Download Quality of service provision in noncooperative networks with ... and more Lecture notes Computer Systems Networking and Telecommunications in PDF only on Docsity!
Quality of service provision in nonco op erative networks
with diverse user requirements
K. Park a;^1 M. Sitharam b;^2 S. Chen a;^3
a (^) Network Systems Lab, Dept. of Computer Sciences, Purdue University, West Lafayette, IN 47907, USA b (^) Dept. of Computer and Information Science and Engineering, University of Florida, Gainesvil le, FL 32611, USA
Abstract
This pap er studies the quality of service (QoS) provision problem in nonco op erative networks where applications or users are sel sh and routers implement generalized pro cessor sharing based packet scheduling. We formulate a mo del of QoS provision in nonco op erative networks where users are given the freedom to cho ose b oth the service classes and trac volume allo cated, and heterogenous QoS preferences are captured by a user's utility function. We present a comprehensive analysis of the nonco op erative multi-class QoS provision game, giving a complete characterization of Nash equilibria and their existence criteria, and show under what conditions they are Pareto and system optimal. We show that, in general, Nash equilibria need not exist, and when they do exist, they need not b e Pareto nor system optimal. For certain \resource-plentiful" systems, however, we show that the world indeed can b e nice with Nash equilibria, Pareto optima, and system optima collapsing into a single class. We study the problem of facilitating e ective QoS in systems with multi-dimensional QoS vectors containing b oth mean- and burstiness-related QoS measures. We extend the game-theoretic analysis to multi-dimensional QoS vector games and show under what conditions the aforementioned results carry over.
Keywords : Quality of service, Multi-class QoS provision, Nonco op erative network game, Heteroge- neous user requirements,
(^1) Corresp onding author. Tel.: (765) 494{7821; fax.: (765) 494{0739; e-mail: park@cs.purdue.edu.
Supp orted in part by NSF grants ANI-9714707, ESS-9806741, and grants from PRF and Sprint. (^2) Tel.: (352) 392{1492; fax.: (352) 392{1220; e-mail: sitharam@cise.u .edu. Supp orted in part by NSF
grant CCR-9409809. (^3) Tel.: (765) 494{0875; fax.: (765) 494{0739; e-mail: chensg@cs.purdue.edu. Supp orted in part by
NSF grant ANI-9714707.
Preprint submitted to Elsevier Preprint 9 March 1999
1 Intro duction
1.1 Background
With the increased deployment of high-sp eed lo cal- and wide-area networks carrying a mul- titude of information from e-mail to bulk data to voice, audio, and video, provisioning ad- equate quality of service (QoS) to the diverse application base has b ecome an imp ortant problem [3,13,27,33]. This pap er describ es a QoS provision architecture suited for b est-e ort environments, based on ideas from micro economics and nonco op erative game theory.
We construct a nonco op erative multi-class QoS provision mo del where users are assumed to b e sel sh, and packets are routed over switches where, as a function of their enscrib ed priority, di erentiated service is delivered. The diverse sp ectrum of application QoS requirements is mo deled using utility functions. Users or applications 4 can cho ose both the service classes and the trac volumes assigned to them. The interaction of users b ehaving sel shly in accordance with their QoS preferences leads to a nonco op erative game whose dynamic prop erties we seek to understand.
The traditional approach to QoS provision uses resource reservations along a route to b e followed by a trac stream so that the stream's data rate and burstiness can b e suitably ac- commo dated. Although research ab ounds [8,9,12,13,18,28,33,35,36,10], analytic to ols for com- puting QoS guarantees rely on shaping of input trac to preserve well-b ehavedness across switches which implement some form of packet scheduling discipline such as generalized pro- cessor sharing (GPS), also known as weighted fair queueing [11,35]. Real-time constraints of multimedia trac and the scale-invariant burstiness asso ciated with self-similar network traf- c [29,39,48,37] limit the shapability of input trac while at the same time reserving bandwidth that is signi cantly smaller than the p eak transmission rate. Thus QoS and utilization stand in a trade-o relationship with each other [37] and transp orting application trac over reserved channels, in general, incurs a high cost.
