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Jin, Jiong; Wang, Wei-Hua; Palaniswami, Marimuthu (2009). Utility max-min fair

resource allocation for communication networks with multipath routing

Originally published in Computer Communications, 32(17), 1802-1809 . Available from: http://doi.org/10.1016/j.comcom.2009.06.014

Copyright © 2009 Elsevier Ltd. This is the author’s version of the work, posted here with the permission of the publisher for your personal use. No further distribution is permitted. You may also be able to access the published version from your library. The definitive version is available at http://www.sciencedirect.com/.

Utility Max-Min Fair Resource Allocation forCommunication Networks with Multipath Routing

Jiong Jin∗, Wei-Hua Wang and Marimuthu Palaniswami

Department of Electrical and Electronic EngineeringThe University of Melbourne

Victoria 3010, Australia

Abstract

This paper considers the flow control and resource allocation problem as applied

to the generic multipath communication networks with heterogeneous applica-

tions. We propose a novel distributed algorithm, show and prove that among all

the sources with positive increasing and bounded utilities (no need to be con-

cave) in steady state, the utility max-min fairness is achieved, which is essential

for balancing QoS (Quality of Service) for different applications. By combin-

ing the first order Lagrangian method and filtering mechanism, the adopted

approach eliminates typical oscillation behavior in multipath networks and pos-

sesses a rapid convergence property. In addition, the algorithm is capable of

deciding the optimal routing strategy and distributing the total traffic evenly

out of the available paths. The performance of our utility max-min fair flow con-

trol algorithm is evaluated through simulations under two representative case

studies, as well as the real implementation issues are addressed deliberately for

the practical purpose.

Key words: Network flow control, Utility max-min fairness, Resource

allocation, Multipath communication networks

∗Corresponding author. Phone: +61-3-83448146. Fax: +61-3-93471094.Email address: j.jin@ee.unimelb.edu.au (Jiong Jin)

Preprint submitted to Computer Communications June 25, 2009

1. INTRODUCTION

Though current communication networks, like the prevailing Internet, have

become a great success in providing efficient data transmission services, e.g.,

electronic mail, web browsing and file transfer, it is still far from sufficient to

support the increasing demand of real-time services, e.g., audio, video and mul-

timedia delivery through the network. These real-time applications usually have

stringent Quality of Service (QoS) requirements, and are sensitive to allocated

bandwidth, time delay and packet loss ratio, which are not easy to be guar-

anteed in the TCP-based Internet service architecture nowadays. Therefore,

future communication networks are expected to support heterogeneous applica-

tions with diverse QoS requirements.

To provide a better traffic management for communication networks than the

traditional TCP does, an extensive study has been carried out in the literature.

Among them, the most successful result in the area of network congestion control

and resource allocation is the “Optimal Flow Control”(OFC) approach proposed

by Kelly [1]. This pioneer work was further advanced by the researches in single

path networks [2, 3, 4, 5, 6], multipath networks [7, 8, 9] and multirate multicast

networks [10, 11, 12].

The main idea of OFC is essentially the same to formulate flow control as

an optimization problem and then maximize the total utilities with network

bandwidth constraints. The utility function of bandwidth associated with each

application mathematically models its QoS performance. Following that, OFC

algorithm is derived by solving the optimization problem distributively. It con-

sists of a link algorithm that measures the congestion in the network and a

source algorithm that adapts the transmission rate by the feedback congestion

signals. This optimization approach not only leads to social utility maximization

at convergence, what is more, the resulting bandwidth allocation in equilibrium

is in a fair manner.

Very popularly by selecting the utility as a logarithmic function, Kelly [1]

shows that the OFC approach achieves a proportional fairness of bandwidth

2

allocation. Also by using the OFC strategy, Mo and Walrand [13] and La and

Anatharam [4] investigate another important fairness criterion called max-min

fair allocation [14] (which emphasizes equal sharing compared with proportional

fairness). In their work, the authors define a family of utility functions to ap-

proximate arbitrarily close to a max-min fair allocation. Unfortunately, the util-

ity function used becomes ill-conditioned when the max-min fairness is reached,

and the related link prices at congested links either turn to 0 or diverge to ∞.

