On Optimal Service Differentiation in Congested Network Markets

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by  Mao Zou

On Optimal Service Differentiation in Congested Network Markets
Mao Zou1,2, Richard T. B. Ma3, Xin Wang1,2, Yinlong Xu1,2
1 School of Computer Science and Technology, University of Science and Technology of China
2 AnHui Province Key Laboratory of High Performance Computing (USTC)
3 School of Computing, National University of Singapore
[email protected], [email protected], [email protected], [email protected]
Abstract—As Internet applications have become more diverse decision-making process. In deciding the optimal strategy,
in recent years, users having heavy demand for online video an ISP needs to answer two questions: what are the service
services are more willing to pay higher prices for better services
qualities to offer and what are the associated prices for them.
7 than light users that mainly use e-mails and instant messages.
For the former question, even the range of the optimal set
1 This encourages the Internet Service Providers (ISPs) to explore
0 service differentiations so as to optimize their profits and al- of service qualities is hard to determine; for the latter one,
2 location of network resources. Much prior work has focused it needs to find the optimal quality-price mapping for the
on the viability of network service differentiation by comparing chosen service qualities. The problem is actually a dynamic
n
with the case of a single-class service. However, the optimal
a optimization problem in a functional space and the domain
service differentiation for an ISP subject to resource constraints
J of the price mapping is to be determined. As the realized
has remained unsolved. In this work, we establish an optimal
5 control framework to derive the analytical solution to an ISP’s quality in a service class depends on the allocated network
1 optimal service differentiation, i.e. the optimal service qualities capacity and the aggregate user demand, which is determined
andassociatedprices.Byanalyzingthestructuresofthesolution, bytheavailableservicequalitiesandthecorrespondingprices,
] we reveal how an ISP should adjust the service qualities and
I anothermajorchallengeistomodeltheISP’scapacityalloca-
N prices in order to meet varying capacity constraints and users’
tionplansubjecttonetworkcapacityconstraintsanddetermine
characteristics. We also obtain the conditions under which ISPs
s. have strong incentives to implement service differentiation and the realized quality in terms of network congestion. In other
c whether regulators should encourage such practices. words, ISPs need to allocate limited network capacity to each
[
service class so as to guarantee the promised service quality.
1 I. INTRODUCTION Inviewofthecomplexityoftheoptimalservicedifferentia-
v Recent years have witnessed increasing commercialization tion,thecommonpracticeofpriorworkistoadoptsimplifying
6 of the Internet and considerably rising diversities of Inter- assumptions. For instance, Jain et al. [8] assumed the ISP
7
net applications and users. Users having heavy demand for only provides two service classes. Shakkottai et al. [20]
0
quality-sensitive services such as online video services are overlookedthecongestionexternalitiesandconsideredaloose
4
0 often willing to pay higher prices for better service qualities, upper bound (under the first degree price discrimination) of
. while light users tend to have low willingness-to-pay for the ISP’s optimal profits instead. Chau et al. [4] considered
1
0 elementary services such as e-mails and instant messages. a finite number of service classes, analyzed the viability of
7 Such heterogeneity in the users’ characteristics has rendered differentiated pricing, but did not solve the optimal strategy.
1 the traditional practice of uniform prices, regardless of the In this work, we solve an ISP’s optimal service differen-
v: quality of service, inefficient in both generating profits and tiation in generic settings by explicitly modelling the ISP’s
i allocating network capacity. In light of this, there has been capacityallocationplansubjecttonetworkcapacityconstraints
X
tremendous interest in designing pricing strategies to provide andincorporatingthedualapproachinmicroeconomics[14]to
r
a differentiated services. Many schemes [17–19] have been transformtheproblemintoatractableoptimalcontrolproblem.
designed, e.g., Paris Metro Pricing (PMP) [17] proposed by As for the network capacity constraints, we consider two
Andrew Odlyzko. In such a scheme, network is partitioned typical scenarios: a) the fixed capacity scenario, where an
into several logically separated channels, each of which uses ISP has limited network capacity, and b) the variable capacity
a fraction of the entire network capacity and a different price, scenario, where an ISP can invest to expand the capacity of
and the channels with higher prices attract fewer users but its network infrastructure. Our findings include the following.
provide better services. • We formulate a profit maximization problem subject to
An important theoretical question naturally arises: what is capacity constraints under an optimal control framework
the optimal service differentiation for an ISP subject to its and derive the analytical optimal service differentiation
capacity constraints so as to maximize its profit? Despite the of an ISP, i.e., the qualities and prices of service classes.
vast literature on network service differentiation, this question • We find that the optimal differentiation strategy offers
has remained unanswered. Prior work [4, 8, 10, 20] mostly service classes with sufficiently high qualities and prices
focusedontheviabilityofPMP-typeofdifferentiationanddid such that users with low value will not join the services,
notinvestigatetheoptimalservicedifferentiation.Totacklethe even under abundant capacity. We characterize an ISP’s
optimal service differentiation, we delve a bit into the ISP’s market share by users’ virtual valuation function [16].
• WeshowthatwhenanISPexpandsitscapacity,itshould studies investigated the competition among service providers.
increase market share by offering more premium service For example, Gibbens et al. [6] analyzed the competition
classes and reducing prices. When there are more high- between two ISP’s and show that in any equilibrium compet-
end users, it is optimal for the ISP to offer a narrower itive outcome, both ISPs offer a single-class service. Ma [10]
range of premium service classes with higher prices. studied usage-based pricing scheme and competition among
• We show that if network capacity is scarce, an ISP has oligopolistic service provides. These studies focused on the
strongincentivestoimplementservicedifferentiationand viability of PMP and its variants by comparing with a single-
regulators should encourage such practices in order to class service. Our work, however, focuses on the theoretical
improve the total user surplus; otherwise, it is optimal problem of solving the ISP’s optimal service differentiation
for the ISP to provide a single-class service. subjecttocapacityconstraints.Weestablishanoptimalcontrol
To the best of our knowledge, our work is the first to frameworktoderivetheanalyticalsolutiontotheISP’soptimal
analytically derive and characterize an ISP’s optimal service service differentiation and characterize its properties.
differentiation subject to network capacity constraints. We
III. MODEL
believethatourfindingswillhelpISPstobetterpricenetwork
servicesandguideregulatorstodesignregulations.Therestof A. ISP and User Model
the paper is organized as follows. Section II discusses related
We consider an ISP and a continuum of users with hetero-
work. Section III presents the model and formulate the ISP’s
geneous characteristics. Suppose the ISP offers a set N of
profit-maximizing problem as a nested optimization problem. service classes and adopts flat-rate1 pricing [1] for all service
In Section IV, we derive the analytical solution of the optimal
classes. For each service class i ∈ N, we denote p as the
i
service differentiation and study the structure of the solution.
price and q as the congestion level, a measure of service
i
We reveal the dynamics of the solution under varying market quality. Correspondingly, we denote p (cid:44) {p : i ∈ N} and
i
environments in Section V and conclude in Section VI. q(cid:44){q :i∈N} as the price and congestion vectors.
i
Once the menu of services (p,q) is decided, users will
II. RELATEDWORK
make choices according to their preferences over the bundles
Much theoretical economics literature has considered prod-
{(p ,q ):i∈N}. We characterize each user by her intrinsic
uctdifferentiation.MussaandRosen[15]analyzedtheoptimal i i
value of the network service θ ∈Θ=[0,1]. We denote f(θ)
product mix for a monopolist offering several qualities of
andF(θ)astheprobabilitydensityandcumulativedistribution
a product to consumers with different preferences for qual-
functionsofusers’value,andassumethatf(θ)iscontinuously
ity. Maskin and Riley [11] characterized the optimal selling
differentiable and strictly positive over Θ. When a user with
strategy of a monopolist, and Champsaur and Rochet [2]
value θ chooses a service class (p,q), we define her utility as
studied the product differentiation for two competing firms.
