A pricing method for elastic services that guarantees the GoS in a scenario of evolutionary demand

Computer Communications 36 (2013) 1317–1328

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A pricing method for elastic services that guarantees the GoS in a scenario of evolutionary demand Marcos Postigo-Boix ⇑, José L. Melús-Moreno Department of Telematics Engineering, Universitat Politècnica de Catalunya (UPC), E-08034 Barcelona, Spain

a r t i c l e

i n f o

Article history: Received 27 July 2009 Received in revised form 2 May 2013 Accepted 3 May 2013 Available online 15 May 2013 Keywords: Pricing Elastic reservations Streaming GoS Mean reserved bandwidth per accepted request

a b s t r a c t Service Providers (SPs), which offer services based on elastic reservations with a guaranteed Grade of Service (GoS), should be interested in knowing how to price these services, i.e. service-i-, how to calculate the associated benefits to this service or, how to know the time until which the price for service-i-could be maintained, when an evolutionary function of the aggregate demand considered is involved and the established GoS for the elastic service is guaranteed. Thus this paper proposes a method that price elastic services (or elastic reservations) with guaranteed GoS in a scenario of evolutionary function of the aggregate demand. The method obtains: first at all, the average rate of the accepted elastic reservations of class-i with guaranteed GoS. Second, according to the accepted reservations, calculates the price that maximizes the selected revenue function. The considered aggregate demand function depends not only on a demand modulation factor, the mean reserved bandwidth, Bres,i, but on the evolution of this aggregate demand function, according to a Bass diffusion model. Third, in a scenario where not plenty access bandwidth Bi is available, evaluates the optimum value of the elasticity of the reservations that maximizes the revenue function for the obtained price. Finally, it is forecasted the time until the SP does not need to change the price or elasticity calculated when the demand increases and the GoS is guaranteed. The paper applies the method to a class-i- of elastic reservations, analyzes the influence of each one of the parameters and could be extended to multiple classes of independent and guaranteed elastic services. Ó 2013 Elsevier B.V. All rights reserved.

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 services 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 elastic 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 compression 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 reduction of the price for this service is significant. Elastic reservations require the support of new signaling mechanisms other than the most commonly used today, the ⇑ Corresponding author. Tel.: +34 934016012; fax: +34 934015981. E-mail addresses: [email protected] (M. Postigo-Boix), [email protected] (J.L. Melús-Moreno). 0140-3664/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.comcom.2013.05.001

resource ReSerVation Protocol (RSVP) [1,2]. As an alternative, the Next Steps in Signaling (NSIS) [3] protocol family allows to reserve bandwidth in a specific range. The Internet Engineering Task Force (IETF) created the NSIS Working Group in 2001 to solve new signaling needs for reservations. Since then, several Internet RFCs and papers have been published [4–6], 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 scenario each one of them will respectively represent the bandwidth that the user wants to reserve and the minimum bandwidth that the user needs in order to work properly. Fig. 1 shows the proposed scenario for the reservation of elastic services using the QoS-NSLP-based signaling mechanism. According to Fig. 1, when the user wants to watch a video he access to the website where the SP lists their offered SLAs. In Fig. 1, the SLA is defined by the Grade of Service (GoS) and other parameters 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 (n). 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 Minimum QoS have the same bandwidth and the elasticity is 1 if the Minimum QoS bandwidth is 0. In the scenario of video content

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Fig. 1. Scenario of the reservation of elastic services based on the QoS-NSLP signaling mechanism.

distribution of Fig. 1 the SP determines both values, the highestquality (H) and the lowest-quality (L). Thus the elasticity for the reservations of class i would be:

ni ¼ 1  Li =Hi

ð1Þ

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  res;i defines the is, between the Hi and Li. In addition, the metric B 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 Fig. 1, the SP allows their clients to request elastic reservations with the same GoSi in an established bandwidth range for each reservation of class i. In this paper, some previous considerations should be done from the scenario described in Fig. 1: First,  res;i are related to the reservations of the the parameters GoSi and B class of service i. The reservations are represented by their length and their reserved bandwidth of the generated session. Thus, each established session is characterized 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 obtaining an accepted reservation within the requested range; And third, all the reservations of class i, have the same priority. Fig. 2 shows the entities involved in the scenario described in Fig. 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 access 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 parameters 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 an aggregate demand function that also depends on a demand modu res;i . This last parameter could be justified by the delation factor, B sire of users of paying more for the reservation when the value of  res;i , is closer to Hi. And finally, the SP obtains the value of the elasB ticity 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 expressions in this paper to each considered class. However this method is unable to evaluate and to analyze the case of dependent services, since to deduce the appropriate aggregate 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 maximum accepted service demand in terms of requests per unit time. 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 procedures to know how to quantify the price of these services and to create the appropriated scenario to offer them. Many questions could arise about the utility of this method for the SPs. Thus, one could be referred to the convenience of implementing this method for elastic services, another, is related with the scenario where the SPs do not have plenty of access bandwidth Bi and therefore, will the users accept some changes, mainly related to the acceptation of any kind of elasticity, for the requested services. In this hypothesis, will they obtained revenue for the elastic reservations be better than for the inelastic case? What should be the size of the available resources-access bandwidth-that better support the use of the elastic or inelastic reservations? How many users could access using elastic or

Fig. 2. Entities considered.

