fairness resource allocation in blind wireless multimedia

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946 IEEE TRANSACTIONS ON MULTIMEDIA, VOL. 15, NO. 4, JUNE 2013

Fairness Resource Allocation in Blind WirelessMultimedia Communications

Liang Zhou, Member, IEEE, Min Chen, Senior Member, IEEE, Yi Qian, Senior Member, IEEE, andHsiao-Hwa Chen, Fellow, IEEE

AbstractTraditional -fairness resource allocation in wirelessmultimedia communications assumes that the quality of experi-ence (QoE) model (or utility function) of each user is available tothe base station (BS), which may not be valid in many practicalcases. In this paper, we consider a blind scenario where the BShas no knowledge of the underlying QoE model. Generally, thisconsideration raises two fundamental questions. Is it possibleto set the fairness parameter in a precisely mathematicalmanner? If so, is it possible to implement a specific -fairnessresource allocation scheme online? In this work, we will givepositive answers to both questions. First, we characterize thetradeoff between the performance and fairness by providing anupper bound of the performance loss resulting from employing

-fairness scheme. Then, we decompose the -fairness probleminto two subproblems that describe the behaviors of the usersand BS and design a bidding game for the reconciliation betweenthe two subproblems. We demonstrate that, although all usersbehave selfishly, the equilibrium point of the game can realizethe -fairness efficiently, and the convergence time is reasonablyshort. Furthermore, we present numerical simulation results thatconfirm the validity of the analytical results.

Index Terms -Fairness, blind communication, multimedia

application, resource allocation.

I. INTRODUCTION

A. Motivation and Goal

I N this work, let us consider a generic wireless multimediacommunication scenario, where one base station (BS) as-signs available resource (e.g., bandwidth) to multi-

Manuscript received October 02, 2011; revised January 29, 2012; acceptedApril 04, 2012. Date of publication January 04, 2013; date of current versionMay 13, 2013. This work was supported in part by the State Key Develop-ment Program of Basic Research of China (2013CB329005), the National Nat-ural Science Foundation of China under Grants 61201165 and 61271240, thePriority Academic Program Development of Jiangsu Higher Education Institu-tions, Nanjing University of Posts and Telecommunications Foundation underGrant NY211032, and the National Science Council of Taiwan under GrantNSC99-2221-E-006-016-MY3. The associate editor coordinating the review ofthis manuscript and approving it for publication was Prof. Monica Aguilar.

L. Zhou is with the Key Lab of Broadband Wireless Communication andSensor Network Technology, Nanjing University of Posts and Telecommunica-tions, Nanjing 210046, China (e-mail: [email protected]; [email protected]).

M. Chen is with the School of Computer Science and Technology, HuazhongUniversity of Science and Technology, China (e-mail: [email protected]).

Y. Qian is with the Department of Computer and Electronics Engineering,University of Nebraska-Lincoln, Omaha, NE 68101 USA (e-mail: [email protected]).

H.-H. Chen is with the Department of Engineering Science, National ChengKung University, Tainan City 70101, Taiwan (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMM.2013.2237895

Fig. 1. A wireless multimedia communication scenario.

media users (see Fig. 1). Essentially, fairness resource allocation

in such a system can be expressed as

for

for

(1)

where is a fairness parameter, denotes the total

available resource, represents the allocated resource for user

, and expresses user s Quality of Experience (QoE)

model (or utility function in a more general sense) that reflects

the functional relationship between Mean Opinion Score (MOS)

value and allocated resource . Note that QoE is basically a

subjective measurement of end-to-end performance, and widely

used in multimedia communications. Specifically, QoE is mea-

sured by MOS which reflects the degree of user satisfaction

from a scale of 1 (unacceptable) to 4.5. (excellent) [1].

