Load Balancing in Processor Sharing Systems


 Dora Davis
 2 years ago
 Views:
Transcription
1 Load Balancing in ocesso Shaing Systems Eitan Altman INRIA Sophia Antipolis 2004, oute des Lucioles Sophia Antipolis, Fance Utzi Ayesta LAASCNRS Univesité de Toulouse 7, Avenue Colonel Roche F3077 Toulouse, Fance Balakishna abhu LAASCNRS Univesité de Toulouse 7, Avenue Colonel Roche F3077 Toulouse, Fance ABSTRACT In this pape, we investigate optimal load balancing stategies fo a multiclass multiseve pocessoshaing system with a oisson input steam, heteogeneous sevice ates, and a sevedependent 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 noncoopeative 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 noncoopeative 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/ pocessoshaing 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 sevefams, 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), FistComeFistSeved (FCFS), etc.) detemines the amount of esouces it needs to deploy in ode to guaantee a cetain QualityofSevice (QoS) to its customes. Thus, an investigation of load balancing o outing stategies in multiseve systems can give guidelines to the sevice povide on dimensioning its system. In this pape we study the optimal load balancing in a multiseve pocessoshaing system with heteogeneous sevice capacities. This configuation is also known as pocessoshaing sevefams, 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 multiseve 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 noncoopeative 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 closedfom expessions fo the load on each seve unde any optimal policy. It is wothwhile to note that fo the distibuted noncoopeative setting this
2 is done by showing the existence of a potential function, which allows us to tansfom the noncoopeative 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 sevefam and FCFS sevefam, since in the case of a FCFS sevefam 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 tadeoff 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 socalled 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 multiseve 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 multiseves 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 multiclass 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 multiclass 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 multiseve 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 nonovelapping, 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 (ollaczekkhinchin fomula), an intevalbased policy can minimize the oveall mean delay in the system. Inteestingly, if the sevice capacities ae heteogeneous an intevalbased stategy need not be optimal [8]. In contast, we show that in the case of a multiseve 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 noncoopeative poblem. We assume that decisions ae openloop: 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 noncoopeative 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 socalled 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 sizebased outing stategy. Notation. We shall use a lowe case boldfaced 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 multistategy 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 multistategy 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 multistategy if and only if >. (2) oof. Fo a multistategy 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 ighthand 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 multistategy. Now, assume (2) and conside the multistategy 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 multistategy. 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 multistategy. Assumption. The taffic intensities and the sevice ates ae such that (2) is always satisfied. Note that if p is a stable multistategy, 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 multistategy elevant to the context. Ou objective is to detemine the multistategy 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 multistategy 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 multistategies that minimize (9) subject to (0)(2). 3. Sizeunawae multistategies The following esult will play a key ole in the est of the pape. It shows that thee exists a sizeunawae multistategy that is optimal. oposition 2. Let p be a multistategy satisfying the constaints (0)(2). The multistategy 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 nonnegative 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 multistategy, thee exists a feasible sizeunawae multistategy such that both these stategies induce the same load on the seves. Since the objective function in the M depends on the multistategy 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 multistategy 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 multistategy 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 multistategy, 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 KaushKuhn 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 socalled 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 nondeceasing in γ G. Theefoe, thee is a unique value of γ G such that constaint (6) is satisfied. Since / is nondeceasing 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 sizeunawae multistategy, 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 watefilling. 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 multistategy, 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 multistategy (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 noncoopeative 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 elementwise 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 noncoopeative 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 classi 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 noncoopeative 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 noncoopeative 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 KaushKuhnTucke (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 nondeceasing 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 watefilling 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 watefilling. 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 noncoopeative setting. This indicates that in the noncoopeative 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 multiseve 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 socalled 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 selfishouting (o noncoopeative 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 multicommodity netwok the oa is uppe bounded by 4/3 [20]. In [] and [22] the authos study a multiseve 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 nondeceasing in j. We fist show that in the globally optimal multistategy 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 lefthand 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 noncoopeative 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 noncoopeative 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 watefilling 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 noncoopeative 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 noncoopeative 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 lefthand 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 lefthand 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 postdoctoal 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 Webseve 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] YC 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 statefee, sizeawae dispatching fo heteogeneous M/G/type systems. efomance Evaluation, 62( 4):36 39, [9] V. Gupta, M. HacholBalte, K. Sigman, and W. Whitt. Analysis of jointheshotestqueue outing fo web seve fams. In oceedings of efomance, page 80, [0] M. HacholBalte, 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 multiseve. 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. SpingeVelag, 997. [4] F. Kelly. Stochastic Netwoks and Revesibility. Wiley, Chicheste, 979. [5] E. Koutsoupias and C.H. apadimitiou. Wostcase 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.
