Pareto Set, Fairness, and Nash Equilibrium: A Case Study on Load Balancing

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1 Pareto Set, Fairness, and Nash Equilibrium: A Case Study on Load Balancing Atsushi Inoie, Hisao Kameda, Corinne Touati Graduate School of Systems and Information Engineering University of Tsukuba, Tsukuba Science City, Japan {inoie, kameda, corinne}@osdp.is.tsukuba.ac. jp Abstract Various fairness objectives are studied in relation to Pareto optimal sets and Nash equilibria. We examine the already discussed general parameterized fairness objective that covers a variety of fairness criteria and the newly introduced Nash-proportionate-fairness objective. We study them mainly numerically on a simple static load balancing model with two identical servers (computers) each of which has an independent arrival process and its own queue. Through the numerical results, several intuitive results are shown. For example, we observe that the points that achieve the general parameterized fairness objectives may cover a part but not all of the Pareto set, and at times, do not cover the Nash-proportionate-fair Pareto optimal point. Keywords Fairness,, noncooperative game, Pareto optimum and inefficiency, mathematical programming / optimization. Introduction There exist many systems where multiple independent users, or players, may strive to optimize their own utility or cost unilaterally, which can be regarded as noncooperative games. The situation where each user attains its own optimum coincidentlyis anashequilibrium. Nashequilibria may, however, be Paretoinefficient. In particular, we call a situation of a system strongly Pareto inefficientif all users have more benefits in another situation than the considered situation. As for the communication and transportation networks, examples of such strong Pareto inefficiency have been shown with respect to noncooperative routing, first by Braess (9), and a number of related studies followed (Murchland 90, Frank 9, Cohenand Kelly 990, Kelly 99, Cohen andjeffries 99, Korilis et al. 99, Korilis et al. 999, Roughgarden 00). As for the non-cooperative load balancing in distributed computer systems, the existence of paradoxes that appear only in the case of a finite number of players but not in the case of infinitesimal players has been shown (Kamedaet al. 000, Kameda and Pourtallier 00). Note that load balancing and routing have mutually similar logical structures (Tantawi and Towsley 9, Kameda et al. 99, Li and Kameda 99, Altman and Kameda 00). On the other hand, there can exist innumerably many Pareto-optimal situations. The choice of one to achieve can be controversial among users. One selection criterion is fairness among users. Various fairness concepts that achieve Pareto optima but are not directly related to Nash equilibria have been already proposed (Bertsekas and Gallager 99, Maulloo et al. 99, Kelly 99, Mazumdar et al. 99, Mo and Walrand 000). In contrast, each is fair on all users in the sense that it is achieved by the fair competition (with no coalition) among users. Then, among the Pareto-optima, only those that are strongly Pareto-superior to the Nash equilibrium could satisfy all users. In particular, as the situations that would make all users to feel fairness similar to that of the, we consider a group of situations where each user s utility is proportionately larger than that of the. We say that such situations are Nash proportionately fair to the. If we identify a Nash-proportionately-fair Pareto optimum, the resulting situation will satisfy all users since it reflects the competitive fairness given by the and is Pareto optimal, at the same time. By the Pareto set of a system, we mean the set of all Pareto optima of the system. We are quite interested in the positions that the already proposed and Nash proportionate fairness objectives occupy in the Pareto set. Since it may seem difficult to study this problem in a general framework from this beginning stage, in this paper, we use a simple model of load balancing in distributed computer systems as the platform of the present research. We numerically obtain the cost (the mean response time) of each user at the points in the Pareto set, at the solutions that achieve various fairness objectives, and at the (which is unique in this case (Altman et al. 00)). This paper characterizes these fairness objectives through numerical results in simple static load balancing model with two identical servers (computers) each of which has an identical arrival and its own queue. In numerical results,

2 we compare the fairness objectives. For example, we observe that the points that achieve the general parameterized fairness objectives generally cover a part of the Pareto set, and at times, do not cover the Nash-proportionate-fair Pareto optimal points. The rest of this paper is organized as follows. Section describes our model and formulates as various types of fair and optimal load balancing problems. Section presents the analytical preliminaries. Section shows some numerical results. Section concludes this article. Model and Assumptions We consider a distributed computer system consisting of two servers (computers), and, with two flows of demands φ and φ arriving from users and at servers and, respectively (Kameda et al. 000). Let a fraction x i (0 x i φ i ) of a flow of jobs be forwarded from server i to the other server j ( i). Denote by x the vector (x, x ). Denote further by β and β, respectively, the resulting loads on nodes and. Then, The system is depicted by Figure. β i = φ i x i + x j, i, j =, (i j). x x φ φ Figure : Load balancing in a distributed system consisting of two servers We assume that the processing time at server i for the load of rate β i is given by. For simplicity, we assume µ i β i that forwarding a job requires a fixed delay t. Therefore, the cost of user i, that is, the delay of each flow arriving from the user i, can be written: T i (x) = [ ] φ i x i + x i (t + ) for i, j =, ( j i). () φ i µ i φ i + x i x j µ j φ j + x j x i Denote by C the feasible region of x,thatis,c = {x 0 x i φ i, i =, andµ i φ i + x i x j > 0, i, j =, (i j)}. Clearly, C is a convex set. For example, we may consider that the utilities of the users are inversely proportional to their costs, that is, U i (T i ) = /T i, i =,. In this system, a x is given as follows: T i ( x) = min x i T i (x i, x j ), s.t. (x i, x j ) C, i, j =, (i j). () For the model in question, there exists a unique (Altman et al. 00). Minimization of weighted sums of costs Minimization of a weighted sum of costs of users gives the following: Ω( x) = min Ω(x) where Ω(x) = ξ p T p (x),ξ i 0, i =,, ξ p > 0. () x C Then, the cost of user i is given by T i ( x), i =,. Clearly, x gives a Pareto optimum. p p

3 Already proposed fairness Already proposed lines of general parameterized fairness objectives are expressed, for example, in the following form (Mo and Walrand 000). F( ˆx) = min F(x) where F(x) = x C α {T p (x)} α. () Similarly as (), we consider maximization of the utilities of the users, and formulate the following fairness objectives: `F( ˆx) = max x C `F(x) where `F(x) = α p {U p (x)} α and U p (x) = /T p (x). () The case of α = 0 shows the simple sum of the costs of the users. The case of α presents a Nash bargaining solution and, in particular, proportional fairness in this system. The case of α corresponds to the Max-Min fairness (Bertsekasand Gallager99, Maulloo et al. 99, Kelly 99, Mazumdaret al. 99, Massoulié and Roberts 999, Boudec 000,Bonald and Massoulié 00, Ya iche et al. 000). Obviously, () also follows the lines of already proposed general parameterized fairness objectives. p Pareto set Denote by Π the Pareto set defined as follows: Π = {(T (x), T (x)) x C, and for any x (x C)ifT i (x ) < T i (x)(i =, ) then T j (x ) > T j (x)( j i)} Nash proportionate fairness The, T i ( x), i =,, may be Pareto inefficient (Kameda et al. 000). Consider P i = ηt i ( x), i =,, By decreasing η, if(p, P ) hits the curve Π, and reaches a Pareto optimal point, ( P, P ), it is the Nash-proportionate-fair Pareto optimum. Analytical Preliminaries Although the model of the system investigated in this paper may look simple, we need to rely mostly on numerical examinations to obtain insight into the problem. We have, however, some preliminary properties obtained analytically. Convexity We have already noted that the strategy set C is convex. On the other hand, T i (x), i =,, x C, is not convex in x while T i (x), i =,, is convex in each of x and x for (x, x ) C. Pareto set and minimization of weighted sums of costs We have the following property (Aubin 99). If the strategy set C is convex, and if the cost functions T i, i =,, are convex, any Pareto optimum is given by a solution x of the minimization of weighted sums of costs (). In fact, however, T i, i =,, are not convex in x as noted in the above paragraph. Therefore, there may exist Pareto optimal points that are not a solution of minimizing a weighted sum of costs, which is actually shown in the numerical examples given in the later section. Already proposed fairness and minimization of weighted sums of costs Recall (). Denote C i (x) = {T i(x)} α α for α 0, while C i (x) = log T i (x) forα =. Then, we have C i (x) C i (x ) T i (x) T i (x ), i =,. That is, a Pareto optimum for user cost C i, i =,, is also a Pareto optimum for user cost T i, i =,, and vice versa. Similarly as () gives a Pareto optimum for user cost T i, i =,, () will give also a Pareto optimum for user cost T i, i =,. The same arguments hold for (), and the objective () will also give a Pareto optimum for user cost T i, i =,. Numerical Results We characterize fair and optimal load balancing problem through some numerical results. For convenience, we add the constraint ξ + ξ = in minimization of weighted-sums without losing generality. We have paid special attention not to catch local optima that are not global optima since the functions to be optimized are not convex in general.

4 . Nash proportionate-fair Pareto optimum. Weighted-sum Pareto border x= x= Figure : Combinations of response times, respectively, T and T,ofusersandinthecasewherethevaluesof system parameters are φ =., µ =, φ =., µ =., and t = Note that the left and right graphs of this figure and the left and right graphs of the following figure show the same case of the system. Only differences lie in that the points shown may be what achieve objectives different among the four graphs. This applies also to the set of graphs in Figs. and and to the set of graphs in Figs. and. We show the points that achieve [Left] Nash proportionate fairness and [Right] minimization of weighted sums of costs. We observe that the points that achieve the weighted-sum objective do not cover all the Pareto optimal points =. =.. x= x= Figure : Combinations of response times, respectively, T and T, of users and in the case where φ =., µ =, φ =., µ =., and t = We show the points that achieve the already proposed fairness objectives () [Left] and () [Right]. A case where weighted-sum objective does not cover all the Pareto optima Figures and show the Nash equilibrium, the part of Pareto set obtained by the weighted-sum optimization and the fairness solutions that achieve () and (). The values of the system parameters are φ =., µ =, φ =., µ =., and t = In the right graph of Figure, we observe that the part of the Pareto set obtained by the weighted-sum objectives are divided into two parts and do not cover all the Pareto set. We note that T and T are nonconvex in x, and, therefore, the optimal solutions to the weighted-sum objectives may not cover the Pareto set since the conditions of the Aubin s theorem are not satisfied. Thus, the above result presents a counter example that shows that if a condition of the Aubin s theorem is not satisfied, the theorem does not hold. In Figure, we observe that, as to the solutions obtained by the already proposed fairness objectives (), the optimal points converges to the point of the Pareto set satisfying T = T as the value of α increases. (Note, however,

5 . Nash proportionate-fair Pareto optimum. Weighted-sum Pareto border x= x=0 Figure : Combinations of response times, respectively, T and T, of users and in the case where φ = 0.9, µ =., φ =, µ =, and t = 0.. We show [Left] the point that achieves the Nash proportionate fairness, and [Right] the points that achieve the minimization of weighted sums of costs. We observe that, in this case, the Nash equilibrium is not Pareto optimal.... = x= x=0 Figure : Combinations of response times, respectively, T and T, of users and in the case where φ = 0.9, µ =., φ =, µ =, and t = 0.. We show the points that achieve the already proposed fairness objectives () [Left] and() [Right]. We note that, in this case, the is not Pareto optimal. that, in this case, we were unable to obtain numerically the optimal values for very large values of α (> 00). This is perhaps because of accumulation of round-off errors.) In particular, some such optimal points are in the part of the Pareto set which the weighted-sum objective cannot cover. On the other hand, the points optimal for the objectives () diverge from the Max-Min fair point as the value of α increases. Superficially thinking, both the objectives () and () would be anticipated to show similar behaves, but, in fact, the objective () is not good as a general fairness objective. It is seen in the above figures that the is almost Pareto optimal, and almost identical to the Nash proportionate-fair Pareto optimum. In this case, however, the Nash proportionate-fair Pareto-optimal point is not in the part of the Pareto set obtained by weighted-sum objectives and any fairness objectives. A case where the is not Pareto optimal Figures and show a case where the values of system parameters are φ = 0.9, µ =., φ =, µ =, and t = 0..

6 0 9 Nash proportionate-fair Pareto optimum 0 9 Weighted-sum Pareto border x=0 x=0 Figure : Combinations of response times, respectively, T and T, of users and in the case where φ = 0., µ =.0, φ = 0.9, µ =., and t =. We show the points that achieve [Left] Nash proportionate fairness and [Right] minimization of weighted sums of costs. We observe that, in this case, only one Pareto optimum point achieves the fairness objective () = = x=0 x=0 Figure : Combinations of response times, respectively, T and T, of users and in the case where φ = 0., µ =.0, φ = 0.9, µ =., and t =. The points achieve the fairness objectives () [Left] and ()[Right]. We note that, in this case, only one Pareto optimum point achieves the fairness objective (). In Figure, we observe that the is not on the Pareto set. The Pareto set and the straight line passing through the origin (0,0) and the intersect at a point, which is the Nash proportionate-fair Pareto-optimal point. In this case, Pareto optimal points that achieve the weighted-sum optimization cover all the Pareto set. The Pareto optimum corresponding to the Nash proportionate fairness is given by ξ 0.9 and ξ 0.0. In this case, the Nash-proportionate-fair optimal point happens to be the point that achieves the fairness objective () for α.. On the other hand, in this case, no points that achieve the fairness objectives () with any values of α can be identical with it. A case where only one Pareto optimum point achieves the fairness objectives () with various values of α Figures and show a case where only one Pareto optimum point achieves the fairness objective () at T. and T., that is, the case of no load balancing (x = 0andx = 0). The values of the system parameters are φ = 0., µ =.0, φ = 0.9, µ =., and t =. Note that in this case, the following relation is satisfied:

7 x=0 0 0 Pareto optimum Figure : Combinations of response times, respectively, T and T, of users and in the case where φ = 0., µ = 0., φ = 0., µ = 0., and t = 0. Only one Pareto optimum point achieves the fairness objectives () with various values of α, and only one Pareto optimum point exists. φ µ = φ µ. Note that, in this case also, the Nash proportionate-fair Pareto-optimal point is different from the Pareto optimum that achieves the fairness objective (). A case where only one Pareto optimum point exists Figure shows a case where only one Pareto optimum point exists. The value of the system parameters are φ = 0., µ = 0., φ = 0., µ = 0., and t = 0. We note that load balancing must be ineffective when job forwarding time t has a large value. In Figure, all optimal points that achieve the weighted-sum optimization for any combinations of the values of ξ and ξ, the points that achieve both fairness objectives () and () with any values of α, and the point happens to be the Pareto optimal point. Concluding Remarks We have numerically examined the generally parameterized fairness objectives and the Nash-proportionate fairness recently introduced. The platform of this research has been simple static load balancing model with two identical servers (computers) each of which has an identical arrival and its own queue. The points that achieve the general parameterized fairness objectives generally cover a part of the Pareto set, and at times, do not covered the Nash-proportionate-fair Pareto optimal point. Since each Pareto optimum may have its own significance, we may wish to have a more generally parameterized fairness objective that all the Pareto optimal points may be achieved with a certain choice of the values of the parameters. We have observed that careful consideration is needed in establishing the concrete form of the fairness objective along the lines of the generally parameterized fairness objectives. Otherwise, we may have an inappropriate objective that would not give us truly fair assignment of resources to users. Future problems that remain to be solved are numerous. For example, we may need to examine other categories of models, and the analytical investigations to reveal the underlying logical structures of the problems. References Altman, E. and H. Kameda (00). Equilibria for multiclass routing in multi-agent networks. In: Proceedings of the 0th IEEE Conference on Decision and Control. Orlando, Florida. pp Altman, E., H. Kameda and Y. Hosokawa (00). Nash equilibria in load balancing in distributed computer systems. International Game Theory Review (), Aubin, J-P (99). Optima and Equilibria: An Introduction to Nonlinear Analysis, nd Ed.. Springer-Verlag. Berlin.

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