From the efficiency point of view, BS aims at achieving the

maximum system performance which is usually measured by

the sum of all users MOS values. Some works (e.g., [2][6])

have demonstrated that the consideration of fairness usually has

a negative impact on the performance. Therefore, how to bal-

ance the tradeoff between the fairness and performance is an

important issue for BS. Recently, there have been a substantial

amount of works done on resolving the aforementioned issue as

shown in (1) in different communication contexts (Section I-C

provides a detail survey of theworks related to this paper).How-

ever, a critical assumption made in most existing studies is that

the QoE model of the multimedia applications is known to the

BS. Clearly, this facilitates the BS to extract structural insights

and makes the underlying problem more tractable. Needless to

say, this assumption may be invalid in many practical scenarios

1520-9210/$31.00 2013 IEEE


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ZHOU et al.: FAIRNESS RESOURCE ALLOCATION IN BLIND WIRELESS MULTIMEDIA COMMUNICATIONS 947

where a priori information describing the QoE model is unavail-

able. Informally, we refer it to a blindscenario when BS has no

knowledge of the QoE model during the whole resource alloca-

tion procedure.

Actually, so far it is still unclear how to achieve the fairness

in a blind communication environment. Two fundamental ques-

tions of interest for such problem are given as follows: 1) How

to set the fairness parameter from the perspective of perfor-

mance-fairness tradeoff? 2) Given a specific fairness parameter

, how to implement the -fairness resource allocation online?

These questions lead to the motivation for this work and our ef-

forts to analyze, whenever possible, and simulate the system of

concern to have a better understanding of this problem.

B. Main Contributions

The purpose of this paper is to provide a family of stylized

-fairness resource allocation schemes for blind multimedia

communications. The primary objective is to quantify the

attributes of such a problem and establish an implementable

procedure. To be more specific, we summarize our contribu-tions along the two dimensions in the sequel:

Qualitative analysis . We derive an exact expression for

the upper bound of the performance loss caused by -fair-

ness, thus characterizing the fairness-performance tradeoff

(Theorem 1). Importantly, this enables a BS to make a

quantitative decision on choosing the right fairness param-

eter and analyze its impact on the system performance

with respect to the total MOS. In particular, we answer the

first question (asked in the previous subsection) by shed-

ding the light on the principle that a higher value of and

a larger number of users will possibly give rise to a higher

performance loss. Technical realization . We decompose a blind fair-

ness-aware resource allocation problem into two sub-

problems to describe the behaviors of the users and BS,

respectively. More precisely, we answer the second ques-

tion by proposing a bidding game for the reconciliation

between the two subproblems (as shown in Table I). In

addition, we show that, although all the users behave self-

ishly, any specific -fairness scheme can be implemented

by the bidding game between the users and BS (i.e.,The-

orem 2). Then, we derive an estimation of the bidding

games convergence time (i.e., Theorem 3), which is of

paramount importance for multimedia communications.

C. Related Works

Although -fairness scheme has been extensively studied

for different kinds of communication systems (e.g., [2][4],

[7][11], and the references therein), the underlying perfor-

mance-fairness tradeoff is still not well understood. Some

recent works have been devoted to theoretically analyzing what

it actually means for a higher value of to lead to a better

fairness [2][4], [6], [8], [10], [12], but the majority of them

only discussed some special cases, such as proportional fairness

and max-min fairness . For the more general

case with varying , which enables a BS to have more flexi-

bility to strike a performance-fairness tradeoff, only empirical

studies or simulation results showed that a higher value of

TABLE IOPTIMAL BIDDING GAME

always results in a larger performance loss. Unfortunately, their

works lack precise mathematical proof [3], [11], [12].

With respect to a technique realizing a specific fairness

resource allocation, roughly speaking, it falls into two broadcategories: convex optimization and game theory. The former

formulates the fairness-aware resource allocation as a convex

optimization problem using a specific fairness criterion [5],

[13][17]. Typically, the utility function or QoE model of each

user is defined according to the characteristics of the transmitted

multimedia sequences and the allocated bit-rate [18], [19]. The

latter designs different resource allocation games to efficiently

and fairly allocate the available network bandwidth to different

multimedia users [20][26]. In particular, [20] established a

general proportional fairness scheme based on the Nash bar-

gaining solutions and coalitions, and it also pointed out that the

max-min approach usually yields the worst performance loss.

