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Pricing and Optimal Resource Allocation in Next

Generation Network Services

Michael G. Kallitsis

Department of Electrical and

Computer Engineering

North Carolina State University

Raleigh, NC 27695

kallitsis@ncsu.edu

George Michailidis

Department of Statistics

University of Michigan

Ann Arbor, MI 48109

gmichail@umich.edu

Michael Devetsikiotis

Department of Electrical and

Computer Engineering

North Carolina State University

Raleigh, NC 27695

mdevets@ncsu.edu

Abstract—In this paper, we introduce a pricing model that

ensures efﬁcient resource allocation that provides guaranteed

quality of service while maximizing proﬁt in multiservice net-

works. Speciﬁcally, a dynamic allocation policy is examined that

relies on online measurements while each service class operates

under a probabilistic bound delay constraint. We present a

rigorous analysis of the properties of the policy that provides

insights into its workings as well as its sensitivity to various

parameters. Finally, its performance is evaluated through an

extensive numerical study.

I. INTRODUCTION

Recent technology advances have led to dramatic changes

in the communications arena. The use of ﬁber optics and the

increased performance of integrated circuits have brought to

the forefront diverse types of networks, such as broadband,

wireless ad hoc and mesh networks, and next generation

cellular systems.

However, these new technologies are not sufﬁcient by

themselves to guarantee business success. Added value and

service differentiation need also be considered, in order for the

service providers to be proﬁtable. Hence, the trend towards

networks providing some degree of value added services

has emerged. Speciﬁcally, Service Oriented Networks is an

evolving architecture that would allow for a priced based

differentiated choice of network services [16]. Along the same

lines is the triple play network architecture, a user-centric

approach in which customers are confronted with a variety of

applications like Voice over IP, IPTV and Video on Demand

and high speed internet services [18].

These emerging network services require enhanced and

diverse quality-of-service (QoS) guarantees. Thus, the devel-

opment of scheduling algorithms that provide differentiated

service guarantees to various classes of trafﬁc is of great

interest. Such generalized schedulers and on demand routers

should dynamically allocate the desired network resources

(e.g., bandwidth, buffer size, CPU capacity) since a static

allocation may result in signiﬁcant under-utilization.

In this paper, we propose a service pricing model that

ensures efﬁcient allocation of resources in a dynamic manner

in the aforementioned multiservice networks. The scheme

requires close on-line monitoring of the incoming trafﬁc. We

Fig. 1. Depiction of the proposed framework: trafﬁc is divided into two

categories; deterministic constraint and ﬂexible constraint services. The system

allocates the excess resources to the latter set.

assume a Fractional Brownian Motion trafﬁc model, because

of its ability to adequately capture characteristics of real

network traces, such as self-similarity and the presence of

heavy tailed marginal distributions [17].

Optimal resource allocation is also studied in [1], [9], [20].

Speciﬁcally, Peng et al. propose a measurement-based re-

source allocation scheme based on a linear pricing model and

average queue delay guarantees. This scheme has the disadvan-

tage of not being scalable to large number of service classes.

Moreover, average queue delay is not always an appropriate

QoS constraint. In [9], they perform a maximization over a

utility function provided from the network users and resources

are shared based on the solution of that optimization problem.

In [20], the authors study the problem of resource allocation

with dynamic pricing in which the network administrator

controls the price of the resources that users demand; based

on the demand the prices are dynamically changed over

different time periods so as to maximize the revenue of the

administrator. Finally, measurement-based resource allocation

has also been studied in different contexts in [10]–[12].

The remainder of this paper is structured as follows. The

proposed modeling framework is described in Section II, while

the optimization problem for a single network element based

on a nonlinear pricing model is formulated in Section III.

Some numerical results illustrating the model’s performance

are presented in Section IV, while some concluding remarks

are drawn in Section V.