M. Postigo-Boix, J.L. Melús-Moreno / Computer Communications 36 (2013) 1317–1328

inelastic reservations in this scenario? What is the value of the elasticity that maximizes the revenue? This method allows SPs to answer some of these questions. In essence, this paper presents several contributions:  A method to price a single class, class-i of elastic streaming services with guaranteed of GoSi according to; the elasticity of the reservations, the resources available (access bandwidth), an evolutionary function of the aggregate demand and maximizing the selected revenue function. This method could be extended to the case of multiple independent and guaranteed elastic streaming services with the same priority. Thus, for each one the price is independently evaluated using the same procedure as for a single class, class-i.  This method is based on an analytical evaluation that obtains a closed-form expression that reduces the computational complexity of other evaluation techniques such as Markov-chain based.  The aggregate demand function will depend not only on its price (€/reservation), pi, as usually is considered, but also on a  res;i that also depends on the access demand modulation factor, B bandwidth, Bi. Although the selected aggregate demand function in the paper is a linear-based function, it could be considered another one. In any case, the price will be finally obtained inverting this demand function.  The evolutionary function of the selected aggregate demand is based on the Bass diffusion model [29,30]. Alternative diffusion models have been used to describe this evolution process and the adopted model is well known in estimating the diffusion of all forms of mobile telecommunication [31–34]. The study and use of the diffusion process could be considered of paramount importance in understanding the factors influencing further development of users demand for elastic services. It could also allow forecasting the time until which the SP does not need to change the price, elasticity or both without diminution of the guaranteed GoSi for the elastic service. In this sense, the SP could achieve a balance among the dynamic behavior of the prices and the resilience of the users to accept continuous change on the price of the requested services.  In scenarios when not enough access bandwidth Bi is available the maximization of the revenue function depends not only about the price but also on the value of elasticity, see expression (4). Thus for the calculated price, which maximized the revenue function, it is possible to further obtain an optimum value of the elasticity that maximizes that revenue function. Therefore the method highlights the importance of both price and elasticity in the maximization process of the revenue 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 a single class of elastic service, class-i-, (that could be extended to the case of multiple classes of independent and guaranteed elastic services). Section 4 applies, evaluates and presents the obtained results of the proposed method for a study case of a single class of elastic reservations, class-i-, considering a linear-based aggregate demand function Di, which evolves according to an evolutionary function of the demand. Also in this section the importance of the elasticity and the available resources (access bandwidth) in the evaluation of the revenue function is highlighted. Section 5 summarizes the main conclusions. Also this paper includes two appendices. In Appendix A, the GoSi and the mean reserved bandwidth per ac res;i are deduced using both a Markov-chain based cepted request B model and an approximate analytical model. The results of both,

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Markov-chain model and approximate model, are very close thus the analysis of the method is based on the analytical model, which allows further to show more clearly the relationship among the involved parameters in the process of pricing these services, such as it was established in Section 4. In Appendix B the linear-based aggregate demand function Di that evolves following a Bass diffusion model is analyzed and evaluated.

2. Related work This paper describes and analyzes a method that could help SPs to price elastic services with guaranteed GoS that have an aggregate demand function, Di that evolves following a Bass diffusion model [29]. As it is known, the demand function establishes the relation between the users that are willing to get the service according to the price that they want to pay for. The price of each class of service is based on several parameters. Some of them are: the average rate of the accepted class of elastic reservations ni with guaranteed GoSi and its mean reserved bandwidth per accepted re res;i . In [7] the parameters Bres,i and B  res;i were introduced quest, B and was established how both vary when different values of the  res;i could be considGoSi are applied. In this sense, the parameter B ered as a demand modulation factor for the price of the elastic res res;i is also considered ervations. Here, in this paper the parameter B as a modulation factor (quantitative component) in the determination of the price of the elastic reservations. Although many methods to price services have been proposed only a few focuses on elastic reservations, however neither of them has jointly tackled the same issues nor has been used at the same manner as this paper shows. Thus no paper treats the GoS as a constraint that affects the assignation of prices to elastic services and further includes  res;i or the evofactors such as a modulation factor for the demand B lutionary function of the aggregate demand Ref. [8] studied and analyzed the quantitative influence of the GoSi in the evaluation of the mean reserved bandwidth for each  res;i and also proposed a method to price two classes reservation B of elastic services. This work fundamentally differs from that paper [8] on the method used to calculate the price, since there the evaluation of the price only depended on the considered revenue that SPs wanted to obtain. In [9] a method to price substitute guaranteed services was described that used an exponential aggregate demand function. The prices were found inverting their demand functions and considering that in equilibrium the average rate of accepted reservations for each class of service, which maximizes the revenue function, is equal to its aggregate demand function Di. That method could be applied to N classes of substitute services, although the graphical analysis there only included two substitute services, the aggregate demand functions of each substitute services depended on the price of the other service and, from the set of pairs of the accepted service demand that accomplished the guaranteed GoSi, the pair that maximized the revenue function was selected. Finally, since a Markov-chain based model was used it was difficult to establish the relationships among the parameters involved. This paper differs from [9] and [8] in: the selected aggregate demand function, linear-based, that depends on the price and  res;i , the evolution of the demand function the modulation factor B that evolves according to a diffusion model and the price, which is obtained maximizing the revenue function subject to the average rate of the accepted reservations, ni that accomplish the GoSi. Also, in scenarios where the access bandwidth is scarce, this paper also calculates the value of the elasticity that maximizes this revenue function with the obtained price Refs. [10–16] present different methods to price elastic and inelastic services with guaranteed QoS and in different scenarios; however the proposed solutions are far from what this paper

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presents. Thus in [10], the authors study the combined study of price competition and traffic control in a congested network where the SPs assign the price to maximize their profits. In [11] the authors present a State Estimation based Internet traffic flow control system where the objective is to maximize the aggregate bandwidth utility of network resources over the transmission rates. In [12], the paper focuses on the SP competition, in a game theoretic setting, the traffic considered is elastic, there are multiple types of traffic, and each one is sensitive to different degrees of Quality of Service (QoS). In [13] 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 [14] authors propose an appropriate prioritization pricing structure where users are provided with incentives and are able to choose between two service classes. In [15] authors present an integrated solution (integrating pricing into QoS routing) for enabling the next generation Internet to achieve the differentiated service and availability guarantee. Ref. [16] describes, in a wireless scenario, an admission control algorithm that optimizes the benefits -revenue- when the QoS is guaranteed. Finally, the authors of this paper show that the price depends on the holding prices (bandwidth reserve), the usage price (average usage, the elasticity of the traffic) and the congestion price. In Refs. [17–19] many survey papers present a classification of the different schemes of pricing related to different scenarios or other features of the offered services. Thus in [17] authors review the state of the art and technological growth of congestion control for integrated service networks since pricing is a proper tool to manage congestion, encourage network growth, and allocate resource to users in a fair manner. Ref. [18] is one of the first books that treat conjunctly technology and pricing and, in Ref. [19] a recent classification of the proposed pricing methods in wireless networks is presented. Ref. [20] surveys recent research work on congestion pricing in Wireless Cellular Networks (WCN). In Refs. [21–23] different methods are proposed to determine the utilization function of the users. Thus, in [21] 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 is presented. In [22] users specify the utility or value they attach to different quantities of resource using a utility function, so the resource allocator knows the utility function 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 [23] each user is assumed to have a utility function which is a concave increasing function of the rate at which 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 capacity constraints. Refs. [24–28] evaluate how admission control affects the obtained GoS (this value is considered as a technical constraint) of the services. In these papers are analyzed not only the case of a single class of service but the case of multiple classes. Thus in [24] authors pay their attention to the interrelation between pricing and Admission Control in QoS-enabled networks and propose a tariff-based architecture framework that flexibly integrates pricing and admission control for multi-domain Diffserv networks. In [25] a comprehensive survey about Call Admission Control (CAC) in wireless networks is shown. In [26] authors say that traditional CAC schemes mainly focus on the tradeoffs between new call blocking probability and handoff call blocking probability. Therefore, they introduce the pricing as an additional dimension of the CAC call process in order to efficiently and effectively control the use of wireless network resources. In [27] authors investigate the appropriate conditions for both Best Effort (BE) traffic and traffic explicitly requiring QoS, with Guaranteed Performance, (GP) and propose three CAC rules for the GP traffic. In [28] authors utilize Admission