[21][23], [25], [26] applied the Nash bargaining solutions to a

multimedia multiuser resource allocation problem, where the

utility function for each user was defined as the inverse of the

distortion. It should be noted that both categories request that

the utility function is available to the BS or the controller, and

this assumption is the most significant difference in comparison

with the work presented in this paper.

D. Organization and Notations

The rest of the paper is organized as follows. Section II fo-

cuses on how to set up parameter factor from the perspec-

tive of performance-loss tradeoff. In Section III, we propose and

analyze a blind resource allocation scheme in the form of bid-

ding game. Then, numerous simulation results are presented in


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Section IV. Finally, Section V concludes this work with a sum-

mary of future works.

The following notations will be used throughout the paper.

Let be the set of real vector space, and let be the set of

positive integers. We write for and for . For a

given positive number , the logarithm of with base 10 is

denoted by . For two functions and , the notation

means that remains bounded

as . denotes that

and , and represents that

as . In addition, means

. We also use the abbreviations

RHS/LHS for right/left-hand side, and iff for if and

only if.

II. HOW TO SET FAIRNESS PARAMETER ?

In this section, we show how to set the fairness parameter

for blind multimedia communications by quantifying what is the

maximum performance loss for a specifi

c . First let us defi

nethe loss function, and then derive a tight upper bound for the

loss.

A. Loss Function

As stated earlier, once a BS uses a fairness mechanism, the

system performance which is measured by the sum of all the

users MOS is likely to decrease, if compared to that without

fairness mechanism. Suppose that the BS adopts a specific

-fairness, the performance loss is the difference between the

performance under the -fairness scheme, noted as , and

the optimal system performance which is achieved when

is equal to zero (no fairness). Formally, we can defi

ne theloss function as

(2)

Specifically, corresponds to the per-

centage performance loss compared to the maximum system

performance. In order to make the analysis for tractable,

we make the following assumption on the QoE model.

Assumption 1: is concave over allocated resource

, and there exist positive constants , , ,

enabling the QoE model to satisfy

1) ;2) is a strictly non-decreasing function on ;

3) for any

two .

Remark 1: The above assumption is natural in QoE-based

multimedia communications. In particular, Assumption 1) is

reasonable since the maximum MOS value for any multimedia

application is 4.5; Assumption 2) is also true since, in general,

the more allocated resource, the higher MOS value will be; As-

sumption 3) is a mild rule targeting at controlling the fluctuation

of the MOS curve as parameters vary. Note that this regularity

is also widely used in utility estimation for the sake of analysis

simplicity [3].

Moreover, the following proposition shows the characteris-

tics of .

Proposition 1: If satisfies Assumption 1, then, the

resulting is compact, monotone, and convex.

Proof:

1) Because is compact,

is continuous and bounded over , and thus

is compact.

2) Let . Then, , such that .

Consider an allocation , such that . For any

, let , for . Because

is continuous and non-decreasing, so is . Given also

that and that , it follows that

, such that . Note

that from the monotonicity of , we get that

and is monotone.

3) Let , then , such that .

Let , by convexity of , .

Due to the concavity of , we have

Because is monotone, it follows that

, and hence is convex.

B. The Bound of the Loss

The following theorem provides an upper bound for

in the framework of QoE-based multimedia

communication.

Theorem 1: If Assumption 1 is held and the BS adopts -fair-

ness, the upper bound the performance loss satisfies

where denotes the number of users.

To prove Theorem 1, we need the following preparations mo-

tivated by [2], [27]. Definition 1 defines a measurement function

to characterize the upper bound of the loss. Lemma 1

shows that is unimodal over for any .

Lemma 2 presents the relationship between the maximum value

of and .

We start by designing a measurement function to analyze the

format of .

Definition 1: (Measurement Function) For any ,

, let be defined as

(3)

which will be used to characterize the bound of .

Remark 2: In fact, how to quantitatively measure

is still a challenging problem even when the QoE information is

available [3], [11], [17], [20]. Equation (3) makes a step forward

to find a way to quantitatively approximate its format when the

maximum MOS value of all the users is the same. In this case,

the exponent part of the variable is by using [2,

Theorem 12] when the QoE model satisfies Assumption 1.Lemma 1: is unimodal over for any

.


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Outline of the proof: To simplify the notation, we drop theparameters and from the argument function. The derivativeof is

(4)

where determines the sign of the derivative. It is positivewhen , and hence . It iseasy to verify that is strictly increasing over . Asa result, if , there exists a unique suchthat if , and if

. Similarly, if , is strictly increasingwhen . It follows that is unimodal.

Lemma 2: Let be the unique value at which

achieves its minimal value over , we have

.

Outline of the proof: The proof consists of two parts. First,

we use the definition of , and to charac-

terize the relationship between and

. Next, by extending the result of [2, Theorem3], we connect and , and link and . In

particular, we get

Hence, we know .

Therefore, , that is,

. B y d efinition, we get:

.

We are now ready to prove Theorem 1.

Proof of Theorem 1: Similar to [27], we can set

. Use the Mean Value Theorem, for every ,

there exists a between and , such that

that is,

Next, we need to validate the following three items for a suffi-

ciently small .

1)

;

2) ;

3) .

Let us check them one by one.

1) We show that for any sufficiently large , it follows that

(5)

where is the unique root of (which is defined in (4))

in the interval . Hence, the dominant of

is . Therefore, we can get

for a sufficiently large . Similarly, since the dominant

part of is , we get

. Therefore, we also can have

.

Since the denominator of is strictly in-

creasing, we can get

(6)

Regarding t o the b ound f or t he n umerator o f ,

according to Lemma 1, we have

(7)

Combining (6) with (7), we get

2) It follows from (5).

3) According to Definition 1, we have

Using the above results, we can get


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Therefore, . Together

with Lemma 2, we complete the proof.

Remark 3: The bound we have established depends only on

the number of users involved in the resource allocation, and

hence it is indeed independent of the QoE model as long as it

satisfi

es Assumption 1. Note that the assumption of equal max-imum achievable MOS value is not overly restrictive, as MOSs

of the users are commonly normalized in a variety of settings

so that the comparison between them is meaningful. Through

scaling, the maximum achievable MOS of each user typically

can be set as 4.5.

III. BLIND -FAIRNESS RESOURCE ALLOCATION SCHEME

In this section, our goal is to realize the -fairness resource

allocation when a BS does not know the QoE model of each

user. Specifically, we propose a bidding game to decompose (1)

into two subproblems that describe the behaviors of the users

and the BS, respectively.

A. Bidding Game

Suppose thatuser is willing to pay anamount of

for a multimedia application, and it receives resource

proportional to , with where is the price.

The objective of the user is

(8)

On the other hand, given that user is willing to pay for a

service, the BS strives to find optimal to maximize its ownobjective function, that is

(9)

where , thereafter we call it control function, is

the objective function for the BS relative to the bidding money

, allocated resource , and fairnessparameter . Similar to

[28], our basic idea is to decompose the original problem (1)into

user behavior (8) and BS behavior (9), and we aim at achieving

the -fairness resource allocation by resolving the two problems

jointly. Specifi

cally, we design a bidding game which is shownin Table I. Theoretically, the fairness can be realized by moder-

ately setting the control function even though

the BS does not know the QoE model of each user. It is noted

that is very important for bidding games, and

we will describe its mechanism in the next subsection.

Remark 4: It is necessary to specify the physical meanings of

a bidding game. In (8), user assumes a linear relation between

the amount it pays , and the resource it receives . Specif-

ically, we assume , where corresponds to the

price. Hence, (8) is equal to maximizing , which

can be viewed as the net profit of user . In short, the goal of

each user is to selfishly maximize its net profit using a first order

liner approximation to the relation between bidding money and

allocated resource.

B. Control Function

In this part, we set the control function to resolve the original

problem (1).

Lemma 3: There exists non-negative , , and

with , such that

1) for such that , is a solution of (8);

2) given that user pays per period, is a solution of(9).

Moreover, if , and are all positive vectors, the vector

is also a solution of (1).

Proof: See Appendix A.

