Load Balancing in Processor Sharing Systems

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1 Load Balancing in ocesso Shaing Systems Eitan Altman INRIA Sophia Antipolis 2004, oute des Lucioles Sophia Antipolis, Fance Utzi Ayesta LAAS-CNRS Univesité de Toulouse 7, Avenue Colonel Roche F-3077 Toulouse, Fance Balakishna abhu LAAS-CNRS Univesité de Toulouse 7, Avenue Colonel Roche F-3077 Toulouse, Fance ABSTRACT In this pape, we investigate optimal load balancing stategies fo a multi-class multi-seve pocesso-shaing system with a oisson input steam, heteogeneous sevice ates, and a seve-dependent holding cost pe unit time. Specifically, we study (i) the centalized setting in which a dispatche outes incoming jobs based on thei sevice time equiements so as to minimize the weighted mean sojoun time in the system; and (ii) the decentalized, distibuted non-coopeative setting in which each job, awae of its sevice time, selects a seve with the objective of minimizing its weighted mean sojoun time in the system. Fo the decentalized setting we show the existence of a potential function, which allows us to tansfom the non-coopeative game into a standad convex optimization poblem. Fo the two afoementioned settings, we chaacteize the set of optimal outing policies and obtain a closed fom expession fo the load on each seve unde any such policy. Futhemoe, we show the existence of an optimal policy that outes a job independently of its sevice time equiement. We also show that the set of seves used in the decentalized setting is a subset of set of seves used in the centalized setting. Finally, we compae the pefomance peceived by jobs in the two settings by studying the socalled ice of Anachy (oa), that is, the atio between the decentalized and the optimal centalized solutions. When the holding cost pe unit time is the same fo all seves, it is known that the oa is uppe bounded by the numbe of seves in the system. Inteestingly, we show that the oa fo ou system can be unbounded. In paticula this indicates that in ou system, the pefomance of selfish outing can be extemely inefficient. Keywods Load balancing, M/G/ pocesso-shaing queues, seve fams, potential game, ice of Anachy. INTRODUCTION emission to make digital o had copies of all o pat of this wok fo pesonal o classoom use is ganted without fee povided that copies ae not made o distibuted fo pofit o commecial advantage and that copies bea this notice and the full citation on the fist page. To copy othewise, to epublish, to post on seves o to edistibute to lists, equies pio specific pemission and/o a fee. GameComm 2008,, Octobe 20, 2008, Athens, GREECE. Copyight 2008 ICST Communication sevices such as web seve-fams, database systems and gid computing clustes, outinely employ multiseve systems to povide a ange of sevices to thei customes. An impotant issue in such systems is to detemine the seve to which an incoming equest should be outed to in ode to optimize a given pefomance citeion. Fom the sevice povide s pespective, this choice of the stategy (centalized o decentalized) and the sevice discipline (ocesso Shaing (S), Fist-Come-Fist-Seved (FCFS), etc.) detemines the amount of esouces it needs to deploy in ode to guaantee a cetain Quality-of-Sevice (QoS) to its customes. Thus, an investigation of load balancing o outing stategies in multi-seve systems can give guidelines to the sevice povide on dimensioning its system. In this pape we study the optimal load balancing in a multi-seve pocesso-shaing system with heteogeneous sevice capacities. This configuation is also known as pocessoshaing seve-fams, and is a popula achitectue in computing centes, used fo example in the Cisco Local Diecto, IBM Netwok Dispatche and Micosoft Shaepoint (see [5] fo a ecent suvey). This configuation can also be used to model a web seve fam, whee equests fo files (o HTT pages) aive to a dispatche ae dispatched immediately to one of the seves in the fam fo pocessing. With each seve, we associate a sevice capacity (i.e., some seves could be faste than the othes) and a holding cost pe unit time. We assume that equests aive as a oisson pocess, and that the sevice equiement of each equest is sampled fom a finite set. Fo such a multi-seve system, we investigate load balancing in two diffeent settings: (i) the centalized setting in which a dispatche assigns the seve to an incoming equest with the objective of minimizing the weighted mean sojoun time of jobs in the system, and (ii) the distibuted non-coopeative setting in which an incoming equest selects a seve in ode to minimize its own weighted mean sojoun time in the system. In both cases we assume that the only infomation available to the decision make (the dispatche o the equest itself) is the sevice time equiement of the equest. This might be the case, fo example, in situations whee not all the seves ae in the same location and it may be costly to gathe infomation on the cuent queue lengths at the vaious seves. The main contibutions of the pesent wok ae as follows. Fo both settings, we chaacteize the set of optimal outing policies, and give closed-fom expessions fo the load on each seve unde any optimal policy. It is wothwhile to note that fo the distibuted non-coopeative setting this

