On Modeling Internet QoS Provisioning from

On Modeling Internet QoS Provisioning from Economic Models* Yufeng Wang 1, Wendong Wang 2 1

Communications Engineering Department, Nanjing University of Posts and Telecommunications (NUPT), Nanjing 210000, CHINA 2

State Key Laboratory of Networking & Switching Technology, Beijing University of Posts and Telecommunications (BUPT),Beijing 100876,CHINA

Abstract. The modeling of Internet quality of service (QoS) provisioning is a multidisciplinary research subject. From the viewpoint of game theory, we propose a model that combines QoS index with price factor. We use the MultiNomial Logit (MNL) to model the choice behaviors of users. Each service class is considered an independent competing entity, which aims at maximizing its own utility. Based on noncooperative game, we demonstrate the existence and uniqueness of equilibria between QoS levels and prices among various service classes, and demonstrate the properties of equilibria. We also verified these results via numerical analysis.

1 Introduction With the increasing development of Internet applications, there exists a demand for multiple QoS levels on the Internet. Over the years, network engineers have developed a number of QoS architectures and mechanisms. A core issue that concerns network users, service providers, and network engineers is the constantly changing traffic behavior. Such behavior depends on the aggregated traffic load, which is the result of many users’ individual decisions on how to make use of the network. These decisions are affected by the incentives that users face. Thus it is important to bring price into network design and to integrate the QoS levels with price factor when considering Internet QoS provisioning. Recently, many researchers studied the provisioning of Internet QoS based on economic mechanisms. The M3I (Market-Managed Multi-services Internet) Project funded by the European Union is aimed at studying Internet resource management system. More specifically, it was proposed that different service classes are charged differently, allowing network users to select from multiple prices and QoS levels [1]. *

Research supported by the NSFC Grants 60472067 and 2003CB314806, and State Key Laboratory of Networking and Switching Technology, BeiJing University of Posts and Telecommunications (BUPT).

In [2], the pricing issue when providing multiple service classes in telecommunications networks was considered. Each service class has different QoS requirements, and is characterized by a (service type, source-destination pair) tuple, where the service type can be voice, video, or data, etc. The optimal pricing problem is formulated as a nonlinear integer expected revenue optimization problem and is solved for prices and resource allocations that are necessary to provide connections with guaranteed QoS. Another body of research has considered prices as an endogenous variable, which is determined as a function of the degree of saturation inside the network. Typically in this approach, the price is called a shadow price; it can be considered the Lagrange multiplier of inequality constraints such as capacity restrictions; it is used to achieve the equilibrium between link bandwidth demand and supply [3][4]. However, little work has been reported on systematically modeling the equilibria between QoS levels and prices on the Internet. Recently, network game has undergone significant development [5]. In [6], the equilibria between QoS levels and prices of several service network providers have been investigated based on linear demand function. The assumption of linear demand function makes the problem easy to solve, but does not have scientific foundation. In this paper, we explicitly model customer choice behavior using a MultiNomial Logit (MNL) model, which is a form of random utility model. MNL is both a theoretically sound and an empirically well-tested model of customer choice behavior; it has been widely used to forecast traffic demand in airlines [7]. It constitutes a natural candidate for a choice-based optimization model. Base on the MNL model, we investigated the equilibria between QoS levels and prices; explicitly considered the effect of demand substitute, that is, the demand for certain traffic class is described as a function of prices and QoS levels of all traffic classes supported by the network. Our method was inspired by [8], in which the authors investigated the equilibria between all retailers’ prices and service levels. We applied their idea to the problem of Internet QoS provisioning. In our model, we do not take into account network topology, but rather model each service class as a single entity. In other words, the price and QoS provided by each service class do not depend on the source, destination, or distance, etc. that underlay each user request. On the other hand, we use proper queuing model to describe the amount of resource requested by service classes. This paper is organized as follows: the service model and general assumptions are described in section 2, which are used to prove several theorems in later sections. In section 3, we systematically characterize the equilibrium behaviors of an Internet QoS provisioning model under two possible scenarios. (1) Price-competition only: in this case, we assume that the QoS levels of all traffic class are exogenously chosen and characterize how the price equilibria vary with the chosen QoS levels. (2) Simultaneous price and QoS level competition: in this case, each of the traffic classes simultaneously chooses a QoS level and a price. We provide three theorems about existence and uniqueness of equilibria, and demonstrate the properties of equilibria. A numerical application of these theorems is given in section 4. Finally, we briefly conclude our paper in section 5.

