Ann. Telecommun.DOI 10.1007/s12243-009-0149-3

An economic model for pricing tiered network services

Qian Lv · George N. Rouskas

Received: 30 October 2008 / Accepted: 24 November 2009© Institut TELECOM and Springer-Verlag 2009

Abstract We consider networks offering tiered servicesand corresponding price structures, a model that hasbecome prevalent in practice. We develop an economicmodel for such networks and make contributions intwo important areas. First, we formulate the problemof selecting the service tiers from three perspectives:one that considers the users’ interests only, one thatconsiders only the service provider’s interests, and onethat considers both simultaneously, i.e., the interests ofsociety as a whole. We also present an approximateyet accurate and efficient solution approach for tacklingthese nonlinear programming problems. Given the setof (near-) optimal service tiers, we then employ game-theoretic techniques to find an optimal price for eachservice tier that strikes a balance between the con-flicting objectives of users and service provider. Thiswork provides a theoretical framework for reasoningabout and pricing Internet tiered services, as well asa practical toolset for network providers to developcustomized menus of service offerings. Our results alsoindicate that tiering solutions currently adopted by

This work was supported by the NSF under grantCNS-0434975.

Q. Lv · G. N. Rouskas (B)Department of Computer Science,North Carolina State University,Raleigh, NC 27695-8206, USAe-mail: rouskas@ncsu.edu

Q. Lve-mail: qlv@ncsu.edu

ISPs perform poorly both for the providers and societyoverall.

Keywords Tiered services · Price structure ·Economic model

1 Introduction

Internet service providers (ISPs) have introduced sev-eral forms of a tiered service, in which users may se-lect from a small set of tiers that offer progressivelyhigher levels of service with a corresponding increase inprice. Multitiered price systems are prevalent for bothbusiness and residential Internet access, and have beenemployed in various forms regardless of whether theunderlying pricing scheme is capacity-based (in whichthe subscription fee is determined solely by the user’saccess speed) or usage-sensitive (in which price is afunction of the actual bytes transferred over a certaintime period, usually 1 month).

If designed and applied appropriately, multitieredpricing schemes have the potential to be a catalyst forInternet service innovation and penetration. On theprovider side, tiered structures can be an effective toolfor ISPs to optimize and specialize their offerings soas to capitalize on the increasing sophistication andrequirements of various segments of Internet users,as well as to differentiate themselves from the com-petition. On the user side, tiered pricing is likely tospur adoption by providing a wide menu of customizedservices from which users may select based on needsand affordability.

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To realize this potential, it is crucial that both the ser-vice tiers and the corresponding prices be determinedin a manner that simultaneously takes into account the(usually conflicting) objectives of users and providers.In current practice, however, there is considerable lackof transparency in how ISPs set their tiered price struc-tures, and it is unclear whether the perspective of usersis even considered in the process. For instance, certainservice tiers for business Internet access are based onthe bandwidth hierarchy of the underlying networkinfrastructure (e.g., DS-1, OC-3, etc.). While this isa natural arrangement for the service provider, it isunlikely that hierarchical rates designed decades agofor voice traffic would be a good match for today’sbusiness data applications. The ADSL tiers availablefrom various providers, on the other hand, appear tohave been determined in some ad-hoc manner; similarobservations apply to the usage-sensitive tiered pricestructure employed by one cable ISP for a recent pilotprogram [6], which sets the tiers at 5, 10, 20, and 40 GBof monthly download traffic. While tier values thatare round numbers may be an appropriate choice formarketing purposes, the relationship between these ex-ponentially increasing levels of service (and their price)and the usage patterns (and willingness or ability topay) of the population of potential subscribers is opento debate.

In this work, we develop an economic model fornetworks with a tiered service and price structure andmake contributions in two important areas. First, wepresent nonlinear programming formulations for theproblem of selecting the service tiers from three per-spectives: one that considers the users’ interests only,one that considers only the service provider’s interests,and one that considers both simultaneously, i.e., the in-terests of society as a whole. We also devise an efficientsolution methodology by developing a dynamic pro-gramming algorithm to optimally solve an approximateformulation of the original service tier selection prob-lem. Given the set of (near-) optimal service tiers, wethen employ game-theoretic techniques based on Nashbargaining to find an optimal price for each service tierthat strikes a balance between the conflicting objectivesof users and service provider. Our work provides atheoretical framework for reasoning about and pricingInternet tiered services, as well as a practical toolsetfor network providers to develop customized menus ofservice offerings that cater to user needs while ensuringthat both parties are satisfied.

The rest of the paper is organized as follows. InSection 2, we describe the tiered-service network weconsider in this study, along with related applications.

In Section 3, we introduce an economic model fortiered-service networks that takes into account theuser’s perspective, the provider’s perspective, or both.We also formulate and solve the corresponding prob-lems for selecting the set of service tiers optimally. InSection 4, we use Nash bargaining theory to determinean optimal price for each of the service tiers. Wepresent numerical results in Section 5, and we pointto extensions of our work in Section 6. We discussrelated work in Section 7, and we conclude the paperin Section 8.

2 Tiered-service networks

We consider a network that offers a service character-ized by a single parameter, e.g., the bandwidth of theuser’s access link or the amount of traffic generated bythe user, and charges users on the basis of the amount ofservice they receive. Users may request any amount ofservice depending on their needs and their willingnessor ability to pay the corresponding service fee. Weassume that the distribution of the size x of user servicerequests is known; such a distribution may be obtainedempirically or extrapolated from observed user behav-ior and application requirements. Let f (x) and F(x) bethe probability density function (pdf) and cumulativedistribution function (cdf), respectively, representingthe population of user requests. The pdf and cdf aredefined in the interval [xmin, xmax], where xmin and xmax

correspond to the minimum and maximum, respec-tively, amount of service requested by any user.

The network offers K levels (tiers) of service, wheretypically, K is a small integer, much smaller than thenumber N of network users (i.e., K � N). We defineZ =< z1, z2, · · · , zK > as the vector of service tiersoffered by the network provider; without loss of gen-erality, we assume that the service tiers are distinct andare labeled such that z1 < z2 < · · · < zK. For notationalconvenience, we also define the “null” service tierz0 = 0.

With tiered service, a user with service requestx, xmin ≤ x ≤ xmax subscribes to service tier z j such thatz j−1 < x ≤ z j. Figure 1 shows a sample mapping ofservice requests to a vector of K = 6 service tiers, underwhich all users with requests x ∈ (z j−1, z j] subscribe toservice tier z j, j = 1, · · · , K. Note also that, in order toaccommodate all user requests, the highest tier mustbe such that zK = xmax, an assumption we will makethroughout this work.

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z = x

f(x)

xmin

1 3 42 5 max6z z z z z

x

Fig. 1 Sample mapping of service requests to service tiers

The network provider incurs a cost for the service itprovides, and consequently, it will be inclined to selectthe service tiers, and the corresponding price to charge,so as to recoup its costs (and make a profit). On theother hand, each user subscribes to a service that is atleast as good as the one requested, but the additionalvalue, if any, that the user receives may be offset bythe higher cost of the service. Our aim is to applyeconomic theory to capture analytically these tradeoffs,and to develop techniques to select the service tiers andprices in a manner that accounts for both the users’ andproviders’ perspectives.

To develop an economic model for tiered-servicenetworks, we assume the existence of three non-decreasing functions of service x, as shown in Fig. 2.The utility function, U(x), is a measure of the value thatusers receive from the service, and it stands for theirwillingness to pay for the service. The cost function,

US$

y

x

Utility U(x)

Cost C(x)

Social surplus

Price P(x)User surplus

Service provider surplus

Service

Fig. 2 Utility, cost, and price functions

C(x), represents the cost incurred by the provider foroffering the service. Finally, the price function, P(x),represents the amount that the service provider chargesfor the service. Figure 2 shows that U(x) lies above P(x)

(otherwise, users would not be willing to pay for theservice), and in turn, P(x) lies above C(x) (otherwise,providers would not be inclined to offer the service).We make the reasonable assumption that utility, cost,and price are all expressed in the same units (e.g.,US$). Note that utility and cost typically depend onlyon the user and service provider, respectively, but thatprice is the result of market dynamics and the relativebargaining power of users and service providers.

In our model, we make two fundamental assump-tions. First, we consider a homogeneous user popula-tion in the sense that all users are characterized by thesame utility function U(x); the case of heterogeneoususers is the subject of ongoing research in our group.Second, we assume that any user with demand x thatis mapped to some tier z j will subscribe to this tier aslong as the price is less than its willingness to pay asrepresented by the utility value U(z j).

The tiered service model we consider in this paperarises naturally under both pricing schemes, capacity-based or usage-sensitive, that are prevalent for Internetservices [8].

Capacity-based pricing Capacity-based schemes relatepricing to usage by setting a price based on the band-width or speed of the user’s connection link. This isaccomplished by charging for the configuration (i.e.,bandwidth) of the connection, but not the actual bitssent or received. Capacity-based pricing is the prevail-ing pricing policy for residential broadband Internetaccess services. This scheme relates to our tiered servicemodel as follows: the service is characterized by theamount of access bandwidth; each of the service tiers,z1, · · · , zK, corresponds to a certain access speed, andusers are charged based on the tier to which they havesubscribed. Currently, the offered tiers are either tiedto the bandwidth hierarchy of the underlying networkinfrastructure (e.g., T1, T3, or higher for virtual privatenetworks) or are determined in some ad-hoc manner(e.g., the various ADSL tiers).

Usage-sensitive pricing Usage-sensitive pricing poli-cies charge users for the actual amount of traffic theygenerate. In current practice, ISPs charge a customer(e.g., a video-on-demand provider) based on theirtraffic volume using the 95-th percentile rule [7, 21].Specifically, the ISP measures the user’s traffic volumeover 5-min intervals during each billing period (e.g.,

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1 month), and charges the user based on the 95-th per-centile value among these measured values. Typically,ISPs have a tiered pricing structure [21] in which eachof the service tiers, z1, · · · , zK, corresponds to a certaintraffic volume, and higher tiers are mapped to higherprices. Such a structure can be mapped to our tieredservice model by considering a customer with a 95-th percentile value x such that z j−1 < x ≤ z j as having“subscribed” to tier z j and charging the customer ac-cordingly.1

3 Economic model for sizing of service tiers

In this section, we use concepts from economics todescribe the relationship between users and serviceproviders, and we propose optimization problems forselecting the service tiers. We also illustrate how tosolve these problems to obtain a set of (near-) optimaltiers. In the following section, we apply Nash bargainingtheory [12, 14] to determine optimal pricing strategiesfor this fixed (near-optimal) set of tiers.

Consider now the demand–supply relationship be-tween the users and network service providers. On theone hand, users want to maximize the utility they obtainfrom the service while keeping the fee they have to payto the service provider as low as possible; in economicterms, users want to maximize the user surplus [1, 4],defined as the difference between the utility they obtainfrom the service and the price they have to pay for it.On the other hand, the network providers’ objective isto charge a high fee so as to offset the cost of offeringthe service and make a profit; in other words, serviceproviders want to maximize the service provider sur-plus [1, 4], defined as the difference between price andcost. The concepts of user surplus and service providersurplus are illustrated in Fig. 2.

From the point of view of the society as a whole,it is preferable to maximize the overall social welfare,defined as the sum of the user surplus plus the providersurplus (see also Fig. 2). We will refer to the socialwelfare as social surplus [1, 4]. Once the maximumsocial surplus has been determined, the users and ser-vice providers may negotiate its division into user and

1Note that, with capacity-based pricing, the tier (e.g., accessspeed) to which a user subscribes does not change over time(except, for instance, when a user upgrades to a higher speed), butwith usage-sensitive pricing, a user may be charged according toa different tier every billing period, i.e., depending on the actualtraffic volume generated during each period. Nevertheless, thisdistinction does not affect the economic model we present in thenext section.

service provider surpluses through bargaining, as weexplain in the next section.

Let us define the user surplus Susr(x) = U(x) − P(x),the provider surplus Spr(x) = P(x) − C(x), and the so-cial surplus Ssoc(x) = U(x) − C(x). In the tiered-servicenetwork under consideration, the problems of maximiz-ing the surplus of users, service providers, or societyamount to appropriately selecting the set of service tiersto be offered, as we discuss next.

3.1 Maximization of expected surplus

Let S(x) be the surplus function (i.e., one ofSusr(x), Spr(x), or Ssoc(x), defined above), and supposefor the moment that the vector Z =< z1, · · · , zK =xmax > of K service tiers is given. In this case (referalso to Fig. 1), all users with requests in the interval2

(z j−1, z j] subscribe to tier z j, incurring a surplus ofS(z j), j = 1, · · · , K. Recalling that f (x) and F(x) arethe pdf and cdf, respectively, of user requests, the ex-pected surplus S̄(z1, · · · , zK) for the given service tiervector Z can be expressed as:

S̄(z1, · · · , zK) =K∑

j=1

(∫ z j

z j−1

S(z j) f (x)dx

)

=K∑

j=1

(S(z j)

∫ z j

z j−1

f (x)dx

)

=K∑

j=1

(S(z j)

(F(z j) − F(z j−1)

)). (1)

Consider now the problem of optimally selecting theservice tiers from the users’ point of view. Based onour earlier discussion, the objective of each networkuser is to maximize its surplus. Considering all the usersin the network as a whole, the objective is to selectthe set of service tiers so as to maximize the expectedaggregate user surplus, i.e., the weighted sum of theindividual user surpluses in expression 1 with Susr inplace of S(x). Similarly, the goal of the service provideris to maximize its expected aggregate surplus; whileconsidering the welfare of the society (i.e., both usersand providers), the objective would be to maximizethe expected aggregate social surplus. These last twoobjectives are obtained by using Spr(x) and Ssoc(x),respectively, in place of S(x) in Eq. 1.

2Note that the leftmost interval is (z0, z1], where z0 = 0 is the“null” service tier we defined earlier. Since F(z0) = 0, the sum-mation in expression 1 is correctly defined for all service tierintervals.

Ann. Telecommun.

These three optimization problems can be formallyexpressed as instances of the following problem, whichwe will refer to as the maximization of expected sur-plus (MAX-ES) problem. Note that the objective func-tion Eq. 2 is nonlinear with respect to the variablesz1, · · · , zK; hence, MAX-ES is a nonlinear program-ming problem.

Problem 3.1 (MAX-ES) Given the cdf F(x) of userrequests, an integer number K of service tiers, and asurplus function S(x), find a service tier vector Z =< z1, · · · , zK > that maximizes the objective function(expected surplus):

S̄(z1, · · · , zK) =K∑

j=1

(S(z j)

(F(z j) − F(z j−1)

))(2)

subject to the constraints:

xmin < z1 < z2 < · · · < zK = xmax. (3)

Let us assume for the moment that an optimal so-lution to MAX-ES can be obtained; we will discussshortly how to find such a solution. Consider the op-timal solution obtained by solving the MAX-ES prob-lem from the perspective of users or providers (i.e.,by using the user Susr(x) or provider Spr(x) surplusfunction in place of S(x), respectively). Such a solutionis unlikely to be of practical value, for two reasons.First, it assumes that users and service providers mayselect the service tiers optimally based only on theirown interests. In reality, a service tier vector that isoptimal for the users may not be acceptable to theservice provider, and vice versa. Therefore, it is im-portant to obtain a jointly optimal solution that takesinto account the perspectives of both users and serviceproviders. Second, both the user and provider surplusfunctions assume the existence of a pricing functionP(x). In general, the price function is the result ofmarketplace dynamics, including negotiation betweenusers and service providers; hence, it may not be knownin advance.

On the other hand, the social surplus function Ssoc(x)

depends only on the cost and utility functions, which aregenerally known in advance. Therefore, considering thewelfare of the society as a whole overcomes the abovedifficulties since (1) it takes into account simultaneouslythe interests of both users (through the utility function)and providers (through the cost function), and (2) al-lows us to determine the optimal service tier vectorwithout knowledge of the pricing function. Therefore,for the remainder of this paper, we will consider theMAX-ES problem from the society’s point of view

only; although, for simplicity, we will continue usingS(x) as the surplus function, the reader should keepin mind that, from now on, we assume that S(x) =Ssoc(x) = U(x) − C(x).

We also note that, from a practical point of view, theprovider is the entity that determines and advertisesthe service tier and price structure. However, as ourdiscussion above implies, the fact that the provider fixesthe tiers does not necessarily imply that it should do thisunilaterally without taking into account the users’ inter-ests (through the use of an appropriate utility functionU(x) and demand function f (x) that can be estimatedusing market surveys and relevant techniques). Forexample, if a provider fixes the tiers and their prices atlevels that do not align well with user demands and/orwillingness to pay, users in a competitive environmentwill opt for the services of another provider.

3.2 Solution through nonlinear programming

If the nonlinear objective function Eq. 2 of the MAX-ES problem is concave, and since the constraints in Eq.3 are convex, we may use the Karush–Kuhn–Tucker(KKT) conditions to find the global maximum [2]. Thefollowing lemma derives sufficient conditions for thefunction Eq. 2 to be concave.

Lemma 3.1 If S(x) and F(x) are continuous and twicedifferentiable in [xmin, xmax] and the two conditions

S′′(x)[F(x) − F(y)] + 2S′(x)F ′(x) + S(x)F ′′(x)< 0 (4)

− [S′(x)F ′(y)]2 − S(x)F ′′(y)

× {S′′(x)[F(x)−F(y)] +2S′(x)F ′(x)+S(x)F ′′(x)

}>0

(5)

are satisfied for all x, y ∈ [xmin, xmax] with y < x, then theMAX-ES objective function S̄ is concave in the feasiblearea xmin < z1 < · · · < zK−1 < zK = xmax.

Proof Define ω(x, y) = S(x)(F(x) − F(y)). We canthen rewrite Eq. 2 as:

S̄(z1, · · · , zK) =K∑

j=1

ω(z j, z j−1). (6)

Since the sum of concave functions is also a concavefunction, a sufficient condition for S̄ to be concave isfor ω to be concave in the feasible area xmin < y < x <

xmax.

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The Hessian of ω is the symmetric matrix

H =(

h1,1 h1,2

h2,1 h2,2

), (7)

where

h1,1 = ∂2ω

∂x2 = S′′(x)[F(x) − F(y)] + 2S′(x)F ′(x)

+ S(x)F ′′(x)

h2,2 = ∂2ω

∂y2 = −S(x)F ′′(y)

h1,2 = h2,1 = ∂2ω

∂x∂y= −S′(x)F ′(y)

If ω is continuous and has continuous first and secondderivatives, then it is concave if its Hessian is negativedefinite in the feasible area x, y ∈ [xmin, xmax] with y <

x, or:

h1,1 < 0 and h1,1h2,2 − h21,2 > 0,

from which the two conditions Eqs. 4 and 5 follow. ��

In general, however, the objective function may notbe concave. For instance, an empirically obtained cdfF(x) may not be continuous, in which case, the Hessianmatrix is not defined. In such cases, it may be possibleto formulate and solve approximate linear program-ming formulations, or apply branch-and-bound tech-niques [2]. One drawback of such solution methodsis that they have to be customized to the specific cdfand surplus functions. More importantly, such methodsmay need a large number of iterations, or they may gettrapped at a local maximum.

3.3 An efficient approximate solution

We now present an approximate yet efficient and accu-rate method for solving general instances of the MAX-ES problem. Rather than developing a sub-optimalalgorithm for solving MAX-ES directly, we take a dif-ferent approach: we provide an approximate formula-tion of MAX-ES that asymptotically converges to theformulation in Eqs. 2–3, along with an algorithm thatsolves this new problem optimally.

3.3.1 An approximate formulation of MAX-ES

We note that it is always possible to create a discreteapproximation of the pdf f (x), regardless of its form,as illustrated in Fig. 3. In particular, we can choose aninteger M > K and partition the interval [xmin, xmax]

m

f(x)

xmin

x max1 2 3 M... ...

x

Fig. 3 Forming a pdf approximation: The area under f (x) overan interval is paired with the right-hand endpoint of the interval

into M intervals, each of length equal to xmax−xminM . The

right-hand endpoint of the m-th interval is em = xmin +m(xmax−xmin)

M ; we associate with em a discrete point massdensity

Pm =∫ em

em−1

f (x) dx . (8)

The M pairs {(em, Pm)} form the approximation of f (x).We also define

Fm =m∑

i=1

Pi , m = 1, · · · , M, (9)

so that the M pairs {(em, Fm)} form the approximationof the cdf F(x).

In order to obtain an efficient solution to the MAX-ES problem, we also impose the additional restrictionthat the K service tiers may only take values from theset {em} of the interval endpoints. Consequently, ourobjective is to solve the following discrete version ofMAX-ES, which we will refer to as discrete-MAX-ES.

Problem 3.2 (Discrete-MAX-ES) Given the M-pointapproximation {em, Pm} of the pdf of user requests,an integer number K < M of service tiers, and asurplus function S(x), find a service tier vector Z =< z1, · · · , zK > that maximizes the objective function(approximate expected surplus):

S̄(z1, · · · , zK) =K∑

j=1

(S(z j)

(Fm j − Fm j−1

))(10)

subject to the constraints:

z j = em j ∈ {em}, j = 1, · · · , K, m = 1, · · · , M (11)

z1 < z2 < · · · < zK = xmax. (12)

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Clearly, as M → ∞, the pdf approximation ap-proaches the original pdf and discrete-MAX-ES re-duces to the original MAX-ES problem.

3.3.2 Optimal solution to discrete-MAX-ES

Define �(m, k) as the optimal value of the objectivefunction Eq. 10 when the number of intervals is mand the number of service tiers is k ≤ m, where para-meter m represents the first such intervals among theM discretized ones. Then, �(m, k) may be computedrecursively as follows:

�(m, 1) = S(em)Fm, m = 1, · · · , M (13)

�(m, k + 1) = maxq=k,··· ,m−1

{�(q, k) + S(em)(Fm − Fq)},

k = 1, · · · , K − 1; m = 2, · · · , M (14)

Expression 13 can be explained by observing that, ifthere is only one tier of service, it must coincide withthe right-hand endpoint of the m-th (i.e., rightmost)interval. The recursive expression Eq. 14 simply statesthat, for k + 1 service tiers, the largest tier must coin-cide with the right-hand endpoint of the m-th interval,and the remaining k tiers must be optimally assigned tothe endpoints of any feasible interval q, k ≤ q ≤ m − 1.

The running time of the above dynamic program-ming algorithm to obtain �(M, K) is O(KM2). Notethat �(M, K) is the value of an optimal solution tothe discrete-MAX-ES problem. The optimal solutionmay not be unique, in which case, the algorithm willrandomly return one of the optimal ones. Sincediscrete-MAX-ES is an approximate formulation of theoriginal MAX-ES problem, �(M, K) represents a solu-tion close to the optimal solution to MAX-ES, regard-less of the shape of the objective function Eq. 2. Clearly,the better the pdf approximation, i.e., the larger thevalue of M, the closer that �(M, K) will be to thetrue optimal solution for a given pdf; the tradeoff isan increase in running time. We have found that thevalue of �(M, K) converges quickly as the value of Mapproaches 50–100 for all the distribution functions wehave considered; thus, a (near-) optimal solution can becomputed efficiently for any instance of MAX-ES.

After solving the MAX-ES problem, we obtain a ser-vice vector Z � =< z1, z2, · · · , zK > that maximizes thesocial surplus and depends only on the utility and costfunctions provided by the users and network provider,respectively. Next, we describe an approach to obtain-ing the optimal price for each service tier in Z � in a

manner that strikes a balance between the conflictingobjectives of users and providers.

4 Optimal pricing based on Nash bargaining

Consider a service vector Z � =< z1, z2, . . . , zK > thatmaximizes the social surplus. We are interested in find-ing an appropriate price P(z j) for each service tierz j, j = 1, · · · , K, so as to satisfy both the users andservice provider. Clearly, the price for each service tierz j should be between the values of the cost and utilityfunctions at service level z j, as illustrated in Fig. 4.

In a free telecommunication market, the price for theservice is typically the result of a negotiation processbetween the users and service providers. This negotia-tion, or bargaining, process can be thought of as a gameduring which each party attempts to maximize its ownsurplus [12–14]; the outcome of the game is an optimalprice for the service that is mutually acceptable forboth parties. This game can be seen as an abstractionof market dynamics, e.g., reflecting the users’ ability tocompare prices from various providers and competitionamong providers. We also emphasize that our focus ison a bargaining game that takes place once, after whichoptimal prices are determined and fixed for a relativelylong time compared to typical flow durations. In otherwords, we do not consider the scenario in which theusers and/or the network attempt to set prices on aper-session basis (e.g., as in [8–10]), a scenario thatwe do not believe is practical. We also note that thenegotiation between users and provider is implicit and

z

y

U(x)

C(x)

x1 2 3 j j+1... K...

......

Negotiated prices

z z z z z

Fig. 4 Optimal price vector

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captured in the game through a bargaining parameter,as explained shortly.

4.1 The single-tier case

Let us first consider the case K = 1, whereby theprovider offers a single service tier z1. For notationalconvenience, in this section, we will simply use U , P,and C, instead of U(z1), P(z1), and C(z1), respectively.

Let Phigh, Phigh ≤ U be the maximum price that theusers would accept as a satisfactory outcome of thenegotiating process (game). Similarly, let Plow, Plow ≥C, be the minimum price that the service providerwould find acceptable. We use P1 to denote the pricethat users pay for the service, and P2 the amount thatthe provider receives for the service. In general, theremay exist a gap G between the P1 and P2; this gap isreferred to as transaction cost in economics. Withoutloss of generality, in this work, we assume that the gapG has a fixed value; clearly, if G = 0, then P1 = P2. Wealso define Y1 = U − P1, and Y2 = P2 − C. Figure 5illustrates the relationship among the game parametersU , Phigh, Plow, C, P1, P2, and G.

The two parties, users and service provider, areinterested in dividing the net social surplus, i.e., thesocial surplus minus the transaction cost, which, fromFig. 5, is equal to (U − C − G). As we can see, thenet social surplus is the sum of Y1 and Y2. Y1 andY2 represent the shares of the good to be divided andstand for the excess utility (or net surplus) of usersand provider, respectively. The objective is to find anoptimal division of the net social surplus such that bothparties feel satisfied. This optimization problem wasintroduced by Nash [12, 14] as a cooperative bargaininggame, and is widely used in the literature for character-izing labor negotiations and a range of other bargainingsituations [17].

Let β, 0 ≤ β ≤ 1 be the bargaining power of theusers, and 1 − β be the bargaining power of the serviceprovider. Bargaining power, as defined here, refers tothe relative ability of each party in the bargaining game

Fig. 5 Relationship amongthe game parameters U , C,G, P1, P2, Phigh, Plow, Y1,and Y2

2

C

0

U

P

P

high

low

P

P

1

2

G

1

2Y

Y

C

0

U

P

P

high

low

G>0 G=0

Y

Y2

1

P = P = P1

to influence the setting of prices. Then, � = Yβ

1 Y1−β

2 isthe Nash product [12, 14] in the bargain. In essence,� is the product of the players’ excess utilities, eachscaled by the corresponding player’s bargaining power.Our objective is to find suitable values for Y1 and Y2

that maximize �. The optimization problem can beformulated as:

maxY1,Y2

� = Yβ

1 Y1−β

2 (15)

subject to the constraints:

Y1 + Y2 ≤ U − C − G (16)

Y1 ≥ U − Phigh (17)

Y2 ≥ Plow − C (18)

Constraints 16–18 can be explained by referring toFig. 5.

Figure 6 plots the curve of the objective function� as a function of Y1 and Y2. The feasible area isrepresented by the shaded triangle formed by linearconstraints 16–18. As the value of � increases, the curvemoves upwards, and vice versa. The maximum value of� occurs when the curve intersects the line Y1 + Y2 =U − C − G at exactly one point, and the coordinatesof this point correspond to the optimal values for Y1

and Y2. To obtain the latter values, we rewrite the opti-mization problem Eqs. 15–18 in the following Lagrangeform:

maxY1,Y2

�′ = Yβ

1 Y1−β

2 + η(Y1 + Y2 − U + C + G), (19)

2

Y1

Y2

U−C−G

U−C−G

P −Clow

U−Phigh

Feasible region

(Y* , Y *)1

Ω

Fig. 6 Optimal price point (Y�1 , Y�

2)

Ann. Telecommun.

where η is the Lagrange multiplier. By taking the partialderivatives of �′ with respect to Y1 and Y2, setting themequal to zero, and using the fact that Y�

1 + Y�2 = U −

C − G, we obtain:

Y�1 = β(U − C − G) (20)

Y�2 = (1 − β)(U − C − G) (21)

From the definition of Y1 and Y2, we finally obtain theoptimal prices as follows:

P�1 = (1 − β)U + β(C + G) (22)

P�2 = (1 − β)(U − G) + βC (23)

In the case of no transaction costs (i.e., G = 0), theprice users pay is exactly the amount that the providerreceives, hence:

P� = P�1 = P�

2 = (1 − β)U + βC (24)

As we can see, the value of β determines the positionof the optimal price along the line segment between theutility and cost values.

Let us now consider three special cases with respectto the value of the bargaining parameter β that providesome insight into the optimal solution of the aboveoptimization problem.

Case 1 β = 0, i.e., the service provider has all thebargaining power; this situation arises whenever thetelecommunications market is dominated by one ser-vice provider (monopoly) or a small number ofproviders (oligopoly). In this case, we have that P� =U ; hence, the service provider enjoys the total socialsurplus by squeezing out the users’ surplus.

Case 2 β = 0.5, i.e., users and service provider haveexactly the same bargaining power. In this case, P� =0.5(U + C), implying that the social welfare is equallyshared by the two parties.

Case 3 β = 1, i.e., the bargaining power resides solelywith the users; such a scenario may arise in the telecom-munications market when the supply greatly exceedsthe aggregate user demand. In this case, we haveP� = C, and the provider has to abandon any benefits(provider surplus) from supplying the service.

4.2 The multiple tier case

Let us now consider the general case of K > 1 tiersof service. We can apply the methodology of the pre-vious subsection to each service tier, z j, j = 1, · · · , K,

to obtain the optimal vector of tier prices P� =<

P�(z1), P�(z2), . . . , P�(zK) >. Let us assume for sim-plicity that the transaction cost G is zero; then, usingexpression 24, we obtain:

P�(z j) = (1 − β)U(z j) + βC(z j), j = 1, · · · , K (25)

Since both the utility U(x) and the cost C(x) are non-decreasing functions of bandwidth x, we have that:

P�(z j) < P�(zk), 1 ≤ j < k ≤ K (26)

In other words, the optimal price increases with theservice tier index, i.e., with the amount of bandwidthoffered to the users, consistent with intuition.

5 Numerical results

To illustrate our methodology for pricing of tieredservices, we consider the market for broadband Inter-net access under either a capacity-based or a usage-sensitive tiered pricing scheme.3

Capacity-based pricing We have used data collectedat the San Diego Network Access Point (SDNAP) andavailable at the CAIDA site [3] to obtain the cdf Facc

of Internet access speeds shown in Fig. 7. We adaptedthe raw SDNAP data so that access speeds are in therange [256 Kb/s, 12 Mb/s], typical of current broadbandspeeds in the USA.

Usage-sensitive pricing We make the assumption thatthe monthly amount of traffic generated by users is inthe range [5MB, 1TB] and follows the bounded Paretodistribution (pdf):

f (x)= αkα

1−(

kp

)α x−α−1, 5=k ≤ x ≤ p=106, 0 < α < 2

(27)

We have selected two values for parameter α, corre-sponding to two distribution functions:

• pdf f15/85 has α = 0.00001 and is such that approx-imately 15% of users generate about 85% of thetotal traffic

• pdf f5/50 with α = 0.03, for which 5% of users gen-erate approximately 50% of the overall traffic.

3We have conducted a large number of experiments with a rangeof distribution, utility, and cost functions. To avoid repetition, inthis study, we investigate the MAX-ES problem only with the in-put functions described next. Nevertheless, these input functionsare characteristic of real-life scenarios and the results shown arerepresentative of what we have observed in our experiments.

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0

0.2

0.4

0.6

0.8

1

0 2000 4000 6000 8000 10000 12000

CD

F

Bandwidth (Kb/s)

F_acc

Fig. 7 Cdf Facc of Internet access speeds (data adapted from [3])

The latter distribution has characteristics similar to theusage patterns reported recently by one major cableISP [6].

For all instances of the MAX-ES problem we inves-tigate in this study, we let the utility function be

U(x) = λxγ log(x) (28)

and the cost function

C(x) = μx; (29)

hence, the social surplus S(x) = U(x) − C(x). The pa-rameters of the utility function Eq. 28 can be selectedsuch that it is an increasing, strictly concave, and con-tinuously differentiable function of service level x; thisfunction has also been considered in the context ofelastic traffic [19]. The parameters λ and γ can be usedto control the slope of U(x). In this work, we use thevalues λ = 12, γ = 0.5, and μ = 0.4 for capacity-basedpricing, and λ = 9, γ = 0.5, and μ = 0.05 for usage-sensitive pricing, to ensure that the surplus functionexhibits similar behavior across the different domainsof the corresponding distributions.

5.1 Service tier selection

Let us first consider the impact of the number M of in-tervals in the pdf approximation (refer to Fig. 3) on theconvergence of the dynamic programming algorithmwe presented in Section 3.3.2. Figure 8 plots the valueof the optimal solution �(M, K) as a function of M forthe cdf Facc of Fig. 7 and the surplus function above.Figure 9 is similar, but shows results for the Pareto cdfF5/50.

We make two important observations from thesefigures. First, for a given number K of service tiers, the

1.119e+06

1.1195e+06

1.12e+06

1.1205e+06

1.121e+06

1.1215e+06

1.122e+06

1.1225e+06

1.123e+06

1.1235e+06

1.124e+06

0 50 100 150 200 250 300 350 400 450 500

Exp

ecte

d S

urpl

us

Number of Intervals, M

K=5K=10K=20

Fig. 8 Expected surplus against M, cdf Facc

solution obtained by the dynamic programming algo-rithm �(M, K) converges quickly as M increases. Wehave run experiments with a wide range of instancesof MAX-ES beyond the ones we report here, and wehave found that, in all cases, M = 200 is sufficient forconvergence; hence, we have used this value for theexperiments we present in the remainder of this section.This result confirms that the dynamic programmingalgorithm provides an accurate and efficient solution tothe MAX-ES problem.

We also observe that the expected surplus increaseswith the number K of service tiers. This behavior isconsistent with intuition: a larger number of tiers im-proves the “resolution” of the final solution and allowsthe dynamic programming algorithm to better tailor thetiers to the given surplus and distribution functions.The figures also demonstrate the (expected) effect ofdiminishing returns, as further increases in the number

1.176e+07

1.177e+07

1.178e+07

1.179e+07

1.18e+07

1.181e+07

1.182e+07

1.183e+07

1.184e+07

1.185e+07

0 50 100 150 200 250 300 350 400 450 500

Exp

ecte

d S

urpl

us

Number of Intervals, M

K=5K=10K=20

Fig. 9 Expected surplus against M, Pareto cdf F5/50

Ann. Telecommun.

K of service tiers provide smaller improvements to theexpected surplus.

We now compare four solutions to the MAX-ESproblem in terms of the expected social surplus theyachieve:

1. Optimal: This is the optimal dynamic programmingsolution to the corresponding discrete-MAX-ESinstance.

2. Optimal-rounded: This is the tier structure derivedby rounding the values of the optimal tiers of theabove solution to the nearest multiple of 256 kb/s(for capacity-based pricing) or 10 GB (for usage-sensitive pricing). The motivation for this solutionarises from considerations related to marketing theservice to customers who do not have intimateknowledge of the manner in which the optimal tierstructure is determined. More specifically, a tier of,say, 100 GB is likely to seem more natural andunderstandable to users compared to the outcome,say, 98.54 GB, of the dynamic programming algo-rithm, which could well be considered arbitrary.

3. Uniform: The K service tiers are spread uniformlyacross the domain [xmin, xmax], i.e.,

zk = xmin + kxmax − xmin

K, k = 1, · · · , K.

4. Exponential: Each tier provides a level of servicethat is twice that of the immediately lower tier:

zk+1 = 2zk, k = 1, · · · , K − 1.

As a result, the tiers divide the domain [xmin, xmax]into intervals of exponentially increasing length.

The uniform and exponential are simple, straightfor-ward solutions that do not involve any optimizationand are along the lines of the structures employed bymajor ISPs.4 We consider them here as baseline casesand to demonstrate that they perform poorly in termsof maximizing the expected social surplus.

Since the raw expected surplus values are not compa-rable across different instances of the MAX-ES prob-lem, we introduce the concept of normalized expectedsurplus to illustrate the relative performance of the fouralgorithms above. For a given problem instance, letSmax be the maximum expected surplus value achievedby any of the four algorithms over all values of thenumber K of tiers evaluated in our experiments. If S̄

4Many ADSL providers offer download speeds that follow anexponential tiering structure, e.g., 384 kb/s, 768 kb/s, 1.5 Mb/s,3 Mb/s, etc. Similarly for the 5/10/20/40 GB tiers of monthly trafficused in the recent pilot program by a cable ISP [6].

is the expected surplus for a given algorithm-K pair, wedefine the normalized expected surplus for this pair as:

S̄norm = S̄Smax

, 0 ≤ S̄norm ≤ 1. (30)

The normalized expected surplus takes values in (0,1)for all instances of MAX-ES and provides insight intothe relative behavior of the four algorithms.

Figures 10, 11, and 12 plot the normalized expectedsurplus as a function of the number K of service tiers forthe three distribution functions Facc, F5/50, and F15/85,respectively. Each figure contains four curves, eachcorresponding to one of the algorithms for the MAX-ES problem described above. We observe that thecurves for the optimal and optimal-rounded solutionsalmost overlap, and exhibit the best performance byfar across all the values of K except very small ones,regardless of the underlying distribution function. Inparticular, the exponential solution decreases rapidlyfor K > 2 to about 30–50% of the optimal expectedsurplus, depending on the distribution (Facc or Pareto).These results demonstrate that exponential groupingof customers, though favored by ISPs, performs farfrom optimal from an economic standpoint. In fact,the uniform tiering structure performs better than theexponential one, but it can also be far from the optimalsolution for other than very small values of K.

The main conclusion from the results shown inFigs. 10–12 is that, by employing our dynamic program-ming algorithm, which has low computational require-ments, it is possible to obtain optimal tiering structuresthat improve the expected surplus over simple solu-tions by a factor of up to 2–3. More importantly, ourapproach makes it possible to re-optimize the tiering

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 5 10 15 20 25 30 35 40 45 50

Nor

mal

ized

Exp

ecte

d S

urpl

us

Number of Service Tiers, K

OptimalOptimal-rounded

UniformExponential

Fig. 10 Normalized social surplus comparison, cdf Facc

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0

0.2

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0.8

1

0 5 10 15 20 25 30 35 40 45 50

Nor

mal

ized

Exp

ecte

d S

urpl

us

Number of Service Tiers, K

OptimalOptimal-rounded

UniformExponential

Fig. 11 Normalized social surplus comparison, Pareto cdf F5/50

structures over time to accommodate evolving userdemands and market conditions.

5.2 Optimal pricing of service tiers

Given a vector < z1, · · · , zK > of K of service tiers,as well as the utility U(x) and cost C(x) functions,we may solve the optimization problem in Section 4to obtain the optimal prices for each of the servicetiers. Figure 13 plots the optimal prices for the K = 5optimal service tiers obtained for cdf Facc. As we cansee, the optimal tier structure does not resemble ei-ther the uniform or exponential structures. Three pricestructures are shown, corresponding to the three valuesof the bargaining power of users β = 0.25, 0.5, 0.75. Asexpected, the lower the bargaining power of users, thehigher the corresponding price. Also, for a fixed valueof β, the price increases with the tier index, consistent

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40 45 50

Nor

mal

ized

Exp

ecte

d S

urpl

us

Number of Service Tiers, K

OptimalOptimal-rounded

UniformExponential

Fig. 12 Normalized social surplus comparison, Pareto cdf F15/85

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 2000 4000 6000 8000 10000 12000

Nor

mal

ized

Pric

e C

urve

Bandwidth (Kb/s)

beta=0.25beta=0.5

beta=0.75

Fig. 13 Optimal prices, cdf Facc, for K = 5 tiers

with our discussion in Section 4.2. Moreover, the priceincrease from one tier to the next is tied directly to theshape of the utility and cost functions, thus reflectingthe perspective of both users and providers.

In the previous subsection, we demonstrated that theexponential and uniform tiering structures are far fromoptimal with respect to the expected social surplus.We now show that these structures are also subopti-mal in terms of the revenue collected by the serviceprovider. Consider a tier vector < z1, · · · , zK >, andlet P(z j), j = 1, · · · , K, be the optimal price structureobtained by applying the methodology of Section 4.Then, the expected revenue R̄ collected by the serviceprovider can be calculated as:

R̄(z1, · · · , zK) =K∑

j=1

(P(z j)

(F(z j) − F(z j−1)

)). (31)

Figures 14 and 15 plot the normalized expected revenueagainst the number K of service tiers for the four so-lutions to the MAX-ES problem we described earlier;the normalized expected revenue is defined similarlyto the normalized expected surplus in expression 30.Note that the highest revenue is obtained when thereis only one tier, in which case all users are mappedto the highest possible service (that also incurs thehighest price); such a solution is unlikely to be adoptedin a market environment, and is included here forillustration purposes only. As the number K of tiersincreases, the expected revenue decreases for a whileand then stabilizes. The curves for the optimal andoptimal-rounded solutions both converge quickly to avalue that is around one-half that of the maximum rev-enue for K = 1. However, the exponential and uniformsolutions drop much more rapidly, eventually reachinga value that is only one-quarter (for Facc) or one-sixth

Ann. Telecommun.

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40 45 50

Nor

mal

ized

Exp

ecte

d R

even

ue

Number of Service Tiers, K

OptimalOptimal-rounded

UniformExponential

Fig. 14 Normalized revenue comparison, cdf Facc

(for the Pareto distribution) of the maximum revenue.Again, the uniform tiering structure outperforms theexponential one, while the optimal solution achieves anexpected revenue that is up to 2-3 times higher than theother two, consistent with the results of the previoussection.

Since the cost of providing the service is the sameregardless of what tiered structure is selected, theseresults indicate that, by adopting simple, suboptimalsolutions, the service provider may end up foregoing asubstantial fraction of potential revenues. More impor-tantly, these additional revenues are not at the expenseof users, but rather due to the larger surplus achievedby the optimal solution. In other words, the optimaltiered structure provides substantially more value toboth users and providers.

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40 45 50

Nor

mal

ized

Exp

ecte

d R

even

ue

Number of Service Tiers, K

OptimalOptimal-rounded

UniformExponential

Fig. 15 Normalized revenue comparison, Pareto cdf F5/50

6 Extensions

We now discuss two directions in which our work maybe extended.

6.1 Optimizing the number of service tiers

The MAX-ES problem takes the number K of servicetiers as input, and its objective function Eq. 2 is basedon the assumption that there is no cost associated withoffering each tier. In general, however, the total costto the network provider of offering K tiers of serviceconsists of two components. The first component is dueto the cost of bandwidth: the higher the access speedor the amount of traffic generated by the users, thehigher the cost. We used the nondecreasing functionC(x) to denote this cost, which may be used to representthe link cost for carrying user traffic, as well as thecost of switching the traffic in the network. The secondcomponent is due to the cost of software and hardwaremechanisms (e.g., queueing structures, policing mecha-nisms, control plane support, etc.) required inside thenetwork for implementing a given number K of servicetiers.

Let Ct(k) be a nondecreasing function representingthe cost of employing k service tiers. In this variant ofthe MAX-ES problem, the objective is to determineboth the optimal number K of service tiers and theirvalues so as to maximize the objective function:

S̄(z1, · · · , zK) − Ct(K), (32)

where S̄(z1, · · · , zK) is the expected surplus in expres-sion 2. This problem can be solved near-optimally withthe dynamic programming algorithm we presented inSection 3.3.2 after modifying expressions 13–14 to ac-count for the cost component Ct(k). The running timeof this algorithm is O(M3), since it has to examine allM possible values for the number of service tiers.

6.2 Tiered structures for bundles of services

In this work, we have considered services that arecharacterized by a single parameter, e.g., access speedor amount of traffic generated. With advances in tech-nology, more sophisticated network providers may beable to bundle and market several services in a singlepackage. For instance, it is conceivable that providersmight offer subscribers storage space on top of broad-band access, or Internet connections that are charac-terized not only in terms of speed but also in terms ofadditional parameters such as availability or reliability.In such a scenario, a tiered structure would consist ofservice tuples, where each tuple is the set of levels of

Ann. Telecommun.

each service in the bundle corresponding to this tier.For instance, the tuple (3 Mb/s, 5 GB) would corre-spond to a tier in which the customer subscribes to aservice bundle that offers access speed of 3 Mb/s andstorage capacity equal to 5 GB. Then, the problem weconsidered in this work generalizes to finding a tieredservice and pricing structure that is jointly optimal forall the services in the bundle. This problem, which isthe subject of ongoing research in our group, is furthercomplicated by the fact that, in general, the utility andcost functions will be different for each of the bundledservices

Finally, we note that the economic model we con-sidered can be extended to include a heterogeneouspopulation of users, in which different user segmentsare characterized by different utility functions, as wellas heterogeneous providers (e.g., in terms of size, powerof negotiation, etc.). Our model only takes into accounttechnical costs, and it is possible to extend it to considernon-technical and fixed costs as well. The implicationsof tiered service as the basis of a universal Internetservice are also worth exploring.

7 Related work

Pricing of Internet services using concepts from eco-nomic theory has been a subject of research for morethan a decade [5, 8–11, 15, 16, 18, 20]. This is a broadarea that encompasses issues from calculating the costof resources to determining the services to offer andsetting appropriate prices, and from dealing with therealities and economics of layered networks to inter-connection agreements between ISPs. An initial focuswas on charging as a mechanism for controlling thebehavior of users, and/or for limiting usage to makeroom for higher-paying users. Early work [9, 10] alsoaddressed the issues of charging, rate control, and rout-ing in communication networks carrying elastic traffic.The main finding of these studies was that the systemreaches an optimum state when the network’s choice ofallocated rates is at equilibrium with users’ choices ofcharges. The Paris metro pricing (PMP) scheme in [16]separates the network into independent subnetworksthat behave similarly but charge their customers atdifferent rates. A mathematical model of PMP wasdeveloped in [18] by viewing each subnetwork as a sin-gle bottleneck queue, and assuming that data packetsmay select the most suitable subnetwork “intelligently”by considering not only the delays but also the pricescharged. The conclusion of the study was that thereexist necessary and sufficient conditions for the systemto attain stability. The issue of charging at the session

or network layer while maintaining a clean separationbetween the underlying technologies was consideredin [8].

More recent work has studied the issues arising inpricing multiple classes of service, especially in the con-text of differentiated services. A game theoretic pricingmechanism for “statistically guaranteed” service in theInternet was proposed in [20]. This mechanism wasshown to offer better service and lower prices to users,and enables the provider to adopt various service andrevenue models. The work in [5] also adopted a game-theoretic strategy to study a simple two-class differenti-ated service model, and found that the system is easy totrap into an undesirable equilibrium whenever pricesdo not properly reflect the quality of the service pro-vided. Accordingly, a new dynamic pricing approachwas proposed in order to avoid this problem. Finally,a free market economic model for ad-hoc wirelessnetworks was proposed recently in [15]. Based on agreedy pricing strategy, the model maximizes the socialwelfare while ensuring non-negative profit for the usersand service provider. This study also developed a non-greedy policy that optimizes a profit fairness metric.

Our work differs from existing literature in thatour focus is on optimizing the service tiers and cor-responding price structures given some informationabout users and providers, regardless of the underly-ing assumptions upon which this information is based.Consequently, our work is quite general in scope andmay be applied to a variety of contexts, independentlyof whether the pricing scheme is capacity-based orusage-sensitive, whether charging is at the network orsession/application layers, or whether the transaction isbetween users and provider or between providers.

8 Concluding remarks

We proposed an economic model for tiered-servicenetworks and developed an efficient algorithm to selectthe service tiers in a manner that optimizes the socialsurplus. We also presented a method, based on Nashbargaining, to determine the optimal price for eachservice tier. Our approach provides insight into theselection and pricing of Internet tiered services, as wellas a theoretical framework of practical importance tonetwork providers.

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