Both adopt a common assumption that the quality of the u(p,q;θ)=θv(q)−p, (1)
product can be unilaterally decided by the seller. However,
which can be interpreted as follows.
these models are inapplicable to network services because the
quality of services are influenced by both the user demand • v(·)isasatisfactiondiscountfunctionwhichcapturesthe
and the network capacity. Chander and Leruth [3] studied the negative effect of congestion on users’ intrinsic values.
best strategy of a monopolist in the presence of congestion • θv(q) is the user’s achieved value over the service under
effects. Nevertheless, they did not provide the solution of the a congestion level q, and thus u(p,q;θ) is the user’s net
optimal strategy and their congestion model does not capture surplus, i.e., the achieved value θv(q) minus the price p.
the scarcity of capacity in network markets. • Without loss of generality, we normalize the range of
Various pricing schemes for providing differentiated net- congestion level q to the interval [0,1] and assume that
work services has been proposed [17–19]. In particular, Paris v(·) : [0,1] → R+ is continuously differentiable and
Metro Pricing (PMP) [17] has attracted considerable attention strictly decreasing, satisfying v(0) = 1 and v(1) = 0.
for its simplicity. Since PMP does not guarantee the quality This models that the user’s value is at maximum without
of service for users, which depends on the spontaneous user congestion, i.e., q = 0, and decreases to zero when the
demand, its viability and effectiveness need to be established. network is heavily congested, i.e., q →1.
Prior studies [4, 6, 8, 10, 20] addressed the viability of PMP. Weassumethatusersareindividuallyrational,i.e.,anyuser
Shakkottai et al. [20] characterized the types of environments with value θ chooses either a) to join the service class k that
in which simple flat-rate pricing is efficient in extracting yields the highest nonnegative utility2:
profits. Lee et al. [9] extended the model of [20] to explicitly
k =argmax u(p ,q ;θ) (2)
considerthecongestionexternalitiesandindicatethatwhenthe j j
j∈N
network is congested, a simple flat-rate pricing is inefficient
in extracting profits. Jain et al. [8] analyzed the ISP’s profit- 1Our analysis can be extended to handle usage-based pricing schemes.
maximizingproblemwhentheISPisrestrictedtoprovidertwo InterestedreadersarereferredtotheAppendixfordetails.
service classes. Chau et al. [4] provided sufficient condition 2Technically, we should write k ∈ argmaxj∈N u(pj,qj;θ). By using
theequalsign,weassumewhentherearemorethanoneserviceclassesthat
of congestion externalities for the viability of PMP. Several maximizetheuser’sutility,theuserwillchoosetheonebyherowncriteria.
or b) to opt out of the ISP’s services if joining any service function w(q) models the capacity required to maintain the
class induces a negative utility, i.e., u(p ,q ;θ) < 0,∀i ∈ N. congestion level q given that the service class has a unit user
i i
To unify the above two cases, we assume the existence of a demand. Under Assumption 1, the implied capacity function
dummy service class 0 which satisfies p = 0 and q = 1, satisfies C((cid:15)d,q) = (cid:15)C(d,q), indicating that users will not
0 0
and therefore, any user can always choose service class 0 to perceive any difference in terms of congestion after service
gain zero utility. We denote the set of user types that choose partitioning or multiplexing. This form is a generalization of
the service class i as Θ (p,q). Notice that if the ISP offers C(d,q) = d/q (to see this, let w(q) = 1/q), which has been
i
two service classes i and j such that p > p and q > q , considered in much prior work [4, 6, 8, 10].
i j i j
no user would choose class i because it is inferior to class j WemodeltheISP’scapacityallocationplanbyavectorc(cid:44)
in terms of both price and congestion. Therefore, without loss {c : ∀i ∈ N}, where c is the capacity allocated to service
i i
of generality, we can sort the indexes of non-dummy service classi.Toguaranteethecongestionlevelofeachserviceclass,
classes in an ascending order of congestion level as follows: the ISP’s capacity allocation plan c must satisfy
p1 >p2 >...>p|N| and q1 <q2 <...<q|N|. ci ≥C(di(p,q),qi), ∀i∈N. (6)
Lemma 1: Given that |N| < +∞, we define a set of Intuitively, when the ISP has limited network capacity and
(cid:8) (cid:9)
vectors θi|i=0,1,...,|N| as some of the inequalities in (6) do not bind, it probably has
 p not taken full advantage of the capacity and can gain more
|N|
θ0 =1, θ|N| = v(q ), profits by readjusting the menu of services. We will strictly
|N| (3)
p −p characterize this in Theorem 1 in Section IV-C.
θi = v(qi)i−vi(+qi1+1), ∀i=1,2,...,|N|−1. We have by far established the relationship between the
If 1≥θ ≥...≥θ , then Θ (p,q) can be characterized by ISP’smenu(p,q)ofservicesandthecapacityallocationplan
1 |N| i
c. We next proceed to analyze the ISP’s limitation in capacity
Θi(p,q)=[θi,θi−1], i=1,2,...,|N|. (4) and formulate the ISP’s profit-maximizing problem.
Lemma13 statesthatifeachserviceclasscancapturesome
users,thesetofusertypescapturedbytheserviceclassiisan
interval[θi,θi−1].Ifthereexistsanyserviceclassthatcaptures C. Profit Maximization Problem
no users, we can always exclude it from the ISP’s menu of
serviceswithoutaffectingitsprofitsandcapacityconsumption.
WefirstanalyzethenetworkcapacityconstraintsoftheISP.