2 Service model and general assumptions We first present the notations that are used in this paper: p=(p1,…,pN): the vector of prices of service classes, pi denotes price of service class i; We denote a vector using bold face, e.g. p and denote its i-th component using pi ; f=(f1,…,fN): the vector of QoS levels of service classes, fi denotes the QoS level of service class i;

d i ( p, f ) : R+N + N → R+ :demand for service class i, which depends on the entire price vector and the entire QoS vector;

µi(fi,di): the amount of resource required by the service class i in order to provide the QoS level fi for demand di ; R i(fi,di): the cost of service class i to provide the QoS level fi for demand di ; Ui(p,f): the net revenue of service class i (the utility function of service class i); Let us consider a network in which there are N service classes, the set of which is denoted by I={1,2,…,N}. Each service class has two parameters with respect to the N +N

service it offers: ( S , I ) ∈ R+ service: d i : R

N +N +

.Each service class i experiences a demand for its

→ R+ ,which is a function of a price vector and a QoS vector.

In this paper, we explicitly consider the effect of demand substitute, that is, the demand function of service class i is dependent not only on its own parameters, pi and fi , but also on the prices and QoS levels offered by its competitor classes. We make the following assumptions regarding the shape of demand function: (I)

For i=1,…,N,

∂d i ( S , I ) ∂d i ( S , I ) ≤ 0, ≥ 0; ∂p i ∂f i

(II)

For all j ≠ i ,

∂d i ( S , I ) ∂d ( S , I ) ≥ 0, i ≤ 0; ∂p j ∂f j

∑

∂d i < 0 (or j =1 ∂p j N

(III) For all i=1,…,N,

∑ ∂∂dp N

j =1

j

< 0 ).

i

These assumptions are intuitive: assumption (I) denotes that the demand for a service class decreases with its own price, and increases with its own QoS level; assumption (II) represents the effect of demand substitute among all service classes, that is, if one service class increases its own price (or QoS level), this will result in an increase (or decrease) in the demand of his competitors; assumption (III) denotes no service

class’s sales are expected to increase under a uniform price increase (similarly, aggregate sales usually decrease if one of the service classes increases its price.) The service model that we study in this paper is shown in Fig. 1: service class i purchases resource µi(fi,di) from the network and offers service to demand di(p,f). The utility function of service class i is represented as the profit of service class i, which equals the revenue obtained from providing service to customers minus the cost of purchasing required resources.

d 1 ( p, f

µ1 ( f , d 1 1)

)

µ2 ( f 2 , d 2 )

d 2 ( p, f ) M

M

M

d N ( p, f

dN)

µN( fN,

)

Fig. 1. Illustration of mutli-service model

Assume the network owner charges each service class a cost per unit of resource requested, v, and the amount of resource µi requested by service class i is dependent on the demand it experiences and on the QoS it wishes to offer ( the higher the demand and the better the QoS, the higher µi will be). For example, if let QoS level fi denote the probability of successful transmission, for M/M/1 queuing model, fi=1−di/µi, then µi=di/(1− fi )Thus, in order to provide the QoS level fi to demand di , the fee paid by service class i can be written as: Ri(fi,di)=vµi(fi,di). Assume that the resource capacity µi required by service class i has the following form:µi(fi,di)=digi(fi), where gi is a convex, two-differentiable and increasing function. This assumption is intuitive: the operational costs increase convexly with the service level. The net revenue of service class i can be represented as: pidi(p,f)−Ri(fi,di), which is defined as the utility function of service class i. We have:

U i ( S , I ) = ( pi − vg i ( f i ))d i ( S , I ) The

strategy

space,

5 i = {( p i , f i ) : 0 ≤ p

gˆ i ( f i ) = vg i ( f i ) .

