Journal of Service Science and Management, 2012, 5, 386-402
http://dx.doi.org/10.4236/jssm.2012.54045 Published Online December 2012 (http://www.SciRP.org/journal/jssm)
Analysis of a Pricing Method for Elastic Services with
Guaranteed GoS
Marcos Postigo-Boix, José L. Melús-Moreno
Department of Telematics Engineering, Universitat Politècnica de Catalunya (UPC), Barcelona, Spain.
Email: marcos.postigo@entel.upc.edu, teljmm@entel.upc.edu
Received September 24th, 2012; revised October 26th, 2012; accepted November 10th, 2012
ABSTRACT
Service Providers (SPs), which offer services based on elastic reservations with a guaranteed Grade of Service (GoS),
should know how to price these services and how to quantify the benefits in different scenarios. This paper analyzes a
method for evaluating the price of a service based on elastic reservations with a guaranteed Grade of Service. The
method works as follows: First, the SP determines the requirements of the service that wants to offer; Second, the SP
evaluates the average rate of the accepted elastic reservations of the service with a guaranteed GoS; Third, the SP calcu-
lates the price that guarantees the GoS with an aggregate demand function that depends on a demand modulation factor
of the elastic reservations that is the mean reserved bandwidth, Bres; and Finally, the SP obtains the optimum value of
the elasticity of the reservations that gives the maximum revenue, and the required access bandwidth in this case. The
paper not only applies the method to a class i of elastic reservations when a linear-based demand and a revenue function
are selected, but it also analyzes the influence of each one of the considered parameters. This method could be extended
to the case of multiple classes of independent and guaranteed elastic services, applying the method to each service with
its estimated demand and revenue functions.
Keywords: Elastic Reservations; Streaming; GoS; Mean Reserved Bandwidth per Accepted Request;
Aggregate Demand Function; Pricing; Revenue
1. Introduction
Service Providers (SPs) want to estimate the revenue of
the services that they provide that usually depends on the
applied price to the offered services. Currently, the ser-
vices that the SPs present treat to cover a wide spectrum
of profiles in the aim to adjust them to the preferences of
their different users. In that sense there exist users that
request elastic services that could be delivered with
variable bandwidth, that is, they assume that not always
they could receive the same bandwidth for the requested
service (the bandwidth reservation for the service is elas-
tic and it fluctuates between a minimum and a maximum
values). One example of this type of elastic service is the
delivery of streaming video flows with different com-
pression levels. Users want to get high-quality for their
reservations, but also they could accept some tolerable
degradation in the quality of their reservations if the re-
duction of the price for this service is significant.
Elastic reservations require the support of new signal-
ing mechanisms other than the most commonly used to-
day, the resource ReSerVation Protocol (RSVP) [1]. As
an alternative, the Next Steps in Signaling (NSIS) [2]
protocol family allows to reserve bandwidth in a specific
range. The Internet Engineering Task Force (IETF) cre-
ated the NSIS Working Group in 2001 to solve new sig-
naling needs for reservations. Since then, several Internet
RFCs and papers have been published [3,4], including
the QoS NSIS Signaling Layer Protocol (QoS-NSLP)
that describes the procedures to signal QoS reservations
between a Desired QoS and a Minimum QoS. In our sce-
nario each one of them will respectively represent the
bandwidth that the user wants to reserve and the mini-
mum bandwidth that the user needs to work properly.
Figure 1 shows the proposed scenario for the reserva-
tion of elastic services using the QoS-NSLP-based sig-
naling mechanism. According to Figure 1, when the user
wants to watch a video he access to the website where
the SP lists their offered SLAs. In Figure 1, the SLA is
defined by the Grade of Service (GoS) and other pa-
rameters such as the Desired and the Minimum QoS,
which are respectively the highest (H) and lowest (L)
bandwidth reservations for the service, and the elasticity
of the reservations (
). In this paper this parameter, for a
class i, is defined according to (1). Thus, if the elasticity
of the reservations is 0 the Desired and the Mini- mum
QoS have the same bandwidth and the elasticity is 1 if
Copyright © 2012 SciRes. JSSM
Analysis of a Pricing Method for Elastic Services with Guaranteed GoS 387
Figure 1. Scenario of the reservation of elastic services based on the QoS-NSLP signaling mechanism.
the Minimum QoS bandwidth is 0. In the scenario of vi-
deo content distribution of Figure 1 the SP determines
both values, the highest-quality (H) and the lowest-qual-
ity (L). Thus the elasticity for the reservations of class i
would be:
1
ii
LH
i
 (1)
In this scenario, another important parameter that
helps users to qualify and to differentiate among SPs is
the reserved bandwidth per accepted request Bres,i. It
represents the effectively reserved bandwidth of class i
for each user within its specified range, that is, between
the Hi and Li. In addition, the metric ,res i
B defines the
mean reserved bandwidth per accepted request, which
establishes the mean size of the reservations of class i in
the requested range. This metric represents a demand
modulation factor of the accepted reservations in the
sense that users would desire that the SP offered the
value of this metric closer to Hi.
According to Figure 1, the SP allows their clients to
request elastic reservations with the same GoSi in an es-
tablished bandwidth range for each reservation of class i.
In this paper, some previous considerations should be
done from the scenario described in Figure 1: First, the
parameters GoS and ,res i
B are related to the reservations
of the class of service i. The reservations are represented
by their length and their reserved bandwidth of the gener-
ated session. Thus, each established session is character-
ized by the time since the user asks for the reservation
until the session ends up, and does not take into account
the features of the packets transmitted during the session;
Second, although many definitions have been used to
evaluate the GoSi for a class i of service, the evaluation
of this parameter here is based on the probability of ob-
taining an accepted reservation within the requested
range; And third, all the reservations of class i, have the
same priority. Figure 2 shows the entities involved in the
scenario described in Figure 1. Thus the SPs, which may
also act as Content Providers (CPs), offer for each class
of service, class i, a guaranteed GoSi. In that sense the
SPs should establish the appropriated agreements with
the Network Providers (NPs) to buy the necessary access
bandwidth that allow them to have the appropriated ac-
cess bandwidth (Bi) in order to guarantee the offered
GoSi.
Therefore, in this paper, it is proposed and analyzed a
method that evaluates the price of a class i of guaranteed
elastic reservations related to some of the described pa-
rameters such as: its elasticity, its guaranteed Grade of
Service, its mean reserved bandwidth and the available
access bandwidth for the reservations. Qualitatively
speaking it works as follows: First, the SP establish the
characteristics of the service that wants to offer; second,
the SP obtains the average rate of the accepted elastic
reservations of this class, class i, with a guaranteed GoSi;
third, the SP calculates the price of these reservations
that guarantee the GoSi with a demand function that also
depends on a demand modulation factor, ,resi
B. This last
parameter could be justified by the desire of users of
paying more for the reservation when the value of ,res i
B,
is closer to Hi. And finally, the SP obtains the value of
the elasticity of the reservations that gives the maximum
revenue and the optimal bandwidth that maximizes the
revenue for this elasticity. The pricing method for a class
of guaranteed elastic service also could be extended to
the evaluation of multiple classes of independent and
guaranteed elastic services, applying the obtained ex-
pressions in this paper to each considered class. However
Copyright © 2012 SciRes. JSSM
Analysis of a Pricing Method for Elastic Services with Guaranteed GoS
388
Figure 2. Entities considered.
this method is unable to evaluate and to analyze the case
of dependent services, since to deduce the appropriate
demand functions or to establish the relations between
the variables that are involved in is very difficult task
with the analytical tools used here.
It is straight to deduce qualitatively speaking some
conclusions, such as, the value of the established GoS for
the reservations determines the accepted demand of the
requested services. So, qualitatively speaking, if the
guaranteed GoS is high the accepted requests will be less
than if the guaranteed GoS is low and consequentially,
the price of the service would increase for the considered
access bandwidth. However, the SPs not only want to get
qualitative results, but they also need to establish proce-
dures to know how to quantify the price of these services
and to create the appropriated scenario to offer them.
Many questions could appear about the utility of this
method for the SPs. Thus, the first one could be for them
to try to identify the convenience of its implementation,
that is, when this method could be appropriated to im-
plement for elastic services? Other, without any specific
order, maybe when the SPs don’t have enough resources
(limited access bandwidth) and also, their users could
