval k. In the following sections, wedemonstrate the utility of this framework byderiving tractable feasibility optimal schedulingpolicies for two different scenarios.

A SCHEDULING POLICY FORUNRELIABLE FADING CHANNELS

We first consider the scenario where rate adapta-tion is not employed. Thus, all transmissions takeone time slot long, and a transmission for client nis successful with probability pc,n, under channelstate c. Further, we assume that all clients requirethe same delay bound, that is, τn ≡ T.

We use the delivery debt, rn(3)(k). The pro-

posed policy is called the joint debt-channel poli-cy.In the joint debt-channel, the AP sorts allclients in descending order of pck,nr(3)(k), at the-beginning of each interval, where ck is the chan-nel state of the kth interval. In each time slotwithin the interval, the AP schedules a transmis-sion for the client whose pck,nr(3)(k) is the largestamong those whose packets are not deliveredyet. This policy is similar to the Longest Con-nected Queue (LCQ) policy proposed by Tassiu-las and Ephremides [13], although they considera much simpler channel model and do not pro-vide delay guarantees. On one hand, this policygives higher priority to clients with larger rn

(3)(k),so as to prevent a client from being starved. Onthe other hand, the policy also takes advantageof current channel state by opportunisticallyassigning higher priority to clients with largerpck,n. We can prove that this simple online policyis feasibility optimal.

Theorem 7 ([5], Theorem 4, and [9], Theorem3): The joint debt-channel policy is feasibilityoptimal.

A SCHEDULING POLICY FOR RATE ADAPTATIONWe next consider the scenario where rate adap-tation is employed. A transmission for client ntakes sc,n time slots under channel state c, andall transmissions are error-free. Also, each clientn specifies its individual delay bound, τn, witheach τn ≤ T.

We use the delivery debt, rn(3)(k). In each

interval, the AP schedules transmissions forsome of its clients. The schedule of transmis-sions can be expressed as an ordered list L ={l1, l2, …}, with the interpretation that the APfirst transmits the packet for client l1, and thenthe packet for client l2, and so on. The packet ofclient lm is then delivered at time Σm

n=1sc,ln. Thus,to meet the delay bound of each client, werequire that Σm

n=1sc,ln ≤ τlm, for all scheduledclients.

Suppose that, in some interval k, the APschedules transmissions for a list L of clients,and delivers all their packets on time. We thenhave μn(k) = 1, for all n ∈ L, and μn(k) = 0,otherwise. Thus, ΣN

n=1E{rn(3)(k)+μn(k)|[rn

(3)(k)],ck, Sk} = Σn∈Lrn

(3)(k)+. Maximizing Σn∈Lrn(3)(k)+

while meeting all delay bounds for all scheduledclients is acctually a variation of the knapsackproblem. The policy shown in Algorithm 1, themodified knapsack policy, can be proved to befeasibility optimal.

UTILITY MAXIMIZATION FORSYSTEMS WITH RATE ADAPTATION

In this section, we address the problem of utilitymaximization for systems that employ rate adap-tation. we will address two complexities. Thefirst is that each client requires a certain mini-mum timely-throughput. The second is thatclients can engage in strategic behavior. That is,they can lie about their utility functions to theAP, in the hope that they can thereby secure ahigher throughput.

Similar to a previous section, we assume thatthe timely-throughput requirements of clientsare elastic. Each client n receives a utility ofUn(qn) when the timely-throughput it receives isqn. The utility functions, Un(⋅), are assumed tobe concave and infinitely differentiable. We willfurther consider the additional complexity that,although the timely-throughput requirements areelastic, eachclient n still requires a guaranteedminimum timely-throughput, which is denoted qni.e., the APneeds to guarantee that qn ≥ qn.

The goal of the AP is to maximize the totalutility of the system, while providing minimumtimely-throughput guarantees for all the clients.This problem can be formally stated as follows:

(9)

s.t. Network dynamics and feasibility constraints,(10)

and qn ≥ qn, ∀n. (11)

In addition, we allow for the selfish behaviorby clients. In practice, as the AP does not haveexact information of utility functions, a strategicclient may lie about its utility function so as togain more service from the AP. Thus, a mecha-nism that can both prevent clients from benefit-

Max U qn nn

N( )

=∑

1

E r k k r k c Sn n n k kn

N( ) ( ) [ ( )], ,+

={ }∑ μ

1

Algorithm 1. Modified knapsack policy [5].