This makes it imp ortant to organize to day's b est-e ort bandwidth, as exampli ed by the Internet, into strati ed services with graded QoS prop erties such that the QoS requirements of a comp endium of applications can b e e ectively met. This is particularly useful for applications that p ossess diverse but|to varying degrees| exible QoS requirements. It would b e overkill to transp ort such trac over reserved channels. On the other hand, relying on homogenous b est-e ort service, characteristic of to day's Internet, would b e equally unsatisfactory. A dual architecture capable of supp oring reserved and strati ed b est-e ort service is needed which, in turn, helps amortize the cost of ineciencies stemming from overprovisioned resources for guaranteed trac through the l ling-in e ect [25].
Recently, micro economic/game-theoretic approaches to resource allo cation have received signi cant interest with application domains spanning a numb er of di erent con-
(^4) We will use the terms users , applications , and sometimes, players , interchangeably.
loss rate), and cj is monotone in qj where qj =
P
i ij^.^ The^ generalization^ to^ multi-dimensional QoS vectors is shown in Section 3.4. Each user is endowed with a utility function Ui (ij ; cj ) which indicates the satisfaction received by user i when sending volume ij of trac receiving QoS level cj through service class j. We assume that Ui is monotone in ij , cj.
The ab ove assumptions are fairly natural given that all that we have said is that the QoS asso ciated with a service class deteriorates when more trac is pump ed 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 ab ove. First, since cj is a function of the allo cation vectors 1 ; 2 ; : : : ; n , by function comp osition, Ui is a function of the allo cation vectors and the latter constitute the only indep endent variables. Second, by comp osition of monotone functions, Ui remains monotone in ij. These implied facts will b ecome relevant later.
1.3 Summary of new results
Before we state the results, three notions are of imp ort to their understanding (de ned formally in Section 2.3): Nash equilibrium, Pareto optimum, and system optimum. Roughly sp eaking, a con guration is a Nash equilibrium if each player cannot improve its individual lot through unilateral actions a ecting its trac allo cations. Thus if every player nds herself in such a \lo cal optimum," then from the nonco op erative p ersp ective, the system is at an impasse|i.e., stable rest p oint. A con guration is a Pareto optimum if in order to improve the lot of some player, the lot of others must b e sacri ced. A con guration is system optimal if the sum of the individual lots is maximized.
Nash equilibria and existence conditions 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-concave 5 but not concave|nature of utility functions arising in the general networking context. For the sp ecial case of unsplittable games, however, where a user's trac ow is prohibited from b eing split into separate sub ows going into di erent service classes, we show that Nash equilibria always exist.
Relationship to Pareto and system optimality We analyze the conditions under which Nash equilibria|if they exist|are Pareto and system optimal. The latter is shown to b e related to the Pareto optimality of a certain normal form con guration 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.
Resource-plentiful systems We show that for certain \resource-plentiful" systems, Nash
(^5) Recall that a (vector) function f (x) is quasi-concave (quasi-convex ) i for all the set fx : f (x) g
(fx : f (x) g) is convex.
equilibria, Pareto optima, and system optimal all conincide collapsing into a single class. This item is interesting from the p ersp ective that it gives a sucient condition under which Nash equilibria are guaranteed to b e desirable in the optimality sense. We also show that for resource- plentiful systems a certain self-optimization pro cedure leads to quick, robust convergence to globally optimal Nash equilibria.
Extension to multi-dimensional QoS vectors We extend the game-theoretic analysis to multi-dimensional QoS vector games containing s 1 di erent QoS measures. The mono- tonicity assumptions describ ed 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.
1.4 Related work
Micro economic approaches to resource allo cation In recent years, there has b een a surge of work in \micro economic approaches to resource allo cation" where ideas and to ols from micro economics and game theory have b een applied in the formulation and solution of problems arising in ow control, routing, le allo cation, load balancing, multi-commo dity ow, and quality of service provision, among others [15,42,22,21,34,24,26,16,45,46,30,7,41,38]. A collection of pap ers covering a broad range of topics can b e found in [6]. A brief survey of some of the literature is provided in [14]. Some standard references to game theory and micro economics include [1,17,40,43,44].
Many of the earlier pap ers, including some recent ones [16,15,26,31,41], have esp oused a co op- erative game theory framework to mo del user interactions and derive results based on Pareto optimality. Although fruitful to investigate due to the p owerful to ols available in co op erative game theory, a p otential drawback of this approach is the assumption that users or applica- tions b ehave cooperatively in networking contexts. For the long-term establishment of virtual circuits or the leasing of telephone lines, the co op erative user mo del may indeed b e viable 6. However, for b est-e ort applications that comprise much of to day's Internet trac, users are largely anonymous with resp ect to thousands of other users who concurrently share network resources at any given time, and a nonco op erative framework where each user is assumed to optimize individual p erformance based on his or her limited available information ab out the network state is b etter suited.
The noncooperative framework can b e traced as far back as '81 to a pap er by Yemini [49] who has since b een more strongly asso ciated with the co op erative approach. The nonco op erative network resource allo cation approach has b een actively pursued by Lazar and his co-workers
(^6) It is also p ossible that intermediaries p erform long-term leasing of network resources which are
then packaged and made available as high-level services to the user. Asp ects of such activities may b e mo deled as coalition b ehavior.
ow/congestion control model. Phrased in the language of the QoS provision mo del de ned in Section 1.2 (a formal de nition is given in Section 2.3), b oth [22] and [42] corresp ond to the situation where n = m, each player i is p ermanently assigned to the xed service class i, and either ii 0 [42] or 0 ii i [22], but in b oth cases, ij = 0 for i 6 = j. That is, a player, b eing tied to a xed service class, has the option of controlling how much trac [42]| or using what time schedule [22]|to send his trac but not where. Since delay or any other p erformance measure will deteriorate with increased trac volume, but volume itself, keeping other things xed, will generally increase utility, there is an optimum volume assignment|i.e., optimal ow or congestion control|that maximizes player i's utility.
In our mo del, there is no a priori xed 1-1 corresp ondence of players to service classes. Indeed, the very essence of the QoS provision problem is to give each player i 2 [1; n] the freedom to cho ose where she wants to send her trac, from service class 1 all the way to service class m. Hence, our QoS provision mo del is more general and fundamentally di erent from the ow control mo dels in its implications, b eing more complex and pro ducing equilibria structures that are di erent from those of [22], [42].
Comparison with Orda et al.'s routing mo del In [34], Orda et al. present a nonco op- erative routing game where a set of users with xed throughput demands have a choice of assigning their ow to a set of paral lel links or routes. Although motivated by di erent con- texts, assuming independence b etween the parallel links|i.e., the p erformance characteristics (e.g., queueing delay) on some link or route dep ends only on the aggregate trac volume as- signed to it|a 1-1 corresp ondence can b e established b etween Orda et al.'s routing mo del and the QoS provision mo del studied here.
Phrased in our language, the set of parallel links corresp ond to the service classes j 2 [1; m], and a user i's average throughput demand i is assigned to the m routes given by the assignment vector i = (i 1 ; i 2 ; : : : ; im ). Orda et al. then de ne a cost function J (^) ji which corresp onds to our utility function Ui (ij ; cj ). Both dep end on the player i as well as the service class (or route) j. Since J (^) ji 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 J (^) ji (called typ e-A, typ e-B, and typ e-C). In typ e-B and typ e-C, the cost function J (^) ji takes on the form ij cj (qj ) thus losing its dep endence on i except for the weighting term ij. As is formally de ned in Section 2.3, in our QoS provision game, the utility function has the form ij Ui (cj (qj )); thus the utility's dep endence on heterogenous user preferences is preserved. Hence the results proved in [34] for typ e-B and typ e-C functions corresp ond to a p opulation of users with homogenous preferences, and thus do not carry over to the more general QoS provision game studied here.
As for typ e-A games where dep endence on individual preferences is preserved, the assumption is made that J (^) ji is convex (concave in our context) in ij. However, as has b een explicated in Section 1.2, the two monotonicity assumptions|cj is increasing in qj and Ui is decreasing in cj |which are basic p ostulates applicable to most networking contexts of interest, are in- compatible with the assumption that J (^) ji is convex is ij. In fact, a simple consequence of the
monotonicity assumption is that J (^) ji is quasi-convex in ij. This is so since the comp osition of the two monotone functions again relates Ui monotonically (decreasing) to ij , and mono- tone functions are trivially quasi-convex. Convexity and quasi-convexity, in the QoS provision context, however, can lead to di erent consequences.