Control Algorithms, with guaranteed QoS, to optimize the revenue and from these to derive optimal pricing of multiple classes of service in Wireless Cellular Networks. A new contribution of this paper is the introduction of an evolutionary function of the selected aggregate demand (linear-based) that is based on a well-known diffusion model (Bass model). The diffusion process is considered as a factor that influences the evolution of the user demand as it is suggested by some papers. In this sense Refs. [29–34] present some research papers on modeling and forecasting the diffusion of innovations. In reference [29], F.M. Bass suggest that individuals are influenced by a desire to innovate (coefficient of innovation p) and by a need to imitate others in the population (coefficient of imitation q). In [30] from the same author, F.M. Bass, discuss some background and history of the development of Ref. [29], explains the reasons why the model has been influential and presents some important extensions. In [31] some factors that affect the diffusion of new generation of mobile telecommunications technologies are identified. Among them competition between firms has been found to be the key determinant of the diffusion speed across all generations. The objective of Ref. [32] is to review the research on the main models of innovation diffusion with an emphasis on their contribution to improving on forecasting accuracy. Ref. [33], uses the evaluation of the most widely used aggregate technology diffusion models, such as the Bass model and others, and examines the diffusion rate of mobile subscriptions in Greece. The estimated values for this country (p = 0.001 and q = 0.63) are the utilized values to the study case presented in Section 4 of this paper. In [34] the Bass model and some of its key extensions are developed. The authors also discuss how to obtain parameter values for these models, review a number of applications and comment on the uses and limitations of aggregate diffusion models.

3. A pricing method for services based on elastic reservations This section describes the pricing method for a single class-i of elastic streaming services with guaranteed GoS according to: the elasticity of the reservations, the resources available (access bandwidth), an evolutionary function of the aggregate demand and maximizing the selected revenue function. This method could be extended to the case of multiple independent and guaranteed elastic streaming services with the same priority and for each one the price is independently evaluated using the same procedure as for a single class, class-i. However, before going on with the method, it is convenient to take into account the difficulty of determining the aggregate demand function Di. 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. 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 always hard to identify the behavior of the users and therefore, the associated demand curve. Thus, the SP has to estimate by whatever means it deems adequate (analytically, by simulation, heuristically, etc.) the demand function for each service. This paper assumes for simplicity in Section 4 that the aggregate demand function is linearbased. In any case, if the demand function applied changes, the quantitative results that the method offers will be different. The outcomes of this method are the price, pi, the value of the elasticity of the service ni and the SP’s revenue. The method consists of the following steps: (1) The SP offers guaranteed and independent elastic services with the appropriated service requirements as they are shown in 0 and 0. Thus, a single service-i is guaranteed with a particular GoSi, elasticity ni, price pi, Desired QoS equal to Hi

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Mb/s and Minimum QoS, Hi  ð1  ni Þ. Other requirement is the access bandwidth that the SP needs, that is, the available access bandwidth, Bi, where the sum of the reservations of class-i should be inferior to Bi. (2) The SP evaluates the maximum service demand (in terms of requests per unit time) that can be allowed for service-i in order to guarantee the selected GoSi. In the case of multiple classes, the value of this maximum service demand will be evaluated for each offered class of independent elastic reservations. This paper, in Appendix A.2, presents an analytical expression to evaluate this maximum service demand of elastic reservations that closely approximates to the obtained results by simulation. (3) The SP obtains the price that maximizes the revenue function subject to the average rate of the accepted reservations, ni accomplish the GoSi. However if the access bandwidth is scarce, the SP could determine the elasticity that maximizes the revenue with the price previously calculated and the service demand is always lower than the maximum service demand with a guaranteed GoSi obtained in step 2. In this paper additionally, the SP models the evolution of the aggregate demand using a diffusion model. As a consequence of the demand evolution, the SP could know how long the service-i will be guaranteed with this GoSi, the price and the elasticity previously calculated and while the demand is under the maximum service demand established in Step 2. The revenue function used in this paper, assumes for simplicity that depends on the price, the rate of accepted requests and the cost of the access bandwidth. However, more complex expressions could also be used. In order to obtain the price for the service, the SP needs to estimate the aggregate demand function Di by whatever means it deems adequate. This paper assumes a linear-based aggregate demand function that evolves according to a Bass diffusion model. This evaluation is described in Appendix B where the aggregate demand function depends on the price, the modulation  res;i and its evolution through factor or mean reserved bandwidth B  res;i also dethe time according to a Bass diffusion model. Since B pends on the access bandwidth Bi, the aggregate demand function will depend on the price and the access bandwidth, Bi. 4. Analysis of the method: Price and elasticity of the reservations that maximizes the revenue and how long the evolutionary function of the demand guarantees the GoSi In this section the pricing method, described in Section 3, is applied to a study case that includes a single class-i of elastic reservations. In addition, it will be calculated how long the SP can keep the price and elasticity for an access bandwidth without to change the GoSi and when an evolutionary function of the aggregate demand is applied to the selected linear-based demand function.

min., maximum required bandwidth to deliver the service, Btop,i = Hi = 1 Mb/s and Pmax,i = 10€, and the used parameters for the diffusion Bass model [33] are, p = 0.001, q = 0.63. 4.2. Step 2: Evaluating the maximum service demand that guarantees the GoS The SP calculates the average rate of the accepted reservations, ni, that guarantees the GoSi. Using approximation (A.4) of the Appendix A.2, the expression for the maximum accepted service demand [2] is obtained.