2 is done by showing the existence of a potential function, which allows us to tansfom the non-coopeative game into a standad convex optimization poblem. We then give an optimal policy in which an incoming equest is outed to a seve with a pobability that is independent of the sevice equiement of the equest. This popety of the S discipline could be useful in systems in which the sevice equiement of equests is not known a pioi and it illustates an impotant diffeence between the optimal load balancing policy in a S seve-fam and FCFS seve-fam, since in the case of a FCFS seve-fam it has been shown that the optimal load balancing does use infomation on the sevice equiement of each equest [0, 8]. Futhe, we show that highe the atio of the holding cost pe unit time to the sevice capacity of a seve the lighte is the load on it, thus defining an index to ode the seves. Fo cetain input paametes (i.e., an aival pocess, sevice time distibution, available sevice capacities, holding cost pe unit time), it is thus possible that some of the seves will not be pocessing any equests. We show that the set of seves pocessing equests in the decentalized setting is a subset of that in the centalized setting. Thus, thee is a tade-off in the pefomance gains and cost of seves to be consideed when choosing between the two settings. We also note that, given the input paametes, this analysis gives the set of seves that a sevice povide should choose in ode to minimize the mean sojoun time in its system. Finally, we compae the pefomance peceived by jobs in the two settings by studying the so-called ice of Anachy (oa), that is, the atio between the selfish decentalized and the optimal centalized solutions. When the holding cost pe unit of time is the same in evey seve it is has been shown that the oa is uppe bounded by the numbe of seves in the system, see fo example [22, ]. Inteestingly, we show that fo ou system the oa is unbounded, that is, it can be abitaily close to infinity. This indicates that unequal holding costs may have a pofound impact on the system s pefomance. In paticula, the pefomance of selfish outing can be unboundedly wose than the pefomance obtained by a centalized outing.. Related wok Load balancing in multi-seve systems has been peviously investigated not only in the context of communications sevices but also in the boade context of queueing systems. Global and Individual optimality in load balancing ae consideed in the monogaph [3], which does not conside decisions based on knowledge of the amount of load. Systems with geneal sevice time distibution and FCFS scheduling discipline wee studied in [7, 2, 3, 8], while [7, ] studied systems with exponential sevice time distibutions and abitay scheduling discipline. In [9] the authos analysed a multi-seves S system whee equests join the seve that has the smallest numbe of equests. In a ecent wok [6] the authos investigate the pefomance of a seve fam whee the scheduling discipline in each seve is SRT (Shotest Remaining ocessing Time Fist). Ou wok is closely elated to [22] and []. The main diffeences ae that (i) we conside a multi-class job aival pocess, allowing the dispatche to use infomation on the size of the equests and (ii) the addition of a heteogeneous holding cost pe time unit in each seve. As we will see, both (i) and (ii) genealizations allow us to daw impotant conclusions, that to the best of ou knowledge wee not known befoe. By consideing a multi-class system, we wish to analyze how the infomation on the sevice equiements of uses impacts the stuctue of the optimal load balancing. Ou esults show that the stuctue of the optimal outing in a system with the S scheduling discipline is adically diffeent with espect to the FCFS case. Fo a multi-seve FCFS system with homogeneous sevice capacities it was conjectued in [0], and poved in [8], that the optimal load balancing scheme consists in assigning to each seve all jobs whose pocessing times fall within non-ovelapping, continuous intevals of pocessing times. The intuitive explanation to this esult comes fom the fact that this stategy educes the vaiability of sevice times fo each queue. Since the mean delay in a FCFS queue is diectly popotional to the vaiability of the sevice time distibution (ollaczek-khinchin fomula), an inteval-based policy can minimize the oveall mean delay in the system. Inteestingly, if the sevice capacities ae heteogeneous an inteval-based stategy need not be optimal [8]. In contast, we show that in the case of a multi-seve S system the optimal load balancing stategy does not take advantage of the sevice time infomation, that is, the pobability that a job joins a given seve is independent of the job s sevice equiement..2 Oganization of the pape The est of this pape is oganized as follows. In Section 2, we descibe the system model, state the assumptions, and give the mathematical fomulation fo the poblem unde consideation. In Section 3, we teat the centalized setting, which is followed by the teatment of the decentalized setting in Section 4. In Section 5, we compae the pefomance of the two settings using vaious measues, such as the seve utilization and the ice of Anachy. 2. MODEL FORMULATION Conside a seve fam consisting of a set of C seves. Let S {, 2,..., C} denote the index set of the set of seves. Seve has a sevice ate, fo all j S. At evey seve, jobs ae seved accoding to the pocesso shaing (S) discipline. Customes aive to the system accoding to a oisson pocess with ate λ. Depending on the application in mind, a custome may coespond to a job with a cetain amount of sevice equiement, o of a file that has to be tansmitted and has a cetain size. In the latte case we shall identify the sevice equiement of the file as being its size. Let {σ k : k K} denote the set of possible sevice equiements (i.e. the job sizes) and assume that K is finite. Let K {, 2,..., K} denote the index set of the set of possible sevice equiement. Customes have independent and identically distibuted sevice equiements which ae sampled fom {σ k : k K} such that the pobability that a custome has sevice equiement σ i is given by β i, fo all i K. As mentioned in the Intoduction, we ae inteested in compaing the pefomance between the globally optimal solution and the distibuted non-coopeative poblem. We assume that decisions ae open-loop: they ae taken without knowledge of the queue sizes. Howeve, we assume that the sevice equiement of an aiving use is known, both to the dispatche in the centalized case and to the use itself in the distibuted non-coopeative setting. The decision on

3 which queue an aival joins is assumed to depend only on that infomation. Since the pocesses geneated by splitting a oisson pocess ae still oisson, each seve can be seen as an M/G/ S queue. We ecall that the mean delay in a S queue depends on the sevice time distibution only though its mean (the so-called insensitivity popety of S [4]), theefoe the mean numbe of jobs in an M/G/ S queue is the same as in an M/M/ queue. All aivals with a given size ae called a class. We thus have K classes of jobs whee jobs of class i have mean size. We associate with class i an aival ate λ i λβ i, and a taffic intensity i λ iσ i. Let σ i i K i denote the total input taffic intensity. Remak. Note that the value of K is abitay. Theefoe ou fomulation allows us to appoximate a continuous distibution abitaily closely, and thus we can investigate the optimal size-based outing stategy. Notation. We shall use a lowe case bold-faced chaacte to denote a vecto. The elements of a vecto will be denoted by the coesponding lowe case chaactes. Fo example, a denotes the m vecto (a, a 2,..., a m) whee m is the size of a. The vectos 0 m and m will denote the m vectos with all elements as 0 and, espectively. We shall use the symbol to denote elementwise inequality fo vectos. Stategies. A stategy fo a class i of customes is defined to be the pobability vecto (p i,..., p ic), whee p ij is the pobability that a class i custome goes to queue j. Note that fo any stategy C j pij. We define a multistategy p (p ij), i K, j C as the matix of stategies of all classes. Fo a multi-stategy p, let ρ i j(p) denote the load on seve j due to class i. The total load on seve is given by ρ j(p) ρ i j(p) ip ij. () i K i K Fom queueing theoy we know that seve is stable if ρ j(p) <. We shall say that p is a stable multi-stategy if all seves ae stable. The next poposition states the necessay and sufficient condition fo the existence of a stable multistategy. oposition. Thee exists a stable multi-stategy if and only if >. (2) oof. Fo a multi-stategy p, fom () we get ρ j(p) i K ip ij, fo all j S. Summing ove all j and intechanging the two summations on the ight-hand side we get i p ij. (3) ρ j(p) i K If j <, then the load on some seve must be lage than fo (3) to hold. Thus, (2) is necessay fo the existence of a stable multi-stategy. Now, assume (2) and conside the multi-stategy defined by p ij k S, fo all i K, and fo all j S. k Due to the splitting popety of oisson pocesses, the aival pocess to each of the queues will also be oisson unde this multi-stategy. Then, each seve can be modeled as an M/G/ queue with ρ j(p) i K ip ij i K i k S k <. (4) and as a consequence evey seve is stable. Thus, (2) is sufficient fo the existence of a stable multi-stategy. Assumption. The taffic intensities and the sevice ates ae such that (2) is always satisfied. Note that if p is a stable multi-stategy, then necessaily C j ρj(p) < C. Since all the queues in ou system ae M/G/ S queues, the mean numbe of jobs at any queue has the insensitivity popety: it depends on the sevice distibution only though its expectation. Fo all j S, the mean numbe of jobs is given by E[N j(p)] ρj(p) ρ j(p), (5) fo ρ j(p) <, and is infinity othewise. The total aival ate to seve is K i λipij. Thus, by Little s law the mean sojoun time at queue j is given by E[T j(p)] E[Nj(p)] K i λipij. (6) Even though sometimes we will not make the dependency explicit, E[N j], ρ j and E[T j], fo all j S, shall be undestood to depend on the multi-stategy elevant to the context. Ou objective is to detemine the multi-stategy p that minimizes the weighted mean numbe of jobs in the system, that is, C agmin E[N j], (7) p j whee ae some constants that depend on the index of the of the queue and that can epesent, fo example, a cost on the holding time. We ecall that in all pevious woks, the case c, fo all j S, was studied. By Little s law, minimizing the weighted mean numbe of jobs is equivalent to minimizing the weighted mean sojoun time in the system. Finally we note that thoughout the pape we will assume the seves ae labeled such that c c2... cc. (8) 2 C Remak 2. Since the objective function defined in (7) depends only on the mean sevice time at each of the seves, we could also intepet that the aival steam is composed of K classes, whee jobs of diffeent classes have diffeent sevice time distibutions. The mean sevice time of class i jobs is σ i, fo i K. All the esults in the pesent pape would hold unde this intepetation as well. Nevetheless, fo conciseness, in the pesent pape we stick to the intepetation expessed in Remak.

