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A Suboptimal Network Utility Maximization

Approach for Scalable Multimedia Applications

Mohammad S. Talebi∗, Ahmad Khonsari†∗, Mohammad H. Hajiesmaili∗†

and Sina Jafarpour‡

∗School of Computer Science, IPM, Tehran, Iran

†ECE Department, University of Tehran, Tehran, Iran

‡Department of Computer Science, Princeton University,

Emails: {mstalebi, ak, hajiesamaili}@ipm.ir, sina@cs.princeton.edu

Abstract—Wired and wireless data networks have witnessed an

explosive growth of inelastic traffics such as real-time or media

streaming applications. Recently, applications relying on layered

encoding schemes appeared in the context of live-streaming and

video and audio delivery applications. This paper addresses the

Network Utility Maximization (NUM) for scalable multimedia

transmission which is relying on layered encoding schemes. Non-

convexity of the NUM problem for such applications makes dual-

based approaches incompetent, whereby achieving optimality

proves quite challenging. We adopt the staircase utility function

and formulate the underlying optimization problem. To tackle the

non-convexity of the problem, we use a smooth approximation of

the staircase utility function and propose a dual-based distributed

algorithm for rate allocation and bandwidth sharing in such

scenarios. Numerical results show that the proposed algorithm

achieves suboptimal yet efficient solution.

I. INTRODUCTION

Over the past few years, the usage of multimedia appli-

cations in computer networks has been growing explosively.

Recently, applications with streaming traffic such as live-

streaming over peer-to-peer and wireless ad-hoc networks are

emerged and expected to continue growing. Thus, in the course

of recent years, the research community has witnessed the

emergence of new demands for QoS-provisioning in different

multimedia applications. In order to tackle this issue, many

technical challenges have to be addressed in the two areas of

video coding and networking.

In the video coding area, QoS-provisioning is efficiently

dealt with in the context of video adaptation paradigms [1].

The problem of video adaptation has been widely addressed

through a number of approaches, such as Scalable Coding [2],

[3], Transcoding [5], [6], and Summarization [7]. In Scalable

Video Coding, the objective is to enable the encoding of a

high-quality video bitstream that contains one or more valid

and decodable subset bitstreams. Transcoding is normally

referred to as techniques where a compressed media bitstream

format is converted into another format. Video Summarization

schemes through content analysis and optimization select a

subset of frames from the video sequence to form a concise

representation of the sequence, while incurring as small of a

loss as possible.

In the area of networking, rate allocation is at the nexus of a

wide variety of paradigms, whose boundaries extents different

scenarios ranging from resource-constrained networks to QoS-

aware ones. This also has been the issue of primary concern in

the research community of multimedia applications. Following

the seminal work by Kelly et al. [8], the optimization flow

control approach was proposed by Low et al. [9], in which the

optimal rate allocation of a wired network under elastic traffic

was modeled and led to a dual-based distributed algorithm

for rate allocation both in synchronous and asynchronous

environments. Within the previous decade, the work by Low et

al. was followed by the network research community and led

to a more general optimization framework known as Network

Utility Maximization (NUM) and its generalized form, GNUM

([10] and references therein). The underlying assumption of

these works is that the network traffic is elastic, whose

characteristics can be modeled by a strictly concave utility

function. Such utility functions, make the problem convex and

thereby tractable for optimality analysis.

On the other hand, Internet has witnessed an explosive

growth of inelastic traffics such as those arising in real-time or

media streaming applications. Such applications are relying on

tight performance characteristics, in terms of rate (bandwidth),

delay, jitter, etc., which make the utility function non-concave

[11]. Non-concave utility functions result in non-convex NUM

problems, whose analysis proves quite challenging. So far,

only few works have tackled the non-convex NUM problems,

e.g. [12]-[14]. In [12], the authors adopted sigmoidal-like util-

ity function that is an appropriate choice for the utility of rate-

adaptive multimedia applications, and proposed a distributed

admission control approach for such utilities, called “self-

regulating” heuristic. Hande et al. in [13] investigated the

optimality conditions for the distributed iterative dual-based

algorithm to converge to global optimal despite using non-

concave utility functions. In [14], an efficient but centralized

method based on sum of square approach is developed to

compute the global optimal rate allocation for types of non-

concave utility functions that can be transformed into polyno-

mial functions.

