Non-clairvoyant Weighted Flow Time Scheduling with Rejection Penalty

Transcription

1 Non-clairvoyant Weighted Flo Tie Scheduling ith Rejection Penalty Ho-Leung Chan University of, Sze-Hang Chan University of, Tak-Wah La University of, ABSTRACT Lap-Kei Lee University of, This paper initiates the study of online scheduling ith rejection penalty in the non-clairvoyant setting, ie, the size (processing tie) of a job is not assued to be knon at its release tie In the rejection penalty odel, jobs can be rejected ith a penalty, and the user cost of a job is defined as the eighted flo tie of the job plus the penalty if it is rejected before copletion Previous ork on iniizing the total user cost focused on the clairvoyant singleprocessor setting [2, 8] and has produced O()-copetitive online algorith for jobs ith arbitrary eights and penalties This paper gives the first non-clairvoyant algoriths that are O()-copetitive for iniizing the total user cost on a single processor and ultiprocessors, hen using slightly faster (ie, ( + ϵ)-speed for any ϵ > ) processors Note that if no extra speed is alloed, no online algorith can be O()-copetitive even for iniizing (uneighted) flo tie alone The ne user cost results can also be regarded as a generalization of previous non-clairvoyant results on iniizing eighted flo tie alone (WSETF [4] for a single processor; WLAPS [4] for ulti-processors) The above results assue a processor running at a fixed speed This paper shos ore interesting results on extending the above study to the dynaic speed scaling odel, here the processor can vary the speed dynaically and the rate of energy consuption is an arbitrary increasing function of speed A scheduling algo- The research of Ho-Leung Chan as partially supported by GRF Grant HKU72E The research of Tak-Wah La as partially supported by HKU Grant Part of the ork as done hen Lap-Kei Lee as orking in MADALGO (Center for Massive Data Algorithics, a Center of the Danish National Research Foundation), Aarhus University, Denark Perission to ake digital or hard copies of all or part of this ork for personal or classroo use is granted ithout fee provided that copies are not ade or distributed for profit or coercial advantage and that copies bear this notice and the full citation on the first page To copy otherise, to republish, to post on servers or to redistribute to lists, requires prior specific perission and/or a fee SPAA 2, June 25 27, 22, Pittsburgh, Pennsylvania, USA Copyright 22 ACM /2/6 $ Jianqiao Zhu University of, jqzhu@cshkuhk rith has to decide job rejection and deterine the order and speed of job execution It is interesting to study the tradeoff beteen the above-entioned user cost and energy This paper gives to O()-copetitive non-clairvoyant algoriths for iniizing the user cost plus energy on a single processor and ulti-processors, respectively Categories and Subject Descriptors C4 [Perforance of Systes]: Perforance Attributes; F2 [Analysis of Algoriths and Proble Coplexity]: General Keyords Online scheduling, copetitive analysis, eighted flo tie, nonclairvoyant scheduling, rejection penalty INTRODUCTION It is coon that servers prioritize their jobs and reject soe lo-priority jobs during peak load to eet the perforance guarantee Serving too any jobs prolongs their individual response tie, yet rejecting jobs ould cause users inconvenience and aste the processing poer already spent on the To study the tradeoff beteen response tie and rejection penalty, Bansal et al [2] and Chan et al [8] considered flo-tie scheduling on a single processor hen jobs can be rejected at soe penalty Jobs arrive online ith arbitrary sizes, eights and penalties, and a scheduler ay reject soe jobs before copletion Each job defines a user cost equal to its eighted flo tie plus the penalty if it is rejected, here the flo tie (or siply flo) of a job is the tie elapsed since the job is released until it is copleted or rejected The objective is to iniize the total user cost of all jobs Bansal et al [2] shoed that any online algorith is Ω(ax{n 4, C 2 })-copetitive, here n is the nuber of jobs and C is the ax-in ratio of job penalties In vie of the loer bound, they consider giving the online algorith a slightly faster processor Using a ( + ϵ)-speed processor for any ϵ >, they gave an O( ϵ (log W +log C)2 )-copetitive algorith here W is the axin ratio of job eights Recently, Chan et al [8] iproved the copetitive ratio to O(( + ϵ )2 ) hen using a ( + ϵ) 2 -speed processor, hich is independent of W and C These to results are based on job rejection policies that kno the size of a job at its re- 246

