Journal of Service Science and Management, 2012, 5, 386402 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 PostigoBoix, José L. MelúsMoreno 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 linearbased 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 highquality 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 (QoSNSLP) 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 QoSNSLPbased 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 QoSNSLP 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 highestquality (H) and the lowestqual 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 closedform 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 linearbased 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 linearbased 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 are deduced by means a Markovchain 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 Markovchain 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 linearbased 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 (linearbased) 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 Markovchain 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 . 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 , 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 [1517] 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 [1820] 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 QoSenabled networks and propose a tariffbased ar chitecture framework that flexibly integrates pricing and admission control for multidomain 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 QoSenabled net works to tackle the problem of appropriately provision ing and allocating connections. In [27] is introduced a service model that provides perflow 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 reservationbased QoS archi tecture that provides guaranteed bounds on packet loss and endtoend 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 onpacket 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 linearbased. 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 linearbased 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 linearbased 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 lowmedium 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 preestablished 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 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 linearbased 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 TEC200914598C0202, 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.rfceditor.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.rfceditor.org/rfc/pdfrfc/rfc4080.txt.pdf [3] J. Manner, G. Karagiannis and A. MacDonald, “NSIS Signaling Layer Protocol (NLSP) for QualityofService Signaling,” RFC 5974, 2010. http://www.rfceditor.org/rfc/pdfrfc/rfc5974.txt.pdf [4] J. Ash, A. Bader and C. Kappler, “QoSNSLP QSPEC Copyright © 2012 SciRes. JSSM
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Li, “Pricing Digital Content Distribution over Het erogeneous Channels,” Decision Support Systems, Vol. 50, No. 1, 2010, pp. 243257. 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 statetransi tion rate diagram. The associated analysis of this loss queuing model is based on a wellknown 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 statetransition 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 Markovchain 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 wellknown 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 (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 . 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 B B B H (12) This approximation has constant computational com plexity and allows clarifying, by means of a closedform 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 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 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 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 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 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 linearbased 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 pB0 . Therefore, the demand for the service decreases with the price. 2) D is nondecreasing 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 linearbased 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 earbased function. In (18) is defined the expression of the demand for a linearbased 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 bBB (24) The absolute maximum demand is achieved when p = 0 and max D res top BB: max 1 0DDpa (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 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 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 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)
