On the Interaction between Load Balancing and Speed Scaling


 Jesse Chase
 3 years ago
 Views:
Transcription
1 On the Interacton between Load Balancng and Speed Scalng Ljun Chen and Na L Abstract Speed scalng has been wdely adopted n computer and communcaton systems, n partcular, to reduce energy consumpton. An mportant queston s how speed scalng nteracts wth other resource allocaton mechansms such as schedulng and routng, etc. In ths paper, we study the nteracton of speed scalng wth load balancng. We characterze the equlbrum resultng from the load balancng and speed scalng nteracton, and ntroduce two optmal load balancng desgns, n terms of tradtonal performance metrc and costaware (n partcular, energyaware) performance metrc respectvely. Especally, we characterze the loadbalancngspeedscalng equlbrum wth respect to the optmal load balancng schemes n processor sharng systems. Our results show that the degree of neffcency at the equlbrum s mostly bounded by the heterogenety of the system, but ndependent of the number of the servers. These results provde nsghts n understandng the nteracton of load balancng wth speed scalng and gudng new desgns. Index Terms Load balancng, Speed scalng, Energy effcency, Effcency loss, Data centers. I. INTRODUCTION The energy consumpton rate of computer and communcaton systems has been ncreasng exponentally. Computer and communcaton systems must make a fundamental tradeoff between performance and energy usage, see, e.g., [], [2]. The addton of energy to standard performance metrcs such as delay, throughput and loss fundamentally changes the problem space of some of resource allocaton desgns. Not only are new mechansms needed to optmze energy usage, exstng algorthms and protocols must be reexamned as a formerly optmal algorthm may now perform poorly wth respect to a new energyaware metrc. Energy management decsons must be decomposed and coordnated spatally as well as temporally, and yet global optmalty must be acheved through local algorthms that are mplementable n a dstrbuted manner. In ths paper we study load balancng and ts nteracton wth speed scalng. Energyaware speed scalng to adapt the speed of the system so as to balance energy and performance metrcs s a wdelyadopted power management technque, see, e.g., [3], [4], [5], [6], [7], [8], [9], [0], [], [2], [3]. Prevous works on speed scalng usually focus on a sngle server and study ts nteracton wth schedulng, see, e.g, [4], [8], [9], [0], [3]. Here we consder a network settng and study the nteracton of L. Chen s wth Computer Scence and Telecommuncatons, Unversty of Colorado, Boulder, CO 80309, USA (emal: N. L s wth Electrcal Engneerng, Harvard Unversty, Cambrdge, MA 0238, USA (emal: Prelmnary result of ths paper has been presented at the INFORMS Annual Meetng, Austn, Texas, 200 and the ITA Workshop, Lo Jolla, Calforna, 20. speed scalng wth load balancng, to provde nsghts nto such ssues as: ) How does the system perform under speed scalng n terms of tradtonal performance metrcs as well as energyaware metrcs? ) How to desgn energyaware optmal load balancng and can we decouple the desgn of load balancng from that of speed scalng? ) How does the sophstcaton of speed scalng mpact the desgn and performance of load balancng? We focus on gatedstatc speed scalng n processor sharng systems, and our results provde useful nsghts nto the frst two questons. Specfcally, we characterze the equlbrum resultng from the load balancng and speed scalng nteracton, and ntroduce two optmal load balancng desgn problems, n terms of tradtonal performance metrc and costaware (n partcular, energyaware) performance metrc respectvely. We study n detal the loadbalancngspeedscalng equlbrum and the optmal load balancng desgns n processor sharng systems wth gatedstatc speed scalng, and propose dstrbuted load balancng algorthms to acheve the correspondng equlbrum and optma. Especally, we characterze the degree of neffcency at the loadbalancngspeedscalng equlbrum, n terms of delay as well as energyaware metrc. We show that the degree of neffcency s mostly bounded by the heterogenety of the system, but ndependent of the number of servers n the system. Our results suggest that, as n many applcatons a loworder polynomal provdes a good approxmaton to power functon, we can decouple the desgn of load balancng from speed scalng wthout ncurrng much neffcency n delay. In terms of poweraware performance metrc, our results suggest that, as long as the heterogenety n the system s small, we can decouple the desgn of load balancng from speed scalng wthout ncurrng much effcency loss; but when the heterogenety n the system s large, we have to do energyaware load balancng f the energy consumpton s a man concern. To summarze, we make the followng man contrbutons n ths paper: ) We formulate three dfferent models to study the nteracton of load balancng and speed scalng: energyoblvous load balancng where the dspatcher mnmzes the delay experenced by a job, delayoptmal load balancng where the dspatcher mnmzes the overall delay ncurred at the servers, and energyaware optmal load balancng where the dspatcher mnmzes the overall energy consumpton at the servers. 2) We characterze the equlbra of the above three load balancng desgns n terms of the set of actve servers, and propose dstrbuted algorthms for achevng the
2 2 equlbra. Our algorthms have low mplementaton complexty, and requre only nformaton that can be estmated or measured locally at the dspatcher and the servers. 3) We characterze the effcency loss of the energyoblvous load balancng (whch domnates the current practce) n delay and energy consumpton, and show that the degree of neffcency s mostly bounded by the heterogenety of the system. These results provde nsghts n understandng the nteracton of load balancng wth speed scalng and gudng new desgns. 4) We provde numercal examples to demonstrate the convergence of the proposed algorthms, and verfy the bounds on the effcency loss. The paper s organzed as follows. The next secton brefly dscusses some related work. Secton III descrbes the system model. Secton IV gves a bref characterzaton of the loadbalancngspeedscalng equlbrum, and ntroduces two optmal load balancng desgn problems. Secton V studes n detal the loadbalancngspeedscalng nteracton n processor sharng systems wth gatedstatc speed scalng. Secton VI provdes numercal examples to complement the theoretcal analyss, and Secton VII concludes wth some dscusson on further research. Notaton: major notaton used n ths paper s summarzed n Table I. N λ λ, s F T c (s ) P (s ) α M e + C e C o D e D o TABLE I NOTATIONS Set of servers Job arrval rate at the dspatcher Job arrval rate and servce rate at server N Performance metrc at server N Delay at server N Operatng cost at speed s of server N Power functon of server N Order of polynomal power functon of server N Costaware performance metrc at server N Superscrpt denotng LBSS equlbrum Superscrpt denotng delayoptmal LB Superscrpt denotng energyaware optmal LB Socal cost n delay at LBSS equlbrum Optmal cost n delay Energyaware socal cost at LBSS equlbrum Optmal energyware cost II. RELATED WORK Power management technques have been ncreasngly adopted n desgns from sngledevce level such as chps to network level such as data centers. It has spurred a new branch of research n ts own rght. In partcular, startng wth Yao et al [5], there s extensve research on analytcal study of speed scalng, see, e.g., [6], [7], [8], [9], [20], [2], [4], [22], [8], [23], [24], [9], [0], [], [2], [3]. Bansal et al [8] show that a speed scalng polcy (SRPT, P (n + )) s 3compettve for regular power functons n the worstcase analyss. Ths result has been tghten and extended to PS schedulng as well as to stochastc analyss by Andrew et al [0]. Especally, Andrew et al [0] provde a comprehensve study of speed scalng and ts nteracton wth schedulng, and show a fundamental tradeoff between optmalty, farness and robustness n speed scalng desgns. Stanojevc and Shorten [] study dstrbuted speed scalng to mnmze energy consumpton subject to performance constrants. Son and Krshnamachar [2] study speedscalngaware load balancng for cellular networks, and ther model has structural smlarty to ours for the energyaware optmal load balancng (see Secton VC). However, ther model ncludes delay and energy consumpton n the networkng components, n addton to those n the computng/processng components. Ther equlbrum characterzaton focuses on user assocaton, whle ours focuses on the set of actve servers. Ther teratve algorthm s based heurstcally on a varant of equlbrum characterzaton (.e., user assocaton at the optmum), whle ours s based on the gradent method. Related work also ncludes [25], [26] that show that the degree of neffcency n delay for load balancng n processor sharng systems wth fxed server speeds scales wth the number of servers n the system. Ths result has been extended to the processor sharng system wth multclass load [27], and to other schedulng polces such as SRPT [28]. In contrast to these results, we show that the degree of neffcency n delay for load balancng n processor sharng systems wth speed scalng s bounded by the heterogenety of the system, but ndependent of the number of servers. III. SYSTEM MODEL Consder a system wth a set N of servers and a Posson arrval process of rate λ > 0; see Fgure. We assume that job sze s..d., and wthout loss of generalty, has a mean of. Assocated wth each server s a servce rate (or speed) s. There s a load balancng dspatcher that probablstcally routes arrvals to severs accordng to certan tradtonal performance metrc F that end users are concerned wth, so that F at each server s the same and mnmal. The metrc F can be, for example, the mean response tme E[T ] at the server, the summaton of E[T ] and propagaton delay τ, and the blockng probablty p, etc. It follows that the resultng arrval process to server s Posson wth rate λ. We assume that server s performance curve F = f (s, λ ) (or ts analytcal approxmaton) s contnuously dfferentable, ncreasng n arrval rate λ, and decreasng n servce capacty s wth f (, λ) = 0. Ths s a rather general assumpton. In order to ensure stablty, we must have λ < s for all N. We can thus assume that f (s, λ ) = when λ s. Besdes performance metrc F that s perceved by end users, each server ncurs certan cost c (s ) per unt tme when t runs at a speed of s. The cost can be, for example, the power expended at the server, or any other types of servce costs. Gven an ncomng rate of λ, let g (s, λ ) = E[c (s )], the average cost. The average cost depends on the speed Here a server can be a sngle server, or represent a cluster of collocated servers n, e.g., a mcrodatacenter.