Contemporary to the research studies carried out in the

context of NUM, a lot of recent studies have dealt with the

inelastic multimedia applications through modeling the traffic

characteristics of such applications. These studies appeared

in different forms. Huang et al. [15], proposed a resource

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allocation solution for multi-user video streaming over cellular

wireless networks. They developed a NUM framework with a

resource pricing algorithm via previous well-established dual-

based algorithms. The resource price is obtained in turn, is

used to derive source content adaptation to each user, using

video summarization techniques [7]. In [16], a content-aware

distortion-fair networking framework with joint video source

adaptation and network resource allocation is developed. A

basic difference in this work is that an explicit utility function

for sources is not considered. Instead, a content-aware time-

varying utility function is chosen that is different per each

frame as well as per video content. Based on the idea of

dropping less important frames, a distributed iterative algo-

rithm is proposed to achieve min-max distortion fairness. The

main superiority of this work is taking into account the special

characteristics of video content such as dependency between

frames.

In this paper, our focus is on the rate allocation for mul-

timedia applications with scalable encoding based on NUM

approaches. Today, a plethora of such applications exist in

video and audio delivery systems and are relying on layered

encoding schemes [2], [3]. For scalable multimedia applica-

tions, rate allocation is limited to distinct levels of utility,

i.e. the utility is increased only when a higher layer can be

delivered due to increase in the available bandwidth. Thus, the

ideal utility for these applications is in the form of a staircase

function, which is shown in Figure 1 [4]. In order to deal with

the nondifferentiable and non-concave behavior of the staircase

utility, we introduce multimodal sigmoid approximation as a

smoothed and well-behaved utility function and to remedy

nondifferentiability of the staircase utility. Moreover, we aim

at approximating the underlying NUM in order to come up

with more amenable formulation, and then propose a dual-

based distributed algorithm as the solution to it. To the best

of our knowledge, this is the first work that addresses NUM

problem for scalable multimedia transmission. Numerical re-

sults present a proper validation of our endeavor in achieving

a suboptimal yet efficient solution.

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

describes the network model and utility approximation. Sec-

tion III is devoted to formulate the underlying NUM problem.

Section IV investigates the optimality condition and optimal

solution to the NUM. The optimal dual-based distributed

algorithm is presented in Section V. Numerical results are

presented in Section VI and VII concludes the paper and

outlines some future directions.

II. SYSTEM MODEL

A. Network Model

We consider a network consisting of a set of sources denoted

by S = {1,...,S} and a set of unidirectional links, denoted

by L = {1,...,L}. Let xsand clbe the source rate for source

s and capacity of link l, both in bps, respectively. Without loss

of generality, we assume that source rate of source s is limited

so as to certify

0 < ms≤ xs≤ Ms< ∞

(1)

where msand Msdenote the minimum and maximum rates

for source s, respectively. We assume that source s, when

submitting at rate xs bps, attains a utility function Us(xs),

which models its benefit.

We associate with source s a path, i.e. a set of links

L(s) ⊆ L, that determines the links that source s passes

through. Similarly, we define S(l) ⊆ S, to be the set of sources

traversing link l. For the sake of simplicity, we define the

routing matrix as R = [Rls]L×S, where Rlsis defined as

?

B. Utility Model

As stated in Section I, the utility function of such ap-

plications can be ideally characterized using a non-concave

and non-differentiable utility function referred to as staircase

utility function, which is shown in Figure 1 in solid line [4].

Non-concavity of the staircase utility functions implies that the

conventional theory of Network Utility Maximization (NUM)

cannot be used for such functions. In order to deal with such

non-concave and non-differentiable utility functions, we use a

smoothed approximation of it. Figure 1 shows the idea behind

such an approximation. In this figure, the curve in dashed

line represents the smoothed approximation of the staircase

function.

In order to construct such a smoothed approximation, we

divide its domain into nonoverlapping intervals, so that a step

transition occurs within the midpoint of each interval. The

step transition i, i.e. the part of the curve in which utility

function jumps from level i, (i.e. U(x) = i), to level i+1, (i.e.

U(x) = i + 1) is smoothed and approximated by a sigmoid-

like function, whose point of inflection corresponds to U(x) =

i+1

2.

A sigmoid-like function has been well studied in the field of

neural networks. The most commonly used form of sigmoid-

like function is the logistic function defined as

Rls=

1

0

if source s passes through link l

otherwise

(2)

F(x,α,β) =

1

1 + e−α(x−β)

(3)

It is easy to show that β is the inflection point of F(x), i.e. for

x < β, F(x) is convex, and for x > β it is concave. Moreover,

α > 0 is a parameter that determines the sharpness of its curve.

It’s worth mentioning that α must be chosen sufficiently large

so as to effectively capture the sharp transition of an increase

in the utility level.