accept some changes in the type of service they have
requested. In this case, does the obtained revenue allow
them to get what they want? Or even, will be the ob-
tained revenue for the elastic reservations not far away or
even better than for inelastic ones? Or, the same question
could be formulated in other words, what should be the
size of the resources (access bandwidth) that prioritizes
the use of the elastic against inelastic reservations? How
many users could access to the service using elastic res-
ervations in comparison to the case of using inelastic
reservations? What is the value of the elasticity of the
reservations that gets the maximum revenue? etc. The
method must allow answering these questions to the SPs
with the aim of getting the solutions that best fit their
requirements to price elastic reservations.
In essence, this paper presents four main contributions:
An evaluation of the proposed pricing method that al-
lows to assign price to multiple classes of independ-
ent and guaranteed elastic streaming services with the
same priority and, according to some parameters such
as the elasticity of their reservations, the available re-
sources (access bandwidth), as well as, other parame-
ters that define and establish the offered services and
their demand functions. Each one of the independent
classes could be evaluated following the same proce-
dure as for a class i.
The analysis highlights the importance of the elastic-
ity when the access bandwidth is limited, as well as
the importance of its appropriated dimensioning, in
order to maximize the revenue of the SP.
The analytical evaluation of the price assignation for
a single class of elastic reservations, class i, is based
on a closed-form expression that reduces the compu-
tational complexity of the Markov models.
The demand function for each service will depend not
only on its price (€/reservation), pi, as usually is con-
sidered, but also on a demand modulation factor,
,res i
B that also depends on the access bandwidth. Al-
though the selected demand function in the paper is a
linear-based function that depends on the price and
the access bandwidth, it could be considered another
Copyright © 2012 SciRes. JSSM
Analysis of a Pricing Method for Elastic Services with Guaranteed GoS 389
one. In any case, the price will be finally obtained in-
verting this demand function.
The remainder of the paper is organized as follows.
Section 2 presents some research in this field and it is
labeled as related work. Section 3 describes the proposed
pricing method for services based on elastic reservations.
Section 4 analyzes and applies the proposed method to
evaluate the price assignation for a single class of elastic
reservations, class i, when a linear-based demand func-
tion Di and a revenue function are selected, quantifying
and highlighting the importance of the elasticity of the
reservations and the available resources (access band-
width) in the evaluation of the SPs revenue. Section 5
summarizes the main conclusions. Also this paper in-
cludes two appendices. In Appendix A, the GoSi and the
mean reserved bandwidth per accepted request ,res i
B
are
deduced by means a Markov-chain based model (based
on quadratic computational complexity) and an approxi-
mate model (based on constant complexity). As the ap-
proximate model is very close to the Markov-chain based,
the analysis of the method is done based on that because
it allows understanding more clearly the relations among
the parameters that use this method in the process of as-
signing prices in Section 4. In Appendix B, the price for
a class i of elastic reservations is analytically deduced,
taking into account the selected linear-based demand
function Di.
2. Related Work
This paper describes and analyzes a method that helps
SPs to price elastic services with guaranteed GoS, se-
lecting an aggregate demand function, D, that estab-
lishes the relation between the number of users that are
willing to get the service and the price they pay for it.
The price of each class of these services is based on: the
average rate of the accepted class of elastic reservations
with guaranteed GoS and their mean reserved bandwidth
per accepted request, ,res i
B. In [5] the parameters Bres,i
and ,res i
B were introduced and was also analyzed how is
the influence of the value of the GoSi in their evaluation
In that sense, the parameter ,res i
Bis considered as a de-
mand modulation factor for the price of the elastic reser-
vations. This paper introduces the parameter ,res i
B as a
new component in the determination of the price of the
elastic reservations and also carries out the analysis and
calculation of the elasticity of the reservations that max-
imizes a chosen revenue function. Although many meth-
ods to price services have been proposed only a few are
focused on elastic reservations but neither of them has
jointly tackled the issues treated in this paper. For exam-
ple, no papers assume the GoS as a constraint that affects
the price of the services based on elastic reservations.
Thus, in this section some papers that share part of the is-
sues related in this paper have been revised.
Reference [6] analyses the quantitative influence of the
guaranteed GoSi in the evaluation of the mean reserved
bandwidth for each reservation,res i
B. Additionally the
paper proposes a method to establish the prices of two
classes of elastic services, but differs from the paper pre-
sented here since the calculated price there didn’t have
into account the influence of the user’s demand, that is,
the price of the service always depend on the considered
aggregate demand function. The proposal presented in
this paper is totally different from the presented there,
since here there is selected an estimated (linear-based)
demand function for the service that establishes a relation
with the price of the service and the ,res i
B. Further, is
also analyzed how the elasticity and the bandwidth affect
the SP’s revenue. In [7] the same authors of this paper
described a method to price substitute guaranteed ser-
vices. There, it was selected an exponential aggregate
demand function. The prices were found inverting their
demand functions and knowing that in equilibrium it is
accomplished that the value of the average rate of ac-
cepted reservations for each class of service, that maxi-
mizes the chosen revenue function, is equal to its aggre-
gate demand function Di. Besides, the considered method,
N classes of substitute services, was only graphically
analyzed for the case of two substitute services, the ac-
cess bandwidth and the elasticity of the reservations were
fixed, the determination of the demand functions of the
two substitute services, each one depending on the price
of the other service, and the attainment of the pairs of the
accepted demand (which were obtained by trial and error
until they match an expression) that accomplished for
both services the guaranteed GoS and maximized the re-
venue. Although in this paper the used method to deduce
the price of the elastic services seems to be the proposed
in [7], there are many substantial differences in its appli-
cation. Thus, in [7] wasn’t presented any kind of analysis
of the obtained results, due to the difficulty of getting
them from the use of a Markov-chain based model. How-
ever, in this paper, the approximate analytical solution
(closed form solution) allows to establish in a clear way
the relationships between the parameters that intervene in
the method in the process of pricing elastic services. This
paper differs from the presented in [7] at least in three
aspects: First, it only treats with one class of service,
although the considered dependencies are more complex
than there. In fact, the graphical results depend among
other parameters on the access bandwidth and the elastic-
ity of the reservations, what allows getting the elasticity
and the access bandwidth that optimize the selected
revenue function. Second, the demand function of the
services depends not only on the price, but on a demand
modulation factor, ,res i
B
. Third, the utilization of the
method is based on an approximate analytical model
(very close to the simulation model) used in the evalua-
Copyright © 2012 SciRes. JSSM
Analysis of a Pricing Method for Elastic Services with Guaranteed GoS
390
tion of the GoS and ,res i
B
, and different from the model
used in [7].
Some papers present different methods to price elastic
services but the proposed solutions that are described
there are clearly separated to what is presented in this
paper. Thus in [8], the authors present a combined study
of price competition and traffic control in a congested
network where the SPs set the prices in the aim to maxi-
mize their profits. In [9] the authors present a State Esti-
mation based Internet traffic flow control system where
the objective is maximize the aggregate bandwidth utility
of network sources over their transmission rates. In [10],
the paper focuses on the provider competition aspect, in a
game theoretic setting, the traffic considered is elastic
and there are multiple types of it and each type of traffic
is sensitive to a different degree to Quality of Service
(QoS). In [11] the authors design a framework that is
composed of feedback signals and the corresponding
source adaptation scheme to provide differentiated
bandwidth service for elastic and inelastic applications.
In [12], authors propose an appropriate prioritization
pricing structure where users are provided with incen-
tives and are able to choose between two service classes.
In [13], authors present an integrated solution (integrat-
ing pricing into QoS routing) for enabling the next gen-
eration Internet to achieve the differentiated service and
availability guarantee. Reference [14] describes in a
wireless scenario an admission control algorithm that
optimizes the revenue when the QoS is guaranteed. The
price depends on the holding prices (bandwidth reserve),
the usage price (average usage, the elasticity of the traffic)
and the congestion price. References [15-17] present the
issue of pricing related to different scenarios and further
features for the offered services. Thus in [15] the authors
briefly review the state of the art and technological
growth of congestion control for integrated service net-
works since pricing is a proper tool to manage congestion,
encourage network growth, and allocate resource to users
in a fair manner. Reference [16] is one of the first books
that treat conjunctly technology and pricing and, in ref-
erence [17] the authors present a recent classification of
the proposed pricing methods in wireless networks. In
references [18-20] different methods are presented to
determine the user utilization function. Thus in [18], the
authors propose a solution for bridging the gap between
the existing theoretical work on optimal pricing and the
unavailability of precise user utility information in real
networks. In [19] users specify the utility or value they
attach to different quantities of resource using a utility
function, so the resource allocator knows the utility func-
tion of users at the time of resource allocation and then
allocates resources based on the objective of maximizing
the aggregate average utility obtained by unit time. In [20]
each user is assumed to have a utility function which is a
concave increasing function of the rate at which she
sends data through the network. The problem is to find
the vector of users’ rates such that the sum of all users’
utility functions is maximized, subject to resource capac-
ity constraints. Other references evaluate how admission
control affects the obtained GoS (considered as a techni-
cal constraint) of the services. These papers analyze not
only the case of a class of service, but for multiple ser-
vice classes. Thus in [21] authors pay their attention to
the interrelation between pricing and admission control
in QoS-enabled networks and propose a tariff-based ar-
chitecture framework that flexibly integrates pricing and
admission control for multi-domain Diffserv networks. In
[22] a comprehensive survey about Call admission con-
trol in wireless networks is shown. In [23] authors say
that traditional CAC schemes mainly focus on the trade-
offs between new call blocking probability and handoff
call blocking probability. Therefore, they introduce the
pricing as an additional dimension of call admission con-
trol process in order to efficiently and effectively control
the use of wireless network resources. In [24] authors
investigate the conditions where both BE traffic and traf-
fic explicitly requiring QoS (Guaranteed Performance,
GP) are present and they propose three CAC rules for the
GP traffic. In [25] authors utilize admission control algo-
rithms designed for revenue optimization with QoS gua-
rantees to derive optimal pricing of multiple service cla-
sses in wireless cellular networks. Other authors analyze
price assignation and propose solutions that work in dif-
ferent behavior. Thus in [26], it is described a scalable
connection management strategy for QoS-enabled net-
works to tackle the problem of appropriately provision-
ing and allocating connections. In [27] is introduced a
service model that provides per-flow bandwidth guaran-
tees, where users subscribe for a guaranteed rate. In [28]
authors consider the problem of pricing for bandwidth
provisioning over a single link. The network administra-
tor controls the resource allocation by setting a price at
every epoch, and each user’s response to the price gov-
erned by a demand function. In [29] authors investigate
the sensitivity of resource allocation and the resulting
QoS to resource prices in a reservation-based QoS archi-
tecture that provides guaranteed bounds on packet loss
and end-to-end delay for real applications. In [30] is con-
sidered the pricing and allocation issues of distributing
digital contents via Web and P2P channels. Utilizing a
game theoretic model, the allocation equilibrium with
respect to various business goals is examined. In [31] is
established a method to assign prices based on-packet
queues sizes in the networks.
3. A Pricing Method for Services Based on
Elastic Reservations
This section describes the pricing method for multiple
classes of independent and guaranteed elastic reserva-
tions. However, before going on with the method, it is
Copyright © 2012 SciRes. JSSM
Analysis of a Pricing Method for Elastic Services with Guaranteed GoS 391
convenient to take into account the difficulty of deter-
mining the aggregate demand function D
i, which is a
similar problem in many proposals of pricing services.
The knowledge of this function in advance is always, as
many researchers have pointed out, a very difficult task
that the SP needs to solve. As it is known, the aggregate
demand function usually represents the sum of individual
demands of each user that have different willingness to
pay for the service. It is hard to identify this behavior and
therefore, the curve that shows the desire of the users to
pay a price for the requested services. Thus, the SP has to
estimate by whatever means it deems adequate (analyti-
cally, by simulation, heuristically, etc.) the demand func-
tion for each service. For simplicity, this paper assumes
that in the evaluation of a service class of elastic reserva-
tions, class i in Section 4, the chosen aggregate demand
function is linear-based. Of course, if the demand func-
tion changes, the quantitative results that the method ob-
tains should be different.
The outcomes of this method are the price pi and the
value of the elasticity of each service ξi and the optimal
bandwidth that maximize the SP’s revenue. The method
consists of the following steps:
The SP determines the service requirements that limit
its feasibility. Figures 1 and 2 illustrate the first require-
ment: the SP wants to offer guaranteed and independent
services based on elastic reservations. This implies that
each service has to guarantee a particular GoSi, for the
service of each elastic reservation with elasticity ξi that is
determined by the SP in order to optimize its revenue.
Also, the service requires a Desired QoS equal to Hi
Mb/s and therefore, from (1), the Minimum QoS will be
equal to i
1
i
H
 . Other requirement could appear
from the available resources of the SP, that is, the access
bandwidth for each independent service (Bi Mb/s). As
each service allocates Bi, the sum of the reservations of
all classes should be below this access bandwidth Bi. Bi
may be limited due to several circumstances, such as the
network access technology used by the SP to offer the
service.
The SP evaluates the maximum demand (in terms of
requests per unit time) that can be allowed for the service
in order to guarantee the requirements of step 1, the value
of the GoSi for every offered class of independent elastic
reservations. In this paper, we present in Appendix A.2
an analytical expression that roughly approximates the
GoSi for the elastic services that are offered using an
access bandwidth Bi. This expression has been validated
by simulation and using the loss system model also in-
cluded in Appendix A.2.
The SP obtains the price of the service i that guaran-
tees the GoSi. In order to obtain the price for the service,
the SP needs to estimate the demand function Di by
whatever means it deems adequate. This paper assumes a
linear-based demand function, explained in Appendix B,
that depends on the price and the mean reserved band-
width ,res i
B, (a demand modulation factor that is calcu-
lated in Appendix A.3). Since ,resi
B is also dependent
on Bi, the demand function is also defined in terms of the
price of the service and Bi.
The SP evaluates the revenue Ri using the obtained
price in step 3 and finds out the elasticity of the reserva-
tions that maximizes the revenue. In addition, if band-
width was not limited, the SP could obtain the bandwidth
that optimally determines the service access bandwidth Bi.
The revenue function, used in this paper, assumes for
simplicity that only depends on the price, the rate of ac-
cepted requests and the cost of the access bandwidth.
However, it is well known that more complex expres-
sions, which may express part of the SP’s business model,
could also be used.
4. Analysis of the Method: The Importance
of the Elasticity of the Reservati o ns and
the Bandwidth on the SP’s Revenue
In this section we apply the pricing method, described in
Section 3, to analyze quantitatively how the price of a
class i of elastic reservations and the revenue change
depending on the elasticity of the reservations and the
access bandwidth of the service.
4.1. Step 1: Determining the Requirements for
the Service
Before the SP applies this method, it should define the
suitable parameters for each class of elastic service. Thus,
in this paper it is assumed a guaranteed GoSi = gi, for all
reservations, the same elasticity, and the highest band-
width of the requested reservation range is Hi that is
equal to the maximum required bandwidth to deliver the
content (Btop,i). Specifically, it is supposed a guaranteed
GoSi of 0.95, Hi = 1 Mb/s, and 3 minutes for the mean
reservation holding time. Also, other parameters for the
linear-based demand function as it is presented in Ap-
pendix B are: Dmax,i = 120 reservations/minute, Btop,i = Hi
= 1 Mb/s and Pmax,i = 10€ .
Step 2: Evaluating the maximum demand that guaran-
tees the GoS.
The SP calculates the average rate of the accepted res-
ervations,
i, that guarantees the GoSi. Using the ap-
proximation (12) in Appendix A.2, the expression for the
maximum accepted demand (2) is obtained.


1
GoS
11
ii
iiii
g
i
BH g

i

 


(2)
Figure 3 shows graphically how the maximum ac-
cepted demand that guarantees a GoSi = 0.95 changes for
different values of elasticity and bandwidth. If the elas-
Copyright © 2012 SciRes. JSSM
Analysis of a Pricing Method for Elastic Services with Guaranteed GoS
Copyright © 2012 SciRes. JSSM
392
ticity tends to 1, the demand tends to +, since the res-
ervation requests always are accepted. On the other hand,
if the elasticity is zero, the demand tends to a minimum
value if bandwidth is fixed, since a reservation with elas-
ticity 0 requires no less than Hi Mb/s. For the rest of
combinations of bandwidth and elasticity,
i increases
slightly for low-medium elasticity values and for high
elasticity values
i increases sharply approaching to a
vertical asymptote to + for elasticity equal to 1. As Bi
increases, the maximum accepted demand also increases
in a linear way with a higher slope as elasticity ap-
proaches to 1. Also it is worth to mention that the values
of the guaranteed GoSi, Hi and the holding time 1
i
have an impact on the maximum accepted demand. Thus,
the lower they are, the higher can be the maximum ac-
cepted demand.
with the maximum accepted demand. Regarding the
guaranteed GoSi and according to (3) the price tends to
be 0 as lower is gi. This is because the SP can guarantee a
GoSi that tends to 0, even if the maximum accepted de-
mand is considered. The Desired QoS,Hi, and the mean
holding time of the reservations, 1
i
, make the price
to increase if bandwidth is limited, since the required
resources (bandwidth) also increases. Finally, the in-
creasing of the maximum accepted demand Dmax,i, im-
plies that the price augments in the aim toguarantee the
GoSi and therefore, the average rate of accepted reserva-
tions decreases. On the other hand, an augment of the
maximum price Pmax , i implies that the price increases
since the willingness to pay of users also increases.
4.3. Step 4: Establishing the Elasticity of
Reservations and the Bandwidth in
Order to Maximize the Revenue
4.2. Step 3: Calculating the Price
Appendix B describes the demand function for the ana-
lyzed service that depends on the users’ willingness to
pay and a demand modulation factor (i.e., the mean re-
served bandwidth per accepted request that is described
in Appendix A). In this step, the SP obtains the price of
the accepted elastic reservations that guarantee a GoSi =
gi. This price can be obtained using expressions (2) and
(19) when Btop,I = Hi.
The SP calculates the revenue from the deduced price of
the service that guarantees the GoS for this class of ser-
vice, class i. In this paper, an intuitive revenue function
Ri(4) is considered that have three terms: the accepted
service’s demand DiGoSi, the price paid for the service
pi and the cost of the allocated resources, which is sup-
posed proportional to the access bandwidth. Alternatively,
other more complex revenue functions [32] could be ap-
plied in order to include other special characteristics of
each SP.
Figure 4 shows graphically how the price pi changes
for different values of elasticity and bandwidth. The
price decreases to 0€ when the elasticity i
approaches
to


GoS Cost
λGoS€ minute
ii iii
iiiii
RD pB
pB
 
  (4)

max,
11342 Mb/s
iii ii
BDHg B
,
or similarly, when Bi approaches to The SP evaluates the revenue substituting in expres-
sion (4) the value of the price that guarantees the GoSi
(3), the value of
i for ,topii
B
 
22
max,
113421 M
iii ii
i
DH g

b/s
.
H(32), and the value of
the GoSi (12), as it is shown in expression (5).
A price of 0€ means that the GoSi is guaranteed even
 

2
max, max,
2
max,
GoS
2
max,
1
1
11
11
1
01
ii
i
ii
ii
ii
g
i
iiii
i
B
PBD
g
HD
p
BDH








iiii
i
Hg
g
(3)
 

2
max, max,
2
max,
2
max,
1
1
11
11
11
11
ii
iiii
i
iii i
i
ii
iiii ii
i
BB
PBBD
g
HHD
R
BBD













iii
Hg
Hg
(5)
Analysis of a Pricing Method for Elastic Services with Guaranteed GoS 393
0
200
400 0
0.5
1
0
1000
2000
3000
i
B
i
(Mb/s)
i
(re q/m in)
Figure 3. Maximum demand for a guaranteed GoSi = 0.95
and Hi = 1 Mb/s.
0
200
400 0
0.5
1
0
5
10
i
B
(Mb/s)
p
i
(€)
Figure 4. Price for a Guaranteed GoSi = 0.95 and Hi = 1
Mb/s.
In the case that the SP has a limited bandwidth the
elasticity of the reservations that maximizes the reve-
nue, *
i
is:
max,
max,
max,
31
13
1
03
ii
ii
ii i
i
ii
i
BBDH
HD g
R
BDH


i
i
g
g
(6)
Figure 5 shows graphically how the revenue Ri(5)
changes for different values of elasticity and bandwidth
and Figure 6 shows how the elasticity that maximizes
the revenue for a Guaranteed GoSi depends on bandwidth.
Analyzing expressions (5) and (6), it can be deduced that
the revenue increases until the value of elasticity given by
1114 Mb
ii
B

0
200
400 0
0.5
1
0
100
200
300
i
B
(Mb/s)
R
i
(€ /min)
Figure 5. Revenue for a Guaranteed GoSi = 0.95 and Hi = 1
Mb/s.
0100 200300 400
0
0.2
0.4
0.6
0.8
1
B
i
(Mb/s)
i
*
Figure 6. Values of the elasticity of the reservations that
maximize the revenue for a Guaranteed GoSi = 0.95 and Hi
= 1 Mb/s.
demand Dmax ,i, makes the revenue and the elasticity that
maximize the revenue to increase and an increment of the
maximum price Pmax, i, forces the revenue to increase,
since the willingness to pay of the users also increases,
but this effect has no impact on the optimum elasticity.
The revenue for the optimum elasticity of the reserva-
tions is the following:
max, max,
max,
max,
max,
max, max,
max,
3
2
9
1
when 3
1
1
11
11
when 3
1
when .
3
ii
ii
ii
ii
iiiii
i
iii
i
ii
ii
ii
ii
ii iii
ii
iii
iii
i
B
RP DgB
H
BDHg
BB
RP g
HHD
DHgB DHg
RB
BDHg













 


s, if 114Mb s
i
B. If band-
width is higher than this value, the elasticity of the reser-
vations will not provide any significant benefit in compari-
son with an inelastic reservation. The optimum value of the
elasticity is different for each considered Bi and decreases as
Bi increases. This behavior is due to the fact that an incre-
ment of Bi implies more resources that allow accepting more
users with less elasticity in their reservations. On the other
hand, if the SP is forced to use a pre-established price, the
use of a lower gi can increase its revenue. The increase of
Hi, and 1i
implies that the optimum elasticity aug-
ments. Finally, an increase of the maximum accepted
ii
B
(7)
In the case that the SP has not limited bandwidth the
value of the bandwidth that gives the best revenue, when
Copyright © 2012 SciRes. JSSM
Analysis of a Pricing Method for Elastic Services with Guaranteed GoS
394
is applied the elasticity of the reservations that maxi-
mizes the revenue. It further allows an optimal dimen-
sioning of the access bandwidth Bi, is *
i
B
and is ex-
pressed according to (8).
*
max,
2
max, max,max,
*
2
max,
max,
max,
0
13
27
11
2
ii
ii
i
i
ii iii
i
i
ii
ii
ii
i
iiii
P
H
PD gP
BH
H
P
H
Dg PH