Theorem 2: The format of the control function

satisfies

for

for(10)

Outline of the proof: We can prove the format of the control

function from the following two aspects:

1) Sufficient condition: Since (10) satisfies Lemma 3, it canbe taken as a sufficient condition.

2) Necessary condition: Similar to the process offinding the

optimal solution of (1), (8) and (9), we can adopt Karush-

Kuhn-Tucker (KKT) condition to achieve the format of the

control function.

Since the proof process is similar to that of Lemma 3, we do not

repeat it here.

Remark 5: To make the proposed bidding game work

smoothly in a realistic blind communication scenario, each user

can not cheat during the whole bidding process. It typically

consists of two cases: 1) Each user has the ability to pay for the

bidding money; 2) Each user should strictly comply with Steps1819 in Table I to update its bidding money.

Remark 6: As to the issue of the user cheating, we can have

the following counter-measures: 1) Each user inform its max-

imum available money to BS for its multimedia application,

and the bidding money should not be more than this threshold;

2) BS sets an penalty strategy to deal with the cheaters. Since

the game is an iterative process, it is implementable to set the

penalty function for all the users. For example, once finding

a cheating user during the bidding process, it will be expelled

from the system.

Proposition 2: Suppose that at a fixed point of the bidding

game, each user pays , and receives resource . If bothand are positive for all , then is the solution of (1).

Proof: At first, let . Then, since and

are derived from the fixed point, maximizes

over all , as described in Step 18 in Table I.

Thus, is a solution to (8) if . Similarly, from

Step 16 in Table I, is a feasible solution that maximizes

, over all feasible . Thus, according to

Theorem 2, we can find that is a solution to (9) given that

each user pays . Finally, by Lemma 3, is the solution

to (1).

C. Convergence Time

Since multimedia communication is very sensitive to delay, it

is necessary and important to investigate the convergence time


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of the proposed bidding game. We should note that if

is the optimal solution, then and the allo-

cated resource stops changing. In order to characterize the dif-

ference between iterations during the bidding process, we define

the potential function as , where

. Our main result on the convergence time

is given as follows.Theorem 3: Let be the number of the users in the system,

and be the number of rounds taken by the bidding game

to reach the optimal solution for the first time. Then,

.

The proof of this theorem proceeds as follows. At first,

Lemma 4 gives an upper bound on , and Corol-

lary 1 implies that there is a such that

. Next, Lemma 5 shows that is

a super-martingale and Corollary 2 provides the characteristic

when is not the optimal solution. Finally, using these

results, we obtain an estimation of the convergence time.

Lemma 4: Let be the allocated

resource at iteration by using the bidding game, we have

.

Proof: Let . By the def-

inition, we can get that

where . Using the upper-bound

of and , we have

Moreover, since , this is

at most . By Cauchy-Schwarz equation,

we get

The proof is complete.

Corollary 1: The relationship between and

satisfies

Lemma 5: Suppose that assignment satis-

fies ( ) and, for all ,

. Let be the assignment with

for . Then

.

Proof: See Appendix B.

Corollary 2: If is not the optimal solution, then

Remark 7: In the above proof precesses, we mainly concen-

trated on the difference of in each iteration . In partic-

ular, we used instead of to characterize the expectation

value of the potential function of and the probability of

update when it is not the optimal solution. Intuitively,

maybe have little impact on the convergence time, and this will

be validated in Section IV.

Now we can provide the proof of Theorem 3.

Proof: According to Corollary 1, let ,

, and

. From Lemma 5, we know

that is a super-martingale, and hence

In addition, from Lemma 5 and Corollary 2, we know that if

, . Thus, if

, we get

Hence, we can have

By the definition, means the stopping time. In this case, we

have either 1) , that is, is the optimal solution,

or 2) . Defining

we find that is a sub-martingale. Let be the probability

that 1) occurs. By the Optional Stopping Theorem [29], we get

, and hence

Moreover, still using the Optional Stopping Theorem, we get

Hence, combining Estimation Theorem [30] with Corollary

1, we have . When

1) occurs, according to Lemma 4, we have

. Sim-

ilarly, when 2) occurs, we have

. Therefore,

we can get . This completes the proof.