Hence, their max-min fair flow control algorithms are impractical from an engi-

neering point of view. On the other hand, in order to cope with heterogeneous

network applications of different QoS requirements, Cao and Zegura [15] propose

a new criterion named utility max-min fairness and the corresponding alloca-

tion algorithm. In their approach, each link requires the information of utility

functions from all the traversed sources, which makes network implementation

difficult.

Even though the optimal flow control approach has made great advances in

dealing with congestion control and resource allocation, it also exposes serious

limitations as pointed out in our paper [16].

• At the current stage, OFC approach is only suitable for elastic traffic,

where each application attains a strictly increasing and concave utility

function to ensure the feasible optimal solution and the convergence of

utility maximization process. It cannot deal with congestion control and

resource allocation for communication networks where real-time applica-

tions are involved.

• In the utility maximization approach, if users select different utility func-

tions based on their real performance requirement, the OFC approach

usually leads to a totally unfair resource allocation for practical use. In

particular, an application with low demand on the contrary is allocated

with a high bandwidth.

As the network topographical complexity increases, there always exist mul-

tiple paths between transmitting source and receiving destination. Indeed, mul-

3

tipath communication networks have already attracted significant research at-

tention since they well cater for traffic load balancing and bandwidth usage

efficiency. For instance, the most common IP networks, which more or less

require single path routing previously, allow the traffic to split across several

paths with the help of MPLS technology. Thus, in this paper, we propose a

novel distributed flow control algorithm for multipath communication networks

to be friendly with both elastic traffic and real-time applications. Especially,

the utility max-min fair resource allocation is achieved among heterogeneous

applications of different QoS requirements.

The rest of the paper is organized as follows. In Section 2, we describe

and formulate the problem. Section 3 proposes the utility max-min fair flow

control algorithm. After that, the actual implementation issues are discussed

in Section 4. Finally, we provide two typical case studies through simulations

to illustrate the algorithm performance in Section 5 and draw conclusions in

Section 6.

2. PROBLEM FORMULATION

For a practical network application, bandwidth allocation may be a concern,

but a more important and direct concern to an application is really the utility or

QoS performance. The utility function of an application is a measure of its QoS

performance based on provided network services. In this paper, we characterize

utility in terms of allocated bandwidth, which is a common modelling approach

in the optimal flow control literature. The model emphasizes the important

relationship between bandwidth allocation and QoS performance.

Referring to the paper [17], traditional data applications such as file transfer

are rather tolerant of throughput and time-delays. They are termed as elastic

applications and have been extensively studied under the OFC framework. The

utility function can be described as a strictly concave function as shown in

Figure 1(a). The utility (performance) increases as the increasing of bandwidth,

but the marginal improvement is decreased.

4

Thanks to the development of multimedia technologies, real-time applica-

tions such as teleconferencing are becoming ubiquitous. These applications are

generally delay sensitive and have strict QoS requirements. Unlike the elastic

traffic, they have an intrinsic bandwidth threshold because the data generation

rate is independent of network congestion. The degradation in bandwidth may

cause serious packet drop and severe performance degradation. Thus, a reason-

able description of the utility is close to a sigmoidal-like function as shown in

Figure 1(b) (solid line), which is convex instead of concave at lower bandwidths.

Certain hard real-time applications may require an exact step utility function

as in Figure 1(b) (dash line).

There exists another class of real-time rate-adaptive applications which ad-

just the transmission rate in response to network congestion. For low and high

bandwidths, their marginal utility increase with additional bandwidth is small

and the utility curve may have a general shape as in Figure 1(c).

There are some applications that may take a stepwise utility function as

shown in Figure 1(d). Such applications can be found in audio and video de-

livery systems via a layered encoding and transmission model [10]. For these

applications, bandwidth allocation is limited to distinct levels. The utility is

increased only when an additional layer is reached provided an increasing in

available bandwidths.

In this context, we consider a network involving a set L = {1, 2, . . . , L} of

links of capacity cl, l ∈ L. The network is shared by a set S = {1, 2, . . . , S} of

sources. Each source s attains a non-negative utility Us(xs), modeling the QoS

performance, when it transmits at a rate xs ∈ [ms,Ms] with ms and Ms the

required minimum and maximum transmission rates respectively. In particular,

the utility function Us(xs) is assumed to be continuous, strictly increasing and

bounded (no need to be concave) in the interval [ms,Ms]. Without the loss of

generality, it can be assumed that Us(xs) = 0 when xs < ms and Us(xs) =

Us(Ms) when xs > MS .1

1For the scalability, it can be further assumed that 0 ≤ Us(xs) ≤ 1 and Us(Ms) = 1.