4 3. THE GLOBAL OTIMIZATION ROB- LEM In this section we conside the global optimization poblem, in which a dispatche decides whee each job will get sevice so as to minimize the weighted mean numbe of jobs in the system. The global optimization poblem can be fomulated in tems of the following Mathematical ogam (M): minimize subject to E[N j(p)] (9) p ij, fo all i K; (0) p 0; () ip ij <, fo all j S. (2) i K We note that if condition (2) is satisfied, then thee exists a multi-stategy which satisfies these constaints and vice vesa. Since the objective function is convex and the constaints ae linea, M is a standad convex pogamme, and its solution can be found in polynomial time in the numbe of unknowns and in the numbe of constaints. We note that thee may exist multiple multi-stategies that minimize (9) subject to (0)-(2). 3. Size-unawae multi-stategies The following esult will play a key ole in the est of the pape. It shows that thee exists a size-unawae multistategy that is optimal. oposition 2. Let p be a multi-stategy satisfying the constaints (0)-(2). The multi-stategy ˆp defined by l K ˆp ij lp lj ρj(p)j, (3) fo all i K and fo all j S, also satisfies the constaints (0)-(2). Moeove, the load on a seve unde ˆp is equal to the load on it unde p. oof. The equality ˆp ij l K lp lj, fo all i K, shows that ˆp satisfies (0). Since i is non-negative fo all i K, and p satisfies (), ˆp also satisfies (). The equality i ˆp ij l K lp lj i l p lj i K i K l K helps us to veify that ˆp indeed satisfies (2). Finally, since i K i ˆpij l K ρ j(ˆp) lp lj ρ j(p), fo all j S, the load on a seve is the same unde both p and ˆp. Fom oposition 2, we can infe that, fo evey feasible multi-stategy, thee exists a feasible size-unawae multistategy such that both these stategies induce the same load on the seves. Since the objective function in the M depends on the multi-stategy only though the induced load (cf. (5)), we can conclude that one may estict oneself without loss of optimality to finding policies that take outing decisions independently of the (known) amount of sevice equiement of a job. The esult of oposition 2 futhe illustates that the optimal load balancing in S seve fams is athe diffeent than in FCFS seve fams, whee the size of jobs is used by the optimal outing policy. Moeove, the value of the mathematical pogamming (9)-(2) can be obtained by optimizing diectly ove the loads. The outing pobabilities can be detemined late fom (3), once the load on each seve is detemined. Let f j(x) j x/( x), fo 0 x < ;, othewise. Fom (5) and oposition 2, we can conclude that an optimal load balancing policy is obtained by applying (3) to the solution of the following Reduced Mathematical ogam (RM): minimize f j(ρ j) (4) subject to 0 ρ ; (5) ρ j. (6) Constaint (6) guaantees that all incoming jobs ae seved. 3.2 Chaacteizing the solution Depending on the values of the sevice ates and the holding costs pe unit time, the optimal multi-stategy may not use all seves, but due to constaint (6) we ae cetain that at least one seve will be used. Let S G S denote the subset of seves that the optimal multi-stategy uses. In the following theoem we chaacteize the solution of (4)-(6). In paticula we note that the solution to (4)- (6) is unique. Theoem. The subset of seves that ae used in the optimal load balancing is S G {,..., j }, whee (! j j j ) sup j C : > k (7) Unde the optimal multi-stategy, the load on seve S G is ρ k S j G k. (8) k S ck G k oof. The Lagangian associated with the RM can be defined as L(ρ, ν, ζ, γ G ) f j(ρ j) + ν j(0 ρ j) + ζ j(ρ j )! + γ G ρ j, (9) whee ν 0, ζ 0 and γ G R. Note that the RM is convex. Fom oposition (see (4)) thee exists a feasible solution. As a consequence by

5 Slate s condition [4, Section 5.2.3] stong duality is satisfied. Then, ρ and (γ G, ν, ζ ) ae pimal and dual optimal with zeo duality gap if they satisfy the Kaush-Kuhn- Tucke (KKT) conditions 0 ρ ; ρ j ; γ G R; ν 0; ζ 0; ν j ρ j 0, ζ j (ρ j ) 0, fo all j S; (20) ( ρ j )2 γg ν j + ζ j 0, fo all j S. (2) Condition (20) ae the so-called complementay slackness, which hold due to stong duality. Since the objective function tends to infinity when ρ j tends to at any seve, it follows that necessaily ρ. Theefoe, fom (20) it follows that ζ 0. Since ν 0, fom (2) we get γ G, fo all j S, (22) ( ρ j )2 and on eliminating the vaiables ν j fom (20), we get «( ρ γg ρ j )2 j 0, fo all j S. (23) Fo a given seve, if γ G is geate than /, then (22) can only be satisfied if ρ j is geate than 0 as well, which togethe with (23) implies that ρ j. (24) γg Assume now that γ G /. If ρ j is geate than 0 then γ G / < ( ρ j )2, which violates the complementay slackness condition (23). Thus, if γ G /, then ρ j is equal to 0. In conclusion, we have ( q q ρ j, if γ G > c γ j/; G (25) 0, othewise. Fom the above equation, we see that ρ j ae non-deceasing in γ G. Theefoe, thee is a unique value of γ G such that constaint (6) is satisfied. Since / is non-deceasing in j, it now follows that S G {,..., j }, whee j can be computed using (22) and is such that < γ G < +. (26) + Fom (24) and (6), we obtain γ k S G k, (27) G k S ck G k which togethe with (26) gives ( j sup j C : < j! 2 ) ck k j, k which is an equivalent condition to the one stated in (7) On combining (26) and (25), we get ρ k S j G k, k S ck G k which is the esult stated in (8). Coollay. The size-unawae multi-stategy, ˆp, is given by ˆp ij ρ j, fo all i K and fo all j S. (28) Remak 3. The solution stuctue of Theoem is known as wate-filling. We will say moe about this in Section 4.4. Fom Theoem we see that ρ j > ρ i, fo any j < i. Since the mean numbe of jobs in a seve inceases with its load, we conclude that, unde any optimal multi-stategy, E[N j] > E[N i] fo any j < i. Inteestingly, in the next poposition we show that, even though ρ j > ρ i, the weighted mean sojoun time in seve will be smalle than the weighted mean sojoun time in seve i. oposition 3. Fo the multi-stategy (28), and fo any two seves j and i in S G, E[T j] < c ie[t i], fo < i. oof. Fom Little s law (see equation (6)) and the multistategy (2) we have E[T j] Substituting (8) we get E[T j] E[Nj] i K λi ˆp ij E[Nj] i K λi ρ j. k S ck G k. k K λi k S G k The esult now follows by noting that fo any j < i, / < c i/ i. 3.3 Altenative chaacteization of the optimal solution In this subsection we wite in vecto fom the KKT conditions that chaacteize the optimal solution to the global optimization poblem. This epesentation will play a cucial ole in detemining the optimal outing stategy in the distibuted non-coopeative setting. Fo simplicity in the exposition, we assume that all seves ae used. Let us fist intoduce the Hadamad poduct fo matices. Fo two abitay matices (x) ij and Y (y) ij of the same dimension, we denote by Y the matix whose (i, j) element is a ijb ij. Thus, the Hadamad poduct just efes to the element-wise poduct of matices. The standad poduct of two matices is denoted by B. Finally fo an abitay matix we denote by T its tanspose matix. Let t(p) be the gadient of the objective function, i.e., t(p) is a matix of dimension K C whose (i, j) element is given by t ij k S f k(p) p ij. (29) Then, simila to the deivation of (22)-(23), p is optimal fo the oiginal poblem (9)-(2) if and only if thee exist

6 Lagange multiplies γ,..., γ C and a matix Γ of dimensions K C whose (i, j) element is given by such that Γ ij γ j, (t + Γ) p 0, (30) t + Γ 0, (3) C p T K, p 0. (32) Note that equations (30) and (3) ae the analogue of equations (23) and (22), espectively. This equivalent chaacteization though complementaity inequalities of a globally optimal solution will be essential fo the next section. 4. THE INDIVIDUAL OTIMALITY We study now the distibuted non-coopeative setting, whee an aiving custome, say of class i, awae of its equied amount of sevice (σ i), wishes to minimize its own weighted expected sojoun time. The weighting is done accoding to the queue to which the file is sent as can be viewed as a picing that may vay fom one queue to anothe. If a class-i use chooses to be seved by seve then its weighted conditional expected sojoun time thee is τ ij(p) E[T j(p) i] σ i ρ. (33) j(p) Definition. We say that customes of class i use queue j if ρ i j > 0; i.e., queue j eceives a stictly positive load fom class i. Definition 2. We say that a stategy p is an equilibium fo the individual optimization poblem if fo each i,..., K, each j,..., C and each queue k used by class i, E[c k T k (p) i] min E[Tj(p) i]. (34) j,...,k Without loss of geneality, we can eplace the equilibium condition in (34) with the condition E[d ic k T k (p) i] min die[tj(p) i]. (35) j,...,k whee d i ae abitay stictly positive constants. Equation (34) chaacteizes the equilibium, since only when (34) is satisfied uses will not have an incentive to deviate fom thei stategy. 4. A potential game appoach to obtain the equilibium Denote by T(p) a K C matix whose (i, j) element is τ ij(p). Let a be the matix of dimensions K C whose (i, j) element is given by a ij a j. We can chaacteize the equilibium by the following elations: p is an equilibium if and only if thee is some a such that the following holds. `T(p) + a p 0, (36) T(p) + a 0, (37) C p T K, p 0. (38) We obseve (36)-(38) and note that they ae the same as the system (30)-(32), povided that we identify the minimum cost vecto a with the Lagange multiplie vecto Γ, and we identify T as a gadient vecto of some potential function G. Since system (30)-(32) wee equivalent to a global minimization, we conclude that (36)-(38) ae equivalent to the equilibium p being the global minimum of the function G subject to the constaints (38). Note that the minimum is unique in tems of ρ j if G is a stictly convex function of ρ j. Games that can be tansfomed into an equivalent optimization poblem with a common function optimized jointly by all uses ae known as potential games. They have been intoduced in [] in the context of oad taffic, see also [8, 6, 9, 2]. In paticula, the existence of a potential function is a sufficient condition fo vaious geedy dynamics of the game to convege to equilibium. oposition 4. The distibuted non-coopeative game can be tansfomed into a standad convex optimization poblem of minimizing C c k log T (ρ k (p)) (39) subject to the constaints (0)-(2) whee T (z) : /( z) fo 0 z < and fo z. oof. Define Then G(p) Thus, C G(p) : Z ρk (p) z0 C Z ρk (p) z0 c k T (z)dz G(p) p ij T j(p) dρj dp ij c k T (z)dz. (40) C c k log T (ρ k (p)) c j ρ λi λ i E[T j(p) i] j(p) σ i We conclude that G is indeed a potential as its gadient coincides with the oiginal costs as given in (35), whee d i λ i. The optimal solution p to (39) is given by the only vecto that satisfies the KKT conditions, which in tun ae pecisely given by (36)-(38), whee a denotes the Lagange multiplie vecto. This implies that indeed the game can be tansfomed into a standad convex optimization poblem of minimizing G subject to the constaints (0)-(2), whose solutions ae equilibia in the oiginal game. As we did in Section 3., we can futhe simplify the above optimization poblem. Indeed, the value is diectly obtained R ρk z0 c kt (z)dz subject to though minimizing G(p) : C (5)-(6). The solution to the game poblem is obtained fom the loads that achieve the minimization by using (3). 4.2 Fainess Let us intepet the meaning of the potential function G. Define k : ρ k to be the excess capacity at seve k. We note that the agument that achieves the minimization of G(p) achieves the maximum of the poduct of ( ) c ( 2) c 2 ( C) c C. We conclude the following:

7 Theoem 2. The individual optimal load balancing solution coincides with the outing stategy that achieves the weighted popotional fai excess capacities between the C seves, whee the weight fo seve k is given by the powes c k. oof. The esult is a diect consequence of (39) and the definition of opotional Fai allocation. 4.3 Chaacteizing the Individual Optimal solution Since we have shown that the individual setting coesponds to a potential game, in equilibium, the optimal outing stategy will minimize (40) subject to (5)-(6). We have the following esult. Theoem 3. The subset of seves that ae used in the optimal outing stategy in the non-coopeative setting is of type S I {,..., j }, whee j sup ( j C : j > Fo evey j S I, the load is ρ j! j k ) (4) j k j c. (42) k oof. The deivation follows the same steps of the poof of Theoem. Fom oposition (see equation (4)) thee exists a feasible solution. As a consequence, by Slate s condition [4, Section 5.2.3] stong duality holds. Then fom the Kaush-Kuhn-Tucke (KKT) conditions if 0 ρ j, j,..., C, I ρ j, γ I R, ν j 0, ζ j 0, j,..., C, ν jρ j 0, ζ j(ρ j ) 0, j,..., C, (43) ( ρ j) γi ν j + ζ j 0, (44) then ρ j, j,..., C and (γ I, ν, ζ) ae pimal and dual optimal with zeo duality gap. Since the objective function tends to infinity if ρ j at some seve, it follows that necessaily ρ j <, j,..., C. Because of (43) this implies that ζ j 0, fo all j. Now note that ν j ae slack vaiables which can be eliminated. Since ν j 0, fom (44) we get γ I ( ρ j), (45) and fom (43) we have «( ρ j) γi ρ j 0. (46) Now, if γ I > /, equation (45) can only be satisfied if ρ j > 0, and fom (46) this implies that ρ j γ I. (47) Assume now that γ I /. If ρ j > 0 then this implies that γ I c / < j ( ρ j ), which violates the complementay slackness condition (46). Thus if γ I / then ρ j 0. In conclusion we have that ( c j ρ j j γ I > c γ j/ I j 0 γ I < /. It follows that ρ j > 0 ae non-deceasing in γ I. Thus thee is a unique value of γ I such that constaint (6) is satisfied. It follows that S I {,..., j }. Fom (45) we have that the index j is such that Substituting (47) in (6) we get < γ I < +. (48) + k S γ I k K i i. (49) I k S I c k This poves equation (42). Fom (48) we get that seve is used if and only if j < c k j k, K i i fom whee (4) follows. We note that a outing stategy that achieves the desied load (42) in evey seve (and as a consequence the same pefomance) can be obtained by (3). Remak 4. Fom (42) it is easy to see that (34) is satisfied fo each i,..., K and each j S I. This can also be seen fom equation (47), which implies that in evey seve j S I that is used the mean cost pe unit of sevice equied at the seve, / ρ j γ I, is independent of the seve. Fom Remak 4 and oposition 3 we obseve the main diffeence between the global and individual optimal solutions. In the individual optimal solution is constained to a solution such that the mean sojoun time is the same in each seve. In the global optimal solution the weighted mean sojoun time vaies acoss the seves, and in fact, it inceases as the index of the seve inceases (see oposition 3). When c i c, i, equation (4) becomes j K + < ( k i) j. (50) Equation (50) has a clea intepetation. Seve+ will not be used if the exceeding capacity pe seve when j seves ae used is lage than The stuctue of the selfish outing We ecall fom (8) that seves ae elabeled in inceasing ode with espect to the atio /, j,..., C. Let thee be M seves with c i/ i c /. Let thee be M 2 seves with c i/ i c M +/ M +. Let thee be M k seves with c i/ i c Mk +/ Mk +. Then, fom (34), the optimal policy has the following wate-filling stuctue. Fo λ sufficiently small, only the fist M seves eceive positive flow. This flow is assigned in a

8 way that equalizes the expected delay among the fist M seves. We incease λ till a point whee c ρ c2. (p) 2 Fom this point, we oute flow to all M + M 2 fist seves in a way that equalizes the expected delays on these seves. No flow is sent to othe seves. This type of solution is often efeed as to wate-filling. 5. COMARING THE GLOBAL AND INDI- VIDUAL OTIMUM SOLUTIONS In this section we compae the optimal load balancing expessed in Theoems and 3. Ou fist esult shows S I S G, that is, the numbe of seves that ae used in the global optimum solution is geate o equal to the numbe of seves used in the distibuted non-coopeative setting. This indicates that in the non-coopeative setting, uses will tend to oveload fast seves, and fail to ecognize the benefits that using a slowe seve can have. A simila popety was poven in [2] fo a exponential multi-seve system. In this section, ρ G j and ρ I j will denote the load in seve in the global and individual optimal solution, espectively. In view of (24) and (47) we will conside that both ρ G j : ρ G j (γ) and ρ I j : ρ I j (γ) ae a function of a common vaiable γ. We stat with the following Lemma. Lemma. Fo 0 < γ /, ρ G j (γ) ρ I j (γ) 0. Fo γ > /, ρ G j (γ) < ρ I j (γ). oof. The case γ / is obvious. Fo the second case, we have γ > / γ γ > γ γ > γ q and fom equations (24) and (47) it follows that ρ G j (γ) < ρ I j (γ). oposition 5. Fo any aival ate and sevice time distibution it holds S I S G oof. Fom Theoems and 3 (equations (27) and (49)) it is sufficient to pove that γ G > γ I. We pove the statement by contadiction. Assume that γ G γ I. If γ I < /, then ρ I j (γ) ρ G j (γ) 0. If γ I > / then ρ I j (γ I ) > 0 and fom Lemma we have ρ I j ρ I j (γ I ) γ I γ G ρ I j (γ G ) Lemma > ρ G j (γ G ) ρ G j. It follows then that C j j(ρi j ρ G j ) > 0, but this is a contadiction with (3), and as a consequence γ G > γ I. In the following theoem we show that the individual optimal oveloads the seves with smallest /. Theoem 4. Thee exists an index i such that j ρ G j < ρ I j j < i ρ G j > ρ I j j i. oof. Due to constain (3), thee exists an index i such that ρ G i > ρi i. Now it suffices to show that ρg j > ρ I j, fo all j > i. Fom (24) and (47) we have that ci i ρ G i > ρi i γ < ci G i γ I γ I ci p < γg. i Since j > i, it follows that / > c i / i. Thus γ I < < ci i p γ G p γ G q p γg, q q and eaanging we get < γ G. Fom (24) and (47) γ I it follows that ρ G j > ρ I j. 5. ice of Anachy We now study the so-called ice of Anachy. Definition. The pice of anachy (oa) is defined as the atio between the pefomance (mean delay) obtained by the Wadop equilibium and the global optimal solution [5] (see also [20]). By Little s law, calculating the atio between the mean delays is equivalent to calculating the atio of the mean numbe of uses. Then fom the objective function (7) and the solution of Theoems and (3) we get (note that ): x x x oa k S c I k k S I k k S I k 2 k SG ck k k S G k k S I c k. (5) k S G c k The ice of Anachy has been studied as a measue of the inefficiency of selfish-outing (o non-coopeative decentalized) in netwoks. This measue has eceived lot of attention in ecent yeas. Fo example, in an impotant geneal esult, it has been shown that when the cost function in evey ac is linea, then fo any abitay multi-commodity netwok the oa is uppe bounded by 4/3 [20]. In [] and [22] the authos study a multi-seve system with the objective of minimizing (7) with equal costs, that is, c, j, and show that oa C, with C denoting the numbe of seves. Note that the uppe bound holds fo any paamete configuation. In addition, in [, Example 3.] it is shown that the uppe bound is tight, i.e., thee exists a netwok configuation such that the oa is abitaily close to C. This esult indicates that the inefficiency of selfish outing is limited. In Theoem 5 we show that this changes damatically when holding costs pe unit of time associated to each seve ae consideed in the objective function. In this case the oa is unbounded, that is, fo evey θ <, thee exist a set of values such that oa > θ. Ou main esult on the ice of Anachy is the following.

9 Theoem 5. Fo evey θ, thee exist and, j S, such that oa > θ. oof. In ode to pove this esult we constuct an example in which oa can be unbounded. Let >, and let fo 2 j C. Let ( ) < c <. (52) Fo this paticula choice of costs and seve speeds, / is non-deceasing in j. We fist show that in the globally optimal multi-stategy all the seves ae used, wheeas in the solution of the individual optimization poblem only the fist seve is used. Global optimization: Note that /, j 2. In view of (7), seve, j 2, will be used if j c + j > + j! j, whee the inequality follows fom the assumption c > ( ) 2. Since this is tue fo evey j C, the load on evey seve is positive. Individual optimization: Fo 2, the left-hand side of (4) we have c + c 2 c + 2 < + 2 ( + 2 ) c2 2, whee the inequality follows fom the assumption c <. Thus, in the non-coopeative setting all the jobs choose to go to the fist seve. Fom (5), the ice of Anachy «c oa c ( k S G ck k ) 2 k S G k c k S G c k + (C ) ( c + (C )) 2 (c + (C ))( + (C )) (53) Since ( ) 2 < c <, let c «( ) ( ) 2 2. (54) Now as, the numeato of (53) tends to 2 (C ), wheeas the denominato tends to 0. Theefoe, by choosing close enough to, the ice of Anachy fo this system can be made to exceed any given eal numbe. Remak 5. We note that examples whee the oa is unbounded have been peviously found. Fo instance, it is easy to detemine an instance of the popula isone s dilemma whee the oa is unbounded. It also follows fom the netwok studied in [2] that the oa is unbounded. 5.. Discussion on Theoem 5 In ode to povide an intuitive idea behind Theoem 5, fist note that a key undelying idea is that in the global optimal all seves ae used, wheeas in the non-coopeative setting only one seve is used. This popety follows diectly fom the the uppe and lowe bounds of (52). Let us conside the lowe bound in (52). Fom equations (26) and (27) and the wate-filling stuctue of the solution, we see that if c ( < c ) 2 2 2, only seve will be used. Seve 2 (and similaly all othe seves), will stat being used exactly when c > ( ) 2 /, which explains the lowe bound on c in (52). Similaly, fom (48) and (49) we can see that the uppe bound in (52) guaantees that only seve is used in the non-coopeative setting. As we have seen, the ice of Anachy is given by oa min Cj p E[Nj I ] min Cj. Let us look to the numeato and denominato sepaately. p E[N j G ] In the non-coopeative solution only seve is used. Thus C j E[N j I ] c E[N I ], and seve is a standad M/G/ queue. Thus, as, E[N I ] tends to infinity, but this is compensated by the fact that c 0, and oveall c E[N I ] /2. Anothe way to see this is fom equation (33), whee we see that τ i c. Thus, with c given fom (54), it tuns out that as, the pefomance (weighted with the cost) that uses joining seve emains unchanged. In the global optimal solution, always all seves ae used. As, the global optimal also tends to oute eveything towads seve, but the key popety is that since all seves ae used, the global optimal can do this in such a way that E[N G ] gows moe slowly than the decease of c, and as a consequence c E[N G ] 0. Moe specifically, this is what happens with the global optimal solution. Fist, fo all j 2, as (and c given by (54)), ρ j 0. Since, j 2, emain constant this implies that C j2 E[N j G ] 0. Concening seve, fom (24), as, ρ o( ), which implies that E[N G ] O(/ ). Since c o( ) as, it tuns out that c E[N G ] 0. Thus, fo the global optimal solution C j E[N j G ] 0 as, which explains why the oa can not be bounded. This esult states that the oa is unbounded fo the load balancing poblem unde consideation. It is in complete contast to finite uppe bounds obtained by [, 22], fo simila models but without holding costs pe unit of time associated to each seve. Thus, when holding costs ae taken into account, a significantly diffeent oa is obtained. 5.2 The case when and / ae not equal Theoem 5 can be extended to the case when not all ae equal and / ae not necessaily equal. Let j be the aggegate available sevice ate of system. Let us assume that we ae given a sequence of seve ates such that >. We wish to show that thee exists a sequence {, j S}, such that / is stictly inceasing and that the following two inequalities ae satisfied c + c 2 < ( + 2 )c 2/ 2, (55) ) p c C/ C, (56) > ( which would imply that only the fist seve is used in the solution of the individual optimization poblem wheeas all

10 the seves ae used in the global solution. Fom (55), we equie c 2 2 > c. Fo 2 j C, let c α2j, which esults in an inceasing sequence {/, j S} povided that α >. We shall show that thee exists an α > such that the two inequalities (55) and (56) ae satisfied. The left-hand side of (56) > c + j 2 c + c c α2 A. j 2 Thus, we need to find an α lage than which satisfies the inequality 0 c c A > ( ) p c C/ C j 2 c ( ) αc. The left-hand side of the above inequality, 0 c c A j 2 0 p ( ) + A j 2 c > ( ) whee the inequality follows fom the fact that p ( ) > p ( )( ). Thus, thee exists an α lage than fo which S I {} and S I S G. As, oa will become unbounded in this case as well. 6. ACKNOWLEDGEMENTS The wok of the thid autho was caied out while he was a post-doctoal fellow with the CWI (Amstedam), TU/e (Eindhoven) and EURANDOM (Eindhoven). He wishes to acknowledge thei suppot. 7. REFERENCES [] M. Beckmann, C. B. McGuie, and C. B. Winsten. Studies in the Economics and Tanspotation. Yale Univesity, 956. [2] C.H. Bell and S. Stidham. Individual vesus social optimization in the allocation of customes to altenative seves. Management Science, 29:83 839, 983. [3] S.C. Bost. Optimal pobabilistic allocation of custome types to seves. In oceedings of ACM SIGMETRICS, pages 6 25, Septembe 995. [4] S. Boyd and L. Vandenbeghe. Convex optimization. Cambidge Univesity ess, [5] V. Cadellini, E. Casalicchio, M. Colajanni, and.s. Yu. The state of the at in locally distibuted Web-seve systems. ACM Computing Suveys, 34(2):263 3, 200. [6] H.L. Chen, J. Maden, and A. Wieman. The effect of local scheduling in load balancing designs. efomance Evaluation Review, to appea. [7] Y-C Chow and W.H. Kohle. Models fo dynamic load balancing in a heteogeneous multiple pocesso system. IEEE Tansactions on Computes, 28(5):354 36, 979. [8] H. Feng, V. Misa, and D. Rubenstein. Optimal state-fee, size-awae dispatching fo heteogeneous M/G/-type systems. efomance Evaluation, 62( 4):36 39, [9] V. Gupta, M. Hachol-Balte, K. Sigman, and W. Whitt. Analysis of join-the-shotest-queue outing fo web seve fams. In oceedings of efomance, page 80, [0] M. Hachol-Balte, M. Covella, and C. Muta. On choosing a task assignment policy fo a distibuted seve system. IEEE Jounal of aallel and Distibuted Computing, 59(2): , 999. [] M. Haviv and T. Roughgaden. The pice of anachy in an exponential multi-seve. Opeations Reseach Lettes, 35:42 426, [2] H. Kameda, E. Altman, O. outallie, J. Li, and Y. Hosokawa. aadoxes in pefomance optimization of distibuted systems. In oceedings of SSGRR 2000 Compute and ebusiness confeence, [3] H. Kameda, J. Li, C. Kim, and Y. Zhang. Optimal load balancing in distibuted compute systems. Spinge-Velag, 997. [4] F. Kelly. Stochastic Netwoks and Revesibility. Wiley, Chicheste, 979. [5] E. Koutsoupias and C.H. apadimitiou. Wost-case equilibia. In oceedings of STACS 999, 999. [6] D. Mondee and L.S. Shapley. otential games. Games and Econ. Behavio, 4:24 43, 996. [7] L.M. Ni and K. Hwang. Optimal load balancing in a multiple pocesso with many job classes. IEEE Tans. Softwae Eng., (5):49 496, 985. [8] M. atiksson. The Taffic Assignment oblem: Models and Methods. VS BV, The Nethelands, 994. [9] R.W. Rosenthal. A class of games possessing pue stategy Nash equilibia. Int. J. Game Theoy, 2:65 67, 973. [20] T. Roughgaden. Selfish Routing and the ice of Anachy. MIT ess, [2] W.H. Sandholm. otential games with continuous playe sets. Jounal of Economic Theoy, 97:8 08, 200. [22] D. Staobinski and T. Wu. efomance of seve selection algoithms fo content eplication netwoks. In IFI Netwoking, 2005.

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