min i

Ri ,

of

service

= gˆ i ( f i ) ≤ p i ≤ p

class max i

i,

;0 ≤ f i

(1) is

min

given

≤ fi ≤ fi

as max

}

follows: Where

max

Assumption A: pi

= max{ p i0 ,1 + vα i g i ( f i )} , where pi0 is the price that can

make demand for service class i negligible, and αi denotes certain constant value. In theory, the demand functions themselves can take on a number of forms, and each has its own consequences upon the resulting equilibria. In this paper, we use MNL to model customer choice behavior, which will be defined further in the following section. Many marketing models characterize the market shares obtained by competing service classes via a vector of attraction value a=(a1,…,aN). The market share achieved by a given service class i is given by its attraction value divided by the industry’s total value, i.e., mi = ai

∑

N j =0

a j , in which a0 is the value of no-

purchase option (in this paper, we let a0=0). Attraction models are among the most commonly used market share models, both in empirical studies and in theoretical models. Assume that the attraction value of service class i is given by a general, twice differentiable function of its price and service level, i.e., ai=ai(pi,fi ), with

∂a ∂ai ≤ 0, i ≥ 0 . ∂pi ∂f i

Most attraction models assume a specific structure. We explicitly model consumer choice behavior using a MultiNomial Logit (MNL) model. Generally, In MNL, ai is represented as the linear function of various attributes (those attributes represent the attraction of service class i in various aspects). Below we pay special attention to the following generalization equation: ai(pi,fi)=exp{bi(fi)-αipi}
where αi>0. Assumption B: bi (fi ) is twice differentiable, increasing and concave. This permits us to represent settings where the marginal increase in a service attraction value due to an increase in its QoS level, is non-negative but decreasing in QoS level. Alternatively, if a service class wants to maintain a given attraction value, it needs to compensate for a price increase with ever larger increases in its service level.

3 Modeling QoS Provisioning based on game theory The goal of this paper is to investigate the equilibria of prices and QoS levels of N service classes, and to provide mathematical foundation to Internet QoS provisioning. We systematically characterize the equilibrium behavior under two possible scenarios. (1) The equilibrium of price-competition only: in this case, we assume that service levels of classes are exogenously chosen, and characterize how the price equilibrium and price strategy vary with the chosen service levels. (2) The equilibrium of simultaneous price and service level competition: in this case, service classes simultaneously choose service levels and price strategies. x = log x . In the rest of the paper, we let ~

3.1 Price-based Nash Equilibrium Let Ui(p,f) denote the utility function of service class i, p denote the price vector of service classes, and the vector of QoS levels, f, be fixed at some predetermined value,

Iˆ . The price-based Nash equilibrium in p at Iˆ is the vector p* that solves the following system for all i

ęI. U ( S ,Iˆ ) = max U ( p ,K, p *

i

i

* 1

* i −1

, p i , p i*+1 ,

K, p

* N

, Iˆ )


This equilibrium corresponds to price equilibrium for a fixed QoS vector Iˆ . Theorem 1: Adopting generalized MNL model, under the conditions of assumptions (III) and (A), the game of price has a unique equilibrium p*(f). The price value in equilibrium is the solution to the following equation: ~ (2) ∂U i d 1 = − α i (1 − i ) = 0 ∂p i p i − vg i ( f i ) M For paper space, we omit the detailed proof, which can be found in [10].

3.2 Price and QoS-based Equilibrium Let Ui(p,f) be the utility function of service class i, when the pair of parameter vectors that are set simultaneously by all service classes is given by (p,f), the price and Qos-based Nash equilibrium is the vector (p*,f*) that solves the following system for all i I.

ę

K

K

K

K

Ui (S ,Iˆ) = maxUi ( p1* , , pi*−1, pi , pi*+1, , p*N , f1* , , f i*−1 , fi , f i*+1, , f N* ) *

Assumption C:

lim g i′ ( f i ) = +∞ This assumption holds for a large number of

f i → f i max

queue systems [6]. Theorem 2: The simultaneous price and QoS-based game has a Nash equilibrium *

(p*,f*) and f i is the solution to the following equations:

vg i′ ( f i ) = bi′ ( f i ) α i , if

f i = fi *

min

,

1 g i′ ( f i min ) ; ≤ min bi′( f i ) vα i

g i′ ( f i min ) 1 if . > min bi′ ( f i ) vα i

pi* is the solution to (2) under the result of f i * .

(3)

(4)

Interesting readers can find the detailed proof in [10].

3.3 Properties of Equilibria In the previous subsection, we demonstrate that there exist equilibria of the QoS and price game. In this subsection, we investigate the properties of these equilibria. We explicitly consider the effect of demand substitute, so a change in one of the service classes’ QoS level will result in an increase or decrease in each of the equilibrium prices. Let

δ i ( f i ) = vg i′ ( f i ) − bi′( f i ) α i

0

and f i denotes the QoS levels obtained

from (3) and (4). Theorem 3: In the price-based game of Internet QoS provisioning, the equilibrium *

price of service class i, pi is strictly increasing in f i ; the equilibrium price of service class j (j≠i), p j is strictly decreasing in f i
for f i < f i , and strictly increasing *

0

in f i for f i > f i . 0

For paper space, we omit the detailed proof. Interesting readers can find the detailed proof in [10]. In the subsection, a numerical example is provided to graphically demonstrate the obtained theorems above.

4 Numerical Analysis We suppose in this section that the measure defining the QoS, fi , corresponds to some function of the loss probability, f i = (1 − Ploss ) i

1

s

i

, where Ploss is the loss

probability of service class i, which can be denoted as Ploss = G ( ρ i ) , where i

ρ i = d i µi

is the traffic intensity, and s ≥ 1 is a scaling coefficient that adjusts the

relative importance of the QoS parameter with respect to price. For s>1, the QoS increases as a concave function of its parameter, that is, we allow for decreasing rates of return on the quality of service provide. We consider the scenario when there are two service classes and use M/M/1/2 queuing system to model the loss probability. This model can be considered a good approximation for the loss probability when these service classes are used at access network, and the bottleneck (in terms of loss) is in the shared input buffering area to the network. Let s=2, v=2, a1=exp(20f1−2p1), a2=exp(12f2−2p2), at equilibrium, we have f1=0.82, f2=0.74, p1=4.98, p2=2.02. The relation between the QoS levels and prices of the two service classes are showed in following figures.

6 5.8

price of traffic class 1

5.6 5.4 5.2 5 4.8 4.6 4.4 0.76

0.78

0.8 0.82 0.84 0.86 QoS index of traffic class 1

0.88

0.9

Fig. 2. The relation between QoS index of traffic class 1 and price of traffic class 1 2.036 2.034

price of traffic class 2

2.032 2.03 2.028 2.026 2.024 2.022 2.02 2.018 0.76

0.78

0.8 0.82 0.84 0.86 QoS index of traffic class 1

0.88

0.9

Fig. 3. The relation between QoS index of traffic class 1 and price of traffic class 2

It can be observed from Fig. 2 (Fig. 3) that, when the QoS level of traffic class 2 is fixed in equilibrium value (i.e. f2=0.74), the relation between QoS level of traffic class 1 and price of traffic class 1 (traffic class 2). From Fig. 2, we found that the price of traffic class 1 increases with its own QoS level. While for f10.82, price of traffic class 2 increases with QoS level of traffic class 1. the reason for this phenomenon is that, when the QoS level of traffic class 1 exceeds the equilibrium value determined by theorem 1, the effect of traffic class 1’s price increase (making the demand for traffic class 1 decrease) surpasses the effect of traffic class 1’s QoS level increase (making the demand for traffic class 1 increase), this results in the relatively increased demand for traffic class 2, thus the price of traffic class 2 increases. Those results illustrated in Fig. 2 and Fig. 3 are consistent with the conclusion drawn in theorem 3.