5

U

Bandwidth

(a) Elastic

U

Bandwidth

(b) Real−time

U

Bandwidth

(c) Rate−adaptive

U

Bandwidth

(d) Stepwise

Figure 1: Utility functions for different classes of applications.

Different from single path networks, in this paper, each source s has ns

available paths or routes from the source to the destination. Denote L × 1

vector Rs,i the set of links used by source s ∈ S for its path i ∈ {1, 2, . . . , ns},whose lth element is equal to 1 if and only if the path passes through link l, and

0 otherwise. Then the set of all available paths of user s is defined by

Rs = [Rs,1, Rs,2, . . . , Rs,ns ]

and the whole set of paths in the network is defined by a L×N routing matrix

R,

R = [R1, R2, . . . , RS ]

where N = n1 + n2 + . . . + nS is the total number of paths.

For each source s, define xs,i be the rate of source s on path Rs,i, and

obviously the source rate xs =∑ns

i=1 xs,i. Let

x = [x1,1, . . . , x1,n1 , x2,1, . . . , x2,n2, . . . , xn,1, . . . , xn,nS]T ∈ RN

+

6

be the path rate vector of all sources. In order to formulate the flow control

problem, we first clarify the notion of feasible (or attainable) path rate alloca-

tion.

Definition 1. A path rate allocation x for all available paths is feasible or

attainable if and only if the corresponding source rate xs for each source s is

within the range [ms,Ms], and no links in the network are congested, i.e.:

ms ≤ xs ≤ Ms, xs =ns∑

i=1

xs,i, s ∈ S (1)

Rx ≤ c, x ≥ 0 (2)

where c = [c1, c2, . . . , cL]T is the vector of link capacities.

When considering different QoS requirements of heterogeneous applications,

it may not be appreciative for the network to simply share the bandwidth as con-

ventional max-min fairness does. Instead, the network should allocate the band-

width to the competing applications according to their different QoS utilities.

This motivates the concept of the utility max-min fairness suggested by [15].

Definition 2. A source rate allocation is utility max-min fair, if it is feasible

and for each user s, its utility Us(xs) cannot be increased while maintaining

feasibility, without decreasing the utility Us′(xs′) for some user s′ with a lower

utility Us′(xs′) ≤ Us(xs).

It is even more complicated in the environment of multipath networks, where

the source rate is made up of constituted path rates. Our objective is to guide

traffic to a feasible path rate allocation, in such a way that the sum-up source

rate is utility max-min fair. In other words, each source is treated in a fair

manner and guaranteed high QoS performance. In the following section, we will

develop a new flow control algorithm to achieve utility max-min fairness within

a given multipath network and study its properties in detail.

7

3. UTILITY MAX-MIN FAIR FLOW CONTROL ALGORITHM

Consider the flow control problem formulated in Section 2. Now, we propose

a distributed algorithm that achieves utility max-min fairness for multipath

communication networks.

3.1. A Distributed Utility Based Flow Control Algorithm

The utility max-min fair flow control algorithm adopts the similar flow con-

trol structure as the optimal flow control approach [2] does, making use of pricing

scheme. There are three price vectors α ∈ RS+, β ∈ RS

+ and p ∈ RL+ associated

with constraint (1) and (2) (Note that constraint (1) ms ≤ xs ≤ Ms is regarded

as two separated constraints xs ≥ ms and xs ≤ Ms) respectively. It is designed

in a fully distributed manner, i.e., a link algorithm is deployed at each link to

update the link price depending on the severity of link congestion, and a source

algorithm is implemented at each source edge to adapt the transmission rate

based on the above three prices.

Both link algorithm and source algorithm are iterative. At time t + 1, each

link l updates its link price pl according to:

pl(t + 1) = [pl(t) + γ(xl(t)− cl)]+ (3)

where γ > 0 is a small step size, and xl(t) = Rl.x is the aggregate path rate at

link l. Equation (3) implies that if the aggregate path rate at link l exceeds the

link capacity cl, the link price will be increased; otherwise it will be decreased.

The projection [z]+ = max{0, z} ensures that the link price is always non-

negative.

For each source s, we use the following first-order Lagrangian algorithm to

update its ith path rate:

xs,i(t + 1) = [xs,i(t) + γ(1

Us(xs(t))+ αs(t)− βs(t)− pr

s,i(t))]+ (4)

and then calculate the source rate:

xs(t + 1) =ns∑

i=1

xs,i(t + 1) (5)

8

where

αs(t + 1) = [αs(t) + γ(ms − xs(t))]+ (6)

βs(t + 1) = [βs(t)− γ(Ms − xs(t))]+ (7)

are the lower bound and upper bound price to enforce the source rate constraint

ms ≤ xs ≤ Ms, and

prs,i(t) = max

l∈Rs,i

pl(t) (8)

is the path price, namely the maximum of the link prices along the particular

route. Algorithm 1 summarizes all the above and list out the utility max-min

fair flow control for multiple paths.

For the multipath networks, the set of feasible path rates xs,i may not be

unique, so that the first-order Lagrangian algorithm usually oscillates. This

is one of the typical difficulties researchers face when dealing with multipath

networks. In order to eliminate the undesirable effect and further improve the

convergence speed, we purposely introduce another augmented variable xs,i,

called the optimal estimation of path rate xs,i. In this way, Equation (4) is

slightly modified by applying the concept of low-pass filtering as

xs,i(t + 1) = [(1− γ)xs,i(t) + γxs,i(t)

+γ(1

Us(xs(t))+ αs(t)− βs(t)− pr

s,i(t))]+

xs,i(t + 1) = (1− γ)xs,i(t) + γxs,i(t). (9)

By the theory of filtering, at optimality, xs,i = xs,i(t + 1), and notice that the

augmented variable is assisted solely to remove the oscillation without changing

the optimal solution of xs,i. This will be clearly illustrated in Case Study 1 of

Section 5.

It is worthy emphasizing that the convergence of the algorithm is guaranteed

by two methods together, i.e., the first-order Lagrangian method and low-pass

filtering method. In particular, the first-order Lagrangian method guides the

flows around the neighborhood of the equilibrium. Meanwhile, due to the non-

uniqueness of feasible path rates in this context, oscillation is observed. Thus,

9

Algorithm 1 Utility max-min fair flow control algorithm

• Link l’s algorithm:

At time t = 1, 2, . . ., each link l:

1. Aggregates flow rates xs,i(t) for all paths Rs,i that contain link l.

2. Computes a new link price

pl(t + 1) = [pl(t) + γ(xl(t)− cl)]+.

3. Communicates the new price pl(t + 1) to all sources whose path Rs,i

contains link l.

• Source s’s algorithm:

At time t = 1, 2, . . ., each source s:

1. Receives from the network the path prices

prs,i(t) = max

l∈Rs,i

pl(t)

for all its paths Rs,i, i = 1, 2, . . . , ns.

2. Updates the path rate xs,i(t + 1) and source rate xs(t + 1) using

Equation (4) and (5).

3. Computes new lower bound and upper bound price α(t + 1) and

β(t + 1) for the next step according to Equation (6) and (7).

4. Communicates the new flow rate xs,i(t+1) to all the links which are

contained in paths Rs,i.

10

the low-pass filtering method is further employed to remove it and accelerate

the convergence. It is critical to choose the parameter γ which has a impact on

the convergence rate. Usually, larger γ will lead to faster convergence rate. Ac-

cording to our earlier results shown in [16, 18], however, it should not be chosen

larger than some positive γ∗, otherwise, the first-order Lagrangian algorithm

will diverge.

3.2. Utility Max-Min Fairness

Recalling Definition 1, the interested region of the source rate is [ms, Ms]

and the associated utility for the source rate outside this region is scaled to 0 or

1. Within the region [ms,Ms], both the lower bound and upper bound prices (α

and β) are equal to 0 at convergence. The path rate algorithm of Equation (4)

simplifies to

xs,i(t + 1) = [xs,i(t) + γ(1

Us(xs(t))− pr

s,i(t))]+. (10)

From Equation (10), at convergence, it is observed that either 1Us(xs(t)) =

prs,i(t) or xs,i(t) = 0. If we define pr∗

s = 1Us(xs(t)) for each source s, the latter case

can be interpreted in another way, that is, when the path price prs,i(t) is greater

than pr∗s , this particular path is too “expensive” to carry any flow (xs,i(t) = 0).

The above fact establishes Theorem 1.

Theorem 1. For multipath communication networks, in steady state, the prices

on paths Rs,i that carry positive flows xs,i > 0 must be minimum, and hence

equal, among all the paths Rs of source s. Moreover, the optimal source rates

are given by

x∗s =∑

R∗s,i∈R∗s

x∗s,i = U−1s

([1

pr∗s

]Us(Ms)

Us(ms)

)

and xs,i = 0 if prs,i > pr∗

s

where [z]ba = max(a, min(b, z)), path R∗s,i has the minimum path price pr∗s,i = pr∗

s ,

and R∗s is the set of all minimum price paths R∗s,i of source s.

11

From this theorem, in steady state, the associated utility Us of source s is

simply equal to 1pr∗

swhen pr∗

s ∈ [ 1Us(Ms) ,

1Us(ms) ], otherwise, it attains a utility

Us(ms) of the minimum rate requirement whose value is greater than 1pr∗

s(It

cannot be decreased anymore due to QoS requirement), or a utility Us(Ms) of

the maximum rate requirement whose value is less than 1pr∗

s(It needs not to be

increased any further).

For this reason, we need to consider the resource allocation among the

sources who attain a normal utility U∗s = 1

pr∗s

. Theorem 1 states that the source

rate will be determined by the minimum path price among all the paths, which

is however defined as the maximum link price along the path. In a nutshell,

each source will be bottlenecked by a particular link.

Let Sl be the set of sources which are bottlenecked at link l. Assume in

steady state, there are K different link prices in the network with

p1 > p2 > · · · > pK−1 > pK .

We first select the links associated with the highest link price p1 and refer

them as lp1 , then all the sources s ∈ Slp1attain the same utility Us = 1/p1,

which are the smallest allocated utilities compared with others. If we apply the

utility max-min condition only to this set of sources (Slp1), we see that they

are utility max-min fair. Because if there is a source s ∈ Slp1that increases the

utility Us by increasing its transmission rate xs, there must be another source

s′ ∈ Slp1to decrease its rate xs′ and further decrease its utility Us′ which is

previously equal to Us. In other words, no source can increase its utility without

decreasing another one’s within Slp1, which is the definition of utility max-min

fairness exactly. We now extend this argument to include sources bottlenecked

by links with price p2.

The lp2 set of links are the links with the second highest link price p2,

p1 > p2 > pk, k 6= 1, 2. All the sources s ∈ Slp2have the same utility Us = 1/p2.

Since we have already shown that the sources in Slp1are utility max-min fair

and the utility for the sources in Slp2are equal, if there is a source s ∈ Slp2

that

increases its rate and utility, there must be another source s′ ∈ Slp2∪ Slp1

to

12

decrease its rate which already has a lower utility Us′ ≤ Us. Thus the utility

max-min fairness holds for all the sources within Slp2∪ Slp1

.

Continuing in this way, selecting all the links with positive link prices in the

order p1, p2, · · · , pK−1, pK , it is concluded by induction that the entire source

rate allocation is utility max-min fair and the global fairness is achieved.

Another interesting finding of Theorem 1 is that the proposed algorithm

is able to implicitly select the optimal paths among all the available routings.

In that regard, we refer to the utility max-min fairest resource allocation with

respect to all possible routings as optimal utility max-min fairness. The path

price can be thought of a kind of information feedback about the congestion

status along the particular path. According to Theorem 1, since only the path

bearing the minimum price, i.e. the least congested path, will carry the data

forwards, the sources dynamically coordinate the path rate allocation and always

pick up the best traffic distribution strategy. In view of the entire network, the

optimal utility max-min fairness is achieved or traffic loads are well balanced.

Case Study 2 of Section 5 specifically exemplifies this nice property.

4. NETWORK IMPLEMENTATIONS

In this section, we discuss about the implementation issues of the proposed

utility max-min fair flow control algorithm. First, a buffer management scheme

is provided to further improve the performance of the link algorithm, aiming

at avoiding the buffer overflow and reducing the delay. It is followed by a real

implementation illustration in the Internet.