ki jðGoSi¼g Þ ¼ i

Bi

ð2Þ

1

li Hi ð1  ni Þg i

Fig. 3 shows graphically how the maximum accepted service demand that guarantees the GoSi = 0.95 changes for different values of elasticity and bandwidth. If the elasticity tends to 1, the maximum service demand tends to +1, since the reservation requests always are accepted. On the other hand, if the elasticity is zero, the maximum service demand tends to a minimum value if bandwidth is fixed, since a reservation with elasticity 0 requires no less than Hi Mb/s. Otherwise, ni increases slightly for low-medium elasticity values and when high elasticity values are considered, ni increases sharply approaching to a vertical asymptote to +1 for elasticity equal to 1. As Bi increases, the maximum accepted service demand also increases in a linear way with a higher slope as elasticity approaches to 1. It is worth to mention that the guaranteed GoSi, Hi and the holding time 1=l1 impact on the maximum accepted service demand. Thus, the lower they are, the higher can be the maximum accepted service demand. 4.3. Step 3: Calculating the price Appendix B describes the evolutionary function of the aggregate demand [29,30] for the analyzed service that depends on the users’ willingness to pay, the demand modulation factor (i.e. the mean reserved bandwidth per accepted request that is obtained in Appendix A) and the dynamics of the evolution of the demand (over time). Thus, at time t0, the SP obtains the price that guarantee a GoSi = gi and maximizes the revenue Ri for an access bandwidth Bi. The revenue function Ri, expression (3) has three terms: the accepted service demand Di n GoSi, the price paid for the service pi and the cost of the allocated resources (which is assumed proportional to the access bandwidth). Alternatively, other revenue functions [33] could be applied in order to include special characteristics of each SP.

Ri ¼ Di  GoSi  pi  CostðBi Þ ¼ ki  GoSi  pi  ai Bi ð€=minuteÞ

ð3Þ

3000

Before the application of this method the SP, should define the suitable parameters for each class of elastic service. In the case of a single class-i of elastic services all the reservations have the same elasticity, the guaranteed GoSi = gi, and the highest bandwidth of the requested reservation is Hi, which is equal to the maximum required bandwidth to deliver the content (Btop,i). Specifically, these parameters are: GoSi = 0.95, Hi = 1 Mb/s, and the mean reservation holding time is 3 min. The parameters for the linear-based evolutionary function of the aggregate demand, Dmax,i(t), according to Appendix B are: maximum aggregate demand, Dmax,i = 120 req./

λi (req/min)

4.1. Step 1: Determining the requirements for the service

2000 1000 0 0

1 200

0.5 400

B (Mb/s)

0

ξi

i

Fig. 3. Maximum accepted service demand for a guaranteed GoSi = 0.95 and Hi = 10 Mb/s.

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The maximization of the revenue is achieved according to expression (4)

s:t 0 6 ki 6 ki jðGoSi¼g Þ ¼ i

Bi 1 l Hi ð1  ni Þg i

ð4Þ

i

and Bi P 0; li P 0; Hi ¼ Btop P 0; 0 6 ni 6 1; 0 6 g i 6 1

Revenue ( /min)

300

maximize Ri ðpi ; Bi Þ

200 100 0

1

200

variable fpi g

0.5

100

8  > B i li > > > 1  Dmax;i ðt0 ÞHi ð1ni Þ2 Pmax;i > > > >2 < P pi ¼ 3 max;i  > > > 1  D Bi lðti0 ÞH Pmax;i > > max;i i > > > :1P 2 max;i

0

B i (Mb/s)

The solution for pi according to expression (4) is pi . Bi 6 31l Dmax;i ðt0 ÞHi ð1  ni Þ2 i

0

ξi

Fig. 5. Revenue for a guaranteed GoSi = 0.95 and Hi = 1 Mb/s.

2

1 3li

Dmax;i ðtÞHi ð1  ni Þ 6 Bi 6 31l Dmax;i ðt0 ÞHi i

1 3li

Dmax;i ðt0 ÞHi 6 Bi 6 21l Dmax;i ðt0 ÞHi i

Bi P 21l Dmax;i ðt0 ÞHi i

1

Fig. 4 shows graphically how the price pi changes for different values of elasticity and bandwidth. In general, the price decreases with elasticity ni and bandwidth Bi. In particular, the price arises a Pmax value if access bandwidth is 0 and a Pmax,i/2 value if is higher than ð1=2li ÞDmax;i ðt 0 ÞHi . The Desired QoS, Hi, and 1=li , forces the price to increase if bandwidth is limited, since the required resources (bandwidth) also increases. On the other hand, an augment of the maximum price Pmax,i implies that the price increases since the willingness to pay also increases. The SP can evaluate the revenue, Ri ðpi ; Bi Þ with a price of pi substituting the value of price [5] in expression as it is shown in [6]. 8   > Bi li Bi li > > > Hi ð1ni Þ Pmax;i 1  Hi ð1ni Þ2 Dmax;i ðt0 Þ  ai Bi > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > Dmax;i ðt 0 Þ <2P Bi li  ai Bi 3Btop;i Ri ðpi ; Bi Þ ¼ 3 max;i   > Bl > Bi l i i i > > P 1   ai Bi max;i > Hi ð1ni Þ Hi Dmax;i ðt 0 Þ > > > > D ðt Þ : max;i 0 P max;i  ai Bi

Bi 6 31l Dmax;i ðt0 ÞHi ð1  ni Þ2 i

1 3li

Dmax;i ðt0 ÞHi ð1  ni Þ2 6 Bi 6 31l Dmax;i ðt 0 ÞHi i

1 3li

Dmax;i ðt0 ÞHi 6 Bi 6 21l Dmax;i ðt 0 ÞHi i

Bi P 21l Dmax;i ðt0 ÞHi i

4

ð6Þ

In scenarios where not plenty access bandwidth Bi is available the optimum value of the elasticity of the reservations that maximizes the revenue function in (6) ni is:

8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3Bi li <1  Bi 6 31l Dmax;i ðt 0 ÞHi Hi Dmax;i ðt 0 Þ  i ni ¼ :0 B P 1 D ðt ÞH i

3li

max;i

0

ð7Þ i

Fig. 5 shows graphically how the revenue Ri [6] changes for different values of elasticity and access bandwidth and Fig. 6 shows how

10

pi* (

8 6 0

0

100 200 B i (Mb/s)

0.5 1

ξi

Fig. 4. Optimal price for a guaranteed GoSi = 0.95 and Hi = 10 Mb/s.