II. M

ODELING FRAMEWORK

The employed modeling framework was introduced in [1],

[2] and is depicted in Figure 1. In its present form it represents

a single network element, which can be either a traditional

network component, such as switch or a router, or a mod-

ern network “service center”, like IBM’s DataPower Service

Oriented appliances [3] or CISCO’s Application Oriented

Network (AON) message routing system [19].

It is assumed that the network element serves two cate-

gories of trafﬁc classes; deterministic delay-bound classes and

ﬂexible delay-bound ones. Due to the fact that deterministic

delay-bound classes have strict requirements, their service

level agreement (SLA) can be satisﬁed only by trafﬁc shaping

and admission control schemes [13], [14]. Thus, an amount of

resources is dedicated to them and these classes are excluded

from subsequent analysis. Examples of these inelastic classes

of service include teleconferencing, remote seminars, real-time

distributed computation/simulation and high-precision medical

imaging.

Therefore, the proposed system is responsible for optimally

allocating the excess resources to the remaining ﬂexible delay-

bound classes. These classes enter the Measurement Based

Optimal Resource Allocation (MBORA) system proposed in

[2] and shown in Figure 2. The MBORA system consists of a

measurement module, an optimization module and a resource

orchestrator module. The statistics of the arrival trafﬁc are

measured by the measurement module. It is assumed that

the trafﬁc can be accurately approximated by a Fractional

Brownian motion model, which can account for the burstiness

and long-range dependence observed in real trafﬁc traces. Such

a model can be fully described by the following parameters:

the Hurst parameter H, the mean arrival rate ¯α and the

variance σ. An algorithm for on-line measurement of these

parameters is discussed in [4].

The optimization module receives the trafﬁc characteristics

of each class and calculates the optimal allocation of resources

by solving the optimization problem discussed in Section III. It

should be noted that the optimization problem is solved only

when there is a signiﬁcant change in trafﬁc characteristics.

The optimal solution is fed to the resource orchestrator which

dynamically updates the allocation of resources for each

trafﬁc class and forwards the packets (or, more generally, the

messages, for example XML) toward their destination.

III. P

RICING MODEL AND OPTIMIZATION PROBLEM

FORMULATION

We start by introducing the pricing model, whose solution

yields the optimal allocation of resources to the network

service node we described in the previous section.

A. Non-Linear Pricing Model

Suppose that the node can provide K different types of

services. The proportions of these services to be allocated are

Fig. 2. The MBORA system: the optimization module receives input trafﬁc

measurements and calculates the optimal resource allocation.

denoted by φ =(φ

1

, ··· ,φ

K

). According to [7], the proﬁt of

a provider is the difference between the revenue r(φ) that is

obtained for providing these services, and the cost c(φ) that

incurs from producing them. The aim of this provider is to

maximize the proﬁt function

π = max

φ

{r(φ) −c(φ)} = max

φ

K

k=1

(r

k

(φ

k

) −c

k

(φ

k

)), (1)

subject to the feasibility constraints: φ

k

≥ 0,k =

1, ··· ,K,

k

φ

k

≤ 1.

The revenue is given by a linear function, while the cost by

a nonlinear one. Speciﬁcally, r

i

(φ

i

)=p

i

· φ

i

while the cost

function has the form c

i

(φ

i

)=b

i

·D

i

(φ

i

)·exp[β(D(φ

i

)−d

i

)].

The coefﬁcient p

i

corresponds to the price that the provider

charges for service i and the parameter b

i

is the amount

that the provider has to reimburse the users whenever the

SLAs are not met. A higher priority class u requires better

service than a lower one v and thus it is charged more (i.e.,

p

u

>p

v

and b

u

>b

v

). The parameter β controls the steepness

of the cost function, while D(φ

i

) denotes the value of the

performance metric experienced by users of service i and d

i

the target level under the SLA. Hence, if D(φ

i

) >d

i

the

users are not receiving adequate resources from the provider,

which would incur a cost, until the situation is rectiﬁed. This

function is monotone in D

i

(φ

i

) and is shown in Figure 3. The

steep increase in the cost observed beyond the desired by the

users SLA value of d

i

would force the provider to adjust the

allocation of resources (if possible), in order to satisfy the QoS

requirements and maximize proﬁt.

Probabilistic Delay Constraints: We employ stochastic de-

lay bounds as the metric for QoS considerations. Speciﬁcally,

we adopt the approach used in [5], [6], where trafﬁc is treated

as Long Range Dependent (LRD) and is characterized by the

Hurst parameter H, the mean ¯α and the variance σ.Itisshown

that the queue length at any given time t is bounded by a value

q

max

with probability >0 related to the desired QoS. It is

3.7 3.8 3.9 4 4.1 4.2 4.3 4.4

0

10

20

30

40

50

60

70

80

90

Delay D

i

(φ

i

)

Cost Funtion c

i

(φ

i

)

Cost Function, delay threshold d

i

=4

← threshold

Fig. 3. Our cost function. Notice that even a small increase of 2.5% above

the delay threshold yields an increase above 100% in the cost function. In

this case parameter β =10.

shown that for a speciﬁc class the following holds:

Pr(Q(t) >q

max

) ≈ (2)

and

q

max

=(C − ¯α)

H/(H−1)

(kσ)

1/(1−H)

H

H/(1−H)

(1 −H) (3)

where C can be interpreted as the resources (e.g., bandwidth)

dedicated to this particular class, is the required QoS and

k =

√

−2ln.

Thus, since the queue length and expected delay are related,

we have the following probabilistic delay bound:

Pr(D(t) >D

max

) ≈ (4)

and

D

max

=

(C − ¯α)

H/(H−1)

(kσ)

1/(1−H)

H

H/(1−H)

(1 − H)

C

(5)

This delay bound is used in the cost function.

B. Convex Optimization

Putting the proﬁt and cost components together, the

provider’s proﬁt problem becomes:

max

φ

{

k

i=1

p

i

φ

i

C −

k

i=1

b

i

D

i

(φ

i

)exp[β(D

i

(φ

i

) − d

i

)]} (6)

subject to the feasibility constraints previously described plus

the constraints φ

i

> ¯α

i

, i =1...k.

In the above expression, D

i

(φ

i

) is given from Eq. 5 by

substituting parameter C with φ

i

C, since we are dealing with

a network element with multiple input classes each of which

is allocated a portion φ

i

of the total C resources. Note also

that D

i

(φ

i

) is actually D

max,i

(φ

i

). In addition, note the last

constraint φ

i

> ¯α

i

: this should always stand true due to the

fact that whenever φ

i

≤ ¯α

i

we have Pr[Q(t) >q

max

]=1.

This implies that we are in an unstable case and the queue

would never be able to accommodate the incoming trafﬁc. This

constraint is introduced, in order to avoid for the network to

operate in this undesirable from a QoS perspective regime.

In the over-provisioned case (i.e., when

i

¯α

i

≤ 1), the

solution of the optimization problem exists since we deal

with a convex optimization problem. An outline of the proof

follows.

It is shown next that the proﬁt function is convex, which

combined with the convexity of the feasibility set guarantees

the existence of a global maximum (maybe at a boundary

point). It is easy to see that the feasibility set is a polyhedronin

R

K

+

and hence convex. We establish next the concavity of the

proﬁt function. Some algebra shows that the second derivative

of D

i

(φ

i

) is positive in the feasibility set and therefore D

i

(φ

i

)

is convex on R. The cost function c

i

(D

i

) is convex and

nondecreasing (see Figure 3). The cost function is then convex

as a composition of a convex and nondecreasing function with

a convex function. Therefore, c

i

(D

i

(φ

i

)) is convex on R.

Further,

c

i

(φ

i

) is convex on R

K

+

, since each component of

the sum is convex. Finally,

p

i

φ

i

C is linear, which together

with the previous result establishes the concavity of the proﬁt

function.

The optimal solution can then be found using standard

algorithms, like the Newton method and its variations. Note

that we are dealing with a constrained optimization problem,

which implies that appropriate methods need to be considered

(e.g., a penalty or barrier function to relax the constraints [8]).

Moreover, we can take advantage of the Karush-Kuhn-Tucker

(KKT) conditions that are necessary and sufﬁcient for primal-

dual optimality of a convex optimization problem. The primal

problem is translated to an equivalent, but easier to solve dual

problem. The primal problem has solution φ

∗

, while its dual

has solution (φ

∗

, λ

∗

). The result that KKT conditions give is

that the optimal solution lies on the hyperplane

k

i=1

φ

i

=1.

The proof and the KKT conditions are given next:

k

i=1

φ

∗

i

− 1 ≤ 0 (7a)

−φ

∗

i

+¯α

i

< 0, ∀i ∈{1 ...k} (7b)

λ

∗

i

≥ 0, ∀i ∈{1 ...k} (7c)

λ

∗

1

(

k

i=1

φ

∗

i

− 1) = 0 (7d)

λ

∗

i+1

(−φ

∗

i

+¯α

i

)=0, ∀i ∈{1 ...k} (7e)

∂π(φ)

∂φ

i

− (λ

∗

1

− λ

∗

i+1

), ∀i ∈{1 ...k} (7f)

Since

∂π(φ)

∂φ

i

> 0 and thus from (7f) it can be seen that

λ

∗

1

− λ

∗

i+1

> 0. From the second complementary slackness

condition (7e) we obtain λ

∗

i+1

=0, since the constraint

−φ

∗

i

+¯α

i

< 0 always holds. Thus, we conclude that λ

∗

1

>

0, which together with the other complementary slackness

condition (7d) gives that

k

i=1

φ

∗

i

− 1=0 (8)

Thus, we can solve for φ

k

(or any other φ

i

) and convert

our constraint problem over k variables to a constrained one

over k − 1 variables, having eliminated the ﬁrst requirement.

The latter combined with a Sequential Quadratic Programming

(SQP) algorithm implemented in Matlab releases the other

constraints. The SQP algorithm is a generalization of Newton’s

method for unconstrained optimization. SQP ﬁnds the next

step away from the current iterate after minimizing a quadratic

approximation of the initial problem. For further details about

SQP and its Matlab implementation the reader is referred

to [15]. The advantage of using an iterative algorithm (like

SQP) is that the proposed framework is scalable to any number

of classes.

Remark: For the under-provisioned case, the problem is not

particularly interesting, since the QoS constraints would be

surely violated. Hence, the service provider would allocate

resources according to the average trafﬁc intensities; further,

it is easy to see that the operation would not be proﬁtable.

Hence, this regime is not studied in this paper.

IV. P

ERFORMANCE EVALUATION

In this section, we evaluate our pricing model in the over-

provisioned case with a numerical case study. It is assumed

that there are two types of service classes and the proﬁt

function becomes:

π(φ

1

,φ

2

)=p

1

φ

1

C + p

2

φ

2

C

− b

1

D

1

(φ

1

)e

β(D

1

(φ

1

)−d

1

)

− b

2

D

2

(φ

2

)e

β(D

2

(φ

2

)−d

2

)

(9)

where

D

i

(φ

i

)=

(φ

i

− ¯α

i

)

H

i

H

i

−1

(kσ

i

)

1

1−H

i

H

H

i

1−H

i

i

(1 −H

i

)

φ

i

,i=1, 2

(10)

Hence, we have to solve the optimization problem:

max

φ

π(φ

1

,φ

2

) subject to

φ

1

+ φ

2

=1

φ

1

> ¯α

1

φ

2

> ¯α

2

(11)

The parameters of the proﬁt function used in the study

are shown in Table I. Its concavity over both arguments is

shown graphically in Figure 4, while over the ﬁrst argument

in Figure 5, by substituting φ

2

=1− φ

1

.