Using the notation for the sigmoid-like function introduced

above, we then represent the approximation shown in Figure

1 in dashed line. Recall the interval division of the domain

introduced above. Then, for the step transition i, i.e. jump

from U(x) = i to U(x) = i + 1, we have,

˜U(x) = F(x,α,ki) + i;

x ∈?ki −k

2,ki +k

2

?

(4)

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where˜U(.) denotes the approximation to the original utility

function and [ki−k

i occurs. It’s worth mentioning that k is the required rate

increase to advance the utility U by 1. Hence, 1/k can be

thought of as the slope of the straight line passing through the

midpoint of step transitions.

Combining all of the intervals, we get

intervals, corresponding to N encoding layer.

In statistics, a sigmoid-like function, which is in possession

of a single point of inflection, is usually referred to as

unimodal function. Our approximated staircase utility function

is comprised of several sigmoid-like functions, and thereby

has several points of inflection. Thus, it is a multimodal

function as opposed to the unimodal case. In this respect, we

refer to this approximation as the multimodal sigmoid. The

multimodal sigmoid approximation presented above is non-

differentiable in general; however, if α is chosen sufficiently

large, discontinuity gap between contiguous steps vanishes and

thereby makes it continuous.

Sources in the network may demand for different QoS

requirements; hence, it makes sense that each source s, would

advance its utility according to its own ksfactor, which may

differ from the others. Moreover, each source s is assigned a

positive weight wswhich can be used to address its priority in

rate allocation. Such weights are normalized so as to satisfy

?

˜Us(xs) = ws˜U(xs,α,ks)

where˜U is defined by (5) and ksand α were omitted from

the notation.

2,ki+k

2] is the interval in which transition

˜U =

F(x,α,k) + 1

.

.

F(x,α,ki) + i

.

.

F(x,α,kN) + N

x ∈ [k −k

2,k +k

2]

x ∈ [ki −k

2,ki +k

2]

x ∈ [kN −k

2,kN +k

2]

(5)

where it is assumed that the domain is divided into N equal

sws= 1. Therefore, the (approximated) utility function of

source s is

(6)

III. PROBLEM FORMULATION

In this paper, we mainly focus on modeling a convex

optimization as an approximation to the non-convex NUM

arising in scalable multimedia applications. Thus, for the sake

of simplicity, we consider the simplest form of the NUM,

i.e. the optimization flow control problem, introduced in the

seminal work of Low et al. [9]. The objective of such a simple

NUM is to choose source rates so as to maximize the aggregate

utility of all sources while satisfying capacity constraints, as

follows

S

?

subject to:

?

max

x∈X

s=1

˜Us(xs)

(7)

s

Rlsxs≤ cl;

l ∈ L

(8)

Rate (bps)

Utility

Multimodal Sigmoid Approximation

Staircase Utility

Utility Level i

i-th interval

i ∈ [ki −k

2, ki +k

2]

Utility Level i + 1

Fig. 1.

Approximation (dashed line)

Staircase Utility Function (solid line) and Its Multimodal Sigmoid

where X = X1×X2×...×XSdenotes the Cartesian product

of all rate domains Xs = [ms,Ms]. In order to come up

with a more amenable formulation, we consider the following

optimization problem

max

x∈X

S

?

s=1

log˜Us(xs)

(9)

subject to:

?

s

Rlsxs≤ cl;

l ∈ L

(10)

The following theorem shows that the problem (9)-(10)

approximates the problem (7)-(8).

Theorem 1: The optimization problem (9)-(10) approxi-

mates the problem (7)-(8).

Proof: Taking the logarithm of the objective of (7) yields

??

Since log(.) function is monotone increasing, maximizing

(11) is equivalent to maximizing (7), and thereby problem (11)

is equivalent to problem (7)-(8). On the other hand, log(.) is

a concave function, for zs> 0, we have

?

provided ws ≥ 0 and

functions,˜Us= ws˜U into (12), we get

?

Therefore, the transformed objective of (7) is lower bounded

by the R.H.S of (13). Thus, to obtain an approximate to

problem (7), we choose its lower bound as the objective

function.

max

x∈Xlog

s

˜Us(xs)

?

(11)

log(

s

wszs) ≥

?

ws˜U(xs)) ≥

?

s

wslog(zs)

(12)

sws = 1. Substituting utility

log(

s

?

s

wslog˜U(xs)

(13)

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We defer solving the optimization problem until the next

section.