2 Weighted flo User cost Weighted flo + energy User cost + energy Single processor + ϵ [4] 2( + ϵ )2 [ ] 8( + ϵ )2 [7] 36( + ϵ )2 [ ] > processors 8( + ϵ )2 [4] 2( + ϵ )2 [ ] 2(log + 2)( + ϵ )2 [ ] (using 2( + ϵ)-speed processors) Table : Copetitive ratios of non-clairvoyant scheduling for different objectives involving eighted flo tie Recall that user cost equals eighted flo plus penalty All results assue using a faster processor and, unless specified otherise, are using ( + ϵ)-speed processors Our ne results are arked ith [ ] lease tie and its reaining size at any tie, ie, they only ork in the clairvoyant setting In this paper, e extend the study of rejection penalty to the non-clairvoyant setting, here the size of a released job is not knon until the job is copleted Such a setting is natural fro the viepoint of operating systes Non-clairvoyant flo-tie scheduling on a single processor For the special case hen each job has infinite penalty, no jobs ould be rejected and the proble reduces to the classic proble of iniizing eighted flo tie only In the non-clairvoyant setting, even for uneighted jobs, any algorith is Ω(n /3 )-copetitive [2] for iniizing the total flo, here n is the nuber of jobs Using a slightly faster processor, Kalyanasundara and Pruhs [] analyzed a non-clairvoyant algorith SETF (Shortest Elapsed Tie First, hich prefers jobs that have been processed the least) for a single processor, and they shoed that it is ( + ϵ)-speed ( + )-copetitive for (uneighted) flo Later, Bansal and Dhadhere [4] generalized this result for eighted jobs, and shoed that ϵ the algorith WSETF is ( + ϵ)-speed ( + )-copetitive for ϵ eighted flo Multi-processor scheduling ith arbitrary parallelizability On a ulti-core chip that provides identical processors, soe jobs ight be processed faster hen using several processors in parallel, hile others ight be inherently sequential In the literature, the degree of parallelizability of a job as odeled as follos (eg, [9,, 4]): A job consists of a nuber of phases, each ith an arbitrary size and arbitrary speedup function that specifies the aount of speedup hen running the job on a given nuber of processors A non-clairvoyant scheduler has no inforation about the phases in advance For iniizing (uneighted) flo in such a ulti-processor odel, Edonds [9] gave a (2 + ϵ)- speed O( )-copetitive algorith, and Edonds and Pruhs [] ϵ later shoed that another algorith LAPS (Latest Arrival Processor Sharing, hich shares the processing poer equally aong a constant fraction of latest-arrival jobs) is (+ +ϵ)-speed O( ++ϵ )- ϵ copetitive, for any ϵ > and < [] Very recently, Zhu et al [4] extended the latter result to eighted jobs and gave an algorith WLAPS (Weighted LAPS) that is (+ϵ)-speed 8(+ ϵ )2 - copetitive for eighted flo Ne results on flo plus penalty In this paper, e extend the non-clairvoyant results on flo-tie scheduling to the rejection penalty odel In particular, e propose a siple job rejection policy RWE (Reject When eighted flo Equals penalty), hich rejects an unfinished job hen the eighted flo incurred equals the job penalty Unlike previous job rejection policies [2, 8], RWE does not require any inforation on job size, and it can be used in different settings We develop a rather general technique for analyzing RWE In the single-processor setting, e sho that WLAPS coupled ith RWE is ( + ϵ)-speed 2( + ϵ )2 -copetitive for iniizing total user cost; in the ulti-processor setting, the copetitive ratio becoes 2( + ϵ )2 For the special case hen jobs ust all be copleted (ie, ith infinite penalty), our ne algorith behaves exactly as WLAPS The first to coluns of Table suarize the results on eighted flo and user cost (eighted flo plus penalty) Dynaic speed scaling and energy The above results assue that processors are running at a fixed speed We sho that RWE also orks ell in the dynaic speed scaling odel, in hich the scheduler can anage the energy consuption by scaling the processor speed dynaically (see [] for a survey) Specifically, the processor speed can vary beteen and a axiu speed T, and its rate of energy consuption (ie, poer) P increases ith the speed s according to a certain given poer function, say, P (s) = s 3 In this odel, a scheduler has to deterine dynaically hich job and at hat speed to execute Tradeoff beteen user cost and energy The past fe years have itnessed several online results on optiizing the tradeoff beteen flo and energy under the dynaic speed scaling odel (see [] for survey) Chan et al [8] have also extended the study of rejection penalty to the dynaic speed scaling odel and considered the tradeoff beteen user cost and energy consuption, but their result is liited to the clairvoyant setting In particular, they gave a clairvoyant algorith that is ( + ϵ) 2 -speed O(( + ϵ )2 )- copetitive for iniizing total user cost plus energy on a single processor [8] In this context, a λ-speed processor (here λ > ) eans that, given poer P (s), it can run at speed λ s In this paper, e use RWE to obtain a non-clairvoyant algorith RAW hich is ( + ϵ)-speed 36( + ϵ )2 -copetitive for iniizing total user cost plus energy (see Table ) Multi-processor result We further sho that RWE also leads to a ne non-clairvoyant algorith for iniizing user cost plus energy on > processors, here each processor can scale its speed independently and jobs coprise phases ith different degrees of parallelizability The only past relevant ork as done by Chan, Edonds and Pruhs [6], ho considered scheduling uneighted jobs non-clairvoyantly on processors ith poer function P (s) = s α for any α >, and axiu speed T = The objective is to iniize total flo plus energy, and jobs ust be all copleted They shoed a strong loer bound of Ω( (α )/α2 ) on the copetitive ratio In vie of this loer bound, they assue that jobs satisfy to properties: () They do not have side effect, ie, the execution of a job does not affect anything external to itself, so ultiple copies of a job can run siultaneously (2) They are checkpointable, ie, each copy can save its state periodically and then restart each copy fro the point of execution of the copy that ade the ost progress Then Chan, Edonds and Pruhs [6] ere able to extend LAPS to a ne algorith Multi- LAPS hich is O(2 α log α2 )-copetitive for iniizing (uneighted) flo plus energy, and shoed that any algorith for such log α checkpointable jobs is Ω(log /α )-copetitive Roughly speaking, previous speed scaling ork scales the speed such that flo and energy are incurred at the sae rate, yet MultiLAPS needs to run to ties faster, leading to the ultiplicative factor of 2 α in the copetitive ratio We extend the above result to the rejection penalty odel, and 247