3
i
i
i






(8)
Figure 7 shows how the revenue *
ii
i
R
(7)
changes for different values of bandwidth. As it can be
seen, the revenue has a maximum equal to 268.16€/min
for i= 165.87 Mb/s that corresponds to B0
i
. The
value of the optimum bandwidth i decreases with the
increment of the price of the access bandwidth (
B
i
).
5. Conclusions
This paper proposes a pricing method that helps SPs to
assign suitable prices to multiple classes of independent
streaming elastic services with guaranteed GoS. The pa-
per determines the price of one class i of elastic stream-
ing services taking into account the accepted average rate
of reservations λi, with a guaranteed GoSi and, assuming
a linear-based function as aggregate demand function Di.
The SPs that want to offer elastic services should have
to calculate their prices. In this process this method could
help them in calculating them by means of defining or
estimating in advance for these services some of the pa-
rameters that best could match their needs for the avail-
able resources. Some of them are: the value of the offered
elasticity of the reservations ξi (it could be what offers
the maximum revenue), the highest value of the reserva-
0100 200 300 400
-100
0
100
200
300
B
i
(Mb/s)
R
i
i
=
i
*
(€/min)
Figure 7. Optimum bandwidth for the elasticity of the res-
ervations that maximizes the revenue for a Guaranteed
GoSi = 0.95 and Hi = 1 Mb/s.
tion Hi, this parameter is related to the maximum quality
of the delivered content that the SP expects to give their
users, the value of the guaranteed GoSi (this value could
be set according to the value offered by other SPs or to-
tally different) and the available resources (i.e., the ac-
cess bandwidth, Bi).
On the other hand an issue that could be difficult to
determine is the aggregate demand function. However, it
is known that the SPs have the appropriate tools to ap-
proximately estimate this function and to overcome this
situation. The accuracy of this estimation is crucial to
evaluate the price using this method.
Currently we are developing simulation tools for esti-
mating and establishing the profile of the users that could
access to this type of services what would make possible
to deduce appropriated aggregate demand functions in
scenarios where the SPs could offer these elastic services.
Also, we are also working on extending this method to
the case of substitute services and to include different
priorities to the reservations based on their elasticity.
Finally it is our challenge to analyze the case of multiple
classes of elastic services that are not independent, in this
sense simulation tools are under investigation since ana-
lytic models to deduce the derived aggregated demand
functions in this case and the relations between the pa-
rameters that intervene are really hard to find out. Future
work should also include a deep revision and proposal of
new revenue functions and consequently in the their eva-
luation, the value of the elasticity that maximizes their
revenues and what should be the relation in each case
among the elasticity, the access bandwidth and other pa-
rameters involved in the aim to get the maximum reve-
nue.
6. Acknowledgements
This work was supported by the Spanish Research Coun-
cil under projects TEC2009-14598-C02-02, and the con-
solidated research group 2009 SGR 1242 funded by the
Generalitat de Catalunya.
REFERENCES
[1] B. Braden, et al., “Resource ReSerVation Protocol (RSVP),
Version 1, Functional Specification,” RFC 2205, 1997.
http://www.rfc-editor.org/rfc/pdfrfc/rfc2205.txt.pdf
[2] R. Hancock, G. Karagiannis, J. Loughney and S. Van den
Bosch, “Next Steps in Signaling (NSIS): Framework,”
RFC 4080, 2005.
http://www.rfc-editor.org/rfc/pdfrfc/rfc4080.txt.pdf
[3] J. Manner, G. Karagiannis and A. MacDonald, “NSIS
Signaling Layer Protocol (NLSP) for Quality-of-Service
Signaling,” RFC 5974, 2010.
http://www.rfc-editor.org/rfc/pdfrfc/rfc5974.txt.pdf
[4] J. Ash, A. Bader and C. Kappler, “QoS-NSLP QSPEC
Copyright © 2012 SciRes. JSSM
Analysis of a Pricing Method for Elastic Services with Guaranteed GoS 395
Template”, RFC 5975, 2010.
http://www.rfc-editor.org/rfc/pdfrfc/rfc5975.txt.pdf
[5] M. Postigo-Boix and J. L. Melús-Moreno, “Influence of
the Grade of Service in the Evaluation of the Mean Re-
served Bandwidth for Elastic Services,” IEEE Communi-
cation Letters, Vol. 12, No. 2, 2008, pp. 143-145.
doi:10.1109/LCOMM.2008.071659.
[6] M. Postigo-Boix and J. L. Melús, “Utilization-Based Pri-
ces for Elastic Services”, Computer Communications, Vol.
31, No. 14, 2008, pp. 3503-3509.
doi:10.1016/j.comcom.2008.06.006.
[7] M. Postigo-Boix and J. L. Melús, “A Proposal for Pricing
Substitute Guaranteed Services,” IEEE Communications
Letters, Vol. 15, No. 1, 2011, pp. 100-102.
doi:10.1109/LCOMM.2010.01.101336.
[8] A. Ozdaglar, “Price Competition with Elastic Traffic,”
Journal on Networks, Vol. 52, No. 3, 2008, pp. 141-155.
doi:10.1002/net.20239.
[9] G. Abbas, A. K. Nagar, H. Tawfik and J. Y. Goulermas,
“Estimation Based Distributed QoS Pricing and Schedul-
ing for Elastic Internet Services,” 2nd International Con-
ference on Developments in eSystems Engineering (DESE),
Abu Dhabi, 2009, pp. 89-98.
doi:10.1109/DeSE.2009.15.
[10] P. Dube and R. Jain, “Diffserv Pricing Games in Multi-
Class Queuing Network Models,” 22nd International
Teletraffic Congress (ITC), 7-9 September 2010, Amster-
dam, pp. 1-8. doi:10.1109/ITC.2010.5608737.
[11] Z. G. Li, C. Chen and Y. C. Soh, “Pricing Based Differ-
entiated Bandwidth Service,” 4th IEEE Conference on In-
dustrial Electronics and Applications ICIEA 2009, Xi’an
25-27 May 2009, pp. 780-787.
doi:10.1109/ICIEA.2009.5138310.
[12] J. L. Van Den Berg, M. R. H. Mandjes and R. Nuñez
Quejia, “Pricing and Distributed QoS Control for Elastic
Network Traffic,” Operations Research Letters, Vol. 35,
No. 3, 2007, pp. 297-307. doi:10.1016/j.orl.2006.03.018.
[13] G. Cheng, N. Ansari and S. Papavassiliou, “Adaptive QoS
Provisioning by Pricing Incentive QoS Routing for Next
Generation Networks,” Journal of Computer Communi-
cations, Vol. 31, No. 10, 2008, pp. 2308-2318.
doi:10.1016/j.comcom.2008.02.019.
[14] X. Wang and H. Schulzrine, “Pricing Network Resources
for Adaptive Applications,” IEEE/ACM Transactions on
Networking, Vol. 14, No. 3, 2006, pp. 506-519.
doi:10.1109/TNET.2006.872574.
[15] X. Chang and D. W. Petr, “A Survey of Pricing for Inte-
grated Service Networks,” Journal on Computer Commu-
nications, Vol. 24, No. 18, 2001, pp. 1808-1818.
doi:10.1016/S0140-3664(01)00327-9.
[16] C. Courcoubetis and R. Weber, “Pricing Communication
Networks Economics, Technology and Modelling,” Wiley,
New York, 2003. doi:10.1002/0470867175.
[17] C. A. Gizelis and D. D. Vergados, “A Survey of Pricing
Schemes in Wireless Networks,” IEEE Communications
Surveys & Tutorials, Vol. 13, No. 1, 2011, pp. 126-145.
doi:10.1109/SURV.2011.060710.00028.
[18] S. Park, S. Han and S. W. Cho, “User Utility Discovery
for Priority-Based Network Resource Pricing,” Com-
puters and Industrial Engineering, Vol. 56, No. 4, 2009,
pp. 1357-1368.doi:10.1016/j.cie.2008.08.010.
[19] S. Kalyanasundaram, E. K. P. Chong and N. B. Shroff,
“Optimal Resource Allocation in Multi-Class Networks
with User-Specified Utility Functions,” Computer Net-
works, Vol. 38, No. 5, 2002, pp. 613-630.
doi:10.1016/S1389-1286(01)00275-4.
[20] J. Kuri and S. Roy, “Pricing Network Resources: A New
Perspective,” International Conference on Wireless Com-
munications, Networking and Mobile Computing, Shang-
hai, 21-23 September 2007, pp. 1937-1940.
doi:10.1109/WICOM.2007.485.
[21] T. Li, D. Y. Iraqi and R. Boutaba “Pricing and Admission
Control for QoS-Enabled Internet,” Computer Networks,
Vol. 46, No. 1, 2004, pp. 87-110.
doi:10.1016/j.comnet.2004.03.020.
[22] M. H. Ahmed, “Call Admission Control in Wireless Net-
works: A Comprehensive Survey,” IEEE Communica-
tions Surveys & Tutorials, Vol. 7, No. 1, 2005, pp. 50-69.
doi:10.1109/COMST.2005.1423334.
[23] J. Hou, J. Yang and S. Papavassiliou, “Integration of
Pricing and Call Admission Control to Meet QoS Re-
quirements in Cellular Networks,” IEEE Transactions on
Parallel and Distributed Systems, Vol. 13, No. 9, 2002,
pp. 898-910. doi:10.1109/TPDS.2002.1036064.
[24] M. Baglietto, R. Bolla, F. Davoli, M. Marchese and M.
Mongelli, “A Proposal of New Price-Based Call Admis-
sion Control Rules for Guaranteed Performance Services
Multiplexed with Best Effort Traffic,” Computer Com-
munications, Vol. 26, No. 13, 2003, pp. 1470-1483.
doi:10.1016/S0140-3664(03)00032-X.
[25] O. Yilmaz and Ing-Ray Chen, “Utilizing Call Admission
Control for Pricing Optimization of Multiple Service
Classes in Wireless Cellular Networks,” Computer Com-
munications, Vol. 32, No. 2, 2009, pp. 317-323.
doi:10.1016/j.comcom.2008.11.001.
[26] E. W. Fulp and D. S. Reeves, “Bandwidth Provisioning
and Pricing for Networks with Multiple Classes of Ser-
vice,” Computer Networks, Vol. 46, No. 1, 2004, pp.
41-52. doi:10.1016/j.comnet.2004.03.018.
[27] J. Elias, F. Martignon, A. Capone and G. Pujolle, “A New
Approach to Dynamic Bandwidth Allocation in Quality
of Service Networks: Performance and Bounds,” Com-
puter Networks, Vol. 51, No. 10, 2007, pp. 2833-2853.
doi:10.1016/j.comnet.2006.12.003.
[28] U. Savagaonkar, E. K. P. Chong and R. L. Givan, “Online
Pricing for Bandwidth Provisioning in Multi-Class Net-
works,” Computer Networks, Vol. 44, No. 6, 2004, pp.
835-853. doi:10.1016/j.comnet.2003.12.011.
[29] N. Jin and S. Jordan, “The Effect of Bandwidth and Buf-
fer Pricing on Resource Allocation and QoS,” Computer
Networks, Vol. 46, No. 1, 2004, pp. 53-71.
doi:10.1016/j.comnet.2004.03.023.
[30] Y.-M. Li, “Pricing Digital Content Distribution over Het-
erogeneous Channels,” Decision Support Systems, Vol.
50, No. 1, 2010, pp. 243-257.
doi:10.1016/j.dss.2010.08.027.
[31] R. Hassin and M. Haviv, “To Queue or Not to Queue,”
Copyright © 2012 SciRes. JSSM
Analysis of a Pricing Method for Elastic Services with Guaranteed GoS
Copyright © 2012 SciRes. JSSM
396
Springer, Boston, 2003.
[32] K. Talluri and G. Van Ryzin, “The Theory and Practice
of Revenue Management,” Kluwer Academic Publishers,
Dordrecht, 2005.
[33] L. Kleinrock, “Queueing Systems, Vol. I: Theory,” John
Wiley and Sons, New York, 1975.
[34] M. Abramowitz and I. A. Stegun, “Handbook of Mathe-
matical Functions with Formulas, Graphs, and Mathe-
matical Tables,” Dover, New York, 1972.
Appendix A. Determining the GoSi and the
mean Reserved Bandwidth per Accepted
Request, res,i
B
where .
represents the standard floor operator. Ni rep-
resents the access number of elastic reservations of class i
with an access bandwidth Bi for each class i.
According to these assumptions, the model that better
fits to perform the evaluation of the GoSi and ,res i
B of
the class i of elastic reservations is the M/M/N/N loss
queuing model [33]. Figure 8 represents the state-transi-
tion rate diagram. The associated analysis of this loss
queuing model is based on a well-known expression [33].
Besides the simplicity of the Figure 8, the difficulty in
this type of analysis is to identify the appropriate queuing
model that performs the evaluation of the GoSi and
,res i
Bfor each class i of elastic reservations. Thus the
state-transition diagram of Figure 8 represents the num-
ber of elastic reservations of class i (Ni) that shares the
access bandwidth Bi.
In this Appendix, we present a Markov-chain based
model to evaluate the GoSi and the mean reserved band-
width per accepted request, ,res i
B. In addition, it is also
presented an approximate model that notably simplifies
the computational complexity of this evaluation and it is
very close to the simulation model. This analytical ap-
proximation has been used to get the quantitative results
of Section 4, as well as to obtain the demand function in
terms of the price and the bandwidth of Appendix B.
A.1. Establishing Assumptions
The following assumptions are about the allocation of Bi
among the requested elastic reservations: Each state of the Markov chain shown in Figure 8
represents the probability of having
0, i
kN ac-
cepted elastic reservations of class i, where the state (k,i)
represents that k bandwidth reservations are sharing the
access bandwidth.
The reservations requests follow a Poisson process
with an average rate of λi (reservations/minute).
The reservation’s holding time—that is, the time
needed to deliver the content—is exponentially distrib-
uted with mean 1/μi (minutes). The mean delivery time
for a particular request remains the same regardless of the
value of the reserved bandwidth (Bres,i). This is the case
for the delivery of audio or video streaming using two
different reserved bandwidths
12
; the reserva-
tion’s holding time for B2 is d = C2/B2 (minutes), where
C2 (bits) represents the data to be delivered. When the
reserved bandwidth is B1, the delivery time remains the
same, since the content size is reduced to C1 = (B1/B2)C2
(bits), and the quality of the received content is reduced
accordingly.
The probability of each state is well known (10) and it
is represented by the vector
 
and each component of this vector is represented by the
well-known formula for loss queuing models [33].

0, 1,,
π,π,,π
iii Ni

BB


1
,
0
π
!!
i
kl
N
ii ii
ki
l
kl
 


(10)
A.2. Evaluation of the GoS
In this paper, the GoSi for the elastic reservations is de-
fined as the probability of accepting a reservation of class
i. Equation(11) shows the value of the GoSi where (a,b)
represents the incomplete Gamma function [34].
The reserved bandwidth for an accepted reservation
request is according to (1) within the interval

,1
iiiii
LH HH



,
.
The server rejects an elastic reservation if no access
bandwidth is available to hold the required minimum
bandwidth Li for this reservation.

 




1
,
0
GoS1 π1!!
1
e1,
ii
i
i
ii
Nl
N
ii ii
iNi
l
i
N
ii
iii
Nl
N

 




 

  
(11)
The content server assigns a bandwidth Bi for reserva-
tions of class i that must share it in competition. The
maximum number of reservations of class i (Ni) that can
share the access bandwidth is calculated as follows: Figures 9 and 10 show the type of relations between
the GoSi and the parameters λi, and Bi for ξi equal to 0

1
iiiiii
NBL BH





(9)
Analysis of a Pricing Method for Elastic Services with Guaranteed GoS 397
0,i 1,i k,i N
i
-1,i N
i
,i
i
i
i
i
i
i
i
2
i
k
i
(k+1)
i
(N
i
-1)
i
N
i
i
Figure 8. Transition diagram.
0
100
200
050 100 150200
0
0.5
1
i
(req/min)
B
i
(Mb/s)
GoS
i
Figure 9. GoSi vs. λi and Bi, for Hi = 1 Mb/s and
i = 0.
0
100
200
050 100 150 200
0
0.5
1
i
(req/min)
B
i
(Mb/s)
GoS
i
Figure 10. GoSi vs. λi and Bi, for Hi = 1 Mb/s and
i = 0.8.
and 0.8 respectively, where the mean holding time is ex-
ponential with a mean of 1/μi = 3 minutes and Hi = 1
Mb/s. Although the model has been validated by simula-
tion, here these results are not presented since them ex-
actly match the evaluation results of (11). The observed
behavior in Figures 9 and 10 for the GoSi allows deduc-
ing some interesting general properties for the GoS,
which is defined in the interval
GoS 0, 1 :
The GoS decreases with
, that is,


GoS ,GoS,BB
 

.
This is because more requests arrive at the system for
the same shared resources (bandwidth Bi).
The GoS increases with B, or similarly,


GoS ,GoS ,BBBB


 .
This is because the resources increase and the average
rate of reservation requests (λi) remains constant.
The GoS increases with
. Equivalently,

GoS ,,GoS ,,BB

 .
The elasticity of the reservations reduces their band-
width consumption requirements. Thus, if the reservation
is more elastic, more requests can be accepted for the
same GoSi, and for the same average rate of reservation
requests the GoSi increases.
For specific B and
parameters,

0,GoS, ,
gBg
 


 .
The g parameter is the guaranteed GoS. This is a con-
sequence of the first property and implies that to guaran-
tee a particular GoS, gi, is necessary to keep the average
rate of reservation requests belowi
g
.
In terms of computational complexity, the use of this
model presents a quadratic complexity with respect to N,
and consequently, the needed number of mathematical
operations to obtain the GoSi increases quadratically with
Bi and i
, and decreases quadratically with Hi. Therefore,
in order to decrease this complexity, our suggestion is to
roughly approximate the value of the GoSi , taking in
account that GoSi is a measure of the probability of using
the available resources to get the service. According to
this, the value of the GoSi can be approximated by the
ratio between the used and requested bandwidth:

 
1if1
GoS if 1
1
i
ii
i
iii
ii
ii
ii
i
i
i
H
B
B
H
B
H


(12)
This approximation has constant computational com-
plexity and allows clarifying, by means of a closed-form
expression, how the different parameters affect the eva-
luation of the GoSi. The relative error of this approxima-
tion is less than the 3% for values of Bi far away from the
condition

1ii
i
i
i
H
B

in which the error is significant. Nevertheless, in this
work, this approximation offers a more intuitive expres-
sion that allows knowing better and quicker its quantita-
tive evaluation and to clarify its qualitative behavior.
Figure 11 shows the relative error (higher than 3%) for
the calculation of the GoSi using the approximation in
(12). The relative error is higher when

1ii
i
i
i
H
B
,
and the value of GoS tends to 0.
Copyright © 2012 SciRes. JSSM
Analysis of a Pricing Method for Elastic Services with Guaranteed GoS
398
0
100
200
050 100150 200
0
10
20
30
i
(req/min)
B
i
(Mb/s)
Relative error (%)
Figure 11. Relative error (higher than 3%) for the calcula-
tion of the GoS of Figure 9 using the approximation in (12).
A.3. Evaluation of the Mean Reserved
Bandwidth per Accepted Request res,i
B
The mean reserved bandwidth per accepted reservation
request for class I [5], is expressed as follow:
,
mean_value
res ires i
B,
B (Mb/s) (13)
The evaluation of the ,res i
B includes the value of each
Bres,i for each state k. Thus, expression

,res i
kB
(14) evaluates,res i
B:
,,
0
π
resiresik
k
k
BB
(14)
When k reservations of class i are accepted in the ac-
cess link and ii
, the value of (Bres,i)k is Hi. Oth-
erwise, all k reservations will share the bandwidth Bi, and
(Bres,i)k is
kH B
i
Bk.

,
0
i
res ikii
ii
i
H
kBH
BBkk BH

(15)
Figures 12 and 13 show the relationship among the
parameters,res i
B, λi, and Bi, with ξi = 0 and ξi = 0.8 re-
spectively, the mean holding time is 1i
= 3 minutes
(exponential) and Hi = 1 Mb/s. The showed curves in
these figures allow deducing some features of the pa-
rameter ,res i
Band its value is defined in the interval

,,1
res iiiiii
BLH HH

 

,
.
The value of ,res i
B tends to Hi when i
i
i
i
H
B
. Ac-
cording to this condition, it is required that the mean re-
served bandwidth less than the total bandwidth Bi to offer
to all the arriving requests a reservation equal to Hi.
The value of ,res i
Btends to Li when i
i
i
LB
i
. Ac-
cording to this condition, it is required that the mean re-
served bandwidth higher than the total bandwidth Bi to
offer to all the arriving requests a reservation equal to Li.
Figure 12. res,i
B vs. λi and Bi, for Hi = 1 Mb/s and
i = 0.
Figure 13. res,i
B vs. λi and Bi, for Hi = 1 Mb/s and
I = 0.8.
In terms of the elasticity of the reservationsi
, the
value of ,res i
Btends to

1
ii i
LH

when

1
ii
iii
ii
LH


i
B

.
Similarly, if
1ii
i
ii
B
H
 ,
the ,res i
Btends to
1
ii i
LH
.
The value of ,res i
B
is placed in the range from Hi to
Liwhen
ii
ii
ii
LB H


 i
.
In particular, the value of ,res i
B can be approximated
using the linear expression, ,
i
res ii
i
BB
.
Therefore, gathering the above results, the value of
,res i
B is summarized in (16).
Copyright © 2012 SciRes. JSSM
Analysis of a Pricing Method for Elastic Services with Guaranteed GoS 399


,
if
if 1
if 1
i
iii
i
ii i
res iiiiii
i
i
i
iiii
i
HBH
BBH B
LBH
 


i
H
(16)
Figure 14 shows the relative error (higher than 3%) in
the calculation of the ,resi
Bwhen its value is the deduced
in expression (16). As it can be observed, the relative
error is higher when

1i
i
i
ii
H
B
, and the value of
,res i
Btends to Li.
Appendix B. The Aggregate Demand
Function
As other pricing methods, this method needs to know the
service demand of the users in order to deduce the price
of the service and therefore the optimal elasticity of the
reservations. The aggregate demand function of each
service is difficult to know for the SPs in advance as
many researchers have pointed out. The aggregate de-
mand function usually determines the quantity of the
product that all customers want to buy at a given price. In
other words, it represents the sum of the individual de-
mands of every user, due to the fact that each user has a
different willingness to pay for the product. The most
common aggregate demand functions are: linear, expo-
nential, constant demand elasticity and logit functions
[32]. The aggregate demand is not always easy to obtain
since it means to know the curb that shows the desire of
the users to pay for the service. In any case, it must be
always estimated by each SP using the appropriated tools.
This demand function could also depend on different
service characteristics other than price such as a quality
factor of the elastic reservations as well as other market-
ing variables. In this paper, an equilibrium condition is
assumed, that is, the average rate of the accepted reserva-
tion requests for each class of service should be equal to
its aggregate demand function (17).
0
100
200
050 100 150 200
0
10
20
i
(req/min)
B
i
(Mb/s)
Relative error (%)
Figure 14. Relative error (higher than 3%) for the calcula-
tion of th e res,i
B of Figure 13 using th e approximat ion in (16) .
This paper also assumes that the aggregate demand
function for each service depends not only on its price
(€/reservation), pi,, but on a demand modulation factor,
,res i
Bthat acts as a quality factor of the elastic reservation.
First are revised some of the general assumptions made
for common aggregate demand functions [32], and then
these assumptions are applied to a linear-based aggregate
demand function, which is the selected demand function
applied to the demand service in Section 4. Finally, the
demand function can be expressed in terms of the price of
the service and the used access bandwidth.
So, in equilibrium, the average rate of the accepted
reservation requests, λi, is equal to the aggregate demand
function Difor each pair of values (pi, ,res i
B).
,,
,where0 and 0
iiiresi iresi
DpBp B
 (17)
In general, an aggregate demand function D must sat-
isfy the following common regularity conditions [32].
1) D is continuously differentiable and strictly de-
creasing on p, 0D
p
, ,
res
pB0
. Therefore, the
demand for the service decreases with the price.
2) D is non-decreasing on
res
B, 0
res
D
B
,
,
res
pB 0
. Therefore, the demand for the service in-
creases or remains constant with the quality of the reser-
vations.
3) D tends to a maximum demand when res
B is high
enough for a fixed price. Therefore, the demand will re-
main constant if the price is fixed and res
Bis higher than
the maximum required bandwidth to deliver the content
().
top
4) D is bounded below and thus tends to zero for high
enough prices or when
B
res
B.
The selection of a linear-based function as the aggre-
gate demand function notably simplifies the involved
equations and their graphical representation. However, it
should be clear that the selection of the aggregate demand
function could be another one and, in any case, it will
always depend on the information that the SP has about
the service and the behavior of their users. In the aim to
evaluate the price in this paper the general conditions
previously deduced are applied to the selected lin-
ear-based function.
In (18) is defined the expression of the demand for a
linear-based function that accomplishes condition 1) such
as expression (19) shows.
12 12
0, 0 and 0Dpa ap aap
 (18)
Since