Many-switch systems In [4,5], we describ e an architecture for nonco op erative multi-class QoS provision in many-switch systems 7 or wide area networks. Motivated by the analytical results and insights of this pap er, we use the single-switch mo del as a building blo ck in con- structing 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 intro duce using distributed control that shields the user from complex computations while preserving the basic premise of sel shness. We show that the network system is able to provide predictable, strati ed service without resource reservation and is adaptive under stationary and nonstationary changes to network state.
The rest of the pap er is organized as follows. In Section 2, we describ e the overall set-up and formulate the network QoS provision mo del. Section 2.3 discusses the di erences b etween our mo del and the mo del of Orda et al. [34], and the impact of heterogenous preferences in bringing ab out 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 pro ofs of our results are contained in a separate App endix for the reader's reference. We conclude with a discussion of our results and future work.
2 Nonco op erative network QoS provision game
2.1 Network model
The network mo del is depicted in Figure 5.1. A switch or router is shared by two trac classes|reserved and nonreserved (or best-e ort )|where the former constitutes background or cross trac and the latter is the aggregate application trac. That is, NR^ =
Pn
i=1 i^ where 1 ; 2 ; : : : ; n are the mean arrival rates of n application trac sources. The service rate of the system is given by and we will assume that the switch implements a form of GPS packet scheduling with service weights 1 ; 2 ; : : : ; (^) m where (^) j 0, j 2 [1; m], and
Pm
j =1 j^ =^ 1. Here, m denotes the numb er of service classes. The total service rate is split b etween the two trac classes = R^ + NR^. Service class j of the nonreserved trac class thus receives a service rate of (^) j NR^.
(^7) Also called network of switches in [42].
Cell Loss Rate
Relative Utility
(^01)
E - mail
Cell Loss Rate
Relative Utility
(^01)
Video Application
Fig. 2.2. Utility functions. E-mail application (left) and video application (right).
2.3 De nition of network QoS provision game
QoS provision problem Assume we are given m service classes and n applications or players represented by their mean arrival rates 1 ; : : : ; n and utility functions U 1 ; : : : ; Un. We arrive at a resource allo cation problem in the following way. Let ij 0, i 2 [1; n], j 2 [1; m], denote the trac volume of the i'th application assigned to service class j. Thus, i =
Pm
j =1 ij^.^ That is, application i is given the freedom to cho ose which service classes to assign her trac to and how much. We also consider the sp ecial case when trac assignments are restricted to b e \all in one bag," i.e., ij 2 fi ; 0 g, for all j 2 [1; m].
Let = (ij : i; j ) denote the resource assignment matrix, and let c 1 ; c 2 ; : : : ; cm b e the packet loss rates of the m service classes. Each packet loss rate is a function of ,
cj = cj (); j 2 [1; m]:
Assuming isolatedness (cf. Section 1.2), we have cj = cj (qj ) where qj =
Pn
i=1 ij^ is^ the^ to- tal trac volume assigned to class j. This relation is only approximate for work conserving switches. The precise mo deling of nonlinearities arising from work conservation, although in- teresting in its own right, is a general issue not sp eci c to our context, and we will ignore its e ect in this pap er.
We will also make the assumption that cj is monotone in qj , i.e., dcj =dqj 0, a prop erty satis ed by virtually all service disciplines of interest 8. We will also assume that dUi =dc 0. That is, making the packet loss rate smaller 9 can never decrease the utility exp erienced by player i.
The mo del can b e extended to the case when application QoS requirements are represented
by multi-dimensional QoS vectors x 2 R s^ , s 1. For example, in addition to packet loss rate,
x may sp ecify delay requirements as well as restrictions on their uctuations such as jitter.
(^8) We sometimes use continuous notation for exp ositional purp oses. Our results do not dep end on cj and Ui b eing smo oth. (^9) An analogous assumption is made in the multi-dimensional QoS vector case (Section 3.4).
It turns out that the analysis of the multi-dimensional case reduces to the scalar case under certain conditions, and we will pro ceed with packet loss rate c as the sole QoS indicator.
The weighted utility of application i, given assignment , is de ned as
Ui () =
X^ m
j =
ij Ui (cj ):
Note that the utility function used in Section 1.2, Ui (ij ; cj ), corresp onds to ij Ui (cj ). Sub ject to the ab ove constraints, the static optimization problem can b e formulated as
max
U^ () =
X^ n
i=
U^ i (): (2.1)
This is a nonlinear programming problem with equality constraints.
Nash equilibria, Pareto optima, and system optima Any ^ that satis es (2.1) is called system optimal. Thus system optimality corresp onds to optimizing the usual resource allo cation ob jective function. An assignment ^ is Pareto optimal if for all ,
8 i : Ui (^ ) Ui () =) 8 i : Ui (^ ) = Ui ():
That is, Pareto optimality states that total utility U can only b e improved at the exp ense of one or more individual utility Ui. In general, Pareto optimality do es not imply system optimality. But, clearly, b eing system optimal implies is Pareto optimal.
The formulation of Nash equilibrium needs a further de nition. Given , let i = (i 1 ; i 2 ; : : : ; im ) denote the i'th player's assignment vector. i is also called the strategy of player i. Let
Li () = f ^0 : ^0 k = k ; k 6 = i; and k^0 i k 1 = i g
where kxk 1 =
Pm
j =1 jxj^ j.^ That^ is,^ Li^ ()^ is^ the^ set^ of^ all^ unilateral^ strategies^ for^ player^ i.
An assignment ^ is a Nash equilibrium if 8 i 2 [1; n], 8 2 Li (^ ),
Ui () Ui (^ ):
That is, in a Nash equilibrium, player i cannot improve its individual utility Ui by unilaterally changing its strategy.
In general, a system optimal assignment need not b e a Nash equilibrium and little can b e said ab out the relation b etween system optimality, Pareto optimality, and Nash equilibria. In the context of the nonco op erative network environment where every player acts sel shly, we are interested in characterizing assignments that are Nash since they represent stable xed p oints
3 Prop erties of nonco op erative QoS provision game
3.1 Nash equilibria and existence conditions
This section investigates the structure of Nash equilibria giving a complete characterization of Nash equilibria in the nonco op erative multi-class QoS provision game as well as their existence conditions. First, let us imp ose a total order on the n players given by
i i^0 () i i^0 :
Unless otherwise stated, we will assume such a xed order in the rest of the pap er. Following is a simple but often used fact on the induced ordering of the trac volume thresholds bij. It is a consequence of the total ordering of i and the monotonicity of cj.
Prop osition 3.1 8 i 2 [1; n 1], 8 j 2 [1; m], bij bi+1j.
Next, we de ne certain subsets of service classes|parameterized by user i|that come into play when characterizing Nash equilibria. Let I (^) i+ = f j 2 [1; m] : qj > bij ; ij > 0 g, I (^) i = f j 2 [1; m] : qj < bij g, and I (^) i^0 = f j 2 [1; m] : qj = bij g. That is, I (^) i+ denotes the set of service class indices where player i has assigned a p ositive ow and the total trac volume allo cated exceeds player i's threshold. Thus user i attains 0 utility in these service classes. Conversely for I (^) i and I (^) i^0 , however, it is not required that user i have a nonzero assignment in these classes. Let q ij =
P
k 6 =i k^ j^.^ That^ is,^ q^ i j is^ the^ trac^ volume^ assigned^ to^ service^ class^ j^ not^ counting player i's contribution (if any). Hence qj = ij + q ij.
Let J (^) i+ = f j 2 [1; m] : q (^) ji bij g and J (^) i = f j 2 [1; m] : q (^) ji < bij g. Hence J (^) i+ is the set of service classes where, irresp ective of player i's actions, player i cannot garner any utility. Let J (^) i = f j 2 [1; m] : bij q (^) ji = mink 2 J (^) i bik q ik g. J (^) i is the subset of service classes of J (^) i where the p ositive utility achievable by user i is minimal.
The next two results give uniform upp er b ounds on the individual utility of a xed player where uniformity is with resp ect to all unilateral strategy changes by the player. Recall that the latter is denoted by Li () where is any con guration.
Prop osition 3.2 Given , i 2 [1; n], let vi =
P
j 2 J (^) i ^ bij^ ^ q^
i j.^ Then
8 ^0 2 Li (); Ui (^0 ) vi :
Prop osition 3.3 Given , i 2 [1; n], let i > vi and J (^) i+ = ;. Then 9 j ^2 J (^) i such that
8 ^0 2 Li (); Ui (^0 ) vi (bij q ij ):
The two prop ositions are used in the pro of of the following theorem which gives a complete characterization of Nash equilibria.
Theorem 3.4 (Nash characterization) is a Nash equilibrium i 8 i 2 [1; n] either
(a) I (^) i+ = ;, or (b) I (^) i = ;, J (^) i+ 6 = ;, J (^) i I (^) i^0 , or (c) I (^) i = ;, J (^) i+ = ;, 9 j ^2 J (^) i such that J (^) i n fj ^ g I (^) i^0.
In words, for each player i, one of three conditions must hold: a user either achieves full individual utility i (part (a)), or partial utility vi =
P
j 2 J (^) i ^ bij^ ^ q^
i j \dumping"^ the^ excess trac i vi into one or more service classes b elonging to J (^) i+ (part (b)), or partial utility vi (bij ^ q (^) ji ) with excess trac b eing assigned to one of the service classes in J (^) i (part (c)). Service classes b elonging to J (^) i+ form the most natural dumping ground for channeling excess trac since player i cannot derive utility from j 2 J (^) i+ no matter what. If J (^) i+ = ;, J (^) i takes on a surrogate role.
The next lemma gives a simple suciency condition for 2-application/2-service class games in which Nash equilibria do not exist.
Lemma 3.5 Consider the family of 2 -application= 2 -service class systems such that the thresholds bij on the total trac volume of the service classes satisfy b 1 j < b 2 j , j = 1 ; 2 (i.e., the ordering of Proposition 3.1 is strict ). Furthermore, assume the fol lowing inequalities hold:
(a) 2 < b 11 + b 12 , (b) 2 + 1 > b 21 + b 22 > b 11 + b 12 , (c) 2 > maxfb 11 ; b 12 g.
Then, for such choices of i , bij , no Nash equilibrium exists.
Games satisfying the ab ove conditions are easy to construct, and the reason that there are no Nash equilibria is b ecause the game leads to a limit cycle. This typ e of b ehavior has also b een observed in simulation studies. Next we generalize the \Nash Non-Existence condition" to n-application/m-service class games. The pro of of Theorem 3.6 can b e reduced to Lemma 3. and is a straightforward consequence.
Theorem 3.6 (Nash non-existence) Consider a n-application/m-service class game where the ordering implied by Proposition 3.1 is strict. If there are players i^0 and i^ with i^ > i^0 satisfying
(a)
P
i 6 =i^0
i <
P
j
bi^0 j , (b)
P
i
i >
P
j
bi (^) j ,
(c)
P
ii^0
i +
P
i>i
i < min j bi^ j ,
then no Nash equilibrium exists.
rem 3.15, we show that for certain \resource-plentiful" systems, there is robust convergence to Nash equilibria from any initial state.
3.2 Relationship to Pareto and system optimality
In this section, we characterize the relationship b etween Nash equilibria, Pareto optimal, and system optima for the multi-class QoS provision game. First, we state a useful lemma that can b e used to relate Pareto optimality of a con guration to system optimality.
For a con guration , an equivalent assignment ^0 can b e found with the same total utility so that the players are partitioned into two sets around a unique, dividing player i^0. The rst set consists of players with indices larger than i^0 with resp ect to the ordering induced by Prop osition 3.1, with all players having full utility. The second set consists of players with smaller indices than i^0 , all of them having zero utility. The third set is the singleton set fi^0 g consisting of the dividing player who has partial utility. We will call such an assignment ^0 a normal form of .
Lemma 3.9 (normal form) Let be a con guration with U () <
Pn
i=1 i^.^ Let^ i^ maxfi : Ui () < i g. Then 9 ^0 with U (^0 ) = U () such that
(a) 8 i < i 0 , Ui (^0 ) = 0 , and (b) 8 i > i 0 , Ui (^0 ) = i.
The usefulness of the normal form of a con guration (including Nash) comes into play when checking for system optimality of a Nash assignment. This is so since, as we shall see, it is sucient to check Pareto optimality of the normal form to establish system optimality of the original con guration. Moreover, a normal form is easy to obtain from the original Nash con guration (construction in the pro of of Lemma 3.9) and checking for Pareto optimality is generally easier than checking for system optimality.
Theorem 3.10 (Pareto & system optimal) Given a con guration , let ^0 be its normal form. Then is system optimal i ^0 is Pareto optimal.
An immediate corollary of the theorem is that a Nash equilibrium is system optimal i its nor- mal form is Pareto optimal. Although Theorem 3.10 gives an interesting relationship b etween Pareto optimality and system optimality and is useful for reasoning ab out Nash equilibria in other contexts, it falls short of further exploiting p otential structure sp eci c to Nash equilibria. It is an op en question whether there is some \indep endence" relation b etween Nash equilibria and system optima for the general multi-class QoS provision game.
Given the form of Theorem 3.10, one may wonder whether all assignments that are Nash and Pareto optimal are also system optimal. The next result gives a counterexample which shows that Theorem 3.10 is \tight" in the sense that, when conditioned with Nash equilibria, there are assignments that are b oth Nash and Pareto but not system optimal.
Prop osition 3.11 There exist Nash equilibria that are Pareto optimal but not system opti- mal.
Next, we characterize those Nash equilibria that are Pareto optimal. First, consider a modi ed game , parameterized by some assignment , de ned as follows. The thresholds for the players remain the same as in the original game. However, for each player i, the mean arrival rates are taken to b e (^) i Ui (). Moreover, there is an additional player 0 whose thresholds b 0 j are all 0, but whose trac demand is 0 =
P
i i^ ^
P
i i^.^ Note^ that^ the^ con^ gurations^ (^0) in
the original game for which 8 i : Ui (^0 ) Ui () corresp ond (many-to-one) to system optimal con gurations M for the mo di ed game. Let ij := mini 6 =0 f (^) ij > 0 g:
Theorem 3.12 (Nash-Pareto characterization) Let be a Nash equilibrium and let i be the player such that 8 i > i^ , Ui () = i ; i.e., i^ is the largest player with incomplete utility. Then is a Pareto optimum if and only if the fol lowing hold:
(a) 8 i i^ , I (^) i+ fj : qj > bi^ j g. (b) 8 j [qj bi (^) j ) 8 i j 62 I (^) i+ ]. Notice since is Nash, it fol lows from the hypothesis above and Theorem 3.4 that qj = bi^ j. (c) The two sets of players S 1 fi > i^ : 9 j ij > 0 ; 9 i^0 i^ j 2 I (^) i+ 0 g and S 2 fi > i^ : 9 j ij > 0 ; qj bi^ j g are disjoint. (d) For any system optimum con guration M of the modi ed game, i.e., U (M )
Pn
i=1 i^ , one of the fol lowing holds for each service class j : (d1)
Pn
i= ij =^ bij j when^ ij is^ de^ ned, (d2)
P
i 6 = ij ^ bi^ j , (d3) 0 > bi^ j
P
i 6 =
ij +^
P
j 0 6 =j
bij 0 j
P
i
P
j 0 6 =j
ij.
Note that in part (c) of Theorem 3.12, an even stronger statement is true: Consider the directed graph G whose vertices are the players i > i^ and whose edges are de ned as follows. An edge (i 1 ; i 2 ) exists in G if and only if fj : i 1 j > 0 ; i 2 j > 0 g 6 = ; or 9 j 1 ; j 2 with i 1 j 1 > 0, i 2 j 2 > 0, and qj 2 bi 1 j 2. Then there is no path from any vertex in S 2 to any vertex in S 1 in the graph G. In other words, for all players i > i^ , there is a path from S 2 to i, or from i to S 1 , or neither, but not b oth.
There are several interesting p oints to note in the ab ove characterization. First, parts (a) and (b) dep end on the combination of facts that is b oth Nash and Pareto. Parts (c) and (d), however, dep end only on the fact that is Pareto. Second, removing the universal quanti er in (d) (\For any con guration M : : : ") is imp ossible for reasons similar to removing the existential quanti er in the statement of Theorem 3.7. The problem of deciding whether a con guration is not Pareto is NP -complete as long as the thresholds of each player are allowed to vary arbitrarily across the classes. Third, the optimization problems that corresp ond to the ab ove decision problems p ossess convex feasible regions but the ob jective functions are highly nonlinear and even discontinuous.
On the other hand, the feasible region can b e naturally partitioned into convex subregions over
and Theorem 3.13 shows that this is indeed the case even when users are sel sh. The next theorem shows that such desirable con gurations can b e realized in a nonco op erative manner starting from any initial con guration.
Theorem 3.15 (convergence) Assume the supposition of Theorem 3.13 holds. Then, start- ing from any initial con guration 0 , the dynamic update process P converges to a Nash equi- librium . Moreover, is attained as soon as the sequence of players (i.e., moves) in the process P includes the subsequence n; n 1 ; : : : ; 1.