ξi

ð5Þ 0.5

0 0

20 100 200

B i (Mb/s)

10 0

t0 (years)

Fig. 6. Values of the elasticity of the reservations that maximize the revenue for a guaranteed GoSi = 0.95 and Hi = 1 Mb/s.

the elasticity depends on the access bandwidth and time t0. The optimum value of the elasticity ni is different for each value of Bi and decreases as Bi increases. This behavior is due to the fact that an increment of Bi implies more resources what allow accepting more users with less elasticity in their reservations. The increase of Hi, and 1/li implies that the optimum elasticity augments. Finally, an increase of the maximum accepted demand Dmax,i,(t0) makes the revenue and the elasticity to increase. On the other side 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. Finally, it is forecasted the time until the SP does not need to change the price or elasticity calculated when the aggregate demand increases and the GoSi is guaranteed. Finally, assuming that the SP keep the price with the same access bandwidth after t0, it is possible to evaluate how long the service will be guaranteed with the GoSi. At time t0, the SP decides the price that maximizes the revenue guaranteeing the GoSi for a particular bandwidth Bi, and also determines the elasticity that maximizes the revenue, ni . As the time goes on, the aggregate demand will increase, due to the diffusion process. Assuming that the demand model of Appendix B evolves according to a Bass diffusion model it is possible to obtain the interval of time while the GoSi is guaranteed (Dtgi ). First, we calculate how the difference between the maximum service demand and the real demand (Dki ) for an optimal price at time t0 evolves over a time interval Dtgi as follows:

Dki ðt 0 þ Dt gi Þ ¼ ki jðGoSi ¼gi Þ  ki ðpi ; Bi ; t0 þ Dtgi Þ

ð8Þ

The interval in which the GoSi will be guaranteed (Dt gi ) can be calculated by solving Dki ðt 0 þ Dt gi Þ ¼ 0. Fig. 7 shows that Dt gi is higher as bandwidth and elasticity increase.

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M. Postigo-Boix, J.L. Melús-Moreno / Computer Communications 36 (2013) 1317–1328

20

of the reservations for specific design parameters, such as the available access bandwidth or the guaranteed GoSi, in the aim to allows the SP not to change these parameters in a fixed time interval. Finally it is also interesting to investigate the impact of this pricing method in competing scenarios.

10

Acknowledgments

i

Δ tg (years)

30

0

This work was supported by the Spanish Research Council under projects TEC2012-38574-C02-02, and the consolidated research group 2009 SGR 1242 funded by the Generalitat de Catalunya.

1

20

0.5

10 0

B i (Mb/s)

0

ξi

Fig. 7. Values of the Dtg i for GoSi = 0.95 and Hi = 1 Mb/s.

Appendix A. Determining the GoSi and the mean reserved  res;i bandwidth per accepted request, B

5. Conclusions

This Appendix presents a Markov-chain based model to evaluate the GoSi and the mean reserved bandwidth per accepted re res;i and an approximate model that notably simplifies the quest, B computational complexity of the evaluation that is very close to the simulation model. This analytical approximation has been used to get the quantitative results of Section 4, as well as to obtain the aggregate demand function in Appendix B.

The SPs that want to offer elastic services should calculate the prices of these services. The method described in this paper analyzes and establishes the complex relations among the parameters involved and could help them in estimating in advance some values such as the price and the elasticity of the reservations ni that maximize the revenue function. These values are obtained from several input parameters such as the maximum reservation value Hi, the guaranteed GoSi for service-i (it should be established according to the own available resources of the SP), the available access bandwidth, Bi and the estimated aggregate demand function. The estimation of this last function is very difficult since it directly depends on the behavior of the users and from the accuracy of this estimation will depend the accuracy on the determination of the price. This paper, in the study case, uses a linear-based function as aggregate demand function Di that evolves according to a Bass diffusion model. Thus, this paper proposes a pricing method that helps SPs to price multiple classes of independent streaming elastic services with guaranteed GoSi, for each class and that maximizes a revenue function. In the paper the method to price one class-i of elastic streaming services that maximizes the revenue is developed, considering an accepted average rate of reservations ki and assuming an evolutionary function of the aggregate demand. The analysis and evaluation of the dynamics of this increasing demand function through the time could also help the SP in estimating the time interval until what the SP neither need to change any of the obtained values (price and elasticity) nor the value of the assumed initial parameters. This interval is the time until the SP could maintain the guaranteed GoSi for the increasing entry of demand due to the evolution of the aggregate demand function. In this sense, the SP could achieve a balance among the dynamic behavior of the prices and the resilience of the users to accept continuous change on the price of the requested services. Future research activities include: to establish an appropriate dimensioning of the available resources, such as the access bandwidth, for guaranteed GoSi and different evolutionary demand functions or, to renegotiate the values of the elasticity or the price

λi

λi

μi

Taking into account that the access bandwidth for class-i is Bi, the following assumptions are assumed for the elastic reservations:  The arrivals of the requested reservations follow a Poisson process with an average rate of ki (req/min)  The reservation’s holding time – that is, the time needed to deliver the content – is exponentially distributed with mean 1/li (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 (B1 < B2); the reservation’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 reserved bandwidth for an accepted reservation request is according to [1] within the interval [Li, Hi] = [Hi  (1  ni), Hi].  The server rejects an elastic reservation if no access bandwidth is available to hold the required minimum bandwidth Li for this reservation.  The content server assigns an access bandwidth Bi for the reservation services of class i. The maximum number of reservations of class i (Ni) that shares the access bandwidth is:

Ni ¼ bBi =Li c ¼

λi

λi

1,i

0,i

A.1. Assumptions

k,i

2μi

k⋅μi



Bi Hi  ðni Þ

 ðA:1Þ

λi

λi

Ni,i

Ni-1,i (k+1)⋅μi

Fig. 8. Transition diagram.

