Static Pricing and Quality of Service in Multiple ... - Semantic Scholar

Static Pricing and Quality of Service in Multiple-Service Networks Luiz A. DaSilva †, David W. Petr ‡, Nail Akar †† † Virginia Polytechnic Institute and State University -- The Bradley Department of Electrical and Computer Engineering - Alexandria, VA 24061 ‡ University of Kansas -- Information and Telecommunication Technology Center - Lawrence, KS 66045 †† Sprint Corporation - Overland Park, KS

This paper discusses static pricing policies for multiservice networks. We demonstrate how, by adopting an appropriate policy, the service provider is able to offer the needed incentives for each user to choose the service that best matches her needs, thereby discouraging overallocation of resources and improving social welfare. Furthermore, by indirectly revealing their QoS or resource requirements, users provide information of great value to the traffic management task. Some of the formal results of our analysis are combined with heuristic arguments to provide guidelines for pricing of ATM services.

1. Introduction With the Internet, awareness of the need for high-speed integrated networks has reached the mainstream. There is increased demand for applications that deliver text, audio, images and video, often in real time and with a high degree of interactivity. Networks that offer multiple service classes provide an efficient solution to the problem of delivering adequate quality of service (QoS) to heterogeneous users. Multiple-service networks require incentives for users to choose the service that is most appropriate to their needs, thereby discouraging overallocation of resources and maximizing statistical multiplexing capabilities. This is most commonly achieved through pricing. Today's most mature multiple-service network architecture is Asynchronous Transfer Mode (ATM). There has also been significant recent progress by the Internet Engineering Task Force (IETF) in defining QoSenabled architectures for the Internet, fueling more interest in how to price quality-differentiated services. Pricing policies can be classified into dynamic, in which prices fluctuate as a result of some network condition (e.g. load or indication of congestion) and static policies, which are independent of current network utilization. While dynamic schemes [4, 10, 11, 13, 15, 16, 17, 20] are capable of achieving some optimization objective through the adaptation of prices to current network load, they may require changes to user applications and often imply significant accounting and real-time computational complexity. Most importantly, dynamic pricing makes it more difficult for users to budget their expenses and may

encounter significant resistance from users. This combination of factors explains why virtually all pricing schemes in place today are of a static nature. Studies of static pricing for network services [3, 6, 9, 18] distinguish themselves not so much for the novelty of the proposed policies but rather for the analysis of how these policies are likely to affect users’ service choices, network utilization, performance, user satisfaction and provider revenue. We argue that static pricing can be used effectively to influence service choices made by users and achieve a more efficient use of resources. The performance obtained by any given network user is determined by all users’ traffic and service choices. This interdependence is well modeled through a gametheoretic framework, discussed next.

2. User, Provider and Pricing Models A primary goal of our work was to determine how users’ service choices can be influenced through the use of an appropriate static pricing policy. Game theory has been used for years as a tool of economic analysis, and it provides a useful framework for modeling users’ decisions in the context of commercial networks. We model the pricing problem as a game between a principal (the service provider) and a finite set of players (the users) Α = {1, ! , N } . Each player is offered the opportunity to allocate a certain amount of bandwidth si (the user’s strategy) for the duration of her call, with the objective of maximizing her payoff Ci. Each player independently decides on a strategy, characterizing a noncooperative game. In this model, the network simply transmits packets and guarantees the availability of some minimum bandwidth si to user i for the duration of her call. In the sequel, we discuss how the results might be interpreted in the context of the services defined by the ATM Forum [1]. A user’s strategy space Si is the set of pure strategies available to user i∈ A. A joint strategy s=[s1,s2,…,sN] is a vector containing the pure strategies of all players; it is an element of the joint strategy space S = × i∈Α S i . A

fundamental concept in game theory is that of a Nash equilibrium (NE), a joint strategy where no player can increase her payoff by unilaterally changing her strategy. The NE is considered a consistent prediction of the outcome of a game [7]. A basic assumption of the model is that of a quasi-stationary environment, in which statistics of the aggregate traffic offered to the network change slowly with time. In this environment, a learning process is expected to lead to convergence to a Nash equilibrium. A measure of the efficiency of a NE is its Pareto optimality; a strategy is Pareto optimal if there exists no other joint strategy that one or more users prefer and to which all others are indifferent. 2.1 – Utility Functions and User Surplus Network users’ preferences are modeled through utility functions Ui(⋅), which describe how sensitive users are to changes in QoS. In the context of this work, it is useful to think of utility as the amount of money a user is willing to pay for a given QoS or, alternatively, for certain resources made available to her. Many of our results are largely independent of specific utility functions; rather, they take advantage of widely adopted assumptions including monotonicity, boundedness and concavity. User surplus Ci(s) is the difference between utility derived from a service and the amount a user pays for it, representing what consumers gain from the trade [14]. As in [11, 13, 15], surplus maximization is assumed to be users’ primary objective. 2.2 – Bandwidth Allocation Recall users are allowed to allocate some bandwidth si for their calls. Denoting the total available bandwidth by L, the joint strategy space is given by N S = {s ∈ [0, L ] : ∑i∈A s i ≤ L} . Call admission control allows no bandwidth overbooking. Unallocated bandwidth is available uniformly to all users. Thus, the amount of bandwidth bi(s) available to a user depends on the joint strategy. Utility is expressed as a function of bi; ~ we assume the existence of an upper bound bi on the intervals for which Ui is strictly increasing. In this paper, we study networks that allocate a certain amount of resources to each call. Lazar et al [12] have analyzed a similar non-cooperative game in which users reserve capacity for their connections with the objective of minimizing some cost function. We follow a similar model here, with a couple of crucial distinctions. In particular, we call attention to our definition of resource allocation. By saying a certain amount of bandwidth was allocated to a user, we mean that bandwidth is guaranteed to be available to that user should she need it. However, resources that are allocated but not used are made available to other users. This model is consistent with current architectures such as ATM, but it also creates stronger interdependence among users' optimal strategies, complicating the problem.

2.3 – Pricing Model The price charged for a network service may be a combination of several factors, including service class, network utilization, call duration, call start time and call set-up, leading to a model such as we have proposed in [6]. In the present work, we consider price to be based upon three factors: amount of bandwidth allocated to a user; amount of bandwidth bˆi effectively utilized by the user; and a fixed call set-up charge.

3. Results Before we make the case that static pricing can be effective in influencing truth-telling on the part of users (regarding their needs for bandwidth), let us first point out that it is not effective when the aggregate demand for bandwidth is much lower than the total available bandwidth. We can formalize this concept as1: ~ Proposition 1 – Let b+ = max i∈A bi . If L ≥ Nb+ , then " s# = 0 is a NE. If, additionally, pricing is a strictly " increasing function of bi , s# = 0 is Pareto optimal. In words, if demand is low enough, not allocating any bandwidth is an equilibrium regardless of the pricing policy. Furthermore, this is a Pareto optimal strategy for all reasonable allocation-based pricing policies. We argue that, if demand is high enough, the three-component pricing model mentioned in the previous section will encourage the desired behavior from users. 3.1 – Connection charges The set-up and tear down of connections generate network management costs that can be defrayed by introducing a connection charge, which may also be used to cover some of the fixed infra-structural costs. This charge by itself is not appropriate for multi-service networks, since this would lead to over-allocation of resources. It can be shown [5] that, under a flat rate, it is always a Nash equilibrium for the aggregate bandwidth allocated by all users to equal the total bandwidth offered in the system, even if the bandwidth allocated by a particular user exceeds her needs. Clearly, in this case the induced NE is not necessarily Pareto optimal. This result can be strengthened further: if users are not bandwidth-limited, then the only NE is for each user to allocate as much as allowed by call admission control. 3.2 – Usage charges Usage-based pricing can be effective in preventing overutilization of resources. It can also be employed for service classes where allocation charges do not make sense, e.g. best-effort service. Finally, it is unlikely that traffic parameters in the service level agreement (SLA) will accurately describe the offered traffic at all times; by 1

Proofs for all the propositions may be found in [5].

strategy Nash equilibrium.

(b) Total Revenue per Unit of Bandwidth 5

250

Revenue/Unit of Bandwidth

Nel = 60

max Nin = 50

N = 40 el

200

N = 20

150

el

100 50 0

0

20 40 60 80 Total Offered Bandwidth

4.5 4

Nel = 60

3.5

N = 20 el

3

N = 40 el

max

Nin = 50

2.5 2

100

0

(c) User Surplus − Inelastic Users


100

(d) User Surplus − Elastic Users

7.5

5 max = 50 in

N 7

Nel = 40 N = 20 el

6.5

max = 50 in

N 4 User Surplus

3.3 – Determining the Nash Equilibrium It is possible to establish sufficient conditions for the existence of a Nash equilibrium: Proposition 2– If surplus is continuous and quasi-concave in si ∀i ∈ A , then the game admits at least one pure

(a) Projected Revenue 300

Revenue

Intuition tells us that usage charges by themselves will not prevent over-allocation of resources. Again, we can provide a formal argument to support the intuition in the case of elastic users2 [5]. On the other hand, under the same assumptions regarding utility functions, as long as price is a strictly increasing function of the bandwidth allocated, it can be shown that over-allocation is never an equilibrium [5]. This argument forms the basis for our choice of a pricing model consisting of three components: fixed charges, usage charges and allocation charges.

if L ≥ Nλ in , s in = s el = 0 .