4.1. On-line Buffer Measurement

When the utility max-min flow control algorithm converges, the aggregate

source rate at each bottleneck link will be equal to the link capacity. Since

there is no mechanism in link algorithm to control the buffer occupancy, due

to the statistical process of packet transmission in the practical network, it will

lead to the serious buffer overflow and significant queuing delay. Hence, we

13

make the following enhancements to the basic link algorithm by using “on-line

measurement” technique [19].

At time t, the buffer backlog bl(t) of link l is updated automatically according

to2

bl(t) = [bl(t− 1) + (xl(t)− cl)]+ (11)

in which we assume the buffer size at each link is sufficiently large and never

induces a buffer overflow.

Multiplying both sides of (11) by γ, the step size in link algorithm (3), we

have

γbl(t) = [γbl(t− 1) + γ(xl(t)− cl)]+ (12)

Comparing Equation (12) with (3), we yield the alternative link price adap-

tation rule based on the buffer backlog information bl(t) at link l

pl(t + 1) = γbl(t) (13)

In the new link algorithm (13), the individual link price is updated by the

local buffer backlog information only. It is much simpler than that of the basic

algorithm (3). Furthermore, with the implementation of source algorithm, the

buffer backlog at each link can be well maintained under such a built-in close

loop feedback system, and the packet loss from overflow is greatly prevented.

4.2. Implementation Issues in the Internet

Here we present an implementation scheme of the proposed algorithm in the

Internet. Recall that, in order to support real-time traffic without disturbing the

current IP structure, IETF adopted a new architecture named “Differentiated

Services” (Diff-Serv) in 1998, in which the first 6 bits (with a potential for all

2Here we use a deterministic approach to estimate the buffer dynamics, and assume the

updating time interval is 1. Otherwise

bl(t) = [bl(t− 1) + time interval(xl(t)− cl)]+

and this only results in a weighting coefficient change from γ to γ/time interval in the new

link algorithm (13).

14

8 bits) of the IPv4 ToS (Type of Service) octet [20] and the IPv6 Traffic Class

octet [21] are reserved as Differentiated Services Code Point (DSCP) [22]. One

suggestion of DSCP is for the priority assignment, in which applications with

strict QoS requirements are intuitively assigned with a high priority and receive

a better and faster service than the low priority classes. Another suggestion,

which this paper fits into, is to treat DSCP as an explicit congestion feedback

mechanism to provide a better solution for congestion control and resource al-

location for the future Internet.

Through the Differentiated Services Code Point (DSCP) in the IP header,

the maximal path price prs,i(t) defined by (8) can be easily informed to the

source s. When a packet is sent by the source, the DSCP field is initialized to

0. As the packet makes its way through the network, each link reads the DSCP

field and sets it to a new value

new DSCP = max(old DSCP, current link price),

i.e., each link examines the packet’s DSCP field and compares it with the current

link price. If the link price is greater than the value in the DSCP field, the link

sets the DSCP field to its current link price, otherwise keeps it unchanged. In

this way, when the packet reaches the destination, it contains the maximal link

price along the path. After the source receives the acknowledgment from the

destination, it is able to use the new path price in the DSCP field to updates

the path / source rate.

In the networking case, the implementation complexity of the algorithm or

protocol mostly concerns whether it is distributive and whether it is scalable.

For the Internet, it is impractical to have the central authority that performs the

centralized control, thus the algorithm must be implementable in a distributed

manner. Also it is desired that the algorithm is scalable to the large-scale net-

work. We then analyze the complexity of the proposed algorithm from the

above two aspects. As discussed, our approach only requires each link (actually

executed by the corresponding router) to estimate the link price based on the

local buffer backlog information and carry on one additional comparison opera-

15

S 1

S 2

S 3

S 4

D 1 D 3

D 2

D 4

l 1

l 2

l 3

l 4

Figure 2: The network topology of Case 1.

tion. The source rate is adjusted by the feedback information embedded in the

acknowledgment. Obviously, both link and source algorithms are run locally

and distributively. On the other hand, say a network of L links and S sources

with N different paths, the total number of additional comparison operations is

within O(LN). Hence, the overhead incurred is low and scalable. Therefore, the

proposed utility max-min fair flow control algorithm is practical and realizable

in the real Internet.