Thus, without loss of generality, we assume that the ISP does
In this work, we focus on two typical scenarios:
not offer such empty service classes and the condition of
Lemma 1 naturally holds, and consequently, the population • (Fixed Capacity) The ISP’s services are supported by an
of users that choose service class i can be calculated as existing network infrastructure and hence face a maxi-
(cid:90) (cid:90) θi−1 mum network capacity, denoted by CM.
di(p,q)(cid:44) f(θ)dθ = f(θ)dθ, (5) • (Variable Capacity) The ISP may invest to expand its
Θi(p,q) θi network infrastructure. In this scenario, the ISP needs
(cid:80)
and the ISP’s revenue can be expressed as p d (p,q). to find out the optimal amount of capacity to consume
i∈N i i
(invest). We capture the cost of consuming capacity c in
B. Network Congestion and Capacity Allocation total by S(c), a continuously nondecreasing function.
SincetheISPusuallyhaslimitednetworkcapacity,itneeds
The ISP aims to find the optimal service menu (p,q)
to design a capacity allocation plan to guarantee the promised
and the capacity allocation plan c subject to the capacity
servicequalityintermsofcongestionineachserviceclass.In
constraints in order to maximize its profits. In the fixed
general, the required amount of network capacity by a service
capacityscenario,theallocatedcapacityshouldnotexceedthe
class is determined by its promised service congestion and (cid:80) (cid:0) (cid:1)
maximumcapacity,i.e., C d (p,q),q ≤C ,andthe
user population. For each service class i ∈ N, we model the i∈N i i M
ISP’s profit maximization problem is formulated as follows.
implied capacity to guarantee a congestion level q under the
i
(cid:88)
aggregate user population di as a function C(di,qi), which maximize J = pidi(p,q) (7)
decreases in q and increases in d . N,p,q
i i i∈N
Assumption 1: The implied capacity function satisfies subject to Θ (p,q)={θ ∈Θ:i=argmax θv(q )−p },
i j j
C(d,q)(cid:44)d·w(q), where w(q):[0,1]→R is continuously j∈N
+ (cid:82)
differentiable, decreasing in q and satisfies w(1)=0. d (p,q)= f(θ)dθ, ∀i∈N,
i Θi(p,q)
The form of implied capacity function in Assumption 1 and (cid:80) C(cid:0)d (p,q),q (cid:1)≤C .
i∈N i i M
modelsthecapacitysharing[4,6]natureofnetworkservices,
under which the service capacity is shared by all users. The Likewise, in the variable capacity scenario, the ISP’s invest-
(cid:80)
mentincapacityischaracterizedbyS( C(d (p,q),q )),
i∈N i i
3Duetospacelimitation,wedeferalltheproofstotheAppendix. and the profit maximization problem is formulated as follows.
defines the maximum utility that can be achieved by users of
(cid:18) (cid:19) type θ under their optimal choice of services. The advantage
(cid:88) (cid:88) (cid:0) (cid:1)
maximize J= pidi(p,q)−S C di(p,q),qi (8) ofthedualapproachisthattheISP’sstrategy(Q,p(·))canbe
N,p,q
i∈N i∈N equivalentlyrepresentedbytheuser’sindirectutilityV(·)and
subject to Θi(p,q)={θ ∈Θ:i=argmax θv(q)−p(q)} optimal choice q(·), both of which have favorable properties
j∈N
(cid:82) and a fixed domain, i.e., Θ=[0,1]. Next, we characterize the
and d (p,q)= f(θ)dθ, ∀i∈N.
i Θi(p,q) properties of q(·) and V(·).
By far, we have formulated the ISP’s profit maximization Lemma 2: The user’s optimal choice function q(·) :
problems as nested optimization problems with equilibrium [0,1]→Q is nonincreasing in θ.
constraints. In the next section, we demonstrate how they can TheLemma2statesthatauserwithhighervaluewouldnot
be transformed into more tractable optimal control problems choose a service class with worse congestion, and whatever
by using a dual approach and derive the analytic solutions. the user’s criteria of breaking ties of optimal choices is, the
monotonicityofq(·)alwaysholds.Animmediateconsequence
IV. OPTIMALCONTROLFRAMEWORK is that q(·) is piecewise continuous.
Two of the main challenges in solving the nested opti- ByDefinition1,anyuseroftypeθ’smaximumutilityV(θ)
mization problems (7) and (8) include: 1) the set of optimal can also be expressed by the following equation:
service classes N∗ is hard to predetermine, and 2) the sets
V(θ)=θv(q(θ))−p(q(θ)), ∀θ ∈Θ. (11)
of user demand Θ (p,q) are subject to the nonlinear user
i
utility maximization problems which require different treat- Furthermore, according to the envelope theorem [12], V(·) is
mentsdependingontheconnectednessofN.Toaddressthese absolutely continuous and hence differentiable almost every-
problems,weneedaunifiedandsuccinctrepresentationofthe where, and thus we have
ISP’s strategy. This section is organized as follows. Subsec-
V(cid:48)(θ)=v(q(θ)), for almost all θ ∈Θ. (12)
tion A introduces a dual representation of the ISP’s strategy
and subsection B reformulates the ISP’s profit maximization
The following lemma allows us to represent an ISP’s pricing
problem into a tractable optimal control problem using a dual
strategy using V(·) and q(·).
approach. In subsection C, we presents the analytic solution
Lemma 3: Let q(·) and V(·) be any two mappings from
to the ISP’s optimal service differentiation. In subsection D, Θ to R and Q the range of q(·). If q(·) is nonincreasing and
we study the characteristics of the optimal solution.
(cid:90) θ
A. Dual Approach V(θ)=V(0)+ v(q(s))ds, ∀θ ∈Θ,
0
WedenoteQ⊆[0,1]asasetofprovidedservicequalitiesin
then there exists an unique pricing p(·):R →R such that
termsofcongestionandtheISP’spricingandcapacityalloca- +
tion plan can be represented by two mappings p(·):Q→R+ V(θ)=max θv(q)−p(q) and q(θ)∈argmax θv(q)−p(q).
and c(·) : Q → R , which map levels of congestion to the q∈Q q∈Q
+
corresponding capacities and prices. As mentioned before, we By the envelope theorem [12], the indirect utility function
assumethatQcontainsadummyserviceclasswithcongestion V(·) and the user’s choice function q(·) meet the condition in
level q = 1 and p(q) = c(q) = 0. Since the optimal service Lemma3.Therefore,giventheuser’schoicefunctionq(·)and
domainQ∗ ofthemappingsisalsoundetermined,thestrategy an initial level V(0) of the indirect utility, we can recover the
space is difficult to handle. To this end, we introduce a dual ISP’s service differentiation strategy as follows.
approach, widely used in economics [2, 14], to describe the • The set of provided service qualities is Q=q([0,1]).
ISP’s decision in terms of the user’s choice, rather than the • The pricing strategy is p(q) = ϑv(q)−(cid:82)ϑv(q(s))ds−
0
levels of congestion and prices. V(0), where ϑ is the type that satisfies q(ϑ)=q.
Definition 1: For any type θ, we define q(θ) as the chosen
Given the above discussions, the ISP’s profit maximization
service quality that maximizes the user’s utility, satisfying
problemboilsdowntofindingthecorrespondinguser’schoice
q(θ)∈argmaxθv(q)−p(q). (9) function q∗(·) and the indirect utility level V∗(0) under the
q∈Q optimal service differentiation scheme Q∗ and p∗(·)5.