User Surplus

including a usage component, the provider can tailor the pricing policy more closely to the actual traffic generated by a user.

6 N = 60

N = 20 el

3

N = 40 el

2

el

Since each user’s utility is a concave function of bi (and, thus, concave in si), the convexity of the pricing function is a sufficient condition for the quasi-concavity of surplus. The problem of determining the NE can be stated as a set of interdependent constrained non-linear programs, as s −i ) min − C i ( si , ~ follows3:

− si   " s.t .  ≤0 ∑ j∈A s j − L 

Notice there is one such non-linear programming problem for each user i. For the sake of concreteness, we shall outline an example where both elastic and inelastic users are present, characterized by piecewise-linear utility functions. Consider λin and λel as the maximum bandwidth that is useful to the Nin inelastic users and the Nel elastic users, respectively; we denote the maximum utility for elastic users as Ael. Furthermore, we consider a linear pricing scheme based on allocation and usage: Pi = c + k f s i + k g bˆi . In this case, we obtain: Proposition 3 – When k f +

N −1 N

kg ≥

Ael ( N − 1) , the Nλ el

following Pareto optimal Nash equilibria are induced: Nλ in − L If L ≤ Nλ in , s in = , s el = 0 ; Nλ el 2

Elastic users are those who can take advantage of variable bandwidth; in contrast, inelastic users need a fixed amount of bandwidth available to their applications. 3 Notation: s-j indicates all components of vector s except the jth component.

5.5

1

5

0

N = 60 el

0


100

0


100

Figure 1 – Revenue and user surplus as functions of offered bandwidth. The following principle is at work: since perfectly elastic users derive value out of any bandwidth available to them, we discourage them from allocating by setting the combined allocation and usage charges above these users’ marginal utility. Notice that the usage-based charge by itself should be set below elastic users’ marginal utility, since otherwise these users will get no benefit from the service. Figure 1 shows how the predicted equilibria affect service provider revenue and user surplus.

4.ATM Pricing ATM service is readily available in the U.S., Europe and most other parts of the world. However, there does not seem to exist any semblance of a standard pricing policy for ATM. In fact, it is unlikely that carriers will converge to a single standard pricing policy, since pricing schemes can be used as a factor of carrier differentiation in a competitive environment. Most providers charge a monthly recurring charge (MRC), usually dependent on the access rate. This MRC can also be dependent on geographic distance or on the number of virtual circuits used [2,8,19]. Many carriers add another component to the pricing policy according to how much bandwidth must be allocated, while others charge according to utilization, without an explicit allocation-based price component. Since QoS in ATM networks is closely linked to the bandwidth available to a connection, our argument in favor of a three-component pricing scheme applies to ATM. In particular, without an allocation-based

component, one would expect users to overestimate their bandwidth needs when declaring their Peak Cell Rate (PCR), Sustainable Cell Rate (SCR) and Minimum Cell Rate (MCR) in traffic contracts. Furthermore, some customers express dissatisfaction with the idea of paying for a service when they are not making use of it, arguing for a usage-based component. Most applications utilizing CBR service are expected to be inelastic. Our results indicate that, as long as total network utilization is sufficiently high, allocation-based pricing will provide the incentives for users to truthfully indicate the amount of bandwidth needed for CBR calls by setting the PCR parameter accordingly. The same holds true for VBR and ABR users, to whom some amount of bandwidth will be allocated, determined by the PCR, SCR and MBS parameters in the case of VBR and by the MCR parameter in the case of ABR. In both cases, users’ traffic is expected to exceed at times the allocated rates, arguing for a combined allocation and usagedependent pricing scheme for these services. UBR traffic is treated by the network in a best-effort manner. By encouraging such traffic (for instance, by imposing a fixed MRC that is independent of utilization), total network utilization can be increased, indirectly providing incentives for users with stricter QoS requirements to rely on one of the other service categories and allocate the appropriate amount of bandwidth. The idea of not pricing UBR service according to usage directly addresses results discussed in the previous section. Since at times of low utilization pricing is not likely to be effective in inducing users to reveal their true bandwidth needs, the provider may wish to encourage best-effort traffic in order to drive overall utilization up. Since no QoS guarantees are made for UBR, it imposes minimal opportunity cost; as demand increases, calls belonging to the other, more profitable service categories can still be accepted, and UBR cells can be discarded as needed to guarantee adequate quality to these calls. This pricing policy for UBR also matches current policies for Internet pricing, and non-critical Internet calls can be transmitted over ATM networks using UBR.