5. CASE STUDIES AND SIMULATIONS

In this section, we have two simple but illustrative case studies to show

the algorithm performance over certain classes of network topologies. To be

concrete, in the first case, we apply the proposed algorithm to a specific network

and demonstrate the dynamic behavior. In the second case, we show that our

algorithm is able to choose the optimal strategy for the traffic distribution.

5.1. Case 1

In this case, we evaluate the dynamics of our utility max-min fair flow control

algorithm for multipath communication networks through simulations. Figure 2

depicts the topology of the network. It consists of four unidirectional links

labelled l1, l2, l3 and l4 with capacities c = (4, 6, 8, 10) in Mbps and are shared

by 4 sources S1, S2, S3 and S4. S1 with a total rate of x1 uses the paths: l1

with rate x1,1 and l2 with rate x1,2. S2 with a total rate of x2 uses the paths:

16

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Allocated bandwidth (Mbps)

Util

ity

U1

U3

U2

U4

Figure 3: Source utility functions

l2 → l4 with rate x2,1 and l3 → l4 with rate x2,2. S3 with a total rate of x3 uses

the paths: l3 with rate x3,1 and l1 with rate x3,2. S4 with a rate of x4 uses a

single path: l4 with rate x4,1 i.e., x4 = x4,1.

The utility function of each source is given by: U1(x1) = 1/(1 + e−2(x1−4)),

U2(x2) = log(x2 + 1)/ log 11, U3(x3) = 1/(1 + e−2(x3−6)) and U4(x4) = 0.1x4.

All the sources have their maximum rate requirement at 10 Mbps. Figure 3

illustrates these utility functions. The logarithmic utility function represents an

elastic data flow application whereas the sigmoidal function approximates the

real-time application. The linear utility function corresponds to the application

whose satisfaction increases linearly.

In the simulation, we run the original algorithm with γ = 0.2. The simulation

results are given in Figure 4. Note that time scale in all the relevant figures

is in terms of number of iterations. As expected, the oscillation is observed,

which motives the modification replacing Equation (4) by Equation (9) in the

algorithm. Figure 5 shows the behavior of the modified algorithm. S2 and S4

share the bottleneck link l4 (p4 = 1.5671) with source rate (3.6188, 6.3812).

Both achieve a utility U = U2 = U4 = 1/p4 = 0.6381. S1 and S3 then equally

17

0 100 200 300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

Util

ity(a)

U2 U

4

U1

U3

0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

6

7

8

9

10

Time

Sou

rce

rate

(b)

x2

x1

x4

x3

Figure 4: Simulation results of original flow control algorithm (a) Source utilities (b) Source

rates.

share the remaining “cheaper” network resource (p1 = p2 = p3 = 1.0125) with

a utility of 0.9877.

This confirms that the flow control algorithm given in this paper can provide

an efficient utility max-min fair resource allocation for multipath communica-

tion networks containing heterogeneous applications, for both elastic traffic and

inelastic real-time applications. More importantly, their utility functions (i.e.,

U1(x1) and U3(x3)) may not need to satisfy the critical strictly concave condition

which is strongly required by optimal flow control approach [2, 6].

Remark 1. To the best of our knowledge, we are first to develop the algorithms

dealing with the heterogeneous traffic in the multipath network environment.

Even in the simple single path network, if we directly apply optimal flow control

algorithms for a multi-application network (with both concave and nonconcave

utility functions), it could lead to instability and high network congestion [23].

Additional admission control [23] or link capacity provisioning techniques [24]

are necessarily involved. Instead of maximizing the total utility, which may lead

to unfairness as we suggested previously, our goal is to allocate the resources

such that utility max-min fairness is achieved among different applications.

18

0 100 200 300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

Util

ity

(a)

U2, U

4

U1, U

3

0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

6

7

Time

Pat

h ra

te

(b)

x1, 1

, x2, 2

x3, 2

x2, 1

x1, 2

x3, 1

x4, 1

0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

6

7

8

9

10

Time

Sou

rce

rate

(c)

x1

x4

x2

x3

0 100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5

Time

Link

pric

e

(d)

p4

p1, p

2, p

3

Figure 5: Simulation results of modified flow control algorithm (a) Source utilities (b) Path

rates (c) Source rates (d) Link prices.