The indirect utility of user type θ is defined by
B. Optimal Control Framework
V(θ)(cid:44)max θv(q)−p(q). (10) We demonstrate that the ISP’s profit maximization problem
q∈Q
can be transformed into an optimal control problem, the solu-
When there exist multiple optimal choices, we assume that
userθ choosesonefromthemaccordingtoherowncriteria.4 5Wetakeadetourtoshowtheexistenceanduniquenessofoptimalstrategy.
Givenafiniteset(ormoregenerallyacompactset)ofchoices, In the rest of the section, we first transform the ISP’s profit-maximizing
problem to an equivalent optimal control problem, and find out the unique
theoptimalchoiceq(θ)alwaysexists.TheindirectutilityV(θ)
candidatesolutionthatsatisfiesthenecessarycondition,i.e.,thePontryagin’s
MaximumPrinciple[5].Finallyweusethesufficientconditionstoprovethat
4AsweexplainintheAppendix,thiswillnotaffectthecorrectnessoflater thesolutionisindeedtheoptimalstrategy.Weleavethedetailstotheproof
analysis. ofTheorem1intheAppendix.
tiontowhichistheuser’schoicefunctionq∗(·)andtheindirect • We first consider a relaxed optimal control problem
utility level V∗(·) under the optimal service differentiation. in which the monotonicity constraint is removed. By
Lemma 4: Under the ISP’s optimal service differentiation using the Pontryagin’s Maximum Principle [5], we find
scheme, the initial level of indirect utility satisfies V∗(0)=0. a candidate optimal control solution q(·).
(cid:101)
Lemma 4 states that under the ISP’s optimal service differ- • We check whether q(cid:101)(·) satisfies the sufficient conditions
entiation, users with the lowest valuation, i.e., θ =0, obtains for optimality. If so, q(cid:101)(·) is indeed an optimal control of
zero utility. Consequently, we can set the indirect utility level the relaxed problem and we proceed to the next step.
V(0) to be zero when searching for the optimal strategy. • If q(cid:101)(·) meets the monotonicity constraint, then it is also
TocalculatethetotalprofitsoftheISPusingV(·)andq(·), anoptimalcontroloftheoriginalproblem(6),i.e.q∗(·)=
we can perform integration with respect to the user type θ q(cid:101)(·);otherwise,weconductfurtheranalysisbasedonq(cid:101)(·).
instead of over the service class domain Q, i.e., Weusethefixedcapacityscenariotoillustratethekeysteps
(cid:32) (cid:33) insolvingtherelaxedoptimalcontrolproblem.Wefirstdefine
(cid:90) (cid:90) (cid:90) 1
p(q) f(θ)dθ dq = [θv(q(θ))−V(θ)]f(θ)dθ. the Hamiltonian H as follows:
Q Φ(q) 0
H[θ,q(·),V(·),W(·),λ (·),λ (·)] (16)
1 2
The consumed capacity can be calculated by integrating the
(cid:44)[θv(q(θ))−V(θ)]f(θ)+λ (θ)v(q(θ))+λ (θ)w(q(θ))f(θ),
unit capacity function w(q) over the user types θ as follows. 1 2
Lemma 5: The total consumption of network capacity in where λ (·) and λ (·) are the so-called co-state variables
1 2
(cid:82)1
the system can be calculated by w(q(θ))f(θ)dθ. satisfying the following transversality conditions:
0
By Lemma 5, we can define a state variable 
dλ ∂H
W(θ)(cid:44)(cid:90) θw(q(s))f(s)ds (13) ddλθ1 =−∂∂VH =f(θ)
0 2 =− =0
to measure the aggregate capacity consumption by users with λdθ(1)=0∂,Wλ (1)≤0, λ (1)[W(1)−C ]=0.
1 2 2 M
valuationlowerthanθ.Inparticular,W(1)isthetotalcapacity
consumption of the entire system. It immediately follows that λ1(θ)=F(θ)−1 and λ2(θ) is a
Lemma 6: The user’s choice function q∗(·) under the op- constant over [0,1]. If all the network capacity is consumed,
timal service differentiation can be determined by solving the i.e.,W(1)−CM =0,theslacknessconditiongivesλ2(1)<0;
following optimal control problems. otherwise λ2(1) = 0. In either case, we define λ2(θ) = −µ
a) For the fixed capacity scenario: where µ is a nonnegative real constant.
TheeconomicinterpretationoftheHamiltonianandtheco-
(cid:90) θM
maximize J = [θv(q(θ))−V(θ)]f(θ)dθ state variables are as follows. The co-state constant µ is the
q(·) 0 shadow price of the network capacity which evaluates the
subject to q(·) nonincreasing, q(θ)∈[0,1] ∀θ, contribution to total profits by consuming per unit capacity.
V(cid:48)(θ)=v(q(θ)), V(0)=0, The Hamiltonian H is the surrogate profit the ISP extracts
from the users of type θ, which consists of three components
W(cid:48)(θ)=w(q(θ))f(θ), W(0)=0,
on the right-hand side of (16). The first component is the
and W(θM)≤CM. (14) directimpactofthecontrolvariableq(·)ontheobjectfunction
[θv(q(θ)) − V(θ)]f(θ). The control variable q(θ) also has
b) For the variable capacity scenario:
an indirect impact on the object function: it influences the
(cid:90) 1
value of V(θ + δθ) in the next infinitesimal type by the
maximize J = [θv(q(θ))−V(θ)]f(θ)dθ−S(W(θ ))
q(·) 0 M differential V(cid:48)(θ)=v(q(θ)), which is captured by the second
subject to q(·) nonincreasing, q(θ)∈[0,1] ∀θ, component. Moreover, the third component characterizes the
V(cid:48)(θ)=v(q(θ)), V(0)=0, cost of capacity consumption.
According to the Pontryagin’s Maximum Principle, if an
W(cid:48)(θ)=w(q(θ))f(θ), W(0)=0. (15)
optimalcontrolq(·)exists,thenalongtheoptimaltrajectories,
(cid:101)
The above problems (14) and (15) are more tractable than q(cid:101)(·) is the point-wise maxima of the Hamiltonian H:
the original ones (7) and (8). In these two optimal control
[θv(q)−V(cid:101)(θ)]f(θ)+[F(θ)−1]θv(q)−µw(q)f(θ), (17)
problems,theuser’schoicefunctionq(·)isthecontrolvariable
in the functional space {q(·)∈R[0,1]|q(·) nonincreasing} and where V(cid:101)(·) is the optimal trajectory of state. The economic
the indirect utility function V(·) and capacity consumption interpretation is that to maximize its total profits, the ISP has
function W(·) are the so-called state variables. to maximize the surrogate profit it extracts from each type of
users. Neglecting the components irrelevant to q in (17), we
C. Optimal Solution see that q(·) is the point-wise maxima of Ψ(θ,q), defined by
(cid:101)
(cid:18) (cid:19)
Notice that the monotonicity constraint on q(·) requires 1−F(θ)
Ψ(θ,q)(cid:44) θ− v(q)−µw(q). (18)
specialtreatments.Inparticular,weadoptthefollowingsteps. f(θ)
Byfarwehaveillustratedthefirststep,i.e.,tousetheMax- The upper bound C¯ defined in Theorem 1 is a threshold
imum Principle to find the possible candidate optimal control that determines whether the system’s capacity is abundance.
q(·). Due to space limitation, we show the remaining steps in Noticethattheparameterizationofourmodelisquiteflexible.
(cid:101)
theproofofTheorem1andproceedtopresentourresults,i.e., When w(0)=+∞, we have C¯ =+∞. In this case, q =0 is
q∗(·)andthecorrespondingoptimalservicedifferentiationQ∗ unreachable,whichmodelsthesituationwheretheuseralways
and p∗(·). We introduce the following Assumption 2 and 3 to perceive the congestion externality. On the other hand, when
facilitate further analysis. w(0) < +∞, we have C¯ < +∞. This setting can be used
Assumption 2: The virtual valuation function G(θ)(cid:44)θ− to model the situation where users perceive no congestion ex-
1−F(θ), which is the coefficient of v(q) in (18), has a unique ternality when congestion level drops below certain threshold.
f(θ)
zero point θ on (0,1) and is strictly increasing on (θ ,1). From Theorem 1, we see that if the ISP’s network capacity
0 0
is limited, i.e. C < C¯, all the inequalities in (6) will be
ThevirtualvaluationfunctionG(·)isfirstdefinedbyMyer- M
binding under the ISP’s optimal service differentiation.
sonin[16]designingrevenue-optimalauctions.Onesufficient
condition for Assumption 2 is the standard monotone hazard Theorem 1 characterizes the optimal user choice function
rateassumptioninthemechanismdesignliteraturesuchas[7]. q∗(·) for the fixed capacity scenario. With necessary modifi-
Assumption2issatisfiedbymanycommondistributions,e.g., cations,wecanapplysimilarprocedurestoderivethesolution
normal,exponentialandbetadistributions.WedenoteG−1(θ) for the variable capacity scenario as follows.
as the inverse function of G(θ) over the interval (θ ,1). Theorem 2: For the variable capacity scenario, if the cost
0
Assumption 3: We define h(q)(cid:44)w(cid:48)(q)/v(cid:48)(q) as a virtual functionS(·)isconvexwithS(0)=0,theoptimaluserchoice
capacity function and assume that it is decreasing in q over functionq∗(·)andtheassociatedcapacityconsumptionW∗(1)
the interval (0,1). are uniquely determined by the following set of equations.
a) The user choice function:
The virtual capacity function h(·) is defined as the ratio of
the marginal unit capacity w(cid:48)(q) needed to support service (cid:40)1, ∀θ ∈[0,θˆ);
quality at a congestion level q to the marginal value of a q∗(θ)=
h−1(G(θ)/S(cid:48)(W∗(1))), ∀θ ∈[θˆ,1].
user θ whose best choice of quality is q. h(q) measures
the marginal capacity needed so as to marginally increase b) The marginal user θˆthat uses the network services:
the user value at the level q of congestion in the system.
Assumption 3 intuitively states that more capacity is required θˆ=G−1(cid:0)S(cid:48)(cid:0)W∗(1)(cid:1)h(1)(cid:1).
of the service class with milder congestion to further improve
c) The capacity equation:
users’ achieved values. Furthermore, we define a function
h−1(·)overtheinterval[h(1),+∞),whosevalueistheinverse (cid:90) 1
w(q∗(θ))f(θ)dθ =W∗(1).
of h(·) over [h(1),h(0)] and zero over (h(0),+∞).
0
Under Assumption 2 and 3, we are able to derive a concise
After we have obtained q∗(·) by the above theorems, the op-
analytic solution of the optimal control problem and simplify
timal service differentiation scheme becomes straightforward.
the presentation of our results as follows.
Theorem 1: For the fixed capacity scenario, let C¯ (cid:44)[1−
Corollary 1: Under the ISP’s optimal service differentia-
F(θ0)]w(0). If CM < C¯, the user choice function q∗(·) and tion, the service classes form an interval Q∗ =[q∗(1),q∗(θˆ)],
the shadow price µ are uniquely determined by the following
and the associated prices satisfy
set of equations.
a) The user choice function: (cid:90) ϑ
p∗(q)=ϑv(q)− v(q∗(s))ds, ∀q ∈Q,
(cid:40)
1, ∀θ ∈[0,θˆ); 0
q∗(θ)= h−1(G(θ)/µ), ∀θ ∈[θˆ,1]. where ϑ is the user type such that q∗(ϑ) = q. Furthermore,
p∗(·) is continuously differentiable and decreasing in q.
b) The marginal user6 θˆthat uses the network services:
θˆ=G−1(µ·h(1)). D. Characteristics of the Optimal Solution
In this subsection, we study the structures and properties of
c) The capacity equation:
theoptimalservicedifferentiationschemeandthecorrespond-
(cid:90) 1 ing user choices. We first concentrate on the fixed capacity
w(q∗(θ))f(θ)dθ =C .
M scenario and the following corollary characterizes the shadow
0
price µ of network capacity.
Otherwise, the ISP’s optimal differentiation is to provide a
single-class service, which satisfies Q={0} and p∗(0)=θ . Corollary 2: For any given user distribution F(·) satisfy-
0
ingAssumption2,theshadowpriceµdecreasesinthesystem
6Themarginaluserθˆisthetypeofuserthatisindifferentbetweenjoining capacity CM with the asymptotic limit limCM→C¯µ=0.
andopting-outoftheservices.Userswithvaluationsbelowθˆdonotjointhe Corollary 2 intuitively states that the marginal value of
serviceswhileuserswithvaluationsaboveθˆdo. capacitydecreaseswhentheISPhasmorecapacity,acommon
diminishingreturneffect,andthismarginalvaluedropstozero the cost function S(·). Therefore, the above discussions can
when the capacity becomes abundant. be readily extended to handle the variable capacity scenario.
This next result explicitly shows the type of users that will Insummary,Corollaries3and4showthatanISP’soptimal
be served under an ISP’s optimal service differentiation. service differentiation scheme always offers service classes
Corollary 3: Under an ISP’s optimal service differentia- withsufficientlylowlevelsofcongestionandhighpricessuch
tion, users with values lower than the marginal user value θˆ that users with low values do not subscribe to the ISP, even
will not use any network service. In particular, the marginal if it has abundant network capacity. If the ISP expands its
user value θˆsatisfies that capacity, its market share should be increased; if the ISP’s
a) if h(1) = 0, θˆ is the unique solution to the equation networkcapacityissufficient,i.e.,morethanCˆ,itisoptimalto
G(θˆ)=θˆ− 1−F(θˆ) =0, i.e. θˆ=θ . bunchupthehigh-endusersinthebestserviceclasswithq =
f(θˆ) 0 0. Furthermore, when the users’ valuations lean towards high
b) if h(1)>0, θˆdepends on both the user distribution F(·) values, the ISP needs to sacrifice part of the low-end market
and the fixed capacity CM. Given any F(·), θˆdecreases share and dedicate its capacity to premium service classes for
in CM with the asymptotic limit limCM→C¯ =θ0. high-end users so as to maximize its profits.
Corollary3statesthattheoptimalservicedifferentiationoffers
V. DYNAMICSOFOPTIMALSERVICEDIFFERENTIATION
the service classes with sufficiently low levels of congestion
and high prices such that the users with low value will not In this section, we study the dynamics of an ISP’s optimal
use any service, even when the network capacity is abundant. servicedifferentiationandthecorrespondinguserbehavior.We
The intuition behind Corollary 3 is that by offering premium firstchoosetheparametersofthesystem.Inparticular,weuse
services with high quality and high prices, the gains from the the exponential form e−q to construct the user’s satisfaction
high-end users outweigh the loss of low-end market share. discount on congestion level. As we normalize the domain of
Next,weturntothestructureoftheoptimalchoicesofusers congestion q to be [0,1], we adopt a normalized form v(q)=
with values θ above θˆ. Letting θ¯(cid:44)min{G−1(µh(1)),1}, we e−q−e−1 which satisfies v(0) = 1 and v(1) = 0. We further
1−e−1
rewrite their choices q∗(θ) as follows. model the implied capacity by the quadratic convex function
w(q)=(1−q)2,underwhichthemarginaldemandofcapacity
 (cid:18)G(θ)(cid:19)
h−1 , ∀θ ∈[θˆ,θ¯]; is increasing as the required service quality becomes better,
q∗(θ)= µ i.e., as q decreases. Under these settings, the virtual capacity
0, ∀θ ∈[θ¯,1]. function satisfies h(0)<+∞ and h(1)=0.
Overtheinterval [θˆ,θ¯],auser oftypeθ joinstheservice class A. Impacts of Network Capacity Constraint
q∗(θ) such that G(θ) = µh(q∗(θ)), i.e., the user’s virtual
We study the dynamics of the ISP’s optimal service differ-
valuation G(θ) equals the cost of maintaining service class
entiation with respect to network capacity constraint, i.e., the
q∗(θ), which is the shadow price multiplied by the virtual
maximum capacity C in the fixed capacity scenario and the
M
capacity. In other words, under the ISP’s optimal service
cost function S(·) in the variable capacity scenario.
differentiation,eachuseroftypeθwillchoosetheserviceclass
with highest quality and price that it can afford. Moreover,
U[0,1] U[0,1]
0.8 1
bhwoihguehnnedrehdt(h0ab)ny<tµhh+is(∞0b)o,,utanhnded.ctIohnsettuhosifsermc’saasivnei,trathiuniagilnhgv-easnluedravtuiiocsneesrsmiswiguihlpltpbbeeer 0.6 CCCCMMMM====0000::::0013256369945505JJJJ$$$$====0000::::0012475254054283 CCCCMMMM====0000::::0013256369945505
)
bunchedupintheserviceclassq =0.Thefollowingcorollary $p0.4 $3(q0.5
characterizes this bunching phenomenon.
0.2
Corollary 4: When h(0) < +∞, there exists a threshold
Cˆ <C¯ (cid:44)[1−F(θ )]w(0) such that 0 0
0 0 0.5 1 0.5 0.6 0.7 0.8 0.9
a) If C ≤Cˆ, q(θ) is strictly decreasing in [θˆ,1], implying q 3
M
that all the users that actually choose the ISP’s services
Fig.1. Optimalpricingunderdifferentmaximumcapacity
are completely differentiated under the optimal strategy.
We first investigate the former one. Fig 1 plots the ISP’s
b) If Cˆ < C < C¯, we have θ¯< 1 and q(θ) = 0 for all
M optimalservicedifferentiation(ontheleft)andthecorrespond-
θ ∈ [θ¯,1], implying that high-end users are bunched up
ing user’s choice function (on the right) when user type θ
in the best service class with q =0.
follows the uniform distribution U[0,1]. Each curve in the
When h(0) = +∞, q(θ) is strictly decreasing on [θˆ,1], figures represents a different C and J∗ is the corresponding
M
implying again that users are completely differentiated. optimal profits. We observe that when the maximum capacity
We can analyze the solution for the variable capacity C increases, 1) both the ISP’s optimal pricing curve and
M
scenarioinasimilarway.InTheorem2,themarginalpriceof the user choice curve shift downwards, 2) only users with
network capacity at the optimal consumption level W∗(1) is valuation θ ∈[0.5,1] choose the services, 3) when capacity is
S(cid:48)(W∗(1)),whichplaysthesameroleastheshadowpriceµin sufficient, high-end users are bunched onto the service class
Theorem1anddependsonboththeuserdistributionF(·)and q =0,and4)theISP’sprofitrises.Thesecondobservationisa
0.5 0.25 1 1
,=0:5 J$=0:0868
0.4 0.8 ,,==12 JJ$$==00::11025423 0.8
0.2 ,=3 J$=0:1388 ,=0:5
$)1(0.3 $J $p0.6 3)(0.6 ,,==12
W 0.4 $q0.4 ,=3
0.15
0.2
0.2 0.2
0.1 0.1 0 0
0 2 4 6 0 2 4 6 0 0.5 1 0.4 0.6 0.8 1
t t q 3
Fig.2. Impactsofthecostfunction
Fig.3. Optimalpricingunderdifferentuserdistributions
demonstrationofCorollary3thatwhenh(1)=0,themarginal
attract high-end users to use them so as to increase profits.
user of type θˆ is independent of C . Besides, the observed
M In the variable capacity scenario, when the cost of capacity is
bunching phenomenon is an illustration of Corollary 4. We
cheaper, the ISP has incentives to purchase more capacity to
summarize the other observations in the following corollaries.
obtain higher profits.
Corollary 5: Under a fixed user distribution F(·), denote
B. Impacts of User Distribution
the ISP’s optimal range of service qualities under the maxi-
mum capacity C and C as Q and Q , respectively. If Inthissubsection,westudythedynamicsoftheISP’sopti-
M1 M2 1 2
C <C , then Q ⊂Q .7 malservicedifferentiationwithrespecttotheuserdistribution.
M1 M2 1 2
Corollary5indicatesthatwhentheISP’smaximumcapacity In particular, we consider a family of distribution functions
is larger, its optimal service differentiation offers more pre- F(θ) = θα for θ ∈ [0,1]. If α = 1, the users’ values follow
mium classes. We also observe that the ISP should decrease theuniformdistributionU[0,1];otherwise,theusersareeither
the prices to maximize its profit under this scenario. leaning towards higher (α>1) or lower (α<1) values.