Load Balancing in Processor Sharing Systems
Load Balancing in ocesso Shaing Systems Eitan Altman INRIA Sophia Antipolis 2004, oute des Lucioles 06902 Sophia Antipolis, Fance altman@sophia.inia.f Utzi Ayesta LAASCNRS Univesité de Toulouse 7, Avenue
More informationEfficient Redundancy Techniques for Latency Reduction in Cloud Systems
Efficient Redundancy Techniques fo Latency Reduction in Cloud Systems 1 Gaui Joshi, Emina Soljanin, and Gegoy Wonell Abstact In cloud computing systems, assigning a task to multiple seves and waiting fo
More informationChapter 3 Savings, Present Value and Ricardian Equivalence
Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,
More informationThe transport performance evaluation system building of logistics enterprises
Jounal of Industial Engineeing and Management JIEM, 213 6(4): 194114 Online ISSN: 213953 Pint ISSN: 2138423 http://dx.doi.og/1.3926/jiem.784 The tanspot pefomance evaluation system building of logistics
More informationQuestions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing
M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow
More informationON THE (Q, R) POLICY IN PRODUCTIONINVENTORY SYSTEMS
ON THE R POLICY IN PRODUCTIONINVENTORY SYSTEMS Saifallah Benjaafa and JoonSeok Kim Depatment of Mechanical Engineeing Univesity of Minnesota Minneapolis MN 55455 Abstact We conside a poductioninventoy
More informationRisk Sensitive Portfolio Management With CoxIngersollRoss Interest Rates: the HJB Equation
Risk Sensitive Potfolio Management With CoxIngesollRoss Inteest Rates: the HJB Equation Tomasz R. Bielecki Depatment of Mathematics, The Notheasten Illinois Univesity 55 Noth St. Louis Avenue, Chicago,
More informationSoftware Engineering and Development
I T H E A 67 Softwae Engineeing and Development SOFTWARE DEVELOPMENT PROCESS DYNAMICS MODELING AS STATE MACHINE Leonid Lyubchyk, Vasyl Soloshchuk Abstact: Softwae development pocess modeling is gaining
More informationHEALTHCARE INTEGRATION BASED ON CLOUD COMPUTING
U.P.B. Sci. Bull., Seies C, Vol. 77, Iss. 2, 2015 ISSN 22863540 HEALTHCARE INTEGRATION BASED ON CLOUD COMPUTING Roxana MARCU 1, Dan POPESCU 2, Iulian DANILĂ 3 A high numbe of infomation systems ae available
More informationModeling and Verifying a Price Model for Congestion Control in Computer Networks Using PROMELA/SPIN
Modeling and Veifying a Pice Model fo Congestion Contol in Compute Netwoks Using PROMELA/SPIN Clement Yuen and Wei Tjioe Depatment of Compute Science Univesity of Toonto 1 King s College Road, Toonto,
More informationAn Approach to Optimized Resource Allocation for Cloud Simulation Platform
An Appoach to Optimized Resouce Allocation fo Cloud Simulation Platfom Haitao Yuan 1, Jing Bi 2, Bo Hu Li 1,3, Xudong Chai 3 1 School of Automation Science and Electical Engineeing, Beihang Univesity,
More informationApproximation Algorithms for Data Management in Networks
Appoximation Algoithms fo Data Management in Netwoks Chistof Kick Heinz Nixdof Institute and Depatment of Mathematics & Compute Science adebon Univesity Gemany kueke@upb.de Haald Räcke Heinz Nixdof Institute
More informationContinuous Compounding and Annualization
Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem
More informationSTUDENT RESPONSE TO ANNUITY FORMULA DERIVATION
Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts
More informationPeertoPeer File Sharing Game using Correlated Equilibrium
PeetoPee File Shaing Game using Coelated Equilibium Beibei Wang, Zhu Han, and K. J. Ray Liu Depatment of Electical and Compute Engineeing and Institute fo Systems Reseach, Univesity of Mayland, College
More informationIlona V. Tregub, ScD., Professor
Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation
More informationSeshadri constants and surfaces of minimal degree
Seshadi constants and sufaces of minimal degee Wioletta Syzdek and Tomasz Szembeg Septembe 29, 2007 Abstact In [] we showed that if the multiple point Seshadi constants of an ample line bundle on a smooth
More informationEffect of Contention Window on the Performance of IEEE 802.11 WLANs
Effect of Contention Window on the Pefomance of IEEE 82.11 WLANs Yunli Chen and Dhama P. Agawal Cente fo Distibuted and Mobile Computing, Depatment of ECECS Univesity of Cincinnati, OH 452213 {ychen,
More informationest using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.
9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,
More informationINITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS
INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in
More informationLife Insurance Purchasing to Reach a Bequest. Erhan Bayraktar Department of Mathematics, University of Michigan Ann Arbor, Michigan, USA, 48109
Life Insuance Puchasing to Reach a Bequest Ehan Bayakta Depatment of Mathematics, Univesity of Michigan Ann Abo, Michigan, USA, 48109 S. David Pomislow Depatment of Mathematics, Yok Univesity Toonto, Ontaio,
More informationAn Introduction to Omega
An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei iskewad chaacteistics? The Finance Development Cente 2002 1 Fom
More informationAn Analysis of Manufacturer Benefits under Vendor Managed Systems
An Analysis of Manufactue Benefits unde Vendo Managed Systems Seçil Savaşaneil Depatment of Industial Engineeing, Middle East Technical Univesity, 06531, Ankaa, TURKEY secil@ie.metu.edu.t Nesim Ekip 1
More informationComparing Availability of Various Rack Power Redundancy Configurations
Compaing Availability of Vaious Rack Powe Redundancy Configuations By Victo Avela White Pape #48 Executive Summay Tansfe switches and dualpath powe distibution to IT equipment ae used to enhance the availability
More informationAN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM
AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,
More informationCloud Service Reliability: Modeling and Analysis
Cloud Sevice eliability: Modeling and Analysis YuanShun Dai * a c, Bo Yang b, Jack Dongaa a, Gewei Zhang c a Innovative Computing Laboatoy, Depatment of Electical Engineeing & Compute Science, Univesity
More informationMULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION
MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION K.C. CHANG AND TAN ZHANG In memoy of Pofesso S.S. Chen Abstact. We combine heat flow method with Mose theoy, supe and subsolution method with
More informationUncertain Version Control in Open Collaborative Editing of TreeStructured Documents
Uncetain Vesion Contol in Open Collaboative Editing of TeeStuctued Documents M. Lamine Ba Institut Mines Télécom; Télécom PaisTech; LTCI Pais, Fance mouhamadou.ba@ telecompaistech.f Talel Abdessalem
More informationComparing Availability of Various Rack Power Redundancy Configurations
Compaing Availability of Vaious Rack Powe Redundancy Configuations White Pape 48 Revision by Victo Avela > Executive summay Tansfe switches and dualpath powe distibution to IT equipment ae used to enhance
More information9:6.4 Sample Questions/Requests for Managing Underwriter Candidates
9:6.4 INITIAL PUBLIC OFFERINGS 9:6.4 Sample Questions/Requests fo Managing Undewite Candidates Recent IPO Expeience Please povide a list of all completed o withdawn IPOs in which you fim has paticipated
More informationTracking/Fusion and Deghosting with Doppler Frequency from Two Passive Acoustic Sensors
Tacking/Fusion and Deghosting with Dopple Fequency fom Two Passive Acoustic Sensos Rong Yang, Gee Wah Ng DSO National Laboatoies 2 Science Pak Dive Singapoe 11823 Emails: yong@dso.og.sg, ngeewah@dso.og.sg
More informationElectricity transmission network optimization model of supply and demand the case in Taiwan electricity transmission system
Electicity tansmission netwok optimization model of supply and demand the case in Taiwan electicity tansmission system MiaoSheng Chen a ChienLiang Wang b,c, ShengChuan Wang d,e a Taichung Banch Gaduate
More informationOverencryption: Management of Access Control Evolution on Outsourced Data
Oveencyption: Management of Access Contol Evolution on Outsouced Data Sabina De Capitani di Vimecati DTI  Univesità di Milano 26013 Cema  Italy decapita@dti.unimi.it Stefano Paaboschi DIIMM  Univesità
More informationQuestions for Review. By buying bonds This period you save s, next period you get s(1+r)
MACROECONOMICS 2006 Week 5 Semina Questions Questions fo Review 1. How do consumes save in the twopeiod model? By buying bonds This peiod you save s, next peiod you get s() 2. What is the slope of a consume
More informationOptimizing Content Retrieval Delay for LTbased Distributed Cloud Storage Systems
Optimizing Content Retieval Delay fo LTbased Distibuted Cloud Stoage Systems Haifeng Lu, Chuan Heng Foh, Yonggang Wen, and Jianfei Cai School of Compute Engineeing, Nanyang Technological Univesity, Singapoe
More informationChannel selection in ecommerce age: A strategic analysis of coop advertising models
Jounal of Industial Engineeing and Management JIEM, 013 6(1):89103 Online ISSN: 0130953 Pint ISSN: 013843 http://dx.doi.og/10.396/jiem.664 Channel selection in ecommece age: A stategic analysis of
More informationNontrivial lower bounds for the least common multiple of some finite sequences of integers
J. Numbe Theoy, 15 (007), p. 393411. Nontivial lowe bounds fo the least common multiple of some finite sequences of integes Bai FARHI bai.fahi@gmail.com Abstact We pesent hee a method which allows to
More informationMETHODOLOGICAL APPROACH TO STRATEGIC PERFORMANCE OPTIMIZATION
ETHODOOGICA APPOACH TO STATEGIC PEFOANCE OPTIIZATION ao Hell * Stjepan Vidačić ** Željo Gaača *** eceived: 4. 07. 2009 Peliminay communication Accepted: 5. 0. 2009 UDC 65.02.4 This pape pesents a matix
More informationFinancing Terms in the EOQ Model
Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad
More informationTowards Automatic Update of Access Control Policy
Towads Automatic Update of Access Contol Policy Jinwei Hu, Yan Zhang, and Ruixuan Li Intelligent Systems Laboatoy, School of Computing and Mathematics Univesity of Westen Sydney, Sydney 1797, Austalia
More informationData Center Demand Response: Avoiding the Coincident Peak via Workload Shifting and Local Generation
(213) 1 28 Data Cente Demand Response: Avoiding the Coincident Peak via Wokload Shifting and Local Geneation Zhenhua Liu 1, Adam Wieman 1, Yuan Chen 2, Benjamin Razon 1, Niangjun Chen 1 1 Califonia Institute
More informationThe Binomial Distribution
The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between
More information4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first nonzero digit to
. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate
More informationMemoryAware Sizing for InMemory Databases
MemoyAwae Sizing fo InMemoy Databases Kasten Molka, Giuliano Casale, Thomas Molka, Laua Mooe Depatment of Computing, Impeial College London, United Kingdom {k.molka3, g.casale}@impeial.ac.uk SAP HANA
More informationAn Efficient Group Key Agreement Protocol for Ad hoc Networks
An Efficient Goup Key Ageement Potocol fo Ad hoc Netwoks Daniel Augot, Raghav haska, Valéie Issany and Daniele Sacchetti INRIA Rocquencout 78153 Le Chesnay Fance {Daniel.Augot, Raghav.haska, Valéie.Issany,
More informationTowards Realizing a Low Cost and Highly Available Datacenter Power Infrastructure
Towads Realizing a Low Cost and Highly Available Datacente Powe Infastuctue Siam Govindan, Di Wang, Lydia Chen, Anand Sivasubamaniam, and Bhuvan Ugaonka The Pennsylvania State Univesity. IBM Reseach Zuich
More informationPAN STABILITY TESTING OF DC CIRCUITS USING VARIATIONAL METHODS XVIII  SPETO  1995. pod patronatem. Summary