3 3 l dspatcher Fg.. A pctoral dagram of the system model. l l 2 l N s s 2 s N servers as well as the schedulng polcy at the server. The cost functon g (s, λ ) (or ts analytcal approxmaton) s assumed to be contnuously dfferentable, ncreasng n s, and nondecreasng n λ. Gven arrval rate λ and schedulng polcy, each server wll choose a speed s to mnmze a costaware performance metrc M : M = g (s, λ ) + β λ f (s, λ ), () where β > 0 s used to characterze the relatve weght of nternal cost and tradtonal performance metrc. By the above model, we have actually assumed some knd of statc speed scalng,.e., choose a sngle speed s for a gven arrval rate λ. Wth more complcated notaton, we can also model dynamc scalng,.e., adapt speed to dfferent states such as the number of jobs n the server. Speed scalng can be broadly defned as any behavor of adaptng speed to load, and can be due to varous reasons, correspondng to dfferent choces of cost functon g (s, λ ). In ths paper, we wll mostly focus on energyaware speed scalng as a concrete system to study the nteracton between load balancng and speed scalng, and consder the followng performance metrc: M = E[P (s )] + β λ E[T ], (2) where P (s ) s the power expended when server runs at speed s. The modelng of the power functon P (s ) s an actve research topc, and measurements have shown t can take on dfferent forms dependng on the system nvolved. In many applcatons a loworder polynomal form P (s ) = k s α, k > 0, α > (3) provdes a good approxmaton. For example, for dynamc power n CMOS P s often assumed to be cubc n prevous works [2]. We wll focus on polynomal power functon (3) n ths paper, as n many prevous works on speed scalng. IV. LOADBALANCINGSPEEDSCALING INTERACTION In ths secton, we characterze the equlbrum resultng from the nteracton between load balancng and speed scalng for the general model descrbed n Secton III. We then ntroduce two optmal load balancng problems, Foptmal load balancng and costaware optmal load balancng, under speed scalng. We ntend to characterze the equlbrum wth respect to those two optmal load balancng problems, as well as proposng dstrbuted load balancng algorthms to acheve the correspondng equlbrum and optma. Gven server speeds (s ) N and denote the set of servers used at load balancng by N b,.e., N b ff λ > 0. At load balancng, the F value at any server N b s thus the same, and not larger than the F j value a job would experence f routed to any unused server j N/N b. Ths can be wrtten mathematcally as f (s, λ ) f j (s j, λ j ), j N, N b, (4) λ = λ, (5) N where (λ ) N s the arrval rates at the servers at load balancng. Denote the F value at server N b at load balancng by γ. The load balancng condton (4)(5) can be equvalently wrtten as: there exsts a γ > 0, such that (f (s, λ ) γ)( λ λ ) 0, λ 0, (6) λ = λ. (7) N To see ths equvalence, note that equatons (6)(7) mply that γ must equal the F value at server N b at load balancng. Assume that speed scalng problem mn s>λ M has a unque soluton s (λ ). Under the aforementoned assumptons on f and g, speed scalng s (λ ) satsfes: g (s, λ ) s + β λ f (s, λ ) s = 0. (8) Notce that the dynamc speed range of a server s usually fnte,.e., s r for some r > 0. For smplcty, we do not consder such a constrant n ths paper. Such a constrant does not change the general structure of our model snce t does not change the convexty and the dstrbuted decomposton structure of the model. But t wll affect the characterzaton of effcency loss n Secton V. However, to remove the speed range constrant s reasonable for two reasons. Frst, one key aspect of ths paper s to study the mpact of speed scalng, but the speed range constrant (that s tght relatve to the job arrval rate) wll lmt the capablty of or even dsable a server s speed scalng. Second, the computng capacty s usually not a constrant; and actually a man motvaton for speed scalng s to scale down dle server capacty n order to save energy. Defnton. The loadbalancngspeedscalng (LBSS) equlbrum s defned as a trple {(λ e ) N, (s e ) N, γ e } that satsfes the varatonal nequaltes (6), (7) and (8). The performance of the system under load balancng and speed scalng s determned by the LBSS equlbrum. At the LBSS equlbrum {(λ e ) N, (s ) e N, γe }, s e = s (λ e ) and (f (s (λ e ), λ e ) γ e )( λ λ e ) 0, λ 0, (9) λ e = λ. (0) N
4 4 The followng result s straghtforward [29]. Theorem 2. The LBSS equlbrum satsfes the local optmalty condton for the followng optmzaton problem: mn λ 0 s.t. f (s (λ ), λ )dλ () λ = λ, (2) and γ e s the correspondng optmal dual varable. Proof. Note that LBSS equlbrum condton (9)(0) s a varatonal nequalty characterzaton of optmalty condton for optmzaton problem ()(2) and ts dual [29]. An optmzaton problem characterzaton of the equlbrum s usually very useful. It captures the global structure of the problem, and often we can easly tell from the optmzaton problem f there exsts an equlbrum, the multplcty of the equlbra, as well as derve dstrbuted or effcent algorthm to the equlbrum. When there s no speed scalng,.e., s s fxed, we recover the optmzaton problem characterzaton of usual load balancng. Under ths stuaton, problem ()(2) s strctly convex as f (s, λ ) s an ncreasng functon of λ, and the equlbrum s unque. In general, there may be no or multple LBSS equlbra, dependng on propertes of performance curve f (s (λ ), λ ) under speed scalng. For example, consder performance metrc (2) wth power functon (3) n a processor sharng system wth gated statc speed scalng (see the next secton). Speed scalng s (λ ) satsfes β (s λ ) 2 = k (α )s α 2. When α < 2, f (s (λ ), λ ) s decreasng. So, problem () (2) becomes a problem of mnmzng a concave objectve functon, whch s usually a hard computng problem and may admt multple solutons. In the above load balancng model, the dspatcher routes the arrvals accordng to tradtonal performance metrc F but does not consder the nternal cost g of the server. We call ths model costoblvous load balancng (e.g., energyoblvous n the case of energyaware speed scalng). It can also be seen as a selfsh routng game where each job chooses a server wth mnmal F value [30]. So, the LBSS equlbrum mght not be socally optmal, n terms of metrc F as well as energyaware metrc M. As we mentoned before, speed scalng brngs addtonal dmenson such as energy nto the desgn objectve. It s of sgnfcant value to study ts nteracton wth the exstng algorthms and protocols, e.g., f t s optmal wth respect to tradtonal performance metrc F as well as a new one M, how to desgn dstrbuted optmal algorthms n terms of new performance metrc, and f we can decouple speed scalng from other resource allocaton mechansms. In order to study these questons for load balancng, we consder two new load balancng models, as follows. Foptmal load balancng: The dspatcher routes arrvals so as to acheve socal optmum n terms of tradtonal performance metrc F : mn λ 0 s.t. λ f (s (λ ), λ ) (3) λ = λ. (4) When F = E[T ], we call t delay optmal load balancng. Costaware optmal load balancng: The dspatcher routes arrvals so as to acheve socal optmum n terms of costaware performance metrc M : mn λ 0 s.t. g (s (λ ), λ ) + β λ f (s (λ ), λ ) (5) λ = λ. (6) We call t energyaware optmal load balancng n the case of energyaware speed scalng. The end users as a whole care about problem (3)(4) and the servers/end users as a whole care about problem (5) (6). We ntend to characterze the LBSS equlbrum wth respect to them, as well as proposng dstrbuted algorthms to acheve the correspondng equlbrum or optma. Agan, the general problems (3)(4) and (5)(6) may be hghly nontrval, dependng on the performance curve f under speed scalng. In the remander of ths paper, we wll focus on load balancng wth energyaware speed scalng n processor sharng systems wth performance metrc (2) wth power functon (3), as a concrete system to study the nteracton between load balancng and speed scalng. We wll leave the general problem to future work. V. LOADBALANCINGSPEEDSCALING INTERACTION IN PROCESSOR SHARING SYSTEMS In ths secton, we consder energyaware speed scalng n processor sharng (PS) systems wth performance metrc (2) and power functon (3). Whle general speed scalng polces can be taken at a server, we focus on gatedstatc speed scalng, n whch the server has a zero speed when there s no job and otherwse runs at a constant speed that balances the response tme and energy usage; see, e.g, [9], [0]. Gatedstatc speed scalng s the smplest nontrval speed scalng. It requres mnmal hardware to support. For example, a CMOS chp may set a constant clock speed but AND t wth the gatng sgnal to set the speed to 0 when there s no job [0]. The gated statc speed scalng captures some essence of dynamc speed scalng whle admts more tractable analyss. As mentoned n Secton IV, when α < 2, the problem under gated statc speed scalng may become hard problem of mnmzng a concave objectve functon. We thus focus on the system wth α 2, n order to obtan a clean characterzaton to gan nsghts. Power functons wth α 2 s also practcally mportant, as n the server wth a power functon wth α 2 energy cost s usually the drvng force n decdng on server speed whle n the server wth a power functon wth
5 5 α < 2 tradtonal performance metrc s the drvng force. Besdes, the results obtaned for gated statc speed scalng wth α 2 are expected to carry over to statc provsonng wth α, n whch the server runs at a constant statc speed that s chosen based on workload to balance the response tme and energy usage. Statc provsonng s the smplest form of speed scalng, and s a model often used n energyaware capacty provsonng n data centers. A. Energyoblvous load balancng Under PS schedulng, the mean response tme at server takes the form: f (s, λ ) = s λ. (7) Under gated statc speed scalng, the energy cost s only ncurred durng the tme when the server s busy. Note that the fracton of the tme when the server s busy s λ /s. So, the server decdes on speed s by solvng the followng optmzaton problem: λ mn β + λ P (s ). (8) s >λ s λ s Thus, the speed scalng s (λ ) satsfes where β = β (s λ ) 2 + sα 2 = 0, (9) β k (α ). By equaton (9), we have s (λ ) = 2s (λ ) α s (λ ) (α 2)λ > 0, (20) s (λ ) = (2α 4)(s (λ ) λ s (λ )) (α s (λ ) (α 2)λ ) 2 0, (2) where the second nequalty follows from the fact that s (λ ), and moreover, s (λ ) = and s (λ ) = 0 f and only f α = 2. Hence, speed scalng s (λ ) s a strctly ncreasng, convex functon of λ. Further, (s (λ )) f (s (λ ), λ ) = = α 2 (22) s (λ ) λ β s also a strctly ncreasng functon of λ. Corollary 3. There exsts a unque LBSS equlbrum for processor sharng systems wth gatedstatc speed scalng. Proof. By Theorem 2, the LBSS equlbrum satsfes the optmalty condtons for optmzaton problem: mn dλ (23) λ s (λ ) λ λ = λ. (24) Snce s (λ ) λ s strctly ncreasng n λ, the above optmzaton problem s strctly convex. The exstence and unqueness of LBSS equlbrum follows from the fact that problem (23) (24) has a unque optmum [29]. Now, let us characterze the equlbrum. For each server, defne the base servce rate s 0 = s (0 + ) = β α. 2 Wthout loss of generalty, we assume that s 0 s 0 2 s 0 N. For later convenence, we also assume that s 0 N + =0. Theorem 4. The set of servers that are used at the equlbrum s N e = {, 2,, n}, wth a unque n that satsfes where n ( f ) ( s 0 ) < λ n = f (λ ) = n ( f ) ( = s 0 n+ ), (25) s (λ ) λ. (26) Proof. By equlbrum condton (9), we have < γ e f s 0 N e and γ e otherwse. Further, s 0 λ e = s e γ e > 0, f λ e = 0, f s 0 s 0 < γ e (27) γ e. (28) Snce s 0 s decreasng n, N e takes the form of {, 2,, n}. Note that s < γ e, and f 0 n s 0 (λ ) s an ncreasng n+ functon. So, n ( f ) ( s 0 ) < n =.e., n ( f ) ( s 0 ) < n = n ( f ) (γ e ) = n λ e = λ = n ( f ) ( = n ( f ) ( = s 0 n+ s 0 n+ The unqueness of n follows from the fact that the LBSS equlbrum s unque. We see that the LBSS equlbrum has a waterfllng structure. If we see load balancng as a selfsh routng problem [30], the arrvals wll aggressvely occupy fast servers wth low delay frst. ) Dstrbuted load balancng algorthm: The (convex) optmzaton problem characterzaton of the LBSS equlbrum also suggests a dstrbuted algorthm to acheve the equlbrum. At kth teraton: Each server estmates the arrval rate λ, and adjusts ts speed s, accordng to s (k) = s (λ (k)). (29) The dspatcher measures delay t (k) = s experenced at each server. Denote by E[t(k)] the (k) λ (k) mnmal t(k) at step k such that t(k) = N(k) ). ), N(k) t (k) wth 2 For a functon f(x) : R R, f(a + ) denotes the rght hand lmt lm x a + f(x).
6 6 N(k) := { λ (k) > 0 or t (k) t(k), N}. 3 The dspatcher adjusts λ to each server, accordng to λ (k + ) = [λ (k) ε(t (k) E[t(k)])] +. (30) where ε s a postve stepsze, and + denotes the projecton onto R +, the set of nonnegatve real numbers. When ε s small enough, the above algorthm converges. Let δ (k) = λ (k + ) λ (k). It s easy to verfy that δ (k) = 0, (3) δ (k)t (k) 0. (32) We see that δ (k)t (k) = 0 only f δ (k) = 0, whch requres t = t, or, λ = 0 and t > t. The above algorthm actually follows the negatve gradent drecton of s (λ ) λ dλ subject to λ = λ [29]. Any algorthms that follow a properlychosen negatve gradent drecton would work, and (30) pcks a specfc gradent drecton that wll facltate the convergence analyss. We skp the convergence proof for brevty. The above dstrbuted algorthm, as well as the other two proposed n Secton V.B.) and Secton V.C.), has low mplementaton complexty. All the nformaton requred n the algorthm can be estmated or measured locally at the dspatcher and ndvdual servers. Such algorthms are hghly desrable n a network settng that may nvolve a large number of servers. B. Delayoptmal load balancng In ths subsecton, we study delay optmal load balancng desgn: λ mn (33) λ 0 s (λ ) λ s.t. λ = λ, (34) and characterze the LBSS equlbrum wth respect to t. By equaton (9), λ = s α 2, (35) s (λ ) λ β whch s strctly ncreasng and convex n s. Note that s (λ ) λ s ncreasng and convex. It follows that s (λ ) λ s a strctly convex functon of λ. 4 So, problem (33)(34) s strctly convex, and has a unque optmum. Denote the optmum by (λ ) N. There exsts a unque γ > 0, such that the optmalty condton can be wrtten as [29] ( s (λ ) λ s (λ ) (s (λ ) γ )( λ λ λ )2 ) 0, λ 0, (36) λ = λ. (37) N 3 t and N can be determned n a recursve way as follows. In the begnnng, let N = N and calculate t = N N t (k), and then exclude from N those servers such that λ = 0 and t > t. Repeat the same procedure wth the new sets N, and when t stops we get E[t]. 4 λ Note that, when α = 2, s not strctly convex but lnear n s (λ ) λ λ. But ths would not change the unqueness of the optmum. Theorem 5. The set of servers that are used at the optmum s N o = {, 2,, }, wth a unque that satsfes where ( ˆf ) ( s 0 ) < λ ( ˆf ) ( = = s 0 + )}, (38) ˆf (λ ) = s (λ ) λ s (λ ) (s (λ ) λ ) 2. (39) Moreover, γ γ e and n. Proof. Note that ˆf (λ ) s an ncreasng functon of λ, and ˆf (0) =. The frst part of the theorem follows the same s 0 proof as n Theorem 4. For the second part of the theorem. Note that s (λ ) by equaton (20). Thus, ˆf (λ ) f (λ ). If γ < γ e, then n and ( ˆf ) (γ ) < ( f ) (γ ) ( f ) (γ e ) λ. = = = Ths contradcts = ( ˆf ) (γ ) = = λ = λ. So, γ γ e, and n follows. Notce that γ e has the nterpretaton as the delay at the energyoblvous load balancng, but γ does not have such an nterpretaton as delay. So, γ γ e does not mply a larger delay at the delay optmal load balancng. In fact, n the delayoptmal load balancng dfferent servers may have dfferent delays and the whole system has the best overall delay performance. ) Dstrbuted load balancng algorthm: The delay optmal load balancng s a convex problem. We can apply smlar dstrbuted algorthm to algorthm (29)(30), to gude the optmal load balancng desgn. At kth teraton: Each server estmates the arrval rate λ, and adjusts ts speed s, accordng to s (k) = s (λ (k)). (40) The dspatcher measures delay t (k) = s experenced at each server, and estmates ˆf, accordng (k) λ (k) to ˆf (k) = ˆf (λ (k)) = α λ (k)(t (k)) 2 + α t (k) 2λ (k)t (k) + α. (4) Denote by E[ ˆf(k)] the mnmal ˆf(k) at step k such that ˆf(k) = N(k) N(k) ˆf (k) wth N(k) := { λ (k) > 0 or ˆf (k) ˆf(k), N}. The dspatcher adjusts λ to each server, accordng to λ (k + ) = [λ (k) ε( ˆf (k) E[ ˆf(k)])] +. (42) where ε s a postve stepsze, and + denotes the projecton onto R +, the set of nonnegatve real numbers. Note that delay optmal load balancng algorthm (40)(42) s more complcated than the smple, energyoblvous load balancng algorthm (29)(30). It requres to estmate ˆf. In
7 7 addton, t requres the dspatcher to know the servers power functon characterstc parameters α and k. 2) Effcency loss n delay at the LBSS equlbrum: Defne the socal cost n delay: C = λ s (λ ) λ, (43) we now characterze the neffcency n delay at the LBSS equlbrum. Lemma 6. Let α = max α. Then, γ e γ α 2 γe. (44) Proof. The frst nequalty has been proved n Theorem 5. It remans to prove the second one. By equaton (35), ˆf can be wrtten as ˆf (λ ) = α 2 s α 2 β s. (45) Note that s (λ ) s ncreasng. Thus, s (λ e ) 2 α by equaton (20). Combnng wth s (λ e ), we get f (λ e ) ˆf (λ e ) α 2 f (λ e ) α 2 f (λ e ). If γ > α 2 γe, then Thus, ( ˆf ) (γ ) ( f ) ( 2 α γ ) > ( f ) (γ e ). ( ˆf ) (γ ) > = n ( f ) (γ e ) = λ. = Ths contradcts the fact that = ( ˆf ) (γ ) = λ (also note that n). So, γ α 2 γe. Theorem 7. Denote the socal cost n delay at the LBSS equlbrum by C e and the optmal cost by C o. Then, C e C o α 2. (46) Proof. The socal cost at the LBSS equlbrum s C e = λγ e. (47) When λ > 0, by equatons (22), (45) and (44), we have s α 2 s (λ ) = = 2γ λ β α s 2γ 2γe α α. (48) So, Thus, C o = λ s (λ ) λ 2γe α λ = 2λγe α. (49) C e C o α 2. (50) We see that the degree of neffcency n delay at the LBSS equlbrum depends only on the order α of the power functons. For example, f α = 2, the LBSS equlbrum acheves the socal optmum. As α s a constant ndependent of the number N of the servers n the system, ths result s very dfferent from the effcency loss of the usual load balancng (wth fxed server speeds), whch scales wth N, see, e.g., [25]. Also, note that α 2 can be seen as a measure of heterogenety n power functons. We can thus say that the degree of neffcency at the LBSS equlbrum s bounded by the heterogenety of the system. As the power functon can usually be well approxmated as a loworder polynomal functon, the above result suggests bengn nteracton between energyoblvous load balancng and poweraware speed scalng, n terms of delay. As energyoblvous load balancng s already employed n practce and smple to mplement, we may need not change t as t does not ncur a large penalty n delay. C. Energyaware optmal load balancng In ths subsecton, we study energyaware optmal load balancng desgn: mn λ,s s.t. λ β + λ P (s ) s λ (5) s λ = λ, (52) and characterze the LBSS equlbrum wth respect to t. By speed scalng (.e., solvng for s frst), the above problem reduces to: mn h (λ ) (53) λ s.t. λ = λ, (54) where Note that λ h (λ ) = β + λ P (s (λ )). (55) s (λ ) λ s (λ ) h β s (λ ) (λ ) = (s (λ ) λ ) 2 + k (s (λ )) α = α β s (λ ) α (s (λ ) λ ) 2, (56) h (λ ) = α β 2s (λ ) (s (λ ) + λ )s (λ ) α (s (λ ) λ ) 3. (57) We see that h > 0 and h > 0, and thus h (λ ) s strctly ncreasng and convex. So, problem (53)(54) s a strctly convex problem, and has a unque optmum. Denote the optmum by (λ + ) N. There exsts a unque γ + > 0, such that the optmalty condton can be wrtten as [29] (h (λ + ) γ+ )( λ λ + ) 0, λ 0, (58) λ + = λ. (59) N
8 8 Note that h (λ ) s strctly ncreasng, and h (λ ) α β α ˆf (λ ) α β α f (λ ). (60) Let d 0 = h (0) = α α β s 0. We can defne a permutaton π : {, 2,, N } {, 2,, N }, such that d 0 s n decreasng order under π. We have the followng characterzaton of the optmum. Theorem 8. The set of servers that are used at the optmum s N s = {π (), π (2),, π (m + )}, wth a unque m + that satsfes (h π () ) ( m + = d 0 π (m + ) ) < λ (h π () ) ( m + = d 0 π (m + +) Proof. It follows the same proof as n Theorem 4. We skp t for brevty. We see that the energyaware optmal load balancng has a smlar waterfllng effect, and the arrvals wll occupy servers wth low margnal cost n energyaware metrc frst. As a result, the jobs wll be consoldated nto a subset of servers that have low energyaware cost. ) Dstrbuted load balancng algorthm: The energyaware optmal load balancng s a convex problem. Agan, we can apply smlar dstrbuted algorthm to algorthm (29)(30), to gude the optmal load balancng desgn. At kth teraton: Each server estmates the arrval rate λ, and adjusts ts speed s, accordng to s (k) = s (λ (k)). (6) The dspatcher measures delay t (k) = s experenced at each server, and estmates h, accordng (k) λ (k) to h (k) = h (λ (k)) = α β α (λ (k)(t (k)) 2 + t (k)). (62) Denote by E[h (k)] the mnmal h (k) at step k such that h (k) = N(k) N(k) h (k) wth N(k) := { λ (k) > 0 or h (k) h (k), N}. The dspatcher adjusts λ to each server, accordng to λ (k + ) = [λ (k) ε(h (k) E[h (k)])] +. (63) where ε s a postve stepsze, and + denotes the projecton onto R +, the set of nonnegatve real numbers. Agan, energy aware optmal load balancng algorthm (6) (63) s more complcated than energyoblvous load balancng algorthm (29)(30). In addton to the servers power functon characterstc parameters, the dspatcher requres to know ther weghts β. ). 2) Effcency loss n energyaware performance metrc at the LBSS equlbrum: Defne the socal cost n energyaware performance metrc M : D = λ β + λ P (s ) s λ s = h (λ ). (64) We now characterze the neffcency n energyaware performance metrc at the LBSS equlbrum. It s complcated to characterze the effcency loss for the system wth arbtrary power functons and loads. Here we gve a partal characterzaton, focusng on the case wth power functons of the same order,.e., P (s ) = k s α for all servers, and n heavy traffc,.e., λ. We leave a complete characterzaton of the effcency loss to future work. The case wth power functons of the same order models a system that employs smlar servers but wth dfferent scalng factors and weghts. Heavy traffc regme s of sgnfcant nterest, as the neffcency of loadbalancngspeedscalng nteracton s ntutvely worst under heavy traffc. Theorem 9. Assume that α = 2. Denote the energyaware socal cost at the LBSS equlbrum by D e and the optmal cost by D o. Under the aforementoned condtons, we have D e D o max k mn k N. (65) Proof. When α = 2, at the LBSS equlbrum (λ e ) N, the arrvals wll be routed to the server that has the maxmal β value. 5 Under heavy traffc, the energyaware socal cost at the LBSS equlbrum s D e k λ 2 max k λ 2. At the socal optmum (λ + ) N, λ + j /k λ. The optmal /kj socal cost s Thus, D 0 k (λ + )2 λ 2 /k mn k λ 2. N D e D o max k mn k N. (66) We see that when α = 2, the degree of neffcency at the LBSS equlbrum scales wth the number of servers n the system. Ths happens because the energyoblvous load balancng uses only the server wth the largest base rate, whch ncurs a huge energy cost at ths server, whle the energyaware optmal load balancng wll spread load across all servers, whch leads to much smaller energy cost at the servers. Ths suggests that we should do energyaware load balancng f the energy consumpton s a man concern. 5 There may exst multple servers that have the maxmal β value. But t s reasonable to expect that the number of such servers s bounded by a constant that does not scale wth the total number of the servers n the system. For smplcty of presentaton, we assume that there s only one server that has the maxmal β value. Ths only brngs n a constant factor to the bound on effcency loss, f there are multple such servers.
9 9 α α 2 Lemma 0. Assume α > 2. Defne ζ = αk β server,. Then, mn ζ (γ e ) 2α 2 α 2 for each γ + max ζ (γ e ) 2α 2 α 2. (67) Proof. By equaton (56), h can be wrtten as h (λ ) = αk (s (λ )) α = ζ ( f (λ )) 2α 2 α 2. (68) If γ + < mn ζ (γ e ) 2α 2 α 2, then (h ) (γ + ) < ( f ) (γ e ). Thus, D e D o k β α α 2 max γ ( + mn j ζ j ) α α k ( γ+ k α ) α α γ ( + mn j ζ j ) α α k β α α 2 ζ k ( γ+ k α ) α α = max( ) α mn j ζ j α ( max ζ mn j ζ j ) α α. (73) Thus, (h ) (γ + ) < ( f ) (γ e ) = λ. Ths contradcts the fact that (h ) (γ + ) = λ. So, γ + mn ζ (γ e ) 2α 2 α 2. The second nequalty can be proved smlarly. Theorem. Assume α > 2. Denote the energyaware socal cost at the LBSS equlbrum by D e and the optmal cost by D o. Under the aforementoned condtons, we have D e D o (max ζ mn ζ ) α α. (69) Proof. Under heavy traffc, λ. By Lemma n [9], we have the followng approxmaton for speed scalng under heavy traffc: β s (λ ) λ + λ α 2 λ. Thus, β λ + λ P (s ) λ β + k λ α k λ α. s λ β s Note that, at the LBSS equlbrum (λ e ) N, γ e (s e = )α 2 (λ e )α 2. β β The energyaware socal cost at the LBSS equlbrum s D e k ( β γ e2 ) α α 2 α 2 k β α α 2 ( where the nequalty follows from (67). At the socal optmum (λ + ) N, γ + ) α α, (70) mn j ζ j γ + = αk s α αk (λ + )α. (7) The optmal socal cost s D o k ( γ+ k α ) α α. (72) We see that when α > 2, the degree of neffcency at the LBSS equlbrum depends only on the degree of heterogenety max ζ mn j ζ j n the system but not the number of servers N. If the degree of heterogenety n the system s small, energyoblvous load balancng nteracts bengnly wth speed scalng, n terms of the energyaware cost. In ths stuaton, we may do not need complcated energyaware load balancng,.e., we can decouple the desgn of load balancng from speed scalng. Otherwse, we must do energyaware optmal load balancng f energy consumpton s a man concern. VI. NUMERICAL EXAMPLES In ths secton, we provde numercal examples to complement the analyss n prevous sectons. We frst show the convergence of the three dstrbuted algorthms proposed n secton V, and then verfy the bounds on effcency loss descrbed n Theorem 7, Theorem 9, and Theorem. We consder a system wth 0 servers wth speed scalng. Half of the servers have a power functon of the form P (s ) = k s 5 2 and the other half have a power functon of the form P (s ) = k s 3. The total load s chosen to be λ = 800 correspondng to a heavy traffc scenaro, and the values for parameter k and β used to obtan numercal results are randomly drawn from [2, 8] and [, 5], respectvely. The key consderaton n choosng the parameter values s to ncorporate enough heterogenety n the system. A. Dstrbuted algorthms Fgures 2, 3 and 4 show the evoluton of the arrval rate and servce rate wth stepsze ε = 0.2 for the energyoblvous load balancng, the delayoptmal load balancng and the energyaware optmal load balancng, respectvely. We see that the arrval rates and servce rates approach the correspondng equlbrum or optmum quckly. The numercal results confrm prevous analyss and ntutons. As we go from the energyoblvous load balancng to the delayoptmal load balancng, the load s spread more across the servers, whch s drven by mnmzng the socal cost n delay. We also see that the changes n the arrval rate and servce rate are not severe, whch ntutvely confrms Theorem 7 that gves a small bound on effcency loss at the LBSS equlbrum. As we move to energyaware optmal load balancng, the load becomes more
10 0 evenly dstrbuted. Ths s drven by mnmzng the energyaware socal cost, and an uneven load dstrbuton wll lead to uneven servce rate dstrbuton, whch may result n large cost n energy at the server(s) wth large speed. We also see large changes n the arrval rate and servce rate. Ths mples a large degree of neffcency at the LBSS equlbrum, whch ntutvely confrms Theorem even though t s a characterzaton for the system wth power functons of the same order. In order to study the mpact of dfferent choces of the stepsze on the convergence of the algorthms, we have run smulatons wth dfferent stepszes. We found that the smaller the stepsze, the slower the convergence, and the larger the stepsze, the faster the convergence but the system may only approach to wthn a certan neghborhood of the equlbrum, whch s a general characterstc of any gradent based method. In practce, the dspatcher can frst choose large stepszes to ensure fast convergence, and subsequently reduce the stepszes once the prce starts oscllatng around some mean value. Arrval rate λ server server 2 server 3 server 4 server 5 server 6 server 7 server 8 server 9 server Number of Iteratons Servce rate s server server 2 server 3 server 4 server 5 server 6 server 7 server 8 server 9 server Number of Iteratons Fg. 2. The arrval rate and servce rate evoluton of energyoblvous load balancng. Arrval rate λ server server 2 server 3 server 4 server 5 server 6 server 7 server 8 server 9 server Number of Iteratons Servce rate s server server 2 server 3 server 4 server 5 server 6 server 7 server 8 server 9 server Number of Iteratons Fg. 4. The arrval rate and servce rate evoluton of energyaware optmal load balancng. C e /C o Number of servers Fg. 5. Rato of C e /C o. D e /D o Number of servers Fg. 6. Rato of D e /D o wth homogeneous α = 2. Arrval rate λ server server 2 server 3 server 4 server 5 server 6 server 7 server 8 server 9 server Number of Iteratons Servce rate s 400 server server 2 server server 4 server server 6 server 7 server 8 00 server 9 server Number of Iteratons Fg. 3. The arrval rate and servce rate evoluton of delay optmal load balancng. D e /D o Number of servers Fg. 7. Rato of D e /D o wth homogeneous α = 3. B. Comparson between the three load balancng algorthms Frstly, as shown n Fgure 2 and 3, we observe that energyoblvous load balancng and delay optmal load balancng generate smlar patterns of load schedulng and servce rate. Ths leads to the smlar socal cost n delay, whch s further confrmed n Fgure 5. In Fgure 5, we smulate dfferent szes of systems wth the number of servers beng n = 0, 4,..., 50. Fgure 5 plots the rato of C e /C o where C o s the socal cost n delay of delayoptmal load balancng and C e s the socal cost n delay of the LBSS equlbrum. We see that the effcency loss of LBSS equlbrum s very small wth respect to socal cost n delay. Ths smulaton results are consstent wth the analyss n Theorem 7. As shown n Fgure 2 and 4, as we go from energyoblvous load balancng to energyaware optmal load balancng, the load s spread more across the servers. Ths s drven by mnmzng the energyaware socal cost, and an uneven load dstrbuton wll lead to uneven servce rate dstrbuton, whch may result n large cost n energy at the server(s) wth large speed. To further verfy Theorem 9 and, we smulate dfferent szes of systems wth the number of servers beng
11 n = 0, 4,..., 50 usng homogeneous α. Fgure 6 plots the rato of D e /D o wth α = 2 and Fgure 7 plots the rato of D e /D o wth α = 3. Here D o s the socal cost n energyaware performance metrc M of delayoptmal load balancng and D e s the socal cost n n energyaware performance metrc M of the LBSS equlbrum. Frstly, we observe that the rato D e /D o s much smaller than the worst case bound provded n Theorem 9 and Theorem. 6 Secondly, we see when α = 2, the rato ncreases as the network sze ncreases; n contrast, when α = 3, the rato s ndependent of the network sze. Ths s consstent wth the theoretcal bound n Theorem 9 and Theorem. VII. CONCLUSION We have studed the nteracton between load balancng and speed scalng. We characterze the equlbrum resultng from the load balancng and speed scalng nteracton, and ntroduce two optmal load balancng desgns, n terms of tradtonal performance metrc and costaware (n partcular, energyaware) performance metrc respectvely. We study n detal the loadbalancngspeedscalng equlbrum and the optmal load balancng desgns n processor sharng systems wth gatedstatc speed scalng, and propose dstrbuted load balancng algorthms to acheve the correspondng equlbrum and optma. Especally, we characterze the degree of neffcency at the loadbalancngspeedscalng equlbrum n terms of delay as well as energyaware metrc, and show that the degree of neffcency s mostly bounded by the heterogenety of the system, but ndependent of the number of the servers. These results provde nsghts n understandng the nteracton of load balancng wth speed scalng and gudng new desgns. Further research stemmng out of ths paper ncludes the followng. We are characterzng the effcency loss n energyaware metrc at the loadbalancngspeedscalng equlbrum for the system wth power functons of dfferent polynomal orders. We are also studyng the load balancng and speed scalng nteracton n the processor sharng system wth general power functons (e.g., nonconvex, dscontnuous, wth possbly a dscrete set of allowable speeds), as well as n the system wth other schedulng polces such as Shortest Remanng Processng Tme (SRPT). We wll further study other speed scalng polces and ther mpact on the desgn and performance of load balancng. Fnally, we wll go beyond energyaware speed scalng, and study other types of speed scalng behavors and ther nteracton wth load balancng n, e.g., date centers or call centers. [3] S. Iran and K. R. Pruhs. Algorthmc problems n power management. SIGACT News, 36(2):63 76, [4] L. Yuan and G. Qu. Analyss of energy reducton on dynamc voltage scalngenabled systems. IEEE Trans. Comput.Aded Des. Integr. Crcuts Syst., 24(2): , [5] Y. Zhu and F. Mueller. Feedback edf schedulng of realtme tasks explotng dynamc voltage scalng. Real Tme Systems, 3:33 63, [6] N. Bansal, T. Kmbrel, and K. Pruhs. Speed scalng to manage energy and temperature. J. ACM, 54(): 39, [7] S. Herbert and D. Marculescu. Analyss of dynamc voltage/frequency scalng n chpmultprocessors. In Proc. ISLPED, [8] N. Bansal, H.L. Chan, and K. Pruhs. Speed scalng wth an arbtrary power functon. In Proc. ACMSIAM SODA, [9] A. Werman, L. L. H. Andrew, and A. Tang. Poweraware speed scalng n processor sharng systems. In Proceedngs of IEEE Infocom, [0] L. L. Andrew, M. Ln, and A. Werman. Optmalty, farness, and robustness n speed scalng desgns. In Proceedngs of ACM Sgmetrcs, 200. [] R. Stanojevc and R. Shorten. Dstrbuted dynamc speed scalng. In INFOCOM, 200 Proceedngs IEEE, pages 5, March 200. [2] Kyuho Son and B. Krshnamachar. Speedbalance: Speedscalngaware optmal load balancng for green cellular networks. In INFOCOM, 202 Proceedngs IEEE, pages , March 202. [3] Maryam Elah, Carey Wllamson, and Phlpp Woelfel. Decoupled speed scalng: Analyss and evaluaton. Performance Evaluaton, 73(0):3 7, 204. [4] N. Bansal, K. Pruhs, and C. Sten. Speed scalng for weghted flow tmes. In Proc. ACMSIAM SODA, [5] F. Yao, A. Demers, and S. Shenker. A schedulng model for reduced cpu energy. In Proceedngs of IEEE Symposum on Foundatons of Computer Scence (FOCS), 995. [6] J. M. George and J. M. Harrson. Dynamc control of a queue wth adjustable servce rate. Operatons Research, 49(5):720 73, 200. [7] K. Pruhs, P. Uthasombut, and G. Woegnger. Gettng the best response for your erg. In Scandnavan Worksh. Alg. Theory, [8] J. R. Bradley. Optmal control of a dual servce rate m/m/ productonnventory model. European Journal of Operatons Research, 6(3):82 837, [9] S. Albers and H. Fujwara. Energyeffcent algorthms for flow tme mnmzaton. Lecture Notes n Computer Scence, 3884:62 633, [20] D. P. Bunde. Poweraware schedulng for makespan and flow. In Proc. ACM Symp. Parallel Alg. and Arch, [2] S. Zhang and K. S. Catha. Approxmaton algorthm for the temperatureaware schedulng problem. In Proceedngs of IEEE Conference on Computer Aded Desgn, [22] N. Bansal, H.L. Chan, T.W. Lam, and L.K. Lee. Schedulng for speed bounded processors. In Int. Colloq. Automata, Languages and Programmng, [23] N. Bansal, H.L. Chan, K. Pruhs, and D. Katz. Improved bounds for speed scalng n devces obeyng the cuberoot rule. In Int. Colloq. Automata, Languages and Programmng, [24] T.W. Lam, L.K. Lee, I. K. K. To, and P. W. H. Wong. Speed scalng functons for flow tme schedulng based on actve job count. In Proc. Euro. Symp. Alg., [25] M. Havv and T. Roughgarden. The prce of anarchy n an exponental multserver. Operatons Research Letters, 35:42 426, [26] T. Wu and D. Starobnsk. On the prce of anarchy n unbounded delay networks. In Proc. of Game Theory for Comm. and Networks, [27] E. Altman, U. Ayesta, and B. J. Prabhu. Optmal load balancng n processor sharng systems. In Proceedngs of GameComm, [28] H. Chen, J. Marden, and A. Werman. On the mpact of heterogenety and backend schedulng n load balancng desgns. In Proceedngs of IEEE Infocom, [29] D. P. Bertsekas and J. N. Tstskls. Parallel and Dstrbuted Computaton. Prentce Hall, 989. [30] N. Nssan, T. Roughgarden, E. Tardos, and V. V. Vazran. Algorthmc game theory. Cambrdge Unversty Press, REFERENCES [] O. S. Unsal and I. Koren. Systemlevel poweraware desgn technques n realtme systems. Proc. IEEE, 97(3): , [2] S. Kaxras and M. Martonos. Computer Archtecture Technques for PowerEffcency. Morgan and Claypool, The worst case bound provded n Theorem 9 s 4n and the worst case bound provded n Theorem s around 353.
12 2 PLACE PHOTO HERE theory and ts engneerng applcaton. Ljun Chen (M 05) s an Assstant Professor of Computer Scence and Telecommuncatons at Unversty of Colorado at Boulder. He receved a Ph.D. n Control and Dynamcal Systems from Calforna Insttute of Technology n He was a corecpent of the Best Paper Award at the IEEE Internatonal Conference on Moble Adhoc and Sensor Systems (MASS) n Hs current research nterests nclude optmzaton and control of networked systems, dstrbuted optmzaton and control, convex relaxaton and parsmonous solutons, and game PLACE PHOTO HERE Na L (M 09) s an assstant professor n the School of Engneerng and Appled Scences n Harvard Unversty. She receved her B.S. degree n mathematcs and appled mathematcs from Zhejang Unversty n Chna and PhD degree n Control and Dynamcal systems from Calforna Insttute of Technology n 203. She was a postdoctoral assocate of the Laboratory for Informaton and Decson Systems at Massachusetts Insttute of Technology. She entered the Best Student Paper Award?nalst n the 20 IEEE Conference on Decson and Control. Her research les n the desgn, analyss, optmzaton and control of dstrbuted network systems, wth partcular applcatons to power networks and systems bology/physology.
On the Interaction between Load Balancing and Speed Scaling
On the Interacton between Load Balancng and Speed Scalng Ljun Chen, Na L and Steven H. Low Engneerng & Appled Scence Dvson, Calforna Insttute of Technology, USA Abstract Speed scalng has been wdely adopted
More informationCommunication Networks II Contents
8 / 1  Communcaton Networs II (Görg)  www.comnets.unbremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP
More information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationPowerofTwo Policies for Single Warehouse MultiRetailer Inventory Systems with Order Frequency Discounts
Powerofwo Polces for Sngle Warehouse MultRetaler Inventory Systems wth Order Frequency Dscounts José A. Ventura Pennsylvana State Unversty (USA) Yale. Herer echnon Israel Insttute of echnology (Israel)
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING
ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 6105194390,
More informationAn Analysis of Central Processor Scheduling in Multiprogrammed Computer Systems
STANCS73355 I SUSE73013 An Analyss of Central Processor Schedulng n Multprogrammed Computer Systems (Dgest Edton) by Thomas G. Prce October 1972 Techncal Report No. 57 Reproducton n whole or n part
More informationEnabling P2P Oneview Multiparty Video Conferencing
Enablng P2P Onevew Multparty Vdeo Conferencng Yongxang Zhao, Yong Lu, Changja Chen, and JanYn Zhang Abstract MultParty Vdeo Conferencng (MPVC) facltates realtme group nteracton between users. Whle P2P
More informationData Broadcast on a MultiSystem Heterogeneous Overlayed Wireless Network *
JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 24, 819840 (2008) Data Broadcast on a MultSystem Heterogeneous Overlayed Wreless Network * Department of Computer Scence Natonal Chao Tung Unversty Hsnchu,
More informationA hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm
Avalable onlne www.ocpr.com Journal of Chemcal and Pharmaceutcal Research, 2014, 6(7):18841889 Research Artcle ISSN : 09757384 CODEN(USA) : JCPRC5 A hybrd global optmzaton algorthm based on parallel
More informationAn InterestOriented Network Evolution Mechanism for Online Communities
An InterestOrented Network Evoluton Mechansm for Onlne Communtes Cahong Sun and Xaopng Yang School of Informaton, Renmn Unversty of Chna, Bejng 100872, P.R. Chna {chsun,yang}@ruc.edu.cn Abstract. Onlne
More informationOptimal Scheduling in the HybridCloud
Optmal Schedulng n the HybrdCloud Mark Shfrn Faculty of Electrcal Engneerng Technon, Israel Emal: shfrn@tx.technon.ac.l Ram Atar Faculty of Electrcal Engneerng Technon, Israel Emal: atar@ee.technon.ac.l
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 /  Communcaton Networks II (Görg) SS20  www.comnets.unbremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationWhen Network Effect Meets Congestion Effect: Leveraging Social Services for Wireless Services
When Network Effect Meets Congeston Effect: Leveragng Socal Servces for Wreless Servces aowen Gong School of Electrcal, Computer and Energy Engeerng Arzona State Unversty Tempe, AZ 8587, USA xgong9@asuedu
More informationOn the Optimal Control of a Cascade of HydroElectric Power Stations
On the Optmal Control of a Cascade of HydroElectrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;
More informationForecasting the Demand of Emergency Supplies: Based on the CBR Theory and BP Neural Network
700 Proceedngs of the 8th Internatonal Conference on Innovaton & Management Forecastng the Demand of Emergency Supples: Based on the CBR Theory and BP Neural Network Fu Deqang, Lu Yun, L Changbng School
More informationA Computer Technique for Solving LP Problems with Bounded Variables
Dhaka Unv. J. Sc. 60(2): 163168, 2012 (July) A Computer Technque for Solvng LP Problems wth Bounded Varables S. M. Atqur Rahman Chowdhury * and Sanwar Uddn Ahmad Department of Mathematcs; Unversty of
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More informationOn File Delay Minimization for Content Uploading to Media Cloud via Collaborative Wireless Network
On Fle Delay Mnmzaton for Content Uploadng to Meda Cloud va Collaboratve Wreless Network Ge Zhang and Yonggang Wen School of Computer Engneerng Nanyang Technologcal Unversty Sngapore Emal: {zh0001ge, ygwen}@ntu.edu.sg
More informationA Lyapunov Optimization Approach to Repeated Stochastic Games
PROC. ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING, OCT. 2013 1 A Lyapunov Optmzaton Approach to Repeated Stochastc Games Mchael J. Neely Unversty of Southern Calforna http://wwwbcf.usc.edu/
More informationThe literature on manyserver approximations provides significant simplifications toward the optimal capacity
Publshed onlne ahead of prnt November 13, 2009 Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste,
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMISP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationPrice Competition in an Oligopoly Market with Multiple IaaS Cloud Providers
Prce Competton n an Olgopoly Market wth Multple IaaS Cloud Provders Yuan Feng, Baochun L, Bo L Department of Computng, Hong Kong Polytechnc Unversty Department of Electrcal and Computer Engneerng, Unversty
More informationThe Development of Web Log Mining Based on ImproveKMeans Clustering Analysis
The Development of Web Log Mnng Based on ImproveKMeans Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationThe Power of Slightly More than One Sample in Randomized Load Balancing
The Power of Slghtly More than One Sample n Randomzed oad Balancng e Yng, R. Srkant and Xaohan Kang Abstract In many computng and networkng applcatons, arrvng tasks have to be routed to one of many servers,
More informationCrossSelling in a Call Center with a Heterogeneous Customer Population
OPERATIONS RESEARCH Vol. 57, No. 2, March Aprl 29, pp. 299 313 ssn 3364X essn 15265463 9 572 299 nforms do 1.1287/opre.18.568 29 INFORMS CrossSellng n a Call Center wth a Heterogeneous Customer Populaton
More informationMultiResource Fair Allocation in Heterogeneous Cloud Computing Systems
1 MultResource Far Allocaton n Heterogeneous Cloud Computng Systems We Wang, Student Member, IEEE, Ben Lang, Senor Member, IEEE, Baochun L, Senor Member, IEEE Abstract We study the multresource allocaton
More informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy Scurve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy Scurve Regresson ChengWu Chen, Morrs H. L. Wang and TngYa Hseh Department of Cvl Engneerng, Natonal Central Unversty,
More informationDynamic Resource Allocation and Power Management in Virtualized Data Centers
Dynamc Resource Allocaton and Power Management n Vrtualzed Data Centers Rahul Urgaonkar, Ulas C. Kozat, Ken Igarash, Mchael J. Neely urgaonka@usc.edu, {kozat, garash}@docomolabsusa.com, mjneely@usc.edu
More informationHow Bad are Selfish Investments in Network Security?
1 How Bad are Selfsh Investments n Networ Securty? Lbn Jang, Venat Anantharam and Jean Walrand EECS Department, Unversty of Calforna, Bereley {ljang,ananth,wlr}@eecs.bereley.edu Abstract Internet securty
More information2008/8. An integrated model for warehouse and inventory planning. Géraldine Strack and Yves Pochet
2008/8 An ntegrated model for warehouse and nventory plannng Géraldne Strack and Yves Pochet CORE Voe du Roman Pays 34 B1348 LouvanlaNeuve, Belgum. Tel (32 10) 47 43 04 Fax (32 10) 47 43 01 Emal: corestatlbrary@uclouvan.be
More information9.1 The Cumulative Sum Control Chart
Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s
More informationRevenue Management for a Multiclass SingleServer Queue via a Fluid Model Analysis
OPERATIONS RESEARCH Vol. 54, No. 5, September October 6, pp. 94 93 ssn 3364X essn 565463 6 545 94 nforms do.87/opre.6.35 6 INFORMS Revenue Management for a Multclass SngleServer Queue va a Flud Model
More informationAnalysis of EnergyConserving Access Protocols for Wireless Identification Networks
From the Proceedngs of Internatonal Conference on Telecommuncaton Systems (ITC97), March 223, 1997. 1 Analyss of EnergyConservng Access Protocols for Wreless Identfcaton etworks Imrch Chlamtac a, Chara
More informationPAS: A Packet Accounting System to Limit the Effects of DoS & DDoS. Debish Fesehaye & Klara Naherstedt University of IllinoisUrbana Champaign
PAS: A Packet Accountng System to Lmt the Effects of DoS & DDoS Debsh Fesehaye & Klara Naherstedt Unversty of IllnosUrbana Champagn DoS and DDoS DDoS attacks are ncreasng threats to our dgtal world. Exstng
More informationCrossSelling in a Call Center with a Heterogeneous Customer Population
OPERATIONS RESEARCH Vol. 57, No. 2, March Aprl 2009, pp. 299 313 ssn 0030364X essn 15265463 09 5702 0299 nforms do 10.1287/opre.1080.0568 2009 INFORMS CrossSellng n a Call Center wth a Heterogeneous
More informationPerformance Analysis of Energy Consumption of Smartphone Running Mobile Hotspot Application
Internatonal Journal of mart Grd and lean Energy Performance Analyss of Energy onsumpton of martphone Runnng Moble Hotspot Applcaton Yun on hung a chool of Electronc Engneerng, oongsl Unversty, 511 angdodong,
More informationJ. Parallel Distrib. Comput.
J. Parallel Dstrb. Comput. 71 (2011) 62 76 Contents lsts avalable at ScenceDrect J. Parallel Dstrb. Comput. journal homepage: www.elsever.com/locate/jpdc Optmzng server placement n dstrbuted systems n
More informationFeasibility of Using Discriminate Pricing Schemes for Energy Trading in Smart Grid
Feasblty of Usng Dscrmnate Prcng Schemes for Energy Tradng n Smart Grd Wayes Tushar, Chau Yuen, Bo Cha, Davd B. Smth, and H. Vncent Poor Sngapore Unversty of Technology and Desgn, Sngapore 138682. Emal:
More informationJoint Scheduling of Processing and Shuffle Phases in MapReduce Systems
Jont Schedulng of Processng and Shuffle Phases n MapReduce Systems Fangfe Chen, Mural Kodalam, T. V. Lakshman Department of Computer Scence and Engneerng, The Penn State Unversty Bell Laboratores, AlcatelLucent
More informationOptimal resource capacity management for stochastic networks
Submtted for publcaton. Optmal resource capacty management for stochastc networks A.B. Deker H. Mlton Stewart School of ISyE, Georga Insttute of Technology, Atlanta, GA 30332, ton.deker@sye.gatech.edu
More informationDynamic Pricing for Smart Grid with Reinforcement Learning
Dynamc Prcng for Smart Grd wth Renforcement Learnng ByungGook Km, Yu Zhang, Mhaela van der Schaar, and JangWon Lee Samsung Electroncs, Suwon, Korea Department of Electrcal Engneerng, UCLA, Los Angeles,
More informationMulticlass MultiServer Thresholdbased Systems: a. Study of Noninstantaneous Server Activation
Multclass MultServer Thresholdbased Systems: a Study of Nonnstantaneous Server Actvaton 1 ChengFu Chou, Leana Golubchk, and John C. S. Lu Abstract In ths paper, we consder performance evaluaton of
More informationDistributed Optimal Contention Window Control for Elastic Traffic in Wireless LANs
Dstrbuted Optmal Contenton Wndow Control for Elastc Traffc n Wreless LANs Yalng Yang, Jun Wang and Robn Kravets Unversty of Illnos at UrbanaChampagn { yyang8, junwang3, rhk@cs.uuc.edu} Abstract Ths paper
More informationBERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationAn MILP model for planning of batch plants operating in a campaignmode
An MILP model for plannng of batch plants operatng n a campagnmode Yanna Fumero Insttuto de Desarrollo y Dseño CONICET UTN yfumero@santafeconcet.gov.ar Gabrela Corsano Insttuto de Desarrollo y Dseño
More information2. SYSTEM MODEL. the SLA (unlike the only other related mechanism [15] we can compare it is never able to meet the SLA).
Managng Server Energy and Operatonal Costs n Hostng Centers Yyu Chen Dept. of IE Penn State Unversty Unversty Park, PA 16802 yzc107@psu.edu Anand Svasubramanam Dept. of CSE Penn State Unversty Unversty
More informationINVESTIGATION OF VEHICULAR USERS FAIRNESS IN CDMAHDR NETWORKS
21 22 September 2007, BULGARIA 119 Proceedngs of the Internatonal Conference on Informaton Technologes (InfoTech2007) 21 st 22 nd September 2007, Bulgara vol. 2 INVESTIGATION OF VEHICULAR USERS FAIRNESS
More informationMultiplePeriod Attribution: Residuals and Compounding
MultplePerod Attrbuton: Resduals and Compoundng Our revewer gave these authors full marks for dealng wth an ssue that performance measurers and vendors often regard as propretary nformaton. In 1994, Dens
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More informationEnergy Efficient Routing in Ad Hoc Disaster Recovery Networks
Energy Effcent Routng n Ad Hoc Dsaster Recovery Networks Gl Zussman and Adran Segall Department of Electrcal Engneerng Technon Israel Insttute of Technology Hafa 32000, Israel {glz@tx, segall@ee}.technon.ac.l
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationDominant Resource Fairness in Cloud Computing Systems with Heterogeneous Servers
1 Domnant Resource Farness n Cloud Computng Systems wth Heterogeneous Servers We Wang, Baochun L, Ben Lang Department of Electrcal and Computer Engneerng Unversty of Toronto arxv:138.83v1 [cs.dc] 1 Aug
More informationResearch Article Enhanced TwoStep Method via Relaxed Order of αsatisfactory Degrees for Fuzzy Multiobjective Optimization
Hndaw Publshng Corporaton Mathematcal Problems n Engneerng Artcle ID 867836 pages http://dxdoorg/055/204/867836 Research Artcle Enhanced TwoStep Method va Relaxed Order of αsatsfactory Degrees for Fuzzy
More informationFeature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College
Feature selecton for ntruson detecton Slobodan Petrovć NISlab, Gjøvk Unversty College Contents The feature selecton problem Intruson detecton Traffc features relevant for IDS The CFS measure The mrmr measure
More informationA Secure PasswordAuthenticated Key Agreement Using Smart Cards
A Secure PasswordAuthentcated Key Agreement Usng Smart Cards Ka Chan 1, WenChung Kuo 2 and JnChou Cheng 3 1 Department of Computer and Informaton Scence, R.O.C. Mltary Academy, Kaohsung 83059, Tawan,
More informationLoop Parallelization
  Loop Parallelzaton C52 Complaton steps: nested loops operatng on arrays, sequentell executon of teraton space DECLARE B[..,..+] FOR I :=.. FOR J :=.. I B[I,J] := B[I,J]+B[I,J] ED FOR ED FOR analyze
More informationValue Driven Load Balancing
Value Drven Load Balancng Sherwn Doroud a, Esa Hyytä b,1, Mor HarcholBalter c,2 a Tepper School of Busness, Carnege Mellon Unversty, 5000 Forbes Ave., Pttsburgh, PA 15213 b Department of Communcatons
More informationCan Auto Liability Insurance Purchases Signal Risk Attitude?
Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? ChuShu L Department of Internatonal Busness, Asa Unversty, Tawan ShengChang
More informationDemand Response of Data Centers: A Realtime Pricing Game between Utilities in Smart Grid
Demand Response of Data Centers: A Realtme Prcng Game between Utltes n Smart Grd Nguyen H. Tran, Shaole Ren, Zhu Han, Sung Man Jang, Seung Il Moon and Choong Seon Hong Department of Computer Engneerng,
More informationJoint Resource Allocation and BaseStation. Assignment for the Downlink in CDMA Networks
Jont Resource Allocaton and BaseStaton 1 Assgnment for the Downlnk n CDMA Networks Jang Won Lee, Rav R. Mazumdar, and Ness B. Shroff School of Electrcal and Computer Engneerng Purdue Unversty West Lafayette,
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationNONCONSTANT SUM REDANDBLACK GAMES WITH BETDEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia
To appear n Journal o Appled Probablty June 2007 OCOSTAT SUM REDADBLACK GAMES WITH BETDEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate
More informationIMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUAREROOT INTERVAL METHOD FOR MULTIPLE ZEROS 1
Nov Sad J. Math. Vol. 36, No. 2, 2006, 009 IMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUAREROOT INTERVAL METHOD FOR MULTIPLE ZEROS Modrag S. Petkovć 2, Dušan M. Mloševć 3 Abstract. A new theorem concerned
More informationActivity Scheduling for CostTime Investment Optimization in Project Management
PROJECT MANAGEMENT 4 th Internatonal Conference on Industral Engneerng and Industral Management XIV Congreso de Ingenería de Organzacón Donosta San Sebastán, September 8 th 10 th 010 Actvty Schedulng
More informationA Constant Factor Approximation for the Single Sink Edge Installation Problem
A Constant Factor Approxmaton for the Sngle Snk Edge Installaton Problem Sudpto Guha Adam Meyerson Kamesh Munagala Abstract We present the frst constant approxmaton to the sngle snk buyatbulk network
More informationEfficient Bandwidth Management in Broadband Wireless Access Systems Using CACbased Dynamic Pricing
Effcent Bandwdth Management n Broadband Wreless Access Systems Usng CACbased Dynamc Prcng Bader AlManthar, Ndal Nasser 2, Najah Abu Al 3, Hossam Hassanen Telecommuncatons Research Laboratory School of
More informationOn Competitive Nonlinear Pricing
On Compettve Nonlnear Prcng Andrea Attar Thomas Marott Franços Salané February 27, 2013 Abstract A buyer of a dvsble good faces several dentcal sellers. The buyer s preferences are her prvate nformaton,
More information行 政 院 國 家 科 學 委 員 會 補 助 專 題 研 究 計 畫 成 果 報 告 期 中 進 度 報 告
行 政 院 國 家 科 學 委 員 會 補 助 專 題 研 究 計 畫 成 果 報 告 期 中 進 度 報 告 畫 類 別 : 個 別 型 計 畫 半 導 體 產 業 大 型 廠 房 之 設 施 規 劃 計 畫 編 號 :NSC 962628E009026MY3 執 行 期 間 : 2007 年 8 月 1 日 至 2010 年 7 月 31 日 計 畫 主 持 人 : 巫 木 誠 共 同
More informationLogistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification
Lecture 4: More classfers and classes C4B Machne Learnng Hlary 20 A. Zsserman Logstc regresson Loss functons revsted Adaboost Loss functons revsted Optmzaton Multple class classfcaton Logstc Regresson
More informationAn Intelligent Policy System for Channel Allocation of Information Appliance
Tamkang Journal of Scence and Engneerng, Vol. 5, No., pp. 6368 (2002) 63 An Intellgent Polcy System for Channel Allocaton of Informaton Applance ChengYuan Ku, ChangJnn Tsao 2 and Davd Yen 3 Department
More informationFisher Markets and Convex Programs
Fsher Markets and Convex Programs Nkhl R. Devanur 1 Introducton Convex programmng dualty s usually stated n ts most general form, wth convex objectve functons and convex constrants. (The book by Boyd and
More informationPOLYSA: A Polynomial Algorithm for Nonbinary Constraint Satisfaction Problems with and
POLYSA: A Polynomal Algorthm for Nonbnary Constrant Satsfacton Problems wth and Mguel A. Saldo, Federco Barber Dpto. Sstemas Informátcos y Computacón Unversdad Poltécnca de Valenca, Camno de Vera s/n
More informationAN APPOINTMENT ORDER OUTPATIENT SCHEDULING SYSTEM THAT IMPROVES OUTPATIENT EXPERIENCE
AN APPOINTMENT ORDER OUTPATIENT SCHEDULING SYSTEM THAT IMPROVES OUTPATIENT EXPERIENCE YuL Huang Industral Engneerng Department New Mexco State Unversty Las Cruces, New Mexco 88003, U.S.A. Abstract Patent
More informationProductForm Stationary Distributions for Deficiency Zero Chemical Reaction Networks
Bulletn of Mathematcal Bology (21 DOI 1.17/s11538195174 ORIGINAL ARTICLE ProductForm Statonary Dstrbutons for Defcency Zero Chemcal Reacton Networks Davd F. Anderson, Gheorghe Cracun, Thomas G. Kurtz
More informationCHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol
CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL
More informationA DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATIONBASED OPTIMIZATION. Michael E. Kuhl Radhamés A. TolentinoPeña
Proceedngs of the 2008 Wnter Smulaton Conference S. J. Mason, R. R. Hll, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. A DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATIONBASED OPTIMIZATION
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationJ. Parallel Distrib. Comput. Environmentconscious scheduling of HPC applications on distributed Cloudoriented data centers
J. Parallel Dstrb. Comput. 71 (2011) 732 749 Contents lsts avalable at ScenceDrect J. Parallel Dstrb. Comput. ournal homepage: www.elsever.com/locate/pdc Envronmentconscous schedulng of HPC applcatons
More informationDownlink Power Allocation for Multiclass. Wireless Systems
Downlnk Power Allocaton for Multclass 1 Wreless Systems JangWon Lee, Rav R. Mazumdar, and Ness B. Shroff School of Electrcal and Computer Engneerng Purdue Unversty West Lafayette, IN 47907, USA {lee46,
More information8 Algorithm for Binary Searching in Trees
8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the
More informationRobust Design of Public Storage Warehouses. Yeming (Yale) Gong EMLYON Business School
Robust Desgn of Publc Storage Warehouses Yemng (Yale) Gong EMLYON Busness School Rene de Koster Rotterdam school of management, Erasmus Unversty Abstract We apply robust optmzaton and revenue management
More informationAddendum to: Importing SkillBiased Technology
Addendum to: Importng SkllBased Technology Arel Bursten UCLA and NBER Javer Cravno UCLA August 202 Jonathan Vogel Columba and NBER Abstract Ths Addendum derves the results dscussed n secton 3.3 of our
More informationSchedulability Bound of Weighted Round Robin Schedulers for Hard RealTime Systems
Schedulablty Bound of Weghted Round Robn Schedulers for Hard RealTme Systems Janja Wu, JyhCharn Lu, and We Zhao Department of Computer Scence, Texas A&M Unversty {janjaw, lu, zhao}@cs.tamu.edu Abstract
More informationProject Networks With MixedTime Constraints
Project Networs Wth MxedTme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa
More informationPeriod and Deadline Selection for Schedulability in RealTime Systems
Perod and Deadlne Selecton for Schedulablty n RealTme Systems Thdapat Chantem, Xaofeng Wang, M.D. Lemmon, and X. Sharon Hu Department of Computer Scence and Engneerng, Department of Electrcal Engneerng
More informationDynamic Fleet Management for Cybercars
Proceedngs of the IEEE ITSC 2006 2006 IEEE Intellgent Transportaton Systems Conference Toronto, Canada, September 1720, 2006 TC7.5 Dynamc Fleet Management for Cybercars Fenghu. Wang, Mng. Yang, Ruqng.
More informationA Note on the Decomposition of a Random Sample Size
A Note on the Decomposton of a Random Sample Sze Klaus Th. Hess Insttut für Mathematsche Stochastk Technsche Unverstät Dresden Abstract Ths note addresses some results of Hess 2000) on the decomposton
More informationEfficient Project Portfolio as a tool for Enterprise Risk Management
Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse
More informationOPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL. Thomas S. Ferguson and C. Zachary Gilstein UCLA and Bell Communications May 1985, revised 2004
OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL Thomas S. Ferguson and C. Zachary Glsten UCLA and Bell Communcatons May 985, revsed 2004 Abstract. Optmal nvestment polces for maxmzng the expected
More informationA heuristic task deployment approach for load balancing
Xu Gaochao, Dong Yunmeng, Fu Xaodog, Dng Yan, Lu Peng, Zhao Ja Abstract A heurstc task deployment approach for load balancng Gaochao Xu, Yunmeng Dong, Xaodong Fu, Yan Dng, Peng Lu, Ja Zhao * College of
More informationGraph Theory and Cayley s Formula
Graph Theory and Cayley s Formula Chad Casarotto August 10, 2006 Contents 1 Introducton 1 2 Bascs and Defntons 1 Cayley s Formula 4 4 Prüfer Encodng A Forest of Trees 7 1 Introducton In ths paper, I wll
More informationEconomic Models for Cloud Service Markets
Economc Models for Cloud Servce Markets Ranjan Pal and Pan Hu 2 Unversty of Southern Calforna, USA, rpal@usc.edu 2 Deutsch Telekom Laboratores, Berln, Germany, pan.hu@telekom.de Abstract. Cloud computng
More informationCLoud computing technologies have enabled rapid
1 CostMnmzng Dynamc Mgraton of Content Dstrbuton Servces nto Hybrd Clouds Xuana Qu, Hongxng L, Chuan Wu, Zongpeng L and Francs C.M. Lau Department of Computer Scence, The Unversty of Hong Kong, Hong Kong,
More informationA Probabilistic Theory of Coherence
A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want
More informationSelfAdaptive SLADriven Capacity Management for Internet Services
SelfAdaptve SLADrven Capacty Management for Internet Servces Bruno Abrahao, Vrglo Almeda and Jussara Almeda Computer Scence Department Federal Unversty of Mnas Geras, Brazl Alex Zhang, Drk Beyer and
More informationEfficient Striping Techniques for Variable Bit Rate Continuous Media File Servers æ
Effcent Strpng Technques for Varable Bt Rate Contnuous Meda Fle Servers æ Prashant J. Shenoy Harrck M. Vn Department of Computer Scence, Department of Computer Scences, Unversty of Massachusetts at Amherst
More information