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Fig. 2. QoE models for different multimedia applications.

Fig. 3. Observed loss versus theoretical upper bound.

IV. NUMERICAL RESULTS

In this section, we will evaluate the performance of the bid-ding game using extensive simulation experiments based on

real-world traces consisting of three multimedia applications:

audio, file, and video. We compare our blind scheme with a

full-information case, where the BS knows the QoE model of

each user in advance. As to the QoE model with full-infor-

mation, we extend the QoE models as introduced in [31] to a

wireless down-link multimedia communication system which

is shown in Fig. 1. The total available resource is normalized to

one, and the underlying function between the allocated resource

and the MOS value is presented in Fig. 2. Note that the proposed

QoE model just corresponds to a special wireless multimedia

communication scenario, which is used only for an illustrationpurpose. Of course, they are flexible to the other models only if

they satisfy Assumption 1. We assume that each user can ran-

domly choose one of thethree multimedia applications at a time,

but cannot change it during the bidding game.

At first, we validate the upper bound of the loss function.

Fig. 3 depicts the given bounds (Theorem 1) and real observed

loss values as the number of users increases. From the given

results, we observe that the real MOS loss value is close to the

given upper bound fordifferent . Fig. 3 also shows that a higher

value of usually yields a higher performance loss, which is

consistent with the conclusion of the previous works [3], [12].

Moreover, it also demonstrates that a large number of the users

also incurs a larger performance loss. For example, when

, the MOS loss is at most 0.17 and 0.40 for and

Fig. 4. Performance gap between the proposed bidding game and the optimalsolution.

Fig. 5. Performance of three users in the iteration process (a) (b).

, respectively. For , these numbers are 0.36 and 0.72,

respectively. This observation suggests a basic operation rule

for BS: when the system has a relatively small number of users,

BS can achieve fair allocations without incurring a significantperformance deterioration. However, in the case with a large

number of users, BS should be careful to employ fairness since

it will easily lead to a large performance loss.

Next, we test the optimality of the bidding game compared to

the centralized resource allocation with known QoE model (i.e.,

full-information case). In the simulations, we assumed that the

initial bidding money is for all , and artifi-

cially set if the allocated resource is zero to avoid

computation error for . Fig. 4 shows the gaps of the total

MOS values between the two schemes. These gaps are obtained

using 100 runs in order to obtain statistically meaningful av-

erage values, and each user changed its multimedia applicationat each run. It can be seen that our proposed bidding game al-

most achieves the same performance compared to the full-infor-

mation case, in particular, when is large (e.g., ). That is

to say, it is possible to realize fairness in blind multimedia com-

munication scenarios. It is noted that, when , there is a

slight difference between the two methods. That is because we

set when the allocated resource is zero. When is

small, there is a higher probability that the allocated resource is

zero.

Finally, Figs. 57 present the MOS value of each user at each

iteration when , , and , respectively.

From the given results, we observe that: 1) the bidding game can

converge at a limited iteration rounds; 2) the convergence time

depends largely on the number of the users, and is independent


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Fig. 6. Performance of four users in the iteration process (a) (b).

Fig. 7. Performance offive users in the iteration process (a) (b).

Fig. 8. The convergence time as the number of users varies from 2 to 20.

of the value. Actually, the above observations conform with

Theorem 3. Moreover, Fig. 8 shows the convergence time as

the number of users varies from 2 to 20 with different values.

Note that when is small, Theorem 3 provides a tight bound;

while as goes large, Theorem 3 tends to become a bit of lose.

For example, when the theoretical upper bound is 5,

and the real convergence round is also 5; and these numbers

change to 402 and 351 when . That is because we use

a conservative estimation of in Lemma 4, and we

refer the readers to [29, Theorems 3.13.4] for more details on

the performance deviation using this estimation method.