1: Sort clients such that τ1 ≤ τ2 ≤ ··· ≤ τN

2: L[0, 0] = φ3: M [0, 0] = 04: for n = 1 to N do5: for t = 1 to T do6: if t > τn then7: M[n, t] = M[n, t – 1]8: L[n, t] = L[n, t – 1]9: else if client n has a packet AND

rn(3)(k) + M [n – 1, t – sc,n] > M [n – 1, t] then

10: M[n, t] = rn(3)(k) + M [n – 1, t – sc,n

11: L[n, t] = L[n – 1, t – sc,n] + {n}12: else13: M[n, t] = M[n – 1, t]14: L[n, t] = L[n – 1, t]15: schedule according to L[N, T]

As the AP does nothave exact informa-

tion of utility func-tions, a strategic

client may lie aboutits utility function soas to gain more ser-

vice from the AP.Thus, a mechanismthat can both pre-vent clients from

benefiting by lyingwhile still achievingthe maximum totalutility in the system

is needed.



ing by lying while still achieving the maximumtotal utility in the system is needed.

A TRUTHFUL AUCTION DESIGN ANDITS OPTIMALITY

Before we introduce our proposed mechanism,we first introduce the concept of truthful auc-tion. We denote the timely-throughput of clientn up to the kth interval by qn(k), that is,

where dn(j) is the indicator function that the APhas delivered a packet for client n in the jthinterval. We assume that, if the system termi-nates, or client n leaves the system, at the end ofthe kth interval, then client n receives a rewardin the amount of kUn(qn(k)), and hence its time-average reward is Un(qn(k)) per interval. Letq+

n(k) and q–n(k) denote the values of qn(k) if

client n is scheduled in the kth interval, and if itis not, respectively. To be more specific, we have

and

If the system terminates at the end of the kthinterval, the amount of reward that client nreceives is either kUn(q+

n(k)) or kUn(q–n(k)),

depending on whether client n is scheduled inthe kth interval or not. We hereafter define themarginal reward of client n at the kth interval tobe kUn(q+

n(k)) – kUn(–n(k)).

In an auction, each client n makes a bid,bn(k), to the AP at the beginning of each inter-val k. Based on these bids, the AP schedulessome clients for transmissions and charges eachclient n an amount of ρn(k). The schedulingand charging decisions of the AP are deter-mined by the employed auction mechanism.The net reward of client n, if the system termi-nates at the end of the kth interval, can hencebe expressed as kUn(qn(k)) – Σk

j=1ρn(j). At thebeginning of each interval k, client n aims togreedily maximize its net reward based on pastsystem history, assuming that the system willterminate at the end of this interval. Thus,client n aims to maximize kUn(qn(k)) –Σk

j=1ρn(j), given qn(k – 1) and ρn(1), ρn(2), …,ρn(k – 1). We say that an auction mechanism istruthful if a strategic client may bid accordingto its real marginal reward.

Definition 8: An auction mechanism is truthfulif choosing

bn(k) = kUn(q+n (k)) – kUn(q–

n(k)) (12)

yields the highest net reward for each client n, ifthe system terminates at the end of the kth inter-val.2

We now present our proposed auct ionmechanism. This mechanism is based on theVickrey- Clarke-Grove auction [2, 3, 14]. Atthe beginning of each interval k , the APannounces a non-negative discount, δn(k), to

each client n . Each client n then offers itsbid, bn(k). The AP chooses a sorted list L ={l1, l2, …} and schedules clients accordingly,such that Σn∈L(bn(k) + δn(k)) is maximizedamong all sorted lists under which the packetof each scheduled client is delivered beforeits delay bound. Let L–n be the schedule thatthe AP would have chosen if client n werenot present in the system. The AP chargeseach scheduled client n an amount of ρn(k) =Σm∈L–n(bm(k) + δm(k)) – Σm∈L,m≠n(bm(k) +δm(k)) – δn(k). The AP does not charge any-thing to clients that are not scheduled. It canbe shown that this auct ion mechanism istruthful [7].

In addition, it can be shown that [7], using adual-decomposition technique proposed by Linand Shroff [11], the total utility in the system canbe maximized by employing this truthful auctionmechanism and adapting δn(k) according to

δn(k + 1) = {δn(k) + 1/k [qn – qn(k)]}+. (13)

Finally, this auction mechanism requires theAP to find the list L that maximizes Σn∈L(bn(k)+ δn(k)). This is actually a similar problem tothe one discussed earlier. By substituting rn

(3)(k)with Σn∈L(bn(k)+δn(k)), Algorithm 1 finds thedesired list L. In sum, the proposed auctionmechanism is both truthful and optimal, whilebeing easy to implement.