(Ni-1)⋅μi

Ni⋅μi

M. Postigo-Boix, J.L. Melús-Moreno / Computer Communications 36 (2013) 1317–1328

ðki =li Þk k! l i ðki =li Þ l¼0 l!

pðk;iÞ ¼ PN

ðA:2Þ

A.2. Evaluation of the GoS In this paper, the GoSi for the elastic reservations is defined as the probability of accepting a reservation of class i. Thus, expression (A.3) shows the value of the GoSi and C(a, b) is the incomplete Gamma function, see Ref. [37]. ðki =li ÞNi Ni !

GoSi ¼ 1  pðNi ;iÞ ¼ 1  P

N i ðki =li Þl l¼0 l!

¼1

eðki =li Þ

ðki =li ÞNi  CðNi þ 1; ðki =li ÞÞ

ðA:3Þ

Figs. 9 and 10 show the relations between the GoSi and the parameters ki, and Bi when ni is 0 and 0.8 respectively, the mean holding time is exponential, such as 1/li = 3 min and Hi = 1 Mb/s. The model has been validated by simulation, however here these results are not presented since them exactly match the analytical results of (A.3). The behavior of the parameter GoSi in Figs. 9 and 10 allows deducing some general properties about this parameter, which will be defined in the interval XGoS ¼ ½0; 1 :  The GoS decreases with k that is, 9k > kjGoSðk; BÞ > GoSðk ; BÞ. This is because more requests arrive at the system for the same resources (bandwidth Bi).  The GoS increases with B, or similarly, 9B > BjGoSðk; BÞ < GoSðk; B Þ. This is because the resources increase and the average rate of the requested reservations (ki) remains constant.

1

i

where bc represents the standard floor operator and Ni is 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  res;i is the M/M/N/N loss perform the evaluation of the GoSi and the B queuing model [34]. The associated analysis of this loss queuing model is based on a well-known expression. Thus the state-transition diagram of Fig. 8 represents the number of elastic reservations of class i (Ni) that shares the access bandwidth Bi. Each state of the Markov chain shows the probability of having k 2 ½0; N i  accepted elastic reservations of class i, where the state (k, i) represents that k reservations are sharing the access bandwidth. The probability of each state is represented by the vector Q i = (p(0, i), p(1, i), . . ., p(N, i)) where each component of this vector, according to Ref. [36] is:

GoS

1324

0.5 0 0 0

100 50

100

150

200

200

λi (req/min)

B (Mb/s) i

Fig. 10. GoSi vs. ki and Bi, for Hi = 1 Mb/s and ni = 0.8.

 The GoS increases with n. Equivalently, 9n > njGoSðk; B; nÞ < GoSðk; B; n Þ. The elasticity of the reservations reduces the access bandwidth consumption. 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 n parameters, 9k 2 ½0; kg jGoSðk; B; nÞ P g. The g parameter is the guaranteed GoS. This is a consequence of the first property and implies that to guarantee a particular GoS, gi, is necessary to keep the average rate of reservation requests below kgi . The computational complexity that this model presents is quadratic with N, and needs a large number of mathematical operations to obtain the GoSi that increases quadratically with Bi and ni, and decreases quadratically with Hi. Therefore, in order to decrease this complexity, in this paper our suggestion is to roughly approximate the value of the GoSi obtained in (A.3) for the approximate value in (A.4). This approximation arises from the fact that the GoSi is a measure of the probability of using the available resources to get the service and according to this, the value of the GoSi can be approximated by:

8 <1 used bandwidth GoSi  ¼ requested bandwidth :

Bi ki =li Hi ð1ni Þ

if

li Hi ð1  ni Þ

ki

6 Bi

if

ki

> Bi

li Hi ð1  ni Þ

ðA:4Þ Thus the computational complexity is constant and since (A.4) is a closed-form expression also it allows to clarify how the different parameters affect the evaluation of the GoSi. In terms of the error made in this approximation, only for values of Bi that accomplish this condition ki =li Hi ð1  ni Þ ¼ Bi the error is higher than 3% and could be significant as it is shown in Fig. 11. A.3. Evaluation of the mean reserved bandwidth per accepted request  res; B The mean reserved bandwidth per accepted reservation request for class i [7], is expressed as follows:

GoS

i

1

 res;i ¼ mean v aluefBres;i gðMb=sÞ B

0.5 0 0 0

100 50

100

150

200

200

λi (req/min)

B (Mb/s) i

Fig. 9. GoSi vs. ki and Bi, for Hi = 1 Mb/s and ni = 0.

ðA:5Þ

 res;i includes the value of each Bres,i for each The evaluation of the B  res;i is: state k (Bres,i)k and B

barBres; i ¼

X ðBres;i Þk  pk

ðA:6Þ

k>0

When k reservations of class i are accepted and k  Hi 6 Bi , the value of (Bres,i)k is Hi. Otherwise, all k reservations will share the bandwidth Bi, and (Bres,i)k is Bi/k.

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 res;i tends to Li if ki Li > Bi . That is, the mean  The value of B li reserved bandwidth is required to be higher than the total bandwidth Bi offered to all the arriving requests with reserve equal to Li.  res;i  In terms of the elasticity of the reservations ni, the value of B tends to Li = Hi(1  ni) if lki Li ¼ lki Hi ð1  ni Þ > Bi . Similarly, if i i  res;i tends to Li = Hi(1  ni). ni < 1  Bi li , B

Relative error (%)

30 20 10

H i ki

0

0 0

 res;i will be in the interval between Hi and Li when  The value of B ki ki  li Li 6 Bi 6 li Hi . In particular, the value of Bres;i can be approxil  res;i ¼ i Bi . mated using the linear expression, B

100 50

100

150

200

200

ki

λi (req/min)

B (Mb/s) i

Fig. 11. Relative error in the evaluation of the GoSi between (A.3) and (A.4).

 res;i is Gathering the above results and conditions the value of B summarized in (A.8).

res;i B

8 > Hi > > < l ¼ kii Bi > > > : Li

if

Bi > lki Hi i

if

li Hi ð1  ni Þ

if

ki

6 Bi 6 lki Hi i

ðA:8Þ

ki

Bi < l Hi ð1  ni Þ i

 res;i when Fig. 14 shows the relative error in the calculation of the B its value is the deduced through expressions (A.7) and (A.8). In terms of the error made in this approximation, only for values Bi that accomplish this condition ki =li Hi ð1  ni Þ ¼ Bi the error is higher than 3% and could be significant as it is shown in Fig. 14, and the  res;i tends to Li. value of B Appendix B. Aggregate demand functions and evolutionary function of the demand 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 demand 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 demands 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, exponential, constant demand elasticity and logit functions [35]. 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 marketing variables. In this paper, an equilibrium condition is assumed, that is, the average rate of the accepted reservation requests for each class of service should be equal to its aggregate demand function as expression (B.1) shows. This paper also

 res;i vs. ki and Bi, for Hi = 1 Mb/s and ni = 0. Fig. 12. B

 res;i vs. ki and Bi, for Hi = 1 Mb/s and ni = 0.8. Fig. 13. B

Hi

0 < k 6 Bi =Hi

Bi =k k > Bi =Hi

ðA:7Þ

Figs. 12 and 13 show the relationship among the parameters  res;i , ki, and Bi, ni = 0 and ni = 0.8 respectively, the mean holding B time is 1/li = 3 min (exponential) and Hi = 1 Mb/s. From the curves  res;i could be deduced. The of these figures, some features about B interval for XBres;i ¼ ½Li ; Hi  ¼ ½Hi  ð1  ni Þ; Hi .  res;i tends to Hi if ki Hi < Bi . That is, the mean  The value of B li reserved bandwidth is required to be lesser than the total bandwidth Bi offered to all the arriving requests with reserve equal to Hi.

20 Relative error (%)

ðBres;i Þk ¼

10 0 0 0

100 50

100

150

200

200

λi (req/min)

B (Mb/s) i

 res;i in (A.7) and (A.8). Fig. 14. Relative error in the evaluation of B

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M. Postigo-Boix, J.L. Melús-Moreno / Computer Communications 36 (2013) 1317–1328

assumes that the aggregate demand function, Di for each class, i.e. class-i of service, depends not only on its price (€/reservation), pi,  res;i (quality factor of the but on the demand modulation factor, B elastic reservation). First of all, some general assumptions made for common aggregate demand functions [33] are revisited, and later those will be applied to the linear-based aggregate demand function utilized in this paper, as it was assumed in Section 4. Second, in equilibrium, the average rate of the accepted reservation requests for class-i, ki, is equal to the aggregate demand func res;i ). tion Di that depends on the pair of values (pi, B

 res;i Þ with p P 0 and B  res;i P 0 ki ¼ Di ðpi ; B i

ðB:1Þ

Thus, the aggregate demand function D, without taking into account any specific class of service, should satisfy the next common regularity conditions [33].

 res P Btop b2 ¼ 0 8B

ðB:8Þ

And the absolute maximum aggregate demand (Dmax) is achieved  res P Btop : when p = 0 and B

Dmax ¼ Dðp ¼ 0Þ ¼ a1

ðB:9Þ

First part of condition (iv) means that the demand will tend to zero as the price approaches to some value Pmax, (B.10).

DðPmax Þ ¼ 0

)

a1  a2 Pmax ¼ 0

)

Pmax ¼

a1 a2

ðB:10Þ

According to second part of condition (iv) it is established (B.11):

 res ¼ 0Þ ¼ 0 DðB

)

b1 þ b2  0 ¼ 0

)

b1 ¼ 0

ðB:11Þ

For each class, class-i of service, this paper assumed that in equilibrium the average rate of the accepted reservation requests, should be equal to its aggregate demand function as expression (B.1)

(i) D is continuously differentiable and strictly decreasing on p, @  res P 0. Therefore, the demand for the service D < 0, 8p; B @p decreases with the price.  res , @ D P 0, 8p; B  res P 0. There(ii) D is non-decreasing with B @ Bres fore, the demand for the service increases or remains constant with the quality of the reservations.  res is high enough for (iii) D tends to a maximum demand when B a fixed price. Therefore, the demand will remain constant if  res is higher than the maximum the price is fixed and B required bandwidth to deliver the content (Btop).  res ¼ 0. (iv) D tends to zero for high enough prices or if B

DðpÞ ¼ a1  a2 p a1 > 0;

a2 > 0 and p P 0

Fig. 15. Aggregate demand function when Dmax(t) = 120 req./min, Pmax = 10€ and Btop = 1 Mb/s.

100 λi (req/min)

The selection of a linear-based function as the aggregate demand function simplifies the involved equations and their graphical representation. However, it should be clear that the selection of this aggregate demand function could be another one and, in any case it will always depend on the information that the SP has. In the aim to apply this method to the study case considered and to finally evaluate the price, it is appropriated to verify the incidence of these conditions over the linear-based function selected. Thus expression (B.2) defines the linear-based demand function considered. As it could be proved, it accomplishes condition (i) as expression (B.3) shows:

ðB:2Þ

0 0

Since

@ DðpÞ ¼ a2 < 0 and p P 0; @p

50

 res P 0 B

200

ðB:3Þ

10

0

p (

B (Mb/s) i

i

 res is defined as follows: The dependency between D and B

 res Þ ¼ b1 þ b2 B res DðB

400

5

ðB:4Þ

Fig. 16. Demand function with ni = 0, Dmax(t) = 120 req./min, Pmax = 10€ and Btop = 1 Mb/s.

Condition (ii) is also accomplished as expression (B.5) shows:

 res P 0 B

ðB:5Þ

According to condition (iii) the demand for the service will achieve  res for values equal or greater than Btop. a maximum with respect to B The maximum depends on the fixed value of the price p, such as expressions (B.6)–(B.9) deduce.

 res ¼ Btop Þ ¼ Dmax ðpÞ ¼ DðpÞ ) b1 þ b2 Btop ¼ DðpÞ DðB a1  a2 p res < Btop  b 1 8B ) b2 ¼ Btop

50

0 0

ðB:6Þ

 res P Btop Further, considering that the demand will be constant if B for a price,

 res P Btop b1 ¼ DðpÞ ¼ a1  a2 p 8B

100 λi (req/min)

@ res Þ ¼ b2 P 0; DðB @ Bres

ðB:7Þ

400

5

200 10

p (

0

B (Mb/s) i

i

Fig. 17. Demand function with ni = 0.8, Dmax(t) = 120 req./min, Pmax = 10€ and Btop = 1 Mb/s.