Corollary 6: UnderafixeduserdistributionF(·),anISP’s Fig 3 plots the ISP’s optimal pricing curves (left) and the
optimal profit J∗ increases with its maximum network capac- corresponding user choice curves (right) under various user
ity C over (0,C¯) and have the following asymptotic limit: distributions and fixed maximum capacity C = 0.1. We
M M
observethatwhenuservaluationshiftstowardsthehigherend,
lim J∗ =θ [1−F(θ )]. (19)
0 0 1) the ISP’s optimal pricing curve and the corresponding user
CM→C¯
choice curve both shift upwards, 2) the ISP offers a narrower
Corollary 6 states that if an ISP’s capacity is not abundant,
range of low congestion services and fewer users choose the
i.e., C < C¯, it can gain more profits under the optimal
M network service, 3) the bunching effect vanishes when α is
service differentiation when its maximum capacity expands.
large, and 4) the ISP’s profit rises. The second observation
Next we study the variable capacity scenario. We focus on
is a demonstration of Corollary 3, i.e., when α increases, the
the impacts of cost function S(·) on the optimal consumption marginal user value θˆincreases. The third observation is due
W∗(1)ofcapacityandtheprofitsJ.Tothisend,wechoosea to the fact that the threshold C¯ (cid:44) [1−F(θ )]w(0) increases
0
family of cost functions parameterized by a single parameter
with the parameter α.
t : S(c) (cid:44) 0,∀c ∈ [0,0.1] and S(c) (cid:44) t(c − 0.1)2,∀c ∈
Summaryofimplications:Whentherearemorehigh-value
(0.1,+∞). Our settings model the scenario where the ISP’s
users in the market, the ISP’s optimal service differentiation
existing capacity equals C = 0.1 and only need to pay for
will focus on these high-value users by offering a narrower
costsoftheadditionalcapacity.Ingeneral,alargertmeasures
range of premium service classes and charging higher prices,
higher costs for expanding capacity.
thus extracting more profits from these high-value users.
Figure 2 plots the optimal consumption W∗(1) of capacity
(on the left) and the optimal profits J∗ (on the right) versus C. Comparison with the Optimal Single-Class Service
theparametertvaryingalongthex-axis.Weobservethatboth We evaluate whether an ISP has the incentive to implement
W∗(1) and J∗ decrease with t, which indicates that when the service differentiation so as to obtain higher profits. To this
cost of expanding capacity is cheaper, it is optimal for the end, we use the optimal pricing strategy of a single-class
ISP to purchase more capacity so as to gain more profits. In service8 (p∗,q∗) as our benchmark and compare the ISP’s
particular, when t→+∞, the ISP will not invest in any extra profits and the total user surplus under our optimal service
capacity and the optimal consumption level is W∗(1)=0.1. differentiation with that under the benchmark. We conduct
Summary of implications: In the fixed capacity scenario, simulations under the various user distributions. Since the
when the ISP increases network capacity, its optimal service general trends are almost the same, we adopt the uniform
differentiation scheme will a) enlarge the range of service distribution U[0,1] as the representative setting.
congestionsbyofferingmorepremiumservices,andb)charge Figure4plotstheISP’sprofitsJ∗ (ontheleft)andthetotal
each service class a lower price. In this scenario, the ISP is user surplus s (cid:44) (cid:82)1V(θ)dθ (on the right) versus the maxi-
0
able to allocate more capacity for the premium services and mum capacity C under the two strategies. We observe that
M
7We also observe that the optimal prices satisfy p2(q) < p1(q) for all 8The optimal single-class pricing strategy (p∗,q∗) can be solved by
q∈Q1∩Q2\{1}.However,thisobservationhasnotbeenrigorouslyproved. restricting|N|=1intheoptimizationproblem(7).
0.25 0.1 low value will not use the services, even when the network
OptimalDi, OptimalDi,
BestSingleClass 0.08 BestSingleClass capacity is abundant. When the ISP expands its capacity, its
0.2 0.06 market share will be increased by offering more premium
$J s service classes and reducing prices. Furthermore, when there
0.04
0.15 exist more high-value users, the ISP should focus on them
0.02
by offering a narrower range of premium service classes and
0.1 0 charging higher prices. Our results also show that the ISP has
0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
CM CM strongincentivestoimplementservicedifferentiationwhenits
Fig.4. BenefitsofServiceDifferentiation capacityisscarceandsuggestthatregulatorsshouldencourage
CM=0.11 CM=0.22 such practices as they increase the total user surplus.
0.2
OptimalDi, OptimalDi,
0.15 BestSingleClass 0.2 BestSingleClass ACKNOWLEDGMENT
0.15
s 0.1 s This work is supported by National Science Foundation of
0.1
China (61379038), the Huawei Innovation Research Program
0.05
0.05 (HIRP) and Ministry of Education of Singapore AcRF grant
0 0 R-252-000-572-112.
0.4 0.6 0.8 1 0.4 0.6 0.8 1
3 3
Fig.5. Surplusofeachuser APPENDIX
A. Technical Proofs
when the capacity is scarce, the ISP has stronger incentives
to differentiate services so as to gain more profits. However, Proof of Lemma 1: If 1 ≥ θ1 ≥ ... ≥ θ|N|, we define
asthecapacitykeepsgrowing,theincreaseinprofitsbecomes |N| intervals as follows:
marginal. This observation is consistent with Corollary 6 and I (cid:44)[θ ,1], ∀i∈N.
Theorem1,whichstatethatwhenC approachesC¯,theISP’s i i
M
profit converges to the upper bound which is reached without Note that for i=1,2,...,|N|−1,
thecapacityconstraints.Figure1alsoillustratesthis:thecurve p −p
with CM = 0.3345 shows that when capacity expands, the θ ∈Ii ⇐⇒ θ ≥θi = v(q )i−vi(+q1 )
i i+1
majority of users are bunched up in the best service class
⇐⇒ θv(q )−p ≥θv(q )−p
q = 0 and the profit gained by service classes q ∈ (0,1] is i i i+1 i+1
small compared to that of the service class q =0. ⇐⇒ θ prefers class i to class i+1
The trend of total user surplus s is similar, except for that
and for i = |N|, θ ∈ I ⇐⇒ θ subscribes to the ISP.
i
the increase is also marginal when capacity is scarce. To see
Therefore, we have
this more clearly, we plot the user surplus s versus the user
type θ when CM =0.11 (on the left) and CM =0.22 (on the θ ∈[θi,θi−1] ⇐⇒ θ ∈IN ∩...∩Ii∩I¯i−1∩...∩I¯1,
right) in Figure 5. We observe that under the ISP’s optimal
which means θ prefers class i to i+1, i+1 to i+2, ..., N
service differentiation, more users subscribe to the services.
to opting-out, and i to i−1, ..., 2 to 1. Hence we have
However, when the capacity is insufficient, mid-tier uers are
sacrificed to subsidize the high-end and low-end users. Θ (p,q)=[θ ,θ ], i=1,2,...,|N|.
i i i−1
Summary of implications: The ISP has strong incentives
to implement service differentiation if its network capacity is
Proof of Lemma 2: Suppose q(·) is not nonincreasing in
scarce and under this scenario, regulators should encourage
θ, then there exist two user types θ and θ such that θ >θ
such practices because more users will have access to the 1 2 1 2
and q(θ ) > q(θ ). Let q = q(θ ), q = q(θ ), p = p(q )
services and the total user surplus is higher. 1 2 1 1 2 2 1 1
and p = p(q ). Since the pricing function p(·) is strictly
2 2
VI. CONCLUSION decreasing,wehavep <p .Bythedefinitionofuser’schoice
1 2
In this work, we study an ISP’s optimal service differen- functionq(·),wehavethefollowingtwoincentiveinequalities:
tiation strategy subject to its network capacity constraints in
θ v(q )−p ≥θ v(q )−p ,
1 1 1 1 2 2
a congested network market. In particular, we consider two
θ v(q )−p ≥θ v(q )−p .
typicalscenarios:thefixedcapacityandvariablecapacitysce- 2 2 2 2 1 1
narios.Byincorporatingthedualapproachinoptimaltaxation And they are equivalent to:
theory [14], we establish an optimal control framework to
p −p
analyze the ISP’s profit-maximizing problem. We derive the θ ≤ 2 1 ,
1 v(q )−v(q )
analytical solution to the ISP’s optimal service differentiation 2 1
p −p
and characterize its structures. Our results show that the ISP’s θ ≥ 2 1 .