3 ore interestingly, alloing eighted jobs and arbitrary poer function P (s) We give an algorith MultiRAW hich uses RWE as job rejection policy and is 2( + ϵ)-speed 2(log + 2)( + ϵ )2 - copetitive for iniizing total user cost plus energy To analyze MultiRAW, e need to restrict the total processor speed of the optial offline algorith to follo soe function depending on total eight of unfinished jobs, and sho that such a restriction does not increase the copetitive ratio by a constant factor To this end, e generalize the offline transforation algorith LLB (Latest Lag Behind) first given in [7] for single processor, and allo it to transfor an offline algorith that ould run ultiple different jobs siultaneously on ultiple processors The transfored offline algorith ay also run different jobs at the sae tie using tie sharing and alays has the processor speed folloing the required function, and e sho that its eighted flo is at ost the eighted flo plus energy of the original algorith Organization of the paper The folloing discussion focuses on the results in the dynaic speed scaling odel Note that the results in the fixed speed odel ould be shon as special cases Section 2 defines the odels, probles and notations forally Results on single processor and ulti-processors are given in Sections 3 and 4, respectively 2 FORMAL PROBLEM DEFINITIONS AND NOTATIONS We study job scheduling on a chip containing identical processors Jobs are arriving online, here each job j has a release tie r(j), a size p(j), a eight (j) and a rejection penalty c(j) Jobs are non-clairvoyant, eaning that the size of a job is unknon until it is copleted Preeption and igration are alloed and free Each processor can run independently at any speed s [, T ] (here T is the axiu speed of the processor hich ay be ) The rate of energy usage is given by an arbitrary poer function P (s) Siilar to [3], e assue that P () =, and P at all speeds in [, T ] is defined, strictly increasing, nonnegative, continuous, strictly convex and differentiable As shon in [3], it is possible to use such a poer function P to eulate any arbitrary poer function ith an arbitrarily sall increase in the copetitive ratio Let Q(y) = in{p (y), T } Note that Q is onotonically increasing and concave Eg, if P (s) = s α for soe α >, then Q(x) = x /α In the single processor setting, each job is processed by at ost one processor and its processing rate is alays the speed of the processor ties the fraction of the processor assigned to this job The processing rate is ore coplicated in the ulti-processor setting due to the varying parallelizability of the job In particular, e consider each job as a sequence of q(j) phases j, j 2,, j q(j) Each phase j k is an ordered pair p(j k ), Γ j k, here p(j k ) is the aount of ork in the phase and Γ j k is a speed-up function specifying the degree of parallelis of the phase More precisely, Γ j k(y) represents the rate at hich ork in the phase j k is processed hen using y processors running at speed If the y processors are running at speed s, then the ork is processed at rate Γ j k(y) s Folloing the previous ork (eg, [6, 9, ]), e assue that each speedup function Γ is non-negative, onotonically increasing and Γ(y) sublinear, ie, Γ(y ) for any y y We assue that y y for any phase, its speed-up function Γ satisfies that Γ(y) = y for y [, ] This assuption corresponds to the fact that hen a phase is processed by tie-sharing on a y fraction of a processor, its processing rate should be y ties the speed of the processor In the non-clairvoyant setting, e assue that the size and speedup function of each phase is not knon to the online algorith In the speed scaling odel, the objective is to iniize the su of the total eighted flo tie, total rejection penalty and total energy usage We call it the cost of a schedule Let OPT be the optial offline algorith that alays iniizes the cost We analyze our algoriths hen they are given faster processors Precisely, a y- speed processor runs at speed sy hen the rate of energy usage is P (s) Previous definitions Our results ake use of previous ork like WLAPS and AJW (eg, [7]) We revie the necessary definitions We say that a job is active at tie t if it has arrived but has not yet finished or rejected by tie t Throughout this paper, e denote n a (t) as the nuber of active job in the online algorith at tie t and a (t) be their total eight DEFINITION (-ADJUSTED WEIGHT) Let (, ] be a paraeter Consider any tie t Let j, j 2,, j na (t) be the active jobs in the online algorith ordered in increasing order of arrival ties Let τ be the largest integer such that the latest arrived jobs {j τ, j τ+,, j na (t)} have total eight at least a(t) Then, the -adjusted eight of j i at tie t, denoted (j i, t) (or siply (j i, t)) hen is clear in context), is defined as follos: if i < τ (j i, t) = a (t) n a(t) i=τ+ (j i) if i = τ (j i ) if i > τ DEFINITION 2 (JOB ASