Quality of service provision in noncooperative networks with diverse user requirements
Introduction
With the increased deployment of high-speed local- and wide-area networks carrying a multitude of information from e-mail to bulk data to voice, audio, and video, provisioning adequate quality of service (QoS) to the diverse application base has become an important problem [3], [13], [27], [33]. This paper describes a QoS provision architecture suited for best-effort environments, based on ideas from microeconomics and noncooperative game theory.
We construct a noncooperative multi-class QoS provision model where users are assumed to be selfish, and packets are routed over switches where, as a function of their enscribed priority, differentiated service is delivered. The diverse spectrum of application QoS requirements is modeled using utility functions. Users or applications4 can choose both the service classes and the traffic volumes assigned to them. The interaction of users behaving selfishly in accordance with their QoS preferences leads to a noncooperative game whose dynamic properties we seek to understand.
The traditional approach to QoS provision uses resource reservations along a route to be followed by a traffic stream so that the stream's data rate and burstiness can be suitably accommodated. Although research abounds [8], [9], [10], [12], [13], [18], [28], [33], [35], [36], analytic tools for computing QoS guarantees rely on shaping of input traffic to preserve well-behavedness across switches that implement some form of packet scheduling discipline such as generalized processor sharing (GPS), also known as weighted fair queueing [11], [35]. Real-time constraints of multimedia traffic and the scale-invariant burstiness associated with self-similar network traffic [29], [37], [39], [48] limit the shapability of input traffic while at the same time reserving bandwidth that is significantly smaller than the peak transmission rate. Thus, QoS and utilization stand in a trade-off relationship with each other [37] and transporting application traffic over reserved channels, in general, incurs a high cost.
This makes it important to organize today's best-effort bandwidth, as examplified by the Internet, into stratified services with graded QoS properties such that the QoS requirements of a compendium of applications can be effectively met. This is particularly useful for applications that possess diverse but — to varying degrees — flexible QoS requirements. It would be overkill to transport such traffic over reserved channels. On the other hand, relying on homogenous best-effort service, characteristic of today's Internet, would be equally unsatisfactory. A dual architecture capable of supporting reserved and stratified best-effort service is needed which, in turn, helps amortize the cost of inefficiencies stemming from overprovisioned resources for guaranteed traffic through the filling-in effect [25].
Recently, microeconomic/game-theoretic approaches to resource allocation have received significant interest with application domains spanning a number of different contexts [7], [15], [16], [19], [21], [24], [26], [30], [34], [38], [41], [42], [45], [46]. The overall goal of this area is to formulate a resource allocation problem in the framework of microeconomics and game theory, and show that under certain conditions, the system achieves “desirable” allocations from stabilibity, fairness, and optimality points-of-view. The latter are important in making stratified best-effort bandwidth practically usable by QoS-sensitive applications: predictable service, both in terms of dynamic stability and the rendering of appropriate QoS, are crucial prerequisites to feasibly realizing such an architecture.
The models and approaches proposed in the literature differ along several dimensions, some of the important ones being whether applications or users are assumed to be cooperative or selfish, whether pricing is used or not, and how much computing responsibility is delegated to the user. Several papers have addressed the issue of multi-class QoS provision in high-speed networks [7], [22], [31], [38], [41], [42]. Some of the works employ a cooperative framework or place significant computing responsibilities on the part of the user [31], [41], some investigate the effect of pricing incentives [7], and others represent flow/congestion control and routing models that only partially address the quality of service problem [22], [34], [42].
Our approach differs from previous works in two significant ways. First, we give a comprehensive noncooperative resource allocation model for multi-class QoS provision where users are endowed with heterogenous QoS preferences and make decisions based on selfish user needs. Second, users are allowed to choose both the service classes and traffic volumes assigned to them at a router or switch and the properties associated with utility functions are derived from the networking context. The latter leads to non-concave utility functions and we analyze its impact on the resulting game structure.