5. Further Research In the future, it is likely that development of intelligent agents will enable the implementation of dynamic pricing schemes for commercial network services. However, at present network service providers adopt static pricing. Our work demonstrates how, even within the constraints of static pricing, it is possible to effectively influence users towards good decisions. Areas for further research include: extending the model to that of an oligopoly (rather than a monopoly); comparing the pricing of competing technologies (e.g., ATM versus QoS-enabled IP); relaxing the assumption of greedy sources; and enhancing the model to capture demand elasticity.

6. References [1] The ATM Forum Technical Committee, Traffic Management Specification v4.0, April 1996. [2] D.Axner, “ATM Carrier Services : A Progress Report,” Telecommunications Magazine, July 1997. [3] R.Cocchi, S.Shenker, D.Estrin, L.Zhang , “Pricing in Computer Networks: Motivation, Formulation and an Example,” IEEE Trans. Networking, 1(6):614-627, 12/93. [4] C.Courcoubetis, V.Siris, G.Stamoulis, “Integration of Pricing and Flow Control for Available Bit Rate Services in ATM Networks,” IEEE Globecom96, pp. 644-648. [5] L.A.DaSilva, “Static Pricing in Multiple-service Networks: A Game-theoretic Analysis,” PhD dissertation, The University of Kansas, 1998. [6] L.A.DaSilva, D.W.Petr and N.Akar, “Equilibrium Pricing in Multiservice Priority-based Networks,” Globecom97, pp. S38.6.1-5, Phoenix, AZ, 11/97. [7] D.Fudenberg, J.Tirole, Game Theory, MIT, 1993. [8] I. Gallagher, “Strategy for ATM Access and Backbone Networks,” ICC’98, Atlanta, GA, 06/98. [9] M.L.Honig and K.Steiglitz, “Usage-based Pricing of Packet Data Generated by a Heterogeneous User Population,” IEEE INFOCOM’95, vol.2, pp. 867-874. [10] H.Jiang and S.Jordan, “A Pricing Model for High Speed Networks with Guaranteed Quality of Service,” INFOCOM’96, pp. 888-895. [11] F.Kelly, “Charging and Rate Control for Elastic Traffic,” European Trans. on Comm., 8:33-37, 1997. [12] A.A.Lazar, A.Orda, D.Pendarakis, “Virtual Path Bandwidth Allocation in Multi-User Networks,” INFOCOM’95, pp. 312-320. [13] S.Low and P.P.Varaiya, “A New Approach to Service Provisioning in ATM Networks,” IEEE/ACM Trans. Networking, 1(5):547-553, 1993. [14] K.B.Monroe, Pricing: Making Profitable Decisions, 2nd ed., McGraw-Hill, 1990. [15] J.Murphy and L.Murphy, “Bandwidth Allocation by Pricing in ATM Networks,” Technical Report, Dublin City University, Ireland, 1994. [16] J.Murphy, L.Murphy, E.Posner, “Distributed Pricing for Embedded ATM Networks,” Intl. IFIP Conf. On Broadband Communications, Paris, March 1994. [17] L.Murphy, J.Murphy, J.MacKie-Mason, “Feedback and Efficiency in ATM Networks,” IEEE ICC’96, pp 1045-1049. [18] C.Parris, S.Keshav, D.Ferrari, “A Framework for the Study of Pricing in Integrated Networks,” Tech Report, Intl Computer Science Institute, Berkeley, CA, 1992. [19] K.Schulz et al, “Taking Advantage of ATM Services and Tariffs : The Importance of Transport Layer Dynamic Rate Adaptation,” IEEE Network, pp. 10-17, March/April 1997. [20] Q. Wang, J.Peha, A.Sirbu, “The Design of an Optimal Pricing Scheme for ATM Integrated Services Networks,” Journal of Electronic Publishing, 1995.