PCE SEMINIUM Z PODSTW ELEKTOTECHNIKI I TEOII OBWODÓW 8  TH SEMIN ON FUNDMENTLS OF ELECTOTECHNICS ND CICUIT THEOY ZDENĚK BIOLEK SPŠE OŽNO P.., CZECH EPUBLIC DLIBO BIOLEK MILITY CDEMY, BNO, CZECH EPUBLIC
More informationPromised LeadTime Contracts Under Asymmetric Information
OPERATIONS RESEARCH Vol. 56, No. 4, July August 28, pp. 898 915 issn 3364X eissn 15265463 8 564 898 infoms doi 1.1287/ope.18.514 28 INFORMS Pomised LeadTime Contacts Unde Asymmetic Infomation Holly
More informationAn Infrastructure Cost Evaluation of Single and MultiAccess Networks with Heterogeneous Traffic Density
An Infastuctue Cost Evaluation of Single and MultiAccess Netwoks with Heteogeneous Taffic Density Andes Fuuskä and Magnus Almgen Wieless Access Netwoks Eicsson Reseach Kista, Sweden [andes.fuuska, magnus.almgen]@eicsson.com
More informationON NEW CHALLENGES FOR CFD SIMULATION IN FILTRATION
ON NEW CHALLENGES FOR CFD SIMULATION IN FILTRATION Michael Dedeing, Wolfgang Stausbeg, IBS Filtan, Industiestasse 19 D51597 MosbachLichtenbeg, Gemany. Oleg Iliev(*), Zaha Lakdawala, Faunhofe Institut
More informationChapter 2 Valiant LoadBalancing: Building Networks That Can Support All Traffic Matrices
Chapte 2 Valiant LoadBalancing: Building etwoks That Can Suppot All Taffic Matices Rui ZhangShen Abstact This pape is a bief suvey on how Valiant loadbalancing (VLB) can be used to build netwoks that
More informationScheduling Hadoop Jobs to Meet Deadlines
Scheduling Hadoop Jobs to Meet Deadlines Kamal Kc, Kemafo Anyanwu Depatment of Compute Science Noth Caolina State Univesity {kkc,kogan}@ncsu.edu Abstact Use constaints such as deadlines ae impotant equiements
More informationChris J. Skinner The probability of identification: applying ideas from forensic statistics to disclosure risk assessment
Chis J. Skinne The pobability of identification: applying ideas fom foensic statistics to disclosue isk assessment Aticle (Accepted vesion) (Refeeed) Oiginal citation: Skinne, Chis J. (2007) The pobability
More informationOptimal Peer Selection in a FreeMarket PeerResource Economy
Optimal Pee Selection in a FeeMaket PeeResouce Economy Micah Adle, Rakesh Kuma, Keith Ross, Dan Rubenstein, David Tune and David D Yao Dept of Compute Science Univesity of Massachusetts Amhest, MA; Email:
More information2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,
3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects
More informationHow Much Should a Firm Borrow. Effect of tax shields. Capital Structure Theory. Capital Structure & Corporate Taxes
How Much Should a Fim Boow Chapte 19 Capital Stuctue & Copoate Taxes Financial Risk  Risk to shaeholdes esulting fom the use of debt. Financial Leveage  Incease in the vaiability of shaeholde etuns that
More informationFinancial Derivatives for Computer Network Capacity Markets with QualityofService Guarantees
Financial Deivatives fo Compute Netwok Capacity Makets with QualityofSevice Guaantees Pette Pettesson pp@kth.se Febuay 2003 SICS Technical Repot T2003:03 Keywods Netwoking and Intenet Achitectue. Abstact
More informationAn application of stochastic programming in solving capacity allocation and migration planning problem under uncertainty
An application of stochastic pogamming in solving capacity allocation and migation planning poblem unde uncetainty YinYann Chen * and HsiaoYao Fan Depatment of Industial Management, National Fomosa Univesity,
More informationSpirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project
Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.
More informationTheory and practise of the gindex
Theoy and pactise of the gindex by L. Egghe (*), Univesiteit Hasselt (UHasselt), Campus Diepenbeek, Agoalaan, B3590 Diepenbeek, Belgium Univesiteit Antwepen (UA), Campus Die Eiken, Univesiteitsplein,
More informationA Capacitated Commodity Trading Model with Market Power
A Capacitated Commodity Tading Model with Maket Powe Victo MatínezdeAlbéniz Josep Maia Vendell Simón IESE Business School, Univesity of Navaa, Av. Peason 1, 08034 Bacelona, Spain VAlbeniz@iese.edu JMVendell@iese.edu
More informationThe Impacts of Congestion on Commercial Vehicle Tours
Figliozzi 1 The Impacts of Congestion on Commecial Vehicle Tous Miguel Andes Figliozzi Potland State Univesity Maseeh College of Engineeing and Compute Science figliozzi@pdx.edu 5124 wods + 7 Tables +
More informationEconomics 326: Input Demands. Ethan Kaplan
Economics 326: Input Demands Ethan Kaplan Octobe 24, 202 Outline. Tems 2. Input Demands Tems Labo Poductivity: Output pe unit of labo. Y (K; L) L What is the labo poductivity of the US? Output is ouhgly
More informationGive me all I pay for Execution Guarantees in Electronic Commerce Payment Processes
Give me all I pay fo Execution Guaantees in Electonic Commece Payment Pocesses Heiko Schuldt Andei Popovici HansJög Schek Email: Database Reseach Goup Institute of Infomation Systems ETH Zentum, 8092
More informationResearch on Risk Assessment of the Transformer Based on Life Cycle Cost
ntenational Jounal of Smat Gid and lean Enegy eseach on isk Assessment of the Tansfome Based on Life ycle ost Hui Zhou a, Guowei Wu a, Weiwei Pan a, Yunhe Hou b, hong Wang b * a Zhejiang Electic Powe opoation,
More informationStatistics and Data Analysis
Pape 27425 An Extension to SAS/OR fo Decision System Suppot Ali Emouznead Highe Education Funding Council fo England, Nothavon house, Coldhabou Lane, Bistol, BS16 1QD U.K. ABSTRACT This pape exploes the
More informationSUPPORT VECTOR MACHINE FOR BANDWIDTH ANALYSIS OF SLOTTED MICROSTRIP ANTENNA
Intenational Jounal of Compute Science, Systems Engineeing and Infomation Technology, 4(), 20, pp. 677 SUPPORT VECTOR MACHIE FOR BADWIDTH AALYSIS OF SLOTTED MICROSTRIP ATEA Venmathi A.R. & Vanitha L.
More informationSemipartial (Part) and Partial Correlation
Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated
More informationTHE DISTRIBUTED LOCATION RESOLUTION PROBLEM AND ITS EFFICIENT SOLUTION
IADIS Intenational Confeence Applied Computing 2006 THE DISTRIBUTED LOCATION RESOLUTION PROBLEM AND ITS EFFICIENT SOLUTION Jög Roth Univesity of Hagen 58084 Hagen, Gemany Joeg.Roth@Fenunihagen.de ABSTRACT
More informationAMB111F Financial Maths Notes
AMB111F Financial Maths Notes Compound Inteest and Depeciation Compound Inteest: Inteest computed on the cuent amount that inceases at egula intevals. Simple inteest: Inteest computed on the oiginal fixed
More informationAn Epidemic Model of Mobile Phone Virus
An Epidemic Model of Mobile Phone Vius Hui Zheng, Dong Li, Zhuo Gao 3 Netwok Reseach Cente, Tsinghua Univesity, P. R. China zh@tsinghua.edu.cn School of Compute Science and Technology, Huazhong Univesity
More informationValuation of Floating Rate Bonds 1
Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned
More informationA framework for the selection of enterprise resource planning (ERP) system based on fuzzy decision making methods
A famewok fo the selection of entepise esouce planning (ERP) system based on fuzzy decision making methods Omid Golshan Tafti M.s student in Industial Management, Univesity of Yazd Omidgolshan87@yahoo.com
More informationConcept and Experiences on using a Wikibased System for Softwarerelated Seminar Papers
Concept and Expeiences on using a Wikibased System fo Softwaeelated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wthaachen.de,
More informationFinancial Planning and Riskreturn profiles
Financial Planning and Risketun pofiles Stefan Gaf, Alexande Kling und Jochen Russ Pepint Seies: 201016 Fakultät fü Mathematik und Witschaftswissenschaften UNIERSITÄT ULM Financial Planning and Risketun
More informationEnergy Efficient Cache Invalidation in a Mobile Environment
Enegy Efficient Cache Invalidation in a Mobile Envionment Naottam Chand, Ramesh Chanda Joshi, Manoj Misa Electonics & Compute Engineeing Depatment Indian Institute of Technology, Rookee  247 667. INDIA
More informationTop K Nearest Keyword Search on Large Graphs
Top K Neaest Keywod Seach on Lage Gaphs Miao Qiao, Lu Qin, Hong Cheng, Jeffey Xu Yu, Wentao Tian The Chinese Univesity of Hong Kong, Hong Kong, China {mqiao,lqin,hcheng,yu,wttian}@se.cuhk.edu.hk ABSTRACT
More informationThings to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request.
Retiement Benefit 1 Things to Remembe Complete all of the sections on the Retiement Benefit fom that apply to you equest. If this is an initial equest, and not a change in a cuent distibution, emembe to
More informationPhysics 235 Chapter 5. Chapter 5 Gravitation
Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus
More informationDual channel closedloop supply chain coordination with a rewarddriven remanufacturing policy
Intenational Jounal of Poduction Reseach ISSN: 753 Pint 1366588X Online Jounal homepage: http://www.tandfonline.com/loi/tps Dual channel closedloop supply chain coodination with a ewaddiven emanufactuing
More informationThe Supply of Loanable Funds: A Comment on the Misconception and Its Implications
JOURNL OF ECONOMICS ND FINNCE EDUCTION Volume 7 Numbe 2 Winte 2008 39 The Supply of Loanable Funds: Comment on the Misconception and Its Implications. Wahhab Khandke and mena Khandke* STRCT Recently FieldsHat
More informationTHE CARLO ALBERTO NOTEBOOKS
THE CARLO ALBERTO NOTEBOOKS Meanvaiance inefficiency of CRRA and CARA utility functions fo potfolio selection in defined contibution pension schemes Woking Pape No. 108 Mach 2009 Revised, Septembe 2009)
More informationIBM Research Smarter Transportation Analytics
IBM Reseach Smate Tanspotation Analytics Laua Wynte PhD, Senio Reseach Scientist, IBM Watson Reseach Cente lwynte@us.ibm.com INSTRUMENTED We now have the ability to measue, sense and see the exact condition
More informationHigh Availability Replication Strategy for Deduplication Storage System
Zhengda Zhou, Jingli Zhou College of Compute Science and Technology, Huazhong Univesity of Science and Technology, *, zhouzd@smail.hust.edu.cn jlzhou@mail.hust.edu.cn Abstact As the amount of digital data
More informationThe Predictive Power of Dividend Yields for Stock Returns: Risk Pricing or Mispricing?