In Tables II, III the optimal solution is shown when the

arrival rate and the price coefﬁcients are varied. In Table

II it can be seen that with equal arrival rates and all the

other parameters the same, the optimal solution allocates the

resource equally amongst the two classes, as expected. On the

other hand, the class with the higher arrival rate is allocated a

larger portion of the resources, especially if the system is not

too stressed (see rows 2 and 3 in the Table). In that situation the

proﬁt does not also ﬂuctuate much. Finally, when the system

becomes stressed (last row in the Table) the class with higher

TABLE I

P

ARAMETERS FOR TWO DIFFERENT CLASSES

Class 1 Class 2

p (cents/Mbps) 1 1

b (cents/ms) 0.1 0.1

d (in delay units) 0.01 0.01

QoS(= ) 10

−6

10

−6

¯α (normalized to C) 0.2 0.2

σ (normalized to C) 0.01 0.01

H 0.70 0.70

0.2

0.3

0.4

0.5

0.6

0.2

0.3

0.4

0.5

0.6

0.7

4

6

8

10

12

14

φ

1

X: 0.5

Y: 0.5

Z: 9.957

Utility function as a function of φ

1

, φ

2

φ

2

Fig. 4. Concavity of our Utility Function as a function of φ

1

, φ

2

arrival rate gets a higher proportion, but the overall proﬁt for

the provider decreases substantially, since violations of the

SLA occur more often and therefore a large cost is incurred.

In Table III, the price coefﬁcient varies, while all other

parameters are held ﬁxed (see Table I). Again, with equal

prices we obtain equal allocations, while the allocation of

resources exhibits a strong sensitivity to the price ratio p

1

/p

2

.

V. C

ONCLUSION

In this paper, we have studied a pricing scheme for next gen-

eration multiservice networks. An optimization problem based

on a nonlinear pricing model was formulated, whose solution

yields the optimal resource allocation in a network/service

node, given the QoS requirements of each service class that

the network element serves. Our non-linear pricing model

responds well to changes of the characteristics in the input

TABLE II

C

HANGING THE ARRIVAL RATES ¯α

i

(¯α

1

, ¯α

2

) (φ

∗

1

,φ

∗

2

) π(φ

∗

1

,φ

∗

2

)

(0.2, 0.2) (0.5, 0.5) 9.96

(0.3, 0.2) (0.5421, 0.4579) 9.93

(0.4, 0.2) (0.5873, 0.4127) 9.89

(0.4, 0.5) (0.4516, 0.5484) 6.72

0.3 0.4 0.5 0.6 0.7

9

9.1

9.2

9.3

9.4

9.5

9.6

9.7

9.8

9.9

10

φ

1

Utility Function

Utility function as a function of φ

1

Fig. 5. Concavity of our Utility Function as a function of φ

1

TABLE III

C

HANGING THE PRICING FACTOR p

i

(p

1

,p

2

) (φ

∗

1

,φ

∗

2

) π(φ

∗

1

,φ

∗

2

)

(1, 1) (0.5, 0.5) 9.96

(2, 1) (0.6917, 0.3083) 16.52

(4, 1) (0.7183, 0.2817) 30.69

(4, 4) (0.5, 0.5) 39.96

(1, 2) (0.3083, 0.6917) 16.52

(1.5,6) (0.2739, 0.7261) 46.52

(4, 8) (0.276, 0.724) 67.90

trafﬁc, pricing parameters and QoS requirements. Further,

the resulting convex optimization problem can easily and

efﬁciently be solved using standard iterative methods and

hence the proposed modeling framework approach is scalable

to any number of service classes.

A

CKNOWLEDGMENT

The authors would like to thank Dr. Peng Xu for useful

discussions on the subject. The work of GM was supported in

part by NSF grants CCR-0325571 and DMS- 0505535.

R

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