IV. OPTIMAL SOLUTION

A. Deriving Primal Optimal

We can solve the optimization problem (9)-(10) using dual-

based approach1. However, compared to the conventional

methods for the NUM proposed so far [10], it demands for

partly more elaboration. We obtain the dual problem using the

definition of Lagrangian. The Lagrangian of (9) is given by

L(x,µ) =

?

s

log˜Us(xs) −

?

l

µl

??

s

Rlsxs− cl

?

where µl is the positive Lagrange multiplier associated to

capacity constraint (10) for link l and µ = (µl,l ∈ L) is

the vector of Lagrange multipliers.

The approximated optimization problem introduced above

is non-convex. In order to come up with a convex formulation,

we use the following transformation

˜ xs= eαxs

(14)

Since this transformation is monotonic increasing, maximizing

L(˜ x,λ) is equivalent to maximizing L(x,λ). Rewriting the

Lagrangian with this transformation, we get

L(˜ x,µ) =

?

s

log˜Us(˜ xs) −

?

l

µl

??

s

Rls

α

logxs− cl

?

(15)

We denote the primal-optimal point of the approximated

problem by x∗= (x∗

at optimal point the following conditions must be satisfied:

s,s ∈ S). Based on KKT Theorem [17],

∇˜ xL(˜ x,µ)|(˜ x∗,µ∗)= 0

Rls

α

(16)

?

s

log ˜ x∗

s≤ cl;

l ∈ L

(17)

µ∗

l≥ 0;

l ∈ L

(18)

µ∗

l

??

s

Rls

α

log ˜ x∗

s− cl

?

= 0;

(19)

where 0 is a vector, all of whose element is zero.

Substituting (15) into (16) yields

∂L

∂˜ xs

=

d

d˜ xs

˜U?

˜Us(˜ xs)−

F?(˜ xs,α,ksis)

F(˜ xs,α,ksis) + is

log˜Us(˜ xs) −

1

α˜ xs

?

l

Rlsµl

(20)

=

s(˜ xs)

1

α˜ xs

?

l

Rlsµl

=

−

1

α˜ xs

?

l

Rlsµl= 0

1Due to space limit, we relegate the proof of the convexity to our future

works.

where it is assumed that x∗

Substituting F(˜ xs,α,ksis) into the above result and doing

some algebraic manipulations, we get

sfalls within the isth interval.

Asi

(˜ xs+Asi)2

˜ xs

˜ xs+Asi+ is

−

µs

α˜ xs

= 0

(21)

where µs=?

lRlsµland Asi= eαksis. Further simplifica-

tion of (21) yields

Asi

(1 + is)(˜ xs+ Bsi)(˜ xs+ Asi)=

where Bsi=

quadratic equation Obtaining x∗

?

Solving (23) yields ˜ x∗

µs

α˜ xs

(22)

is

is+1Asi. The above result leads to the following

sdemands for solving (21).

?

sas follows

?

2(1 + is)

Since the above equation must have a real solution, we deduce

that

i∗

˜ x2

s+

Asi+ Bsi−

αAsi

(1 + is)µs

˜ xs+ AsiBsi= 0

(23)

˜ x∗

s= Asi

α

µs− 2is− 1 +(1 −

α

µs)2−4isα

µs

(24)

s= ?µs

4α(1 −α

µs)2?

(25)

Optimal source rate can be obtained simply by taking the

inverse transformation as follows

s=?1

where [.]Xsis the projection operator on the Xs.

We then proceed to solve the problem through its dual. The

dual problem is defined as

x∗

αlog ˜ x∗

s

?

Xs

(26)

max

µ≥0D(µ)

(27)

where D(µ) is dual function and is defined as the maximum

of the Lagrangian over x, i.e.

D(µ) = min

˜ x∈˜

XL(˜ x,µ)

(28)

Problem (28) is an unconstrained optimization problem and is

already solved by ˜ x∗. Therefore, for the dual function we get

D(µ) = L(˜ x∗,µ)

(29)

B. Solving Dual Problem

Now we are ready to solve the dual problem (27). In order

to obtain a distributed solution, we will solve the dual problem

using gradient projection method [19]. The gradient projection

method, iteratively steps toward the opposite direction of the

gradient of the objective of the optimization (minimization)

problem. Therefore, for the dual problem (27), we get

µ(t+1)= [µ(t)− γ∇D(µ(t))]+

(30)

or equivalently,

µ(t+1)

l

=

?