The Pedictive Powe of Dividend Yields fo Stock Retuns: Risk Picing o Mispicing? Glenn Boyle Depatment of Economics and Finance Univesity of Cantebuy Yanhui Li Depatment of Economics and Finance Univesity
More informationEvaluating the impact of Blade Server and Virtualization Software Technologies on the RIT Datacenter
Evaluating the impact of and Vitualization Softwae Technologies on the RIT Datacente Chistophe M Butle Vitual Infastuctue Administato Rocheste Institute of Technology s Datacente Contact: chis.butle@it.edu
More informationPatent renewals and R&D incentives
RAND Jounal of Economics Vol. 30, No., Summe 999 pp. 97 3 Patent enewals and R&D incentives Fancesca Conelli* and Mak Schankeman** In a model with moal hazad and asymmetic infomation, we show that it can
More informationarxiv:1110.2612v1 [qfin.st] 12 Oct 2011
Maket inefficiency identified by both single and multiple cuency tends T.Toká 1, and D. Hováth 1, 1 Sos Reseach a.s., Stojáenská 3, 040 01 Košice, Slovak Republic Abstact axiv:1110.2612v1 [qfin.st] 12
More informationConverting knowledge Into Practice
Conveting knowledge Into Pactice Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 2 0 1 0 C o p y i g h t s V l a d i m i R i b a k o v 1 Disclaime and Risk Wanings Tading
More informationNBER WORKING PAPER SERIES FISCAL ZONING AND SALES TAXES: DO HIGHER SALES TAXES LEAD TO MORE RETAILING AND LESS MANUFACTURING?
NBER WORKING PAPER SERIES FISCAL ZONING AND SALES TAXES: DO HIGHER SALES TAXES LEAD TO MORE RETAILING AND LESS MANUFACTURING? Daia Bunes David Neumak Michelle J. White Woking Pape 16932 http://www.nbe.og/papes/w16932
More informationOptimal Capital Structure with Endogenous Bankruptcy:
Univesity of Pisa Ph.D. Pogam in Mathematics fo Economic Decisions Leonado Fibonacci School cotutelle with Institut de Mathématique de Toulouse Ph.D. Dissetation Optimal Capital Stuctue with Endogenous
More informationOn Some Functions Involving the lcm and gcd of Integer Tuples
SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A: APPL. MATH. INFORM. AND MECH. vol. 6, 2 (2014), 91100. On Some Functions Involving the lcm and gcd of Intege Tuples O. Bagdasa Abstact:
More informationCONCEPTUAL FRAMEWORK FOR DEVELOPING AND VERIFICATION OF ATTRIBUTION MODELS. ARITHMETIC ATTRIBUTION MODELS
CONCEPUAL FAMEOK FO DEVELOPING AND VEIFICAION OF AIBUION MODELS. AIHMEIC AIBUION MODELS Yui K. Shestopaloff, is Diecto of eseach & Deelopment at SegmentSoft Inc. He is a Docto of Sciences and has a Ph.D.
More informationReview Graph based Online Store Review Spammer Detection
Review Gaph based Online Stoe Review Spamme Detection Guan Wang, Sihong Xie, Bing Liu, Philip S. Yu Univesity of Illinois at Chicago Chicago, USA gwang26@uic.edu sxie6@uic.edu liub@uic.edu psyu@uic.edu
More informationSaturated and weakly saturated hypergraphs
Satuated and weakly satuated hypegaphs Algebaic Methods in Combinatoics, Lectues 67 Satuated hypegaphs Recall the following Definition. A family A P([n]) is said to be an antichain if we neve have A B
More informationSupply chain information sharing in a macro prediction market
Decision Suppot Systems 42 (2006) 944 958 www.elsevie.com/locate/dss Supply chain infomation shaing in a maco pediction maket Zhiling Guo a,, Fang Fang b, Andew B. Whinston c a Depatment of Infomation
More informationAdaptive Queue Management with Restraint on NonResponsive Flows
Adaptive Queue Management wi Restaint on NonResponsive Flows Lan Li and Gyungho Lee Depatment of Electical and Compute Engineeing Univesity of Illinois at Chicago 85 S. Mogan Steet Chicago, IL 667 {lli,
More informationExperimentation under Uninsurable Idiosyncratic Risk: An Application to Entrepreneurial Survival
Expeimentation unde Uninsuable Idiosyncatic Risk: An Application to Entepeneuial Suvival Jianjun Miao and Neng Wang May 28, 2007 Abstact We popose an analytically tactable continuoustime model of expeimentation
More informationInsurance Pricing under Ambiguity
Insuance Picing unde Ambiguity Alois Pichle a,b, a Univesity of Vienna, Austia. Depatment of Statistics and Opeations Reseach b Actuay. Membe of the Austian Actuaial Association Abstact Stating fom the
More informationThe Role of Gravity in Orbital Motion
! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State
More information