V. CONCLUSIONS

This paper has attempted to make a step forward to under-

stand the fairness in blind multimedia communications. At first,

we characterized the tradeoff between performance and fair-

ness by providing a upper bound of the MOS loss incurred in

using -fairness scheme. Then, we decomposed the -fairness

problem into two subproblems that describe the behaviors of theusers and the controller, and designed a bidding game for the

reconciliation between the two subproblems. We showed that,

although all users behave selfishly in the game, the equilibrium

point of the game can solve the two subproblems jointly, and the

convergence time is limited. It is our belief that, in a blind envi-

ronment, the fairness parameter can be chosen precisely and

the -fairness resource allocation can be realized efficiently.

Moving forward, we believe that one fruitful direction for fu-

ture research is identifying specialized bidding game that of-

fers a shorter convergence time. On the application front, there

are a number of practical systems wherein it is highly desirable

that resource allocations are fair, i.e., cloud-based multimediaplatforms, multimedia services over Internet of things, etc. Cur-

rently, we are working to apply the proposed fairness-aware al-

location scheme to the real world applications of such systems.

APPENDIX A

PROOF OF LEMMA 3

Motivated by [24], we use the Lagrangian method to prove

this lemma. In what follows, we concentrate mainly on the case

of . For , the proof procedure is same as that of

.

The Lagrangian of (1) is

(A1)

where and are the Lagrange multipliers.

From the KKT condition, is the optimal solution to

(1) when there exist and satisfying

(A2)

(A3)

(A4)

, and .

The Lagrange of (8) is


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where is the Lagrange multiplier to (8). Similar to the above

case, by the KKT condition, is the optimal solution to (8)

when there exists satisfying

The Lagrangian of (9) is

(A5)

where and are the Lagrange multipliers.

Likewise, a vector is the optimal solution to (9)

when there exist vectors and satisfying

(A6)

represents the solution of (1), and , denote the La-

grange multipliers that satisfy (A2)(A4). In addition, let

, , and , for all

. In what follows, our goal is to prove satisfying

conditions 1) and 2).

We first check condition 1). It is obvious that

and , since . Moreover, let the Lagrange

multiplier of (8), , be . Then, we can

have

(A7)

(A8)

where (a) and (b) follow from (A2) and (A4), respectively.

Therefore, meets the KKT conditions for (8), and

is a solution to (8).

We then check condition 2). Since is the solution to

(1), we denote the Lagrange multipliers of (9) by and

, respectively. Given that each user pays , we get

Therefore, meets the KKT condition for (9), and is

a solution to (9).

Assume are positive with , for all ,

satisfying conditions 1) and 2). Our aim is to prove that is a

solution for (1). Denote the Lagrange multiplier for (8) by .Due to for all , the problem of (8) is also feasible for

all . Let and be the Lagrange multipliers for (9). Since

for all , we have for all . By (A6), we have

and hence . Let and .

We can get that is the optimal solution to (1) with Lagrange

multipliers and . Since is a solution to (9), we can get that

satisfies the KKT condition for (1), and thus is asolution to (1).

APPENDIX B

PROOF OF LEMMA 5

Let . We need to prove that

1) ;

2) .

Then,

.

First, we prove item 1). Let . Note

that the derivation of is

where is the average value of each element of . Further-

more, the second derivative is . Thus,

is minimized at . Now note that


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Next, we prove item 2). We assume that and

. Then we get

(B1)

Let , and thus we have for .

Additionally, let for , and we have

and for . Similarly, let

. Hence, the expression of

in terms of is the same as the one of except that is

reduced by one. Therefore, we have

However, since , the upper bound on the

RHS is

(B2)

Specifically, the RHS of (B2) is at most because the max-

imal value can be achieved at . That is,

. Hence, we can get

(B3)

In the case of , the proof is similar to what follows

above. Therefore, the proof is complete.

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Liang Zhou (M09) received his Ph.D. degree majorat Electronic Engineering both from Ecole NormaleSuprieure (E.N.S.), Cachan, France and ShanghaiJiao Tong University, Shanghai, China in March2009. From 2009 to 2010, he was a postdoctoralresearcher in ENSTA-ParisTech, Paris, France.From 2010 to 2011, he was a Humboldt ResearchFellow in Technical University of Munich, Munich,Germany. Now, he joins in Nanjing University ofPosts and Telecommunications, Nanjing, China.His research interests are in the area of multimedia

communications and signal processing.