BROADCASTING REAL-TIME FLOWS WITHNETWORK CODING

In this section, we discuss the problem of broad-casting real-time flows. The major differencebetween wireless broadcasting and wireless uni-casting is that, since ACKs are not implementedfor broadcasting, and it is costly to obtain feed-back information from all clients, the AP usuallydoes not have immediate feedback informationon whether a client receives a transmission cor-rectly.

It turns out that network coding can be fruit-fully employed in such situations. In this section,we discuss a model for broadcasting such real-time flows. We then introduce a framework fordesigning scheduling policies. The utility of thisframework is demonstrated by deriving policiesfor several various scenarios, including thosethat employ different network coding tech-niques.

SYSTEM MODELWe consider a system where the AP broadcastsI real-time flows to N clients. Time is dividedinto slots, and further grouped into intervals,where an interval consists of T time slots. At thebeginning of each interval, each flow i may ormay not generate a packet. The packet genera-tions of flows are assumed to constitute a sta-tionary irreducible Markov chain with finitestates. All packets require a delay bound of Ttime slots. Each client n has an inelastic timely-throughput requirement for each flow i, which isdenoted by qi,n.

For wireless broadcasting without ACKs, itis helpful for the AP to use network coding. In

q kk

kq kn n

− =−

( ) ( ).1

q kk

kq k

kn n+ =

−+( ) ( ) ,

1 1

q kd j

knjk

n( ) :( )

,= =∑ 1

2 A broader and moredetailed discussion oftruthful auction mecha-nisms can be found in[12].

The major differencebetween wirelessbroadcasting andwireless unicasting isthat, since ACKs arenot implemented forbroadcasting, and itis costly to obtainfeedback informationfrom all clients, theAP usually does nothave immediatefeedback informationon whether a clientreceives a transmis-sion correctly.



each time slot, the AP broadcasts a packet,either a raw packet from a real-time flow, or acoded packet that involves packets from multi-ple flows. We do not make any assumptions onthe actual coding scheme that the AP canemploy. When the AP broadcasts a packet,each client n receives the packet with probabili-ty pn. However, the AP has no knowledge onwhether a particular client receives the packetof a particular transmission. To enhance theprobability of packet delivery, the AP maytransmit the same packet multiple times in aninterval, which, in turn, may result in someclients receiving several duplicate packets. Inthis case, redundant duplicate packets aredropped by the client, as they do not carry anynew useful information.

A FRAMEWORK FOR SCHEDULINGREAL-TIME BROADCAST FLOWS

We now introduce a framework for designingscheduling policies. This framework is similarto the one for unicast flows from earlier.Although the AP does not have per-transmis-sion feedback, it can still estimate the numberof packet deliveries. For example, suppose insome interval, the AP broadcasts the packetfrom flow 1 twice, and the packet from flow 2once, then the probabilities that a client nreceives the packet from flow 1 and that fromflow 2 are 1 – (1 – pn)2, and pn, respectively.We hereafter use ξi,n(k) to denote the probabil-ity that client n obtains the packet from flow iin the kth interval. The law of large numbersshows that the long-term timely-throughput ofclient n on flow i is

We can then define the expected delivery debtas follows:

Definition 9: The expected delivery debt, r̂ i,n(k+ 1), of client n on flow i at the beginning ofthe(k + 1)th interval is defined as

(14)

We can obtain a sufficient condition for ascheduling policy to be feasibility optimal withrespect to all policies using some specified net-work coding scheme:

Theorem 8 ([8], Theorem1): Given past sys-tem history and the set of flows that generatepackets at the beginning of the kth interval, apolicy that maximizes

for all k, among all policies using a specified net-work coding scheme, is feasibility optimal underthis network coding scheme.

In the following sections, we demonstrate theutility of this framework by deriving policies forthree scenarios, one that does not employ net-work coding, one that employs XOR coding, andone that employs linear coding.

BROADCASTING WITHOUT NETWORK CODING

We first discuss the scenario when the systemdoes not employ network coding, that is, the APcan only broadcast a raw packet from a singleflow in each time slot. If the AP broadcasts thepacket from flow i a total of σi times in an inter-val, a client n will successfully receive this packetwith probability 1 – (1 – pn)σi. We define themarginal delivery probability of the ς ith transmis-sionfor flow i and client n as πi,n,ςi := [1 – (1 –pn)] – [1 – (1 – pn)ςi –1] = pn(1 – pn)ςi–1, which isthe additional delivery probability that flow i andclient n can benefit from the ςith transmission.

We proposed a greedy policy for broadcastingwithout network coding. Suppose that each flowi has been transmitted ςi times before a time slotin the (k + 1)th interval. The AP then schedulesa transmission for the flow that has generated apacket in the interval and maximizes ΣN

n=1r̂ i,n(k)+πi,n,ςi for this time slot. The followingexample illustrates this scheduling policy.

Example 3: Consider a system with one client,whose channel reliability is p1 = 0.6, two flows,and T = 3. Suppose that, at the beginning ofsome interval k, the expected delivery debts ofthe two flows are r̂1,1 = 1 and r̂2,1 = 0.8. In thefirst time slot of the interval, the marginal deliv-ery probabilities for both flows are 0.6. The APthus broadcasts the packet from the first flow. Inthe second time slot of the interval, as the pack-et for flow 1 has been already broadcast once,the marginal delivery probabilities for the twoflows become 0.4 × 0.6 = 0.24, and 0.6, respec-tively.As r̂1,1 × 0.24 < r̂2,1 × 0.6, the AP broad-casts the packet from flow 2. In the last time slotof the interval, the marginal delivery probabili-ties for both flows are 0.24, and the AP broad-casts the packet from flow 1. �

Theorem 2 in [8] shows that this policy is fea-sibility optimal among the class of all policieswhich do not employ network coding.

BROADCASTING WITH XOR CODINGIn this section, we discuss the scenario where theAP may employ XOR coding as a specific net-work coding mechanism. In this scenario, inaddition to broadcasting raw packets of flows,the AP can also broadcast a packet that containsthe XOR of packets from two flows. We use i ⊕h to denote the packet that contains the XOR ofpackets from flow i and flow h. A client canobtain the packet from flow i by either receivinga raw packet from flow i, or receiving both a rawpacket from flow h and a coded packet i ⊕ h.The following example shows that we can indeedimprove system performance by incorporatingXOR coding.

Example 4: Consider a system with two flowsthat generate one packet in each interval, andonly one client whose channel reliability is p1 =0.5. Assume that there are six time slots in aninterval. Suppose that the AP transmits eachpacket three times in an interval. Then we haveξ1,1(k) = ξ2,1(k) = 0.875. Thus, a system withtimely-throughput requirements q1,1 = q2,1 >0.875 is not feasible when network coding is notemployed. On the other hand, a system thatemploys XOR coding can transmit each of thethree different types of packets, the raw packet

ˆ ( ) ( ), ,r k ki n i nn

N

i

I+

==

+∑∑ ξ 111

ˆ ( ) : ( )., , ,r k kq ki n i n i nj

k+ = −

=∑1

1

ξ

lim inf( )

.,K

kK

i n k

K→∞=∑ 1ξ

To enhance the prob-ability of packet

delivery, the AP maytransmit the same

packet multipletimes in an interval,which, in turn, may

result in some clientsreceiving several

duplicate packets. Inthis case, redundant

duplicate packets aredropped by the

client, as they donot carry any newuseful information.



from each flow and the encoded packet 1 ⊕ 2,twice in each interval, which achieves q1,1 = q2,1= 0.890625. �

Designing a feasibility optimal policy forXOR coding may be computationally complicat-ed. Instead, we aim to design a policy that isboth tractable and has better performance thanpolicies that do not employ network coding. Letσi be the number of times that the packet fromflow i is transmitted under the greedy policy insome interval, given the expected delivery debtsof flows and packet generations at the beginningof this interval. We first sort all flows so that σ1≥ σ2 ≥ …. We enforce two mild restrictions forour policy with XOR coding:• In addition to raw packets, we only allow

encoded packets of the form (2i – 1) ⊕ (2i).The intuition behind this restriction is that weonly combine two packets which have eachbeen transmitted a similar number of timesunder the greedy policy, which implies thatthey have similar importance.

• The total number of transmissions scheduledfor the raw packets from flow 2i – 1 and flow2i, as well as the encoded packet (2i – 1) ⊕(2i), equals σ2i–1 + σ2i. The intuition behindthis restriction is that we aim to enhance theperformance of flows 2i – 1 and 2i by XORcoding without hurting other flows.These two restrictions are called the pairwise

combination restriction and the transmission con-servation restriction for XOR coding, respectively.By enforcing these restrictions, designing a poli-cy for XOR coding breaks down to determiningthe number of transmissions for each of thethree packets, raw packet from flow (2i – 1), rawpacket from flow 2i, and coded packet (2i – 1) ⊕(2i), respectively, for all i, so that the resultingΣI

i=1ΣNn=1r̂ i,n(k)+ξi,n(k + 1) is maximized. This

problemcan be optimally solved in polynomialtime. In [8] it is proved that this policy is feasi-bility optimal among all policies that employXOR coding and follow the above two restric-tions. Furthermore, it should be noted that thispolicy fulfills all systems that can be fulfilledwithout network coding.

BROADCASTING WITH LINEAR CODINGWe now discuss the scenario where network cod-ing with linear coding is employed. When linearcoding is employed, the AP divides all flows intoseveral groups at the beginning of each interval. Ineach time slot within the interval, the AP choosesa group of flows and broadcasts a packet that con-tains a linear combination of packets from flows inthis group. For each group G, if the number ofcoded packets that a particular client has receivedfor the group G is at least the size of G, then thisclient can obtain all packets from flows in Gthrough Gaussian elimination; otherwise, theclient cannot obtain any packets from flows in thisgroup. The following example suggests that linearcoding may improve performance.

Example 5: Consider a system with one client,whose channel reliability is p1 = 0.5, three flowsthat generate one packet in each interval, andnine time slots in an interval. A similar argu-ment as that in Example 4 shows that q1,1 = q2,1= q3,1 > 0.875 is not feasible when network cod-ing is not employed. On the other hand, if the

AP employs linear coding and broadcasts a lin-ear combination of the three flows in each timeslot, the client can decode all packets from thethree flows if it receives at least three packetsout of the nine transmissions in an interval,which has probability 0.91015625. �

To reduce the computational overhead of lin-ear coding, we also enforce two restrictions simi-lar to those used for XOR coding. Let σi be thenumber of times that the packet from flow i istransmitted under the greedy policy in someinterval, given the expected delivery debts offlows and packet generations at the beginning ofthis interval. We sort all flows so that σ1 ≥ σ2 ≥….

The two restrictions are then:• Flows are grouped into subsets as G1 = {1, 2,

…, g1}, G2 = {g1 + 1, …, g2}, …. In each timeslot, the AP broadcasts a linear combinationof packets from flows in one of the subsetsG1, G2, …. The intuition behind this restric-tion is that we only combine packets that havebeen scheduled similar numbers of times.

• The AP broadcasts linear combinations ofpackets from the subset Gh = {gh–1 + 1, gh–1+ 2, …, gh} a total number of Σgh

i=gh–1+1σ itimes, where we set g0 = 0. The intuitionbehindthis restriction is that we aim toenhance the performance of flows within Ghwithout hurting other flows.The two restrictions above are called the adja-

cent combination restriction and the transmissionconservation restriction for linear coding, respec-tively. Under these two restrictions, the problemof finding the optimal schedule for linear codingcan be solved by dynamic programming. In [8], itis proved that the resulting policy is feasibilityoptimal among policies that employ linear codingand follow the above two restrictions. Further,the resulting policy fulfills every system that canbe fulfilled without network coding.

CONCLUDING REMARKS

We have introduced a theory for real-time com-munication over unreliable wireless links andhave surveyed some of its important results. Thecore of this theory is a model that incorporatesthree important aspects of wireless real-timecommunications: a per-packet delay bound, atimely-throughput requirement of each flow, andthe presence of unreliable and heterogeneouswireless channels. This model can be extendedto different scenarios, including those whereflows generate packets under different patternsand have different delay bounds, those wherechannel qualities vary over time, and thosewhere rate adaptation is employed. We haveaddressed three important problems vis-a-visserving real-time flows: admission control, pack-et scheduling, and utility maximization. Further,we have also considered the problem of broad-casting real-time flows. We have incorporatedthe optional usage of network coding andderived policies for broadcasting.

There remain several open problems con-cerning serving real-time flows through wirelessnetworks. We have proposed two schedulingpolicies that provide long-term timely-through-put guarantees for clients. However, the short-

Designing a feasibili-ty optimal policy forXOR coding may becomputationally com-plicated. Instead, weaim to design a poli-cy that is bothtractable and hasbetter performancethan policies that do not employ network coding.



term performance of these policies, that is,whether the timely-throughput a client receivesmay fluctuate greatly in a small time scale, is notaddressed. We also proposed a bidding proce-dure and shown that the total utility in the sys-tem is to converge is not understood. Moreover,the models discussed in this article all focus onthe scenario where there is one AP that sched-ules transmissions for all clients. In many deploy-ments, there may be multiple APs that contendfor wireless resources. Also, in wireless ad-hocnetworks, packet deliveries may require multi-hop relaying. How to provide per-packet delayguarantees for these networks is a challengingopen problem.

REFERENCES[1] Cisco, Cisco Visual Networking Index: Forecast and

Methodology, 2010–-2015. [2] E. Clarke, “Multipart Pricing of Public Goods,” Public

Choice, vol. 11, no. 1, Sept. 1971, pp. 17–33. [3] T. Groves, “Incentives in Teams,” Econometrica, vol. 41,

no. 4, July 1973, pp. 617–31. [4] I-H. Hou, V. Borkar, and P. R. Kumar. “A Theory of QoS

in Wireless,” Proc. IEEE INFOCOM, 2009, pp. 486–94. [5] I-H. Hou and P. R. Kumar, “Scheduling Heterogeneous

Real-Time Traffic Over Fading Wireless Channels,” Proc.IEEE INFOCOM, 2010.

[6] I-H. Hou and P. R. Kumar, “Utility Maximization forDelay Constrained QoS in Wireless,” Proc. IEEE INFO-COM, 2010.

[7] I-H. Hou and P. R. Kumar. “Utility-Optimal Scheduling inTime-Varying Wireless Networks with Delay Con-straints,” Proc. ACM MobiHoc, 2010, pp. 31–40.

[8] I-H. Hou and P. R. Kumar, “Broadcasting Delay-Con-strained Traffic over Unreliable Wireless Links With Net-work Coding,” Proc. ACM MobiHoc, 2011, pp. 33–42.

[9] I-H. Hou et al., “Optimality of Periodwise Static PriorityPolicies in Real-Time Communications, Proc. CDC, 2011.

[10] F.P. Kelly, A.K. Maulloo, and D. K. H. Tan, “Rate Con-trol in Communication Networks: Shadow Prices, Pro-portional Fairness and Stability,” J. Op. Research Soc.,vol. 49, 1998, pp. 237–52.

[11] X. Lin and N. Shroff, “Joint Rate Control and Schedul-ing in Multihop Wireless Networks,” Proc. IEEE CDC,2004, pp. 1484–89.

[12] N. Nisan et al., Algorithmic Game Theory, CambridgeUniv. Press, 2007.

[13] L. Tassiulas and A. Ephremides, “Dynamic Server Allo-cation to Parallel Queues with Randomly Varying Con-nectivity,” IEEE Trans. Info. Theory, vol. 39, no. 2, 1993,pp. 89–103.

[14] W. Vickrey, “Counterspeculation, Auctions and Com-petitive Sealed Tenders,” J. Finance, vol. 16, 1961, pp.8–37.

BIOGRAPHIESI-HONG HOU ([email protected]) is currently an assistantresearch engineer at Texas A&M University. He obtained hisB.S. degree in electrical engineering from National TaiwanUniversity in 2004, and M.S. and Ph.D. degrees in comput-er science from the University of Illinois at Urbana-Cham-paign in 2008 and 2011, respectively. His research interestsinclude wireless networks, wireless sensor networks, real-time systems, distributed systems, and vehicular ad hocnetworks. He received the C. W. Gear Outstanding Gradu-ate Student Award from the University of Illinois at Urbana-Champaign.

P. R. KUMAR ([email protected]) obtained his B. Tech. degreein electrical engineering (Electronics) from the Indian Insti-tute of Technology Madras in 1973, and M.S. and D.Sc.degrees in systems science and mathematics from Wash-ington University, St. Louis, Missouri, in 1975 and 1977,respectively. From 1977 to 1984 he was a faculty memberin the Department of Mathematics at the University ofMaryland, Baltimore County. From 1985 to 2011 he was afaculty member in the Department of Electrical and Com-puter Engineering and the Coordinated Science Laboratoryat the University of Illinois. Currently he is at Texas A&MUniversity, where he holds the College of Engineering Chairin Computer Engineering.

In many deploy-ments, there may be

multiple APs thatcontend for wireless

resources. Also, inwireless ad-hoc net-works, packet deliv-

eries may requiremulti-hop relaying.

How to provide per-packet delay guaran-

tees for thesenetworks is a chal-

lenging open problem.


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