19

A

B

C

D

Figure 6: The network topology of Case 2.

5.2. Case 2

In Case 2, for comparison purpose, we assign all the utility functions U(x) =

x. In this sense, the utility max-min fairness degenerates to (bandwidth) max-

min fairness. The same example as [25] is adopted to show how our algorithm

achieve the optimal max-min fair distribution of the traffic. Consider the net-

work consisting of four unidirectional links AB, BD, AC, CD with capacities

fixed respectively at 1 MB/s, 1 MB/s, 3 MB/s and 2 MB/s as shown in Fig-

ure 6. There are 5 sources S1 − S5 with utility function U(xi) = xi, for any

i = 1...5. S1 − S4 are routed through A → B, B → D, A → C and C → D

individually. S5 has two alternative paths A → B → D and A → C → D.

In the simulation, we choose γ = 0.2 and the results are given in Figure 7.

The utilities (or source rates in this scenario) allocated to S1−S5 are (1, 1, 2, 1, 1)

and S5 automatically chooses the path A → C → D. It is indeed the optimal

max-min rate allocation suggested by [25]. Meanwhile, Table 1 lists out the

resource allocation results of different strategies in the literature. If S5 is routed

through the fixed path A → B → D, the rate allocation (0.5, 0.5, 3, 2, 0.5)

is obtained consequently. The traffic is obviously biased regarding the load

distribution. Moreover, let us consider the popular OFC approach [2, 6] with

the aim to maximize the overall throughput. We will get the rates for source

20

0 100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5

Time

Util

ity

(a)

U1, U

2, U

4, U

5

U3

0 100 200 300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

1

1.2

1.4

Time

Pat

h ra

te

(b)

x5,1

x5,2

Figure 7: Simulation results of Case 2 (a) Source utilities (b) Path rates.

Table 1: Comparison of resource allocation for different strategies.

Fixed Routing Our Algorithm OFC

Source Path Flow Path Flow Path Flow

S1 A → B 0.5 A → B 1 A → B 1

S2 B → D 0.5 B → D 1 B → D 1

S3 A → C 3 A → C 2 A → C 3

S4 C → D 2 C → D 1 C → D 2

S5 A → B → D 0.5 A → C → D 1 − 0

Total 6.5 6 7

21

S1, S2, S3 and S4 of 1 MB/s, 1 MB/s, 3 MB/s and 2 MB/s respectively. The

total throughput is 7 MB/s. However, source S5 is totally prohibited from

transmission as the bandwidth allocated is 0 MB/s. It reconfirms that the utility

maximization derived OFC approach can lead to a seriously unfair situation

for network resource allocations. On the other hand, different from [25] that

takes account of bandwidth allocations merely, our algorithm emphasizes the

actual performance of heterogeneous applications, i.e., the utility. Of course, the

bandwidth consideration can be easily realized by assigning the utility functions

U(x) = x as this example does. Therefore, we show by comparison that the

proposed algorithm has the desirable property of possessing the optimal utility

(or bandwidth) max-min fair resource allocation, or equivalently, balancing the

traffic loads evenly.

6. CONCLUSIONS

In this paper, we have developed a distributive flow control algorithm for

networks with multiple paths between source-destination pairs, and the objec-

tive is to achieve the utility max-min fair resource allocation among competing

users. We have shown that in steady state, the algorithm does meet the goal for

any choice of utility functions and leads to very desirable results. The utility

max-min fair flow control algorithm presented in this paper only requires that

each source utility function is positive, strictly increasing and bounded over the

bandwidth. It is more suitable for practical networks where the utility functions

of real-time applications usually do not satisfy the strict concavity condition

that is strongly assumed by the standard optimal flow control approach. The

simulation reveals that the means we have taken to speed up the convergence

and remove the oscillation in multipath networks is effective. Furthermore, the

proposed algorithm well balances the traffic and eventually leads to the optimal

utility max-min fair resource allocation. For our future work, more factors in-

cluding delays will be incorporated to model the utility functions, as well as the

dynamic behavior such as stability will be studied and analyzed in all details.

22

ACKNOWLEDGMENT

This work was supported by the Australian Research Council under Grant

DP0985322 and Grant DP0559131, and ARC Research Networks on Intelligent

Sensors, Sensor Networks and Information Processing.

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