3

price of traffic class 2

2.8 2.6 2.4 2.2 2 1.8 1.6 0.65

0.7 0.75 0.8 QoS index of traffic class 2

0.85

Fig. 4. The relation between QoS index of traffic class 2 and price of traffic class 2 5.12

5.1

price of traffic class 1

5.08

5.06

5.04

5.02

5

4.98 0.66

0.68

0.7

0.72 0.74 0.76 0.78 QoS index of traffic class 2

0.8

0.82

0.84

Fig. 5. The relation between QoS index of traffic class 2 and price of traffic class 1

From Fig. 4 and Fig. 5, we observe the similar results with Fig. 2 and Fig. 3. We briefly describe them as follows: when QoS level of traffic class 1 is fixed in the equilibrium value determined by theorem 1 (i.e. f1=0.82), the price of traffic class 2 increases with its own QoS level. For f20.74, price of traffic class 1 increases with the QoS level of traffic class 2. These results also verify the theorems obtained in section 3.

5 Conclusion In this paper, we study the modeling of Internet QoS provisioning from the viewpoint of game theory, and obtain the equilibria between prices and QoS levels when there are multiple traffic classes. The motivation of this research is that price factors can provide proper incentive for customers to use network resources rationally, so we can introduce the price factor into the field of network engineering as certain control signal, and simultaneously consider QoS level and price factor in Internet QoS pro-

visioning. We adopt MNL to model the customers’ choice behavior. MNL is both a theoretically sound and empirically well-tested model of consumer choice behavior; it has been widely used to forecast traffic demand. It is a natural candidate for a choice-based optimization model. We assume that the service classes are independent, competitive entities which try to maximize their own utilities (In this paper, the utility function of service class is defined as the revenue obtained from providing the service to the customers minus the cost paid to network owner for purchasing network resources). Based on noncooperative game theory, we prove the existence and uniqueness of equilibria between prices and QoS levels among multiple service classes, and demonstrate the properties of equilibria. In our analysis, the attraction functions of service classes are important; they determine the equilibrium values between prices and QoS levels. In practice, user consumption survey and the Maximum Likelihood Estimate (MLE) can be used to obtain the MNL model. Those methods have been proved to be robust in practice. In conclusion, we have made an initial investigation on modeling Internet QoS provisioning through integrating QoS levels with the price factor. We believe that our work is useful to future research into this area.

References [1]

J. Altmann, H. Daanen, H. Oliver etc., “How to market-manage a QoS network,” In Proc. of IEEE INFOCOM 2003. [2] Keon, G. Anandalingam, “Optimal pricing for multiple services in telecommunications networks offering Quality of Service guarantees,” IEEE/ACM Transactions on Networking, February 2003. [3] F.P. Kelly, A. Maulloo, D. Tan, “Rate control for communication networks: shadow prices, proportional fairness and stability,” J. Oper. Res. Soc., 1998, vol. 49, no. 3, pp. 237-252. [4] R.J. La, V. Anantharam, “Utility-based rate control in the Internet for elastic traffic,” IEEE Transactions on Networking, 2002, vol. 10, no. 2, pp. 272-286. [5] X.-R. Cao, H.-X. Shen, R. Milito, and P. Wirth, “Internet Pricing with a Game Theoretical Approach: Concepts and Examples,” IEEE/ACM Transactions on Networking, 2002, vol. 10, no. 2, pp. 208-216. [6] R. El Azouzi E. Altman and L. Wynter , “Telecommunications Network Equilibrium with Price and Quality-of-Service Characteristics,” Proc. of ITC, Berlin, Sept 2003. [7] K. Talluri, G.V. Ryzin, “A Discrete Choice Model of Yield Management”, unpublished, Available at http://www1.gsb.columbia. edu/faculty/cmaglaras/B9801-001/. [8] F. Bernstein, A. Federgruen, “A general equilibrium model for industries with price and service competition,” unpublished, Available at http://faculty.fuqua.duke.edu/~fernando/bio/. [9] P. Milgrom, J. Roberts, “Rationalizability, learning and equilibrium in games with strategic complementarities,” Econometirca, 1990, vol. 58, pp. 1255~1277. [10] Yufeng Wang, “Study on game theory in mobile Internet”, Technical report, 2004.