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M. Postigo-Boix, J.L. Melús-Moreno / Computer Communications 36 (2013) 1317–1328

shown. Therefore, and according to the expressions (B.1)–(B.11) the aggregate demand function Di is expressed as in (B.12). Fig. 15 shows the linear-based aggregate demand function of the sum of all reservations, when Dmax,i = 120 req./min., Pmax,i = 10€ and Btop,i = 1 Mb/s.   8 pi > > Dmax;i  1Pmax;i >  res;i p 6 P max;i ; 0 6 B  res;i < Btop;i > B < i Btop;i    ki ¼ Di ðpi ; Bres;i ; tÞ ¼  res;i P Btop;i > Dmax;i  1  P pi pi 6 P max;i ; B > > max;i > : 0 pi > P max;i

ðB:12Þ In order to consider the evolutionary behavior of the demand, a Bass diffusion model [29,30] is used and Dmax,i will increase over time as expression (B.13) shows:

Dmax;i ðtÞ ¼ Dmax;i

1  eðpþqÞt 1 þ pq eðpþqÞt

ðB:13Þ

According to condition (i), the expression for Di must be an invertible function. Therefore, by means of expression (B.12) 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, ki (B.14).

8  B > < 1  ki B Dtop;i ðtÞ  Pmax;i res;i max;i  res;i ; t ¼   pi ki ; B > : 1  D ki ðtÞ  Pmax;i



max;i

Fig. 15 shows the aggregate demand function in terms of price and  res is determined by the  res . The price is selected by the SP, but B B real demand (the average rate of the accepted reservation requests, ki), the bandwidth (Bi) and the elasticity of reservations (ni), as it is explained in Appendix A.2. Therefore, Fig.15 shows the general  res , but users’ aggregate behavior with respect to the price and B cannot describe the real demand with respect to the price for a particular bandwidth and elasticity. Accordingly, we particularize the aggregate demand function of Fig. 15 to the particular service that  res perceived by the the SP is offering that will determine the B users. To make this particularization, we search a value of ki that  res to be the same that the one obtained for a given price makes B using the model of Appendix A. Figs. 16 and 17 show the particularized aggregate demand function with respect to price and bandwidth Bi, for ni = 0 and ni = 0.8 respectively, where the mean holding time is 1/li = 3 min (exponential) and Hi = 1 Mb/s. In addition, we can approximate the particularized aggregate demand function by using (A.8) and (B.14), when Btop;i < Hi ð1  ni Þ:

( ki ¼ Di ðpi ; Bi ; tÞ ¼

  pi pi 6 Pmax;i Dmax;i ðtÞ  1  Pmax;i 0

ðB:15Þ

pi P Pmax;i

res;i < Btop;i ; 0 6 ki 6 Dmax;i ðtÞ Bres;i 06B B

top;i

ðB:14Þ

 res;i P Btop;i ; 0 6 ki 6 Dmax;i ðtÞ B

If Hi ð1  ni Þ 6 Btop;i 6 Hi :  8 D ðtÞ  pi max;i > Hi ð1  ni Þ 1  Pmax;i > Btop;i > > > q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < Dmax;i ðtÞ ð1  P pi Þli Bi Btop;i ki ¼ Di ðpi ; Bi ; tÞ ¼ max;i > > > Dmax;i ðtÞ  ð1  P pi Þ > > max;i > : 0

  p H2i ð1  ni Þ2 1  Pmax;i     D ðtÞ pi 6 Pmax;i ; l1 max;i 1  P p H2i ð1  ni Þ2 6 Bi 6 l1 Dmax;i ðtÞ 1  P p Btop;i Btop;i max;i max;i i i Dmax;i ðtÞ Btop;i

pi 6 Pmax;i ; Bi 6 l1 i

ðB:16Þ

p pi 6 Pmax;i ; Bi P l1 Dmax;i ðtÞð1  Pmax;i ÞBtop;i i pi P Pmax;i

If Btop;i P Hi : 8   Dmax;i ðtÞ > 1  P pi Hi ð1  ni Þ > > B top;i max;i > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >   > > < Dmax;i ðtÞ 1  P pi li Bi Btop;i max;i ki ¼ Di ðpi ; Bi ; tÞ ¼ >   > > pi > > > Dmax;i ðtÞ  1  Pmax;i > > : 0

  1  P p H2i ð1  ni Þ2 max;i i   p 1 Dmax;i ðtÞ 1P pi 6 P max;i ; l B H2i ð1  ni Þ2 6 Bi 6 l1 top;i max;i i i   2 p 1 Dmax;i ðtÞ pi 6 P max;i ; Bi P l Btop;i 1  Pmax;i Hi Dmax;i ðtÞ Btop;i

pi 6 P max;i ; Bi 6 l1

Dmax;i ðtÞ ð1 Btop;i

P

p max;i

ÞH2i

ðB:17Þ

i

pi P Pmax;i

When Btop;i 6 Hi ð1  ni Þ, the value of the price could be expressed as:

( pi ðki ; Bi ; tÞ ¼

  ki Pmax;i  1  Dmax;i ki 6 Dmax;i ðtÞ 0

ki P Dmax;i

ðB:18Þ

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M. Postigo-Boix, J.L. Melús-Moreno / Computer Communications 36 (2013) 1317–1328

if Hi ð1  ni Þ 6 Btop;i 6 Hi ; the value of the price could be expressed as:

8   D ðtÞ ki > ki 6 max;i Hi ð1  ni Þ; Bi 6 lki Hi ð1  ni Þ > > Pmax;i 1  Dmax;i ðtÞHi ð1ni Þ Btop;i Btop;i i > > q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi   > > k2i Btop;i Dmax;i ðtÞ ki

> > Pmax;i  1  D ki ðtÞ ki 6 Dmax;i ðtÞ; Bi P lki Btop;i > > max;i i > > : 0 otherwise

ðB:19Þ

And, for Btop,i > Hi, the value of the price could be expressed as:

8   ki > > Pmax;i 1  D ðtÞH Btop;i > max;i i ð1ni Þ > > >   > k2 B < Pmax;i 1  l1 Dmax;ii ðtÞ top;i B i i pi ðki ; Bi ; tÞ ¼ > Btop;i > ki > P  ð1  Þ > max;i > Dmax;i ðtÞ Hi > > : 0

ki 6

Dmax;i ðtÞ Hi ð1 Btop;i

ki 6

Dmax;i ðtÞ Hi ; Bi Btop

 ni Þ; Bi 6 lki Hi ð1  ni Þ i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dmax;i ðtÞ ki li Bi ; l Hi ð1  ni Þ 6 Bi 6 lki Hi ki 6 Btop;i i

i

ðB:20Þ

P lki Btop;i i

otherwise

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