Our model, although principally intended to model resource sharing and arbitration at a router — the building block of wide area networks — in the context of QoS provision, is more general in nature and can be applied to other settings including the delivery of packaged network services by an Internet Service Provider (ISP). Specifically, assuming that a service provider exports a number of different services — platinum, gold, silver, bronze — to the user, it is generally the case that the more users subscribe to a particular service class the less the quality of service experienced in that class due to congestion effects. The behavioral characteristics of such a system when users have heterogenous preferences and act selfishly to optimize individual utility falls within the framework of the model studied here.
Our results rely on a set of elementary assumptions which are described next. The formal network QoS provision game is defined in Section 2. We are given n applications or users and m service classes where each user i∈[1,n] has a traffic demand given by its mean data rate λi. Each user can choose where and how much of its traffic to apportion to the m service classes given by its allocation vector Λi=(λi1,λi2,…,λim)T where λij≥0 and Σjλij=λi.
The QoS achieved in service class j∈[1,m] is determined by a QoS function cj (e.g., packet loss rate), and cj is monotone in qj where qj=Σiλij. The generalization to multi-dimensional QoS vectors is shown in Section 3.4. Each user is endowed with a utility function Ui(λij, cj) which indicates the satisfaction received by user i when sending volume λij of traffic receiving QoS level cj through service class j. We assume that Ui is monotone in λij, cj.
The above assumptions are fairly natural given that all that we have said is that the QoS associated with a service class deteriorates when more traffic is pumped into it, users disapprove of bad service quality, and users don't mind sending more if the “cost” is the same. Two simple observations follow from the above. First, since cj is a function of the allocation vectors Λ1, Λ2,…, Λn, by function composition, Ui is a function of the allocation vectors and the latter constitute the only independent variables. Second, by composition of monotone functions, Ui remains monotone in λij. These implied facts will become relevant later.
Before we state the results, three notions are of import to their understanding (defined formally in Section 2.3): Nash equilibrium, Pareto optimum, and system optimum. Roughly speaking, a configuration is a Nash equilibrium if each player cannot improve its individual lot through unilateral actions affecting its traffic allocations. Thus, if every player finds herself in such a “local optimum,” then from the noncooperative perspective, the system is at an impasse — i.e., stable rest point. A configuration is a Pareto optimum if in order to improve the lot of some player, the lot of others must be sacrificed. A configuration is system-optimal if the sum of the individual lots is maximized.
We give a complete characterization of Nash equilibria and their existence conditions. We show that Nash equilibria need not exist and we show that this is attributable to the non-concave — in particular, quasi-concave5 but not concave — nature of utility functions arising in the general networking context. For the special case of unsplittable games, however, where a user's traffic flow is prohibited from being split into separate subflows going into different service classes, we show that Nash equilibria always exist.
We analyze the conditions under which Nash equilibria — if they exist — are Pareto- and system-optimal. The latter is shown to be related to the Pareto optimality of a certain normal form configuration derived from Nash equilibria. We also show that there are Nash equilibria that are Pareto- but not system-optimal, and that there exist Nash equilibria that are not Pareto-optimal and vice versa.
We show that for certain “resource-plentiful” systems, Nash equilibria, Pareto optima, and system optima all coincide collapsing into a single class. This item is interesting from the perspective that it gives a sufficient condition under which Nash equilibria are guaranteed to be desirable in the optimality sense. We also show that for resource-plentiful systems a certain self-optimization procedure leads to quick, robust convergence to globally optimal Nash equilibria.
We extend the game-theoretic analysis to multi-dimensional QoS vector games containing s≥1 different QoS measures. The monotonicity assumptions described in Section 1.2 are generalized to the s-dimensional QoS vector case. We show that the main results carry over if a uniformity assumption is placed either on application preference or on QoS vector functions.