3.4 Extension of game-theoretic analysis to multi-dimensional QoS vectors
In Section 2, we formulated a nonco op erative QoS provision game based on singleton QoS vectors, x = (c), where c was a b ound on packet loss rate. Here, we will extend the mo del
to multi-dimensional QoS vectors x 2 R s^ , s 1, and show that the singleton vector analysis
carries over unchanged.
Let x = (x 1 ; x 2 ; : : : ; xs )T^ , and let xj^ = (xj 1 ; xj 2 ; : : : ; xjs )T^ denote the quality of service rendered to service class j 2 [1; m]. As b efore, we make the monotonicity assumption dxjr =dqj 0, r 2 [1; s], j 2 [1; m], which is satis ed by most packet scheduling p olicies of interest including weighted fair queueing. Each player's utility function Ui (x), i 2 [1; n], has the form
Ui (x) =
1 ; if 8 r 2 [1; s], xr ir , 0 ; otherwise,
where i^ = ( 1 i ; i 2 ; : : : ; is )T^ 0 is the multi-dimensional threshold vector that represents the i'th application's preference.
In order to deal with the multi-dimensional QoS vectors and thresholds uniformly, we hence- forth make one of two uniformity assumptions: either assume that the thresholds (^) ri can b e ordered such that the ordering is uniform over r , i.e.,
8 r 2 [1; s]; 8 i 2 [1; n] : ir i r+1 ; (3.16)
or we assume that the functional forms xjr are uniform over r for each j , i.e.,
8 j 2 [1; m] : xj 1 = xj 2 = = xjs : (3.17)
By isolatedness, xjr = xjr (qj ), r 2 [1; s], j 2 [1; m], and just as in Prop osition 3.1, the condition xjr (qj ) ir can now b e stated as qj brij using the de nition
brij = (xjr ) ^1 ( (^) ri ):
Let bij b e the minimum over r , i.e., bij = minr 2 [1;s] brij.
We can now rephrase Ui (xj^ ) as
Ui (xj^ ) =
1 ; qj bij , 0 ; otherwise.
Moreover, under the assumption that the functional forms xjr are uniform over r for each j where xj satis es 8 j 2 [1; m], 8 r 2 [1; s], xjr = xj , and using the monotonicity of xj , it can b e observed that the following identity holds:
bij = min r 2 [1;s]
(xj ) ^1 ( ir ) = (xj ) ^1 ( min r 2 [1;s]
ir ): (3.18)
That is, the min op erator commutes with (xj ) ^1.
Now we are ready to state a total ordering on bij for xed j corresp onding to its counterpart Prop osition 3.1.
Prop osition 3.19 For the multi-dimensional QoS vector model with assumption (3.16) or (3.17), there exists an ordering of the players i 2 [1; n] such that 8 i 2 [1; n 1], 8 j 2 [1; m],
bij bi+1 j :
Prop osition 3.20 The game-theoretic results of Section 3 hold for the multi-dimensional QoS vector model with assumption (3.16) or (3.17).
The pro of structure of our game-theoretic results rely on Prop osition 3.1 to order application QoS preferences. The QoS vectors (i.e., scalar packet loss indicator) and their functions a ect the pro of only through Prop osition 3.1. Thus, under either of the uniformity assumptions, and with Prop osition 3.19 in hand, it is straightforward to check that the pro ofs carry over unchanged giving Prop osition 3.20.
4 Conclusion
We have presented a study of the quality of service provision problem in nonco op erative multi- class network environments where applications or users are assumed to b e sel sh. Users are endowed with heterogenous QoS preferences, and they are allowed to cho ose b oth where and how much of their trac to send. Our framework and its conclusions are b est suited|but not exclusively so|for b est-e ort trac environments where the network is not required to provide stringent QoS guarantees which can only b e accomplished, currently, by employing conservative resource reservations. Rather, service classes with di erentiated QoS levels matching the needs of constituent applications are induced by the latter's sel sh interactions, providing reasonably stable and predictable QoS levels as a function of network state.
We have formulated a nonco op erative multi-class QoS provision mo del and given a comprehen- sive analysis of its prop erties. We have shown that Nash equilibria|which corresp ond to stable