2 v(q )−v(q )
optimal service differentiation scheme offers service classes 2 1
with sufficiently high qualities and prices such that users with which contradicts with that θ >θ .
1 2
Proof of Lemma 3: Since the utility function (1) has capacity C¯ (cid:44) [1 − F(θ )]w(0), the corresponding capacity
0
the strict and smooth single crossing differences properties consumption level.
[13], we can apply the constraint simplification theorem (see If W(1)−C = 0, the slackness condition gives µ > 0.
M
Theorem 4.3 in [13]) to derive this lemma. Then q(·) is the point-wise maxima of
(cid:101)
Proof of Lemma 4: Since u(p,q;θ = 0) = −p ≤ 0 and
u(0,1;θ = 0) = 0 (choosing the dummy service class), we Ψ(θ,q)=G(θ)·v(q)−µ·w(q).
have V∗(0)=0.
When θ ∈ [0,θ ), G(θ) < 0 and Ψ(θ,q) is decreasing in q.
0
Proof of Lemma 5: Define the set of users that choose
Therefore q(·) = 1 for all θ ∈ [0,θ ). When θ > θ , we
(cid:101) 0 0
service class q as
calculate the partial derivative with respect to q to find the
Φ(q)(cid:44){θ ∈Θ:q(θ)=q}. maxima.
∂Ψ(θ,q)
We perform summation in a user-wise fashion instead of the =G(θ)v(cid:48)(q)−µ·w(cid:48)(q)=−µ·v(cid:48)(q)[h(q)−G(θ)/µ]
∂q
class-wise one:
(cid:90) (cid:0)(cid:90) f(θ)dθ(cid:1)w(q)dq =(cid:90) (cid:0)(cid:90) f(θ)w(q)dθ(cid:1)dq Since −µ·v(cid:48)(q) > 0, the sign of ∂Ψ∂(θq,q) is determined by
h(q)−G(θ)/µ. Then under Assumption 3, we have
Q Φ(q) Q Φ(q)
(cid:90) (cid:90) (cid:40)
= (cid:0) f(θ)w(q(θ))dθ(cid:1)dq 1, θ ∈[0,θˆ)
q(·)=
Q Φ(q) (cid:101) h−1(G(θ)/µ), θ ∈[θˆ,1];
(cid:90) 1
= w(q(θ))f(θ)dθ where the marginal user is θˆ=G−1(µ·h(1)).
0
By far we have completed the first step, i.e. to solve the
candidate optimal control q(·) for the relaxed optimal control
Proof of Lemma 6: The envelop theorem [12] together (cid:101)
problem.Nextweprovethatq(·)isindeedtheoptimalcontrol
with Lemma 3 have established the one-one correspondence (cid:101)
of the relaxed optimal control problem. This is guaranteed by
between the any service differentiation strategy and the cor-
theArrowsufficiencytheorem[5]duetothattheHamiltonian
responding user choice function q(·) together with an indirect
is concave in V. As for the third step, it is easy to show that
utilitylevelV(0).FromLemma4wehavesetV(0)=0.From
(13) we obtain W(cid:48)(θ) = w(q(θ))f(θ) and the total capacity q(cid:101)(·) is nonincreasing under Assumption 3. Hence we have
q∗(·)=q(·) and the proof of Theorem 1 is completed.
consumption is W(1) by Lemma 5. The first order derivative (cid:101)
of V(·) is given by the envelop theorem. The ISP’s profits is
(cid:32) (cid:33) Proof of Theorem 2: For the variable capacity scenario,
(cid:90) (cid:90) (cid:90) 1
p(q) f(θ)dθ dq = p(q(θ))f(θ)dθ the transversality conditions become
Q Φ(q) 0 
dλ ∂H
=(cid:90) 1[θv(q(θ))−V(θ)]f(θ)dθ  dθ1 =−∂V =f(θ)
dλ ∂H
0 2 =− =0
ThenPLroeomfmofaT6heisoraemsu1m:mIanriSzeactitoionnoIfV-thCe,wabeohvaevreeisnutlrtosd.uced λd1θ(1)=0∂,Wλ2(1)=−S(cid:48)(W(cid:102)(1)).
thethreestepsoftacklingtheoptimalcontrolproblem(14).In
Therefore, the costate variable associated with W(·) is a
particular, we have proved that the candidate optimal control
q(·) for the relaxed optimal control problem is the point-wise constantλ2(θ)≡−S(cid:48)(W(cid:102)(1)).InparallelwithTheorem1,the
(cid:101) marginal cost S(cid:48)(W(cid:102)(1)) at the optimal capacity consumption
maxima of
(cid:18) (cid:19) level W(cid:102)(1) here plays the same role as the shadow price µ in
1−F(θ)
Ψ(θ,q)(cid:44) θ− v(q)−µ·w(q). the fixed capacity scenario. Assuming S(·) is nondecreasing
f(θ)
and convex, the three steps in the proof of Theorem 1 can be
The shadow price µ satisfies the slackness condition: readily applied in this scenario and we omit the details here.
µ≥0, µ·[W(1)−C ]=0
M
Proof of Corollary 1: Note that the indirect utility level
If W(1)−C <0, then we have µ=0, q(·) is the point- isV∗(0)=0,thenCorollary1isstraightforwardfromLemma
M (cid:101)
wise maxima of 3.
Proof of Corollary 2: By Theorem 1, If C <C¯, then
Ψ(θ,q)=G(θ)·v(q) M
the user choice function q∗(·) and the shadow price µ are
Under Assumption 2, G(θ) is negative on the interval [0,θ0) uniquely determined by the following set of equations.
and positive on the interval (θ0,1]. Since v(·) is decreasing a) The user choice function:
in q, it naturally follows that q(cid:101)(·) = 1 for all θ ∈ [0,θ0) and (cid:40)1, θ ∈[0,θˆ)
q(cid:101)(·) = 0 for all θ ∈ [θ0,1]. Then we have the second part q∗(θ)=
of Theorem 1. Moreover, we have obtained the abundance h−1(G(θ)/µ), θ ∈[θˆ,1];