In recent years, there has been a surge of work in “microeconomic approaches to resource allocation” where ideas and tools from microeconomics and game theory have been applied in the formulation and solution of problems arising in flow control, routing, file allocation, load balancing, multi-commodity flow, and quality of service provision, among others [7], [15], [16], [21], [22], [24], [26], [30], [34], [38], [41], [42], [45], [46]. A collection of papers covering a broad range of topics can be found in Ref. [6]. A brief survey of some of the literature is provided in Ref. [14]. Some standard references to game theory and microeconomics include [1], [17], [40], [43], [44].
Many of the earlier papers, including some recent ones [15], [16], [26], [31], [41], have espoused a cooperative game theory framework to model user interactions and derive results based on Pareto optimality. Although fruitful to investigate due to the powerful tools available in cooperative game theory, a potential drawback of this approach is the assumption that users or applications behave cooperatively in networking contexts. For the long-term establishment of virtual circuits or the leasing of telephone lines, the cooperative user model may indeed be viable.6 However, for best-effort applications that comprise much of today's Internet traffic, users are largely anonymous with respect to thousands of other users who concurrently share network resources at any given time, and a noncooperative framework, where each user is assumed to optimize individual performance based on his or her limited available information about the network state is better suited.
The noncooperative framework can be traced as far back as 1981 to a paper by Yemini [49] who has since been more strongly associated with the cooperative approach. The noncooperative network resource allocation approach has been actively pursued by Lazar et al. beginning in the late 1980s [2], [20] with more recent work carried out jointly with Korilis and Orda [21], [22], [23], [24], [34]. Their main work has revolved around an optimal flow control problem, and the development of techniques needed to show the existence of Nash equilibria [22]. Korilis et al. [23], [24] have also looked at the problem of using interventions by an impartial external entity — the network manager — to steer a system toward Nash equilibria that are system-optimal. Of special interest is Orda et al.'s work on routing games [34] which is intimately related to the multi-class QoS provision model studied in this paper. This is further explicated below.
Another significant thrust in noncooperative network games is due to recent work by Shenker [42] where it is shown how choosing a packet scheduling discipline can influence the nature of the Nash equilibria attained. In the context of a congestion control model, it is shown that for a large class of packet scheduling disciplines, a configuration being Nash need not imply that it is Pareto-optimal. A packet scheduling discipline called Fair Share is described and it is shown to lead to Nash equilibria with desirable properties including uniqueness and reachability by a class of self-optimization procedures.
On the implementation side, the work of Waldspurger et al. [45] deserves attention since it is one of the few works that have built a nontrivial working system — CPU allocation and load balancing in workstation networks — and demonstrated that a system based on microeconomic principles can indeed work in practice. Other implementations worth noting include Wellman's work on multicommodity flow problems [46], [47].
Several papers have addressed the specific issue of multi-class QoS provision in high-speed networks using microeconomic models [7], [19], [31], [41]. In [31], [41], utility functions are defined with link bandwidth and switch buffers acting as substitutable resources. Pareto-optimal allocation of resources among service classes is affected either by the network exercising admission control [41] or by users performing purchasing decisions [31]. In both approaches, it is assumed that QoS guarantees are computable, given specific resource reservations. As stated earlier, an important goal of our approach is to shield the user from having to make complex computations to estimate service quality.
In Ref. [7], a general framework for investigating pricing in networks is proposed with service discipline and pricing policy acting as design variables. Simulation results are shown that depict the existence of “desirable” price ranges related to system optimality. The simulations were carried out using a 2-service class packet scheduling algorithm where a shared FIFO queue was partitioned into two segments with high-priority packets being queued at the front and low-priority packets being queued at the back. Four types of applications with different QoS requirements were tested with priority settings set either to 1 or 2.