µ(t)

l

− γ∂D(µ(t))

∂µl

?+

(31)

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where µ(t)= (µ(t)

step, γ is a constant step size and [z]+= max(z,0). For the

derivatives of D(µ), we get

l,l ∈ L) is the value of µ at tth iteration

∂D(µ(t))

∂µl

= cl−

?

s

Rlsx(t)

s

(32)

Substituting (32) into (31) yields

µ(t+1)

l

=

?

µ(t)

l

− γ

?

cl−

?

s

Rlsx(t)

s

??+

(33)

where x(t)

above form an iterative algorithm as the solution to the (9),

which will be discussed in the next section.

s is given by (26) The two update equations obtained

V. OPTIMAL ALGORITHM

In this section, we propose a distributed algorithm based on

the iterative solution obtained in Section IV.

In this subsection, we propose a distributed algorithm based

on the iterative solution obtained above.

Optimal source rate equations, i.e. (26) and (24), and La-

grange multiplier update (33), derived in the previous section,

can be used in conjunction with each other to form an iterative

algorithm as the solution to the optimization problem (9)-(10).

Lagrange multiplier is usually referred to as shadow price

owing to the economic interpretation of its role to adjust the

source rate [8], and hence thereafter we use this term instead.

For each time slot t (or iteration step t), the following key

steps exist in the algorithm:

1) Each link l calculates its corresponding Lagrange mul-

tiplier (shadow price) for the next time slot, i.e. µ(t+1)

based on its previous shadow price and its aggregate

traffic in the current time slot.

2) Each source s calculates its rate based on the aggregate

shadow price in its path.

3) Each source s transmits the packets based on the allo-

cated rate.

l

,

The rate control algorithm can be described as follows. For

each link l, shadow price µl is updated according to (33)

and the new shadow price result is communicated to sources

traversing this link. Each source s receives from the network

the shadow prices for links on its path and calculates µsusing

(33), chooses a new source rate using (26) and (24), and

communicates this new rate to all the links in its path. The

procedures at the links and the video sources are repeated until

the algorithm converges to the optimal video rates and optimal

shadow prices. The iterative algorithm for solving (27) is listed

as Algorithm 1.

Algorithm 1. Dual-based Rate Control Algorithm

for Scalable Multimedia

Do until maxs|x(t+1)

At each link l,

1. Update the shadow price as following:

s

− x(t)

s | < ?

µ(t+1)

l

= [µ(t)

l

− γ

?

cl−?

sRlsx(t)

s

?

]+

At each source s,

1. Obtain the path price µs(t)=?

lRlsµ(t)

l

2. Update x(t)

s

according to the following equation:

a. i(t+1)

s

b. ˜ x(t+1)

s

= ?µs(t)

=

4α(1 −

α

µs(t))2?

Asi

−2i(t+1)

s

−1+

α

µs(t)+

2(1+i(t+1)

?

(1−

α

µs(t))2−4i(t+1)

s

µs(t)

α

s

)

where A(t+1)

si

= eαksi(t+1)

s

c. x(t+1)

s

=

?

1

αlog ˜ x(t+1)

s

?

Xs

where [.]Xsis the projection operator on the Xs.

VI. NUMERICAL RESULTS

In this section, we present the numerical results of the pro-

posed iterative algorithm. Numerous validation experiments

have been established, however, for the sake of specific

illustration, validation results are presented.

In our scenario we consider a network with a single bot-

tleneck link. There are 5 sources in the network, all passing

through a shared link with capacity c = 15 Mbps. Different

sources use different values of k as a result of different QoS

metrics and rate requirements. Recall that ksis the required

rate increase for source s to advance the utility Us by one.

In this scenario we chose: (k1,...,k5) = (2.8,2,1.3,0.8,1).

For the sake of illustration, utility functions corresponding to

different values of ksare displayed in Figure 2. Different α for

all sources is set to 5 and step size is chosen to be γ = 0.001.

Also, weight factors of all users are assumed to be equal.

Staircase utility functions for sources are depicted in Figure

2.

The evolution of source rates is depicted in Figure 3. The

evolution of the shadow price (µ) is depicted in Figure 4. Both

figures reveal that the convergence is relatively fast. In order

to get more insight, the rate allocation is summarized in Table

1.

VII. CONCLUSION

In this paper we addressed the Network Utility Maximiza-

tion (NUM) for applications relying on scalable multimedia

transmission using layered encoding schemes. The utility

function for such applications was shown to be non-concave,

which makes the NUM non-convex. Thereby dual-based NUM

approaches fail to reach the optimal solution. In order to tackle

this issue, we smooth the non-concave utility function using