Min Chen (SM09) is a professor in School ofComputer Science and Technology at HuazhongUniversity of Science and Technology (HUST). Be-fore he joined HUST, he was an assistant professorin School of Computer Science and Engineering atSeoul National University (SNU) since Sep. 2009.He has worked as a Post-Doctoral Fellow in Dept. ofElectrical and Computer Engineering at Universityof British Columbia (UBC) for three years sinceMar. 2006. Before joining UBC, he was a Post-Doc-

toral Fellow at SNU for one and half years. He haspublished more than 150 technical papers. Dr. Chen received the Best PaperRunner-up Award from QShine 2008. He serves as editor or associate editorfor Wiley I. J. of Wireless Communication and Mobile Computing, Wiley I.J. of Security and Communication Networks, Journal of Internet Technology,KSII Transactions on Internet and Information Systems, IJSNet, etc. He is amanaging editor for IJAACS. He was a TPC co-chair of BodyNets 2010. Heis a symposia co-chair and workshop chair of CHINACOM 2010. He worksas symposia co-chair for IEEE ICC 2012 and IEEE ICC 2013. Currently, he isGeneral Co-Chair for the 12th IEEE International Conference on Computer andInformation Technology (IEEE CIT 2012). His research focuses on multimediaand communications, such as multimedia transmission over wireless network,wireless sensor networks, body sensor networks, RFID, ubiquitous computing,intelligent mobile agent, pervasive computing and networks, E-healthcare,medical application, machine to machine communications and Internet ofThings, etc. He is an IEEE Senior Member since 2009.

Yi Qian (SM07) received a Ph.D. degree in elec-trical engineering from Clemson University. Heis an Associate Professor in the Department ofComputer and Electronics Engineering, Universityof Nebraska-Lincoln (UNL). He is a passionateteacher and a devoted educator. He received theHenry Y. Kleinkauf Family Distinguished NewFaculty Teaching Award in 2011, and HollingFamily Distinguished Teaching Award in 2012, bothfrom the College of Engineering, UNL. Prior tojoining UNL, he worked in the telecommunications

industry, academia, and the government. Some of his previous professionalpositions include serving as a senior member of scientific staff and a technicaladvisor at Nortel Networks, a senior systems engineer and a technical advisor atseveral start-up companies, an Assistant Professor at University of Puerto Ricoat Mayaguez, and a senior researcher at National Institute of Standards andTechnology. His research interests include information assurance and networksecurity, network design, network modeling, simulation and performanceanalysis for next generation wireless networks, wireless ad-hoc and sensornetworks, vehicular networks, broadband satellite networks, optical networks,high-speed networks and the Internet. He has a successful track record to leadresearch teams and to publish research results in leading scientific journals andconferences. Several of his recent journal articles on wireless network designand wireless network security are among the most accessed papers in the IEEEDigital Library. Dr. Qian is a member of ACM and a senior member of IEEE.

Hsiao-Hwa Chen (F10) is currently a Distin-guished Professor in the Department of EngineeringScience, National Cheng Kung University, Taiwan.He obtained his B.Sc. and M.Sc. degrees fromZhejiang University, China, and a Ph.D. degreefrom the University of Oulu, Finland, in 1982,1985 and 1991, respectively. He has authored orco-authored over 400 technical papers in majorinternational journals and conferences, six booksand more than ten book chapters in the areas ofcommunications. He served as the general chair,

TPC chair and symposium chair for many international conferences. He servedor is serving as an Editor or/and Guest Editor for numerous technical journals.

He is the Editor-in-Chief of IEEE Wireless Communications and the foundingEditor-in-Chief of Wileys Security and Communication Networks Journal(www.interscience.wiley.com/journal/security). He is the recipient of the bestpaper award in IEEE WCNC 2008 and a recipient of IEEE Radio Communica-tions Committee Outstanding Service Award in 2008. He is a Fellow of IEEE,a Fellow of IET, and a Fellow of BCS.

fairness resource allocation in blind wireless multimedia

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