The flow or congestion control models of Korilis and Lazar [22] and Shenker [42] represent a form of quality of service provision and we explicate the differences between our model and theirs, given that all three follow the noncooperative framework. The main difference between the models by Korilis et al. and Shenker, and the model studied in the paper is that, indeed, theirs is a flow/congestion control model. Phrased in the language of the QoS provision model defined in Section 1.2 (a formal definition is given in Section 2.3), both [22], [42] correspond to the situation where n=m, each player i is permanently assigned to the fixed serviceclass i, and either λii≥0 [42] or 0≤λii≤λi [22], but in both cases, λij=0 for i≠j. That is, a player, being tied to a fixed service class, has the option of controlling how much traffic [42] — or using what time schedule [22] — to send his traffic but not where. Since delay or any other performance measure will deteriorate with increased traffic volume, but volume itself, keeping other things fixed, will generally increase utility, there is an optimum volume assignment — i.e., optimal flow or congestion control — that maximizes player i's utility.
In our model, there is no a priori fixed 1–1 correspondence of players to service classes. Indeed, the very essence of the QoS provision problem is to give each player i∈[1,n] the freedom to choose where she wants to send her traffic, from service class 1 all the way to service class m. Hence, our QoS provision model is more general and fundamentally different from the flow control models in its implications, being more complex and producing equilibria structures that are different from those of [22], [42].
In Ref. [34], Orda et al. present a noncooperative routing game where a set of users with fixed throughput demands have a choice of assigning their flow to a set of parallel links or routes. Although motivated by different contexts, assuming independence between the parallel links — i.e., the performance characteristics (e.g., queueing delay) on some link or route depends only on the aggregate traffic volume assigned to it — a 1–1 correspondence can be established between Orda et al.'s routing model and the QoS provision model studied here.
Phrased in our language, the set of parallel links correspond to the service classes j∈[1,m], and a user i's average throughput demand λi is assigned to the m routes given by the assignment vector Λi=(λi1,λi2,…,λim). Orda et al. then define a cost function Jji which corresponds to our utility function Ui(λij,cj). Both depend on the player i as well as the service class (or route) j. Since Jji is interpreted as a cost function, their's is a minimization problem.
Orda et al. study the routing game under three successively more restrictive assumptions on the cost function Jji (called type-A, type-B, and type-C). In type-B and type-C, the costfunction Jji takes on the form λijcj(qj), thus losing its dependence on i except for the weighting term λij. As is formally defined in Section 2.3, in our QoS provision game, the utility function has the form λijUi(cj(qj)); thus, the utility'sdependence on heterogenous user preferences is preserved. Hence the results proved in Ref. [34] for type-B and type-C functions correspond to a population of users with homogenous preferences, and thus do not carry over to the more general QoS provision game studied here.
As for type-A games where dependence on individual preferences is preserved, the assumption is made that Jji is convex (concave in our context) in λij. However, as has been explicated in Section 1.2, the two monotonicity assumptions — cj is increasing in qj andUi is decreasing in cj — which are basic postulates applicable to most networking contexts of interest, are incompatible with the assumption that Jji is convex is λij. In fact, a simple consequence of the monotonicity assumption is that Jji is quasi-convex in λij. This is so since the composition of the two monotone functions again relates Ui monotonically (decreasing) to λij, and monotone functions are trivially quasi-convex. Convexity and quasi-convexity, in the QoS provision context, however, can lead to different consequences.
In [4], [5], we describe an architecture for noncooperative multi-class QoS provision in many-switch systems7 or wide area networks. Motivated by the analytical results and insights of this paper, we use the single-switch model as a building block in constructing a scalable architecture for multi-class QoS provision in WAN environments. We solve the end-to-end QoS provision problem in many-switch systems and the inter-switch couplings they introduce using distributed control that shields the user from complex computations while preserving the basic premise of selfishness. We show that the network system is able to provide predictable, stratified service without resource reservation and is adaptive under stationary and nonstationary changes to network state.