20and 0Dp ap
p
 
, (19)
Copyright © 2012 SciRes. JSSM
Analysis of a Pricing Method for Elastic Services with Guaranteed GoS
400
the dependence between D and res
B is defined as fol-
lows:

12 0
resres res
DBbbBB (20)
Condition 2) is accomplished as expression (21) shows:

20
res res
res
DB bB
B
 
0, (21)
Condition 3) states that the demand for the service will
reach a maximum with respect to res
B for values equal
to or greater than top . This maximum depends on the
fixed value of p, expressions (22) to (25).
B

 
max1 2
12
21
top top
res top
top
DBDpDpb bBDp
aap
bbBB
B

(22)
Also considering that, for a given price, the demand
will be constant when res top
BB, we can deduce:

112res top
bDp aapBB  (23)
20res top
bBB
(24)
The absolute maximum demand is achieved
when p = 0 and
max
D
res top
BB:

max 1
0DDpa (25)
The first part of condition 4) means that the demand
will tend to zero as the price approaches some value
, (26).
max
P

1
max12 maxmax
2
00
a
DPa aPPa
 (26)
The second part of condition 4) states (27):
12 1
0000 0
res
DBb bb   (27)
This paper assumes that in equilibrium the average rate
of the accepted reservation requests for each class of ser-
vice should be equal to its aggregate demand functionas
expression (17) shows. Therefore, the analytical demand
function Di can be expressed as in (28). Figure 15 repre-
sents the linear function that specifies the sum of all res-
ervations from all users that arrive at the SP’s server,
when Dmax = 120 requests/minute, Pmax = 10€ and Btop = 1
Mb/s.

,
,
max,max,, ,
max, ,
max,max,, ,
max,
max,
,
1,0
1,
0,
iiiresi
res i
i
iiiresit
itop i
i
iiiresitopi
i
ii
DpB
B
p
DpPB
PB
p
DpPBB
P
pP

 






opi
B
(28)
According to condition 1), the expression for Di must
be aninvertible function. Therefore, by means of expres-
sion (28) the price pi is determined and this value is what
adapts the service demand to the value of the average rate
of the accepted reservation requests,
i (29).
,
,,
max,, ,max,
,
,max,
max,, ,max,
max,
,
10,0
1,0
iiresi
top ires i
iiresitopii
top i
res ii
i
iresitopiii
i
pB
BB
PBBD
B
BD
PBBD
D







 



i
(29)
Figure 15 shows the demand function in terms of price
and res
B. The price is selected by the SP, but res
B is
determined by the real demand (the average rate of the
accepted reservation requests, λi), the bandwidth (Bi) and
the elasticity of reservations
i
, as it is explained in
Appendix A.2. Therefore, Figure 15 shows the general
users’ aggregate behavior with respect to the price
and res
B, but cannot describe the real demand with re-
spect to the price for a particular bandwidth and elasticity.
Accordingly, we particularize the demand function of
Figure 15 to the particular service that the SP is offering
that will determine the res
B perceived by the users. To
make this particularization, we search a value of λi that
for a given price makes res
B to be the same that the one
obtained using the model of Appendix A.
Figures 16 and 17 show the particularized demand
function with respect to price and bandwidth Bi, for ξi= 0
and ξi = 0.8 respectively, where the mean holding time is
1/μi = 3 minutes (exponential) and Hi= 1 Mb/s. In addi-
tion, we can approximate the particularized demand func-
tion by using (29) and (32), when

,1
top iii
BH

:

max, max,
max,
max,
1
,
0
i
ii
i
iiii
ii
p
Dp
P
DpB
pP

 



i
P
(30)
If
,
1
iitopii
H
BH
:
Figure 15. Demand function when Dmax = 120 requests/
minute, Pmax = 10 € and Btop = 1 Mb/s.
Copyright © 2012 SciRes. JSSM
Analysis of a Pricing Method for Elastic Services with Guaranteed GoS 401
0
5
10 0200 400
0
50
100
B
i
(Mb/s)
p
i
(€)
i
(req/min)
Figure 16. Particularized demand function for i
= 0, Dmax
= 120 requests/minute, Pmax = 10 € and Btop = 1 Mb/s.
0
5
10 0200 400
0
50
100
B
i
(Mb/s)
p
i
(€)
i
(req/min)
Figure 17. Particularized demand function fori
= 0.8, Dmax
= 120 requests/minute, Pmax = 10€ and Btop = 1 Mb/s.
 



max,
,max,
2
max, 2
max,
,max,
max,
,max,
max,
2
max, 2
,max,
,11
1
when ,11,
,1
when ,
11
ii
iiiii i
top ii
ii
iii ii
itopi i
ii
iii ii
top ii
ii
ii
iii
itopi i
Dp
DpB H
BP
Dp
pP BH
BP
Dp
DpB B
BP
pP
DpHB
BP


 



 










max,
,
max,
1,
ii
top i
ii
DpB
P





max,
max,
max, max,,
max,
max,
,1
1
when ,1,
,0
when .
i
iii i
i
i
iii ito
ii
iii
ii
p
DpB DP
p
pP BDB
P
DpB
pP





 


If :
,top ii
BH
 



max,
,max,
2
max, 2
max,
,max,
max,
,max,
max,
2
max, 2
,max,
,11
1
when ,11,
,1
when ,
11
ii
iiiii i
top ii
ii
iii ii
itopi i
ii
iii ii
top ii
ii
ii
iii
itopii
Dp
DpB H
BP
Dp
pP BH
BP
Dp
DpB B
BP
pP
DpHB
BP






 












max, 2
max,
max,
,max,
max, 2
max,
,max,
max,
1,
,1
1
when ,1,
,0
when .
ii
i
ii
ii
iii i
top ii
ii
iii i
itopi i
iii
ii
Dp
H
P
Dp
DpB H
BP
Dp
pP BH
BP
DpB
pP











(32)
Also, we can express the value of price as following. If
,1
top iii
BH
:

max, max,
max,
max,
1
,
0
i
ii
i
iii
ii
PD
D
pB
D

 



i
(33)
If
,
1
iitopii
H
BH
:
 
 



max, ,
max,
max,
,
2
,
max,
max,
max,
,
,
max,
max
,1
1
when 1,1,
1
,1
when ,1,
,1
i
iiiitopi
ii i
ii
iiiiii
top ii
top i
i
iiii
iii
iii
iiiiii
top iii
i
iiii
pBP B
DH
DHBH
B
B
pBP DB
DBHBB
B
pBP D
 















topi

,
max, ,
when ,,
,0 otherwise.
i
i
iiitopi
i
iii
DB B
pB




(34)
pi
(31)
If :
,top ii
BH
Copyright © 2012 SciRes. JSSM
Analysis of a Pricing Method for Elastic Services with Guaranteed GoS
Copyright © 2012 SciRes. JSSM
402
 
 



max, ,
max,
max,
,
2
,
max,
max,
max,
,
max,
max,
,1
1
when 1,1,
1
,1
when ,1,
,1
i
iiiitopi
ii i
ii
iiiiii
topii
top i
i
iiii
iii
iii
iiiiii
topiii
t
i
iiii
i
pBP B
DH
DHBH
B
B
pBP DB
DBHB H
B
B
pBP D
 

 













i

,
max,
,
when ,,
,0otherwise.
op i
i
ii
iiii
topii
iii
H
DHB H
B
pB




(35)