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IEEE Wireless Communications • February 2012

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INTRODUCTION

With the advance of broadband wireless trans-

mission, the increasing popularity of mobile

devices, and the deployment of wireless sensor

networks, wireless networks are increasingly

used to serve real-time flows that require strict

per-packet delay bounds. Such applications

include VoIP, video streaming, real-time surveil-

lance, and networked control. For example, a

study by Cisco [1]has predicted that wireless

traffic will grow exponentially, and mobile video

will dominate wireless traffic in the near future,

accounting for 62 percent of wireless traffic by

2015. Serving real-time flows is also a key com-

ponent of many cyber-physical systems. In one

example of a centralized cyber-physical system, a

server may gather surveillance data from wire-

less sensors, based on which it makes control

decisions and disseminates them to actuator

units. Both surveillance data and control deci-

sions need to be delivered in a timely manner, or

the performance of the system may be degraded.

In addition to delay bounds, such applications

also 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 is

particularly challenging since wireless transmis-

sions are subject to shadowing, fading, and inter-

ference, and thus usually unreliable. Further, the

channel qualities may differ from link to link. A

desirable solution for serving real-time flows

thus needs to explicitly take into account the

stochastic and heterogeneous behavior of packet

losses due to failed transmissions.

In this article, we introduce a theory for serv-

ing real-time flows over wireless links. The core

of this theory is a model that takes into account

all the aforementioned challenges, the delay

bounds and timely throughput requirements of

applications, and the unreliable and heteroge-

neous nature of wireless transmission. We

address three important problems concerning

the service of real-time flows: admission control,

packet scheduling, and utility maximization when

timely-throughput requirements are elastic. We

derive a sharp necessary and sufficient condition

that characterizes when it is feasible to fulfill the

demands of all real-time flows, as well as a poly-

nomial-time algorithm for admission control. We

also introduce two on-line scheduling policies

that are proven to fulfill every feasible system.

For the scenario where timely-throughput

requirements are elastic, we formulate a utility

maximization problem that aims to maximize the

total utility over all flows. We introduce a bid-

ding procedure and an online scheduling algo-

rithm that together achieve the maximum total

utility.

We extend the model to incorporate various

real world complexities. We consider the sce-

nario where different applications may gener-

ate traffic with different patterns and require

different delay bounds. We also consider the

situation where channel quality may vary over

time, and where rate adaptation may be

employed to enhance transmission reliability.

We introduce a framework for designing

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

ABSTRACT

Wireless networks are increasingly used to

serve real-time flows. We provide an overview of

an emerging theory on real-time wireless com-

munications and some of its main results. This

theory is based on a model that jointly considers

the delay bounds and throughput requirements

of clients, as well as the unreliable and heteroge-

neous nature of wireless links. This model can

be 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 be

extended to consider broadcast of real-time

flows, and to incorporate network coding.

REAL-TIME COMMUNICATION OVER

UNRELIABLE WIRELESS LINKS:

A THEORY AND ITS APPLICATIONS

This material is based on work partially supported by

USARO under Contract Nos. W911NF-08-1-0238 and W-

911-NF-0710287, NSF under Contracts CNS-1035378,

CNS-0905397, CNS-1035340, and CCF-0939370, and

AFOSR under Contract FA9550-09-0121. Any opinions,

findings, and conclusions or recommendations expressed

in this publication are those of the authors and do not

necessarily reflect the views of the above agencies.

The authors provide

an overview of an

emerging theory on

real-time wireless

communications and

some of its main

results. This theory is

based on a model

that jointly considers

the delay bounds

and throughput

requirements of

clients, as well as

the unreliable and

heterogeneous

nature of wireless

links.

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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 and

rate adaptation. We also address the problem

of maximizing the total utility when the con-

tending flows are elastic.

In addition to unicast flows, we also address

the problem of scheduling real-time flows that

are broadcasts. The major challenge for broad-

casting is that due to the lack of acknowledg-

ment (ACK), the sender may not have

immediate feedback information on whether a

transmission is successful. We extend our model

to address this challenge. The extended model

also allows the optional usage of network cod-

ing. We introduce a framework for designing

scheduling policies for any arbitrary network

coding scheme, and derive policies for various

coding schemes.

The rest of the article is organized as follows.

We describe the basic model for real-time wire-

less communication. We establish a necessary

and sufficient condition for feasibility. We intro-

duce two online scheduling policies that are fea-

sibility optimal. We address the problem of

serving real-time flows when their timely-

throughput requirements are elastic. We extend

the basic model to incorporate various real

world complexities. We develop a framework for

designing scheduling policies, which is then used

to derive policies for two different scenarios. We

address the utility maximization problem when

rate adaptation is incorporated. We study the

problem of broadcasting real-time flows. We

then conclude the article.

BASIC MODEL FOR

REAL-TIME WIRELESS COMMUNICATIONS

Consider a system with one access point (AP)

and N wireless clients. Figure 1 illustrates an

example of such a system. Each client is associat-

ed with one real-time flow that requires service

from the AP. We assume that time is slotted and

slots are numbered by t ∈ {1, 2, 3, …}. The

length of a time slot is set to be the time needed

for the AP to transmit one packet to a client,

including all possible overheads. Time slots are

further grouped into intervals, where an interval

consists 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 model

naturally applies to downlink traffic. When both

downlink traffic and uplink traffic are present,

this model can still be applied by incorporating

server-centric access mechanisms, such as 802.11

PCF, WiMAX, and Bluetooth.

At the beginning of each interval, that is, at

times 1,T + 1, 2T + 1, …, each real-time flow

generates one packet. We assume that packets in

all real-time flows need to be delivered within a

delay bound of T time slots if they are to be use-

ful. In other words, packets that are generated at

the beginning of an interval are only useful if

they are delivered no later than the end of the

interval. If a packet is not delivered before its

delay bound, we say that the packet expires, and

it is dropped from the system. By dropping

expired 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 successful

with probability pn. The values of pnmay differ

from client to client, as the channel reliabilities

for different clients may be different. The AP

has instant feedback information on whether a

transmission is successful (e.g., by enforcing

ACKs).1

We measure the performance of a client by

its timely-throughput, which is defined as the

long-term average number of successfully deliv-

ered packets for the client per interval. We

assume 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 how

to characterize whether it is feasible to fulfill the

requirements of all clients in the system, and

how to design a feasibility optimal policy that ful-

fills the requirements of any set of clients as long

as the set is feasible. These terms are formally

defined as follows.

Definition 1: A system is said to be fulfilled by

some scheduling policy η, if, under η, the timely-

throughput provided to each client n is at least

qn.

Definition 2: A system is feasible if there

exists some scheduling policy that fulfills it.

Definition 3: A scheduling policy is feasibility

optimal if it fulfills every feasible system.

A NECESSARY AND

SUFFICIENT CONDITION FOR FEASIBILITY

In this section, we characterize when a system is

feasible, given the channel reliabilities, pn, and

timely-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 the

long-term average number of time slots the

AP spends on transmitting packets for client n

per interval. Lemma 1 in [4] shows that the

timely-throughput of client n is at least qnif

and only if the long-term average number of

time slots the AP spends on the client is at

least qn/pn. We hereafter define wn= qn/pn

and 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 of

the figure shows the timeline of each client. We use an arrow to indicate the

beginning of an interval.

T

p1

p2

p3

AP

1

2

3

1In this case, a slot is the

time to deliver the packet

as well as receive the

ACK. The quantity pncan

then be regarded as the

probability that the trans-

mission from the AP to

client n is successful, as

well as the ACK from

client n to the AP.

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IEEE Wireless Communications • February 2012

50

fying whether a system is fulfilled is now equiv-

alent to verifying whether the average number

of time slots the AP spends on client n is at

least wnfor all n.

Since the AP can make at most one transmis-

sion in each time slot, we can immediately obtain

that a system is feasible only if ΣN

other words, the total loads imposed by all

clients cannot exceed the number of time slots

available in an interval if the set of clients’

requirements is to be feasible. However, while

this condition is necessary for feasibility, it is not

sufficient, because it may be impossible for the

AP to utilize all the T time slots in every inter-

val. For example, consider a system with two

clients and T = 3. Suppose in a certain interval,

the AP transmits the two packets for the two

clients in the first two time slots, respectively,

and both transmissions are successful. Now, as

the 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 forced

to be idle in the third time slot. This example

suggests that we need to take the amount of

unavoidable 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 the

same for all work-conserving policies [4, Lemma

3].

Definition 4: A scheduling policy is a work-

conserving policy if it schedules some packet for

transmission as long as there are any undeliv-

ered packets, and only idles when all unexpired

packets have been delivered.

It can be shown that, for every feasible sys-

tem, there exists a work-conserving policy that

fulfills it [4, Lemma 4]. Thus, we can limit our

attention to work-conserving policies.

Let I be the long-term average number of

idle time slots in an interval under any work-

conserving policy. Since, on average, the AP can

only make T – I transmissions in an interval, we

obtain that a system is feasible only if ΣN

T – I.

It turns out that this condition is also only

necessary and not sufficient for feasibility. It can

be further refined. Observe that, if we remove

some clients from a feasible system, resulting in

a system that only consists of a subset of clients

of the original one, the resulting system must

also be feasible. Thus, by letting ISdenote the

long-term average number of idle time slots in

an interval when only a subset S ⊆ {1, 2, …, N}

of clients is present, we obtain an even more

stringent 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 ΣN

and Σn∈Swn≤ T – IS, for all S ⊆ {1, 2, …, N} are

not equivalent. The following exxample demon-

strates that the condition in Lemma 1 is indeed

a stronger one.

Example 1: Consider a system with two clients

and interval length T = 3. The reliabilities for

both clients are p1= p2= 0.5. Client 1 requires

a timely-throughput of q1= 0.876, while the

timely-throughput requirement of client 2 is q2

= 0.45.

Now, we have:

n=1wn≤ T. In

n=1wn≤

n=1wn≤ T – I

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 all

clients {1, 2}, we have w1+ w2= 2.66 < 2.75 =

T – I{1, 2}. However, if we evaluate the condition

for the subset of S = {1}, we find w1= 1.76 >

1.75 = T – I{1}. This indicates that the set of

clients is not feasible. Thus, merely evaluating

the condition for the set of all clients is not

enough. ?

It turns out that the more stringent necessary

condition in Lemma 1 is actually both necessary

and sufficient.

Theorem 1 ([4], Theorem 4): A system is fea-

sible if and only if Σn∈Swn≤ T – IS, for all S ⊆

{1, 2, …, N}. ?

While the above theorem provides a sharp

characterization of feasibility, it involves testing

the condition Σn∈Swn≤ T – ISfor all subsets S

⊆ 1, 2, …, N. Sine there are 2Nsuch subsets, this

appears to require exponentially many tests.

However, it turns out that this condition can be

greatly simplified.

Theorem 2 ([4], Theorem 5): Sort all clients

so that q1≥ q2≥ · · · ≥ qN. Let Sm:= {1, 2, …,

m} be the subset that consists of clients 1

through m. A system is then feasible if and only

if Σm

This theorem reduces the number of tests for

checking feasibility to N. Each such test can be

performed efficiently by using the Fast Fourier

Transform, using which [4] has developed a poly-

nomial time algorithm based on this theorem.

FEASIBILITY OPTIMAL SCHEDULING POLICIES

n=1 wn≤ T – ISm, for all m ∈ {1, 2, …, N}. ?

In this section, we introduce two scheduling poli-

cies that are feasibility optimal (i.e., they fulfill

every feasible systems). Both policies are what

we call the largest debt first scheduling policies.

When employing a largest debt first scheduling

policy, the AP calculates the debt it owes to each

client at the beginning of each interval. The AP

then prioritizes all clients according to the debts

owed to them, such that clients with larger debts

get higher priority. In each time slot of the inter-

val, the AP transmits the packet for the client

with the largest debt among those whose packets

are not delivered yet.

Figure 2 shows an example of scheduling

using the largest debt first scheduling policy. In

this example, we assume that at the beginning of

some interval, client 1 has the largest debt, client

3 has a medium debt, and client 2 has the small-

est debt. The AP first transmits the packet for

client 1, and keeps retransmitting the packet in

case the previous transmission fails. The packet

for client 3 is only transmitted after the packet

for client 1 is successfully delivered. Finally, the

packet for client 2 is only transmitted after both

packets for clients 1 and 3 are successfully deliv-

ered.

w

q

p

w

q

p

1

1

1

2

2

2

1 76.

0 9. ,

==

==

,

When employing a

largest debt first

scheduling policy, the

AP calculates the

debt it owes to each

client at the begin-

ning of each interval.

The AP then priori-

tizes all clients

according to the

debts owed to them,

such that clients with

larger debts get high-

er priority.

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IEEE Wireless Communications • February 2012

51

The two scheduling policies differ in their

definitions of debt. The first kind of debt, the

time-based debt, is derived from the concept of

load wndefined earlier.

Definition 5: Let un(k) be the number of time

slots that the AP spends transmitting the packet

for client n in the kthinterval. The time-based

debt of client n at the beginning of the (k + 1)th

interval is defined as rn

un(j). The largest debt first policy that applies

the time-based debt is called the largest time-

based debt first scheduling policy.

The second kind of debt, the weighted-delivery

debt, is derived directly from the timely-through-

put requirement qnof a client.

Definition 6: Let dn(k) be the indicator func-

tion of the event that the AP delivers a packet

for client n in the kthinterval. The weighted-

delivery debt of client n at the beginning of the

(k + 1)thinterval is defined as

(1)(k + 1) := kwn– Σk

j=1

The largest debt first policy that prioritizes

according to the weighted-delivery debt is called

the largest weighted-delivery debt first scheduling

policy.

It has been shown that both largest debt first

policies fulfill every feasible systems, and are

hence feasibility optimal.

Theorem 3 ([4, Theorems 2 and 3]): Both the

largest time-based debt first policy and the

largest weighted-delivery debt first policy are

feasibility optimal. ?

SIMULATION RESULTS

We now present simulation results when applying

the two largest debt first policies for VoIP traffic.

We follow the G.711 codec, which is an ITU-T

standard for audio compression, and the IEEE

802.11b to decide parameters for the simulation.

Important parameters are summarized in Table 1.

We assume that there are two groups, A and

B, of clients. The timely-throughput requirement

of each client in group A is 0.99 packets per

interval, while that of each client in group B is

0.8 packets per interval. The channel reliability

of 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 6

group A clients and 5 group B clients is feasible,

while a system with 6 group A clients and 6

group B clients is not.

We compare the two largest debt first poli-

cies against two standard techniques. One is the

IEEE 802.11 DCF mechanism, and the other is

a centralized policy where the AP assigns priori-

ties to clients randomly at the beginning of each

interval. The two policies are referred to as DCF

and Random, respectively. We use the total time-

ly-throughput deficit, defined as Σn(qn– actual

timely-throughput of client n)+, to evaluate the

performance of a policy. A system is fulfilled by

a policy if, under that policy, the total timely-

throughput deficit converges to zero.

The simulation results for the feasible set is

shown in Fig. 3. It can be seen that both largest

debt first policies fulfill the feasible set, as the

total timely-throughput deficits converge to zero

over time under these two policies. On the other

hand, both the DCF policy and the Random pol-

icy fail to fulfill this feasible set. This result

shows these two policies are not adequate for

serving real-time flows.

The simulation result for the infeasible set is

shown in Fig. 4. In this setting, the total time-

lythroughput deficits remain strictly positive and

bounded 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 debt

first policies achieve smaller total timely-through-

put deficits than the other two policies. This sug-

gests that the largest debt first policies have

better performance even for infeasible systems.

UTILITY MAXIMIZATION FOR

ELASTIC TRAFFIC

In previous sections, we have supposed that the

timely-throughput requirement of each client, qn,

is specified by the client and inelastic. In this

section, we discuss scenarios where the timely-

throughput requirements are elastic.

PROBLEM FORMULATION

Suppose now that the value of qnis tunable by

the AP, for each n. Suppose also that each client

n receives some utility based on the timely-

rk

kqdj

p

n

nj

k

n

n

( )( ):

( ).

2

1

1

+=

−

=

∑

Figure 2. An example illustrates the largest debt first scheduling policy. The size

of 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

3

FS

FS

Table 1. Simulation setup.

Bit rate64 kb/s

Packetization interval20 ms

Payload size per packet 160 bytes

Time slot length610 ps

Transmission data rate11 Mb/s

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IEEE Wireless Communications • February 2012

52

throughput that is provided. The relation

between the utility derived and the timely-

throughput provided is characterized by each

client’s utility function, Un(qn). The utility func-

tion, Un(·), is assumed to be concave and infinite-

ly differentiable. Different clients may have

different 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 feasible

vector [qn] so as to maximize the total utility of

all clients. Since Theorem 1 already has estab-

lished a necessary and sufficient condition for

feasibility, 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 A

BIDDING PROCEDURE

Since the utility functions of clients are all con-

cave, the above optimization problem is indeed a

convex one, and standard techniques for solving

convex optimization problems can be employed to

determine the optimal solution that maximizes the

total utility of the system. However, this problem

involves 2Nfeasibility constraints, as in Eq. 2,

resulting in a huge computation overhead for stan-

dard techniques. Further, in practice, the AP may

not even know the utility functions of all clients.

To alleviate these challenges, we can employ a

decomposition technique introduced in [10], and

decompose the SYSTEM problem into two sub-

problems. The total utility in the system can be

maximized 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 beginning

of each interval, each client n pays the AP a cer-

tain amount of money, ρn. The AP then chooses

a feasible vector [qn] to achieve weighted pro-

portional fairness among clients, where the

weight of client n is ρn. It is known that weighted

proportional fairness can be achieved by maxi-

mizing ΣN

solve the following problem:

n=1ρnlog qn. Thus, the AP aims to

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 its

price per unit of timely-throughput as ψn:=

ρn/qn. Client n assumes a linear relation between

payment and timely-throughput, and aims to

find a new ρnso as to maximize its own net utili-

ty, that is, the difference between its utility and

the amount of its payment. More formally, client

n aims to solve the following problem:

CLIENTn:

(7)

over 0 ≤ ρn≤ ψn. (8)

Under this bidding procedure, the AP aims to

achieve weighted proportional fairness among

Max Un

n

n

n

ρ

ψ

ρ

,

,

,

⎛

⎝⎜

⎞

⎠⎟−

s.t.

q

p

TISN

n

n

S

n S

∈

≤− ∀ ⊆…

∑

,{ , , 1 2 , }.

Max

ρnn

n

N

∑

q log,

=

1

s.t.

q

p

TISN

n

n

S

n S

∈

≤− ∀ ⊆…

∑

, { , , 1 2, }.

MaxU q

n

1

n

n

N

∑

( ),

=

Figure 4. Performance of an infeasible set.

Time (s)

10

1

0

Total timely throughput deficit

2

3

4

5

6

7

8

2345

Weighted-delivery

Time-based

Random

DCF

Figure 3. Performance of a feasible set.

Time (s)

10

1

0

Total timely throughput deficit

2

3

4

5

6

7

8

2345

Weighted delivery

Time-based

Random

DCF