The rest of the paper is organized as follows. In Section 2, we describe the overall set-up and formulate the network QoS provision model. Section 2.3 discusses the differences between our model and the model of Orda et al. [34], and the impact of heterogenous preferences in bringing about non-concave utilities. This is followed by Section 3 which gives a game-theoretic analysis of the QoS provision game structure. Section 3.3 discusses the resource-plentiful case and Section 3.4 extends the game-theoretic analysis to multi-dimensional QoS vectors. The proofs of our results are contained in a separate Appendix A Proofs of, Appendix B Proofs of, Appendix C Proofs of for the reader's reference. We conclude with a discussion of our results and future work.
Section snippets
Network model
The network model is depicted in Fig. 1. A switch or router is shared by two traffic classes — reserved and nonreserved (or best-effort) — where the former constitutes background or cross traffic and the latter is the aggregate application traffic. That is, λNR=Σi=1nλi where λ1,λ2,…,λn are the mean arrival rates of n application traffic sources. The service rate of the system is given by μ and we will assume that the switch implements a form of GPS packet scheduling with service weights α1,α2,…,
Nash equilibria and existence conditions
This section investigates the structure of Nash equilibria giving a complete characterization of Nash equilibria in the noncooperative multi-class QoS provision game as well as their existence conditions. First, let us impose a total order on the n players given by
Unless otherwise stated, we will assume such a fixed order in the rest of the paper. Following is a simple but often used fact on the induced ordering of the traffic volume thresholds bij. It is a consequence of the total
Conclusion
We have presented a study of the quality of service provision problem in noncooperative multi-class network environments where applications or users are assumed to be selfish. Users are endowed with heterogenous QoS preferences, and they are allowed to choose both where and how much of their traffic to send. Our framework and its conclusions are best suited — but not exclusively so — for best-effort traffic environments where the network is not required to provide stringent QoS guarantees which
Kihong Park received his BA from Seoul National University, Korea, and his PhD in Computer Science from Boston University (1996). Presently, he is an assistant professor of computer science at Purdue University. His research centers around design and control issues in high-speed multimedia networks including congestion control, quality of service provision, routing, and the facilitation of adaptive, fault-tolerant computing on large-scale distributed systems. He has over 40 technical
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Kihong Park received his BA from Seoul National University, Korea, and his PhD in Computer Science from Boston University (1996). Presently, he is an assistant professor of computer science at Purdue University. His research centers around design and control issues in high-speed multimedia networks including congestion control, quality of service provision, routing, and the facilitation of adaptive, fault-tolerant computing on large-scale distributed systems. He has over 40 technical publications and has served on several international program committees. He was a Presidential University Fellow at Boston University, is a recepient of the NSF CAREER Award, Fellow-at-Large of the Santa Fe Institute, and is a member of several professional societies including ACM and IEEE.
Meera Sitharam received a B. Tech. from the Indian Institute of Technology, Madras, India, in 1984 and a PhD in Computer Science in 1990 from the University of Wisconsin, Madison, in 1990. She joined the faculty of the Department of Mathematics and Computer Science at Kent State University in 1990, and served as a Humboldt Fellow at the University of Bonn in 1990–1991. She was a visiting associate professor at Purdue University in 1997–1998. Currently she is associate professor at the CISE department of University of Florida at Gainesville.
Shaogang Chen is a PhD student in Electrical and Computer Engineering at Purdue University. He received his B. Eng. in Electrical Engineering from Tsinghua University, China, in 1990, and a M.S. from Ohio State University in 1995. His research interests include quality of service provision architectures for wide area networks, microeconomic approaches to network resource allocation, and traffic control for bursty traffic.
- 1
Supported in part by NSF grants ANI-9714707, ANI-9875789 (CAREER) ESS-9806741, and grants from PRF and Sprint.
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Tel.: +1-352-392-1492; fax: +1-352-392-1220. Supported in part by NSF grant CCR-9409809.
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Tel.: +1-765-494-0875; fax: +1-765-494-0739. Supported in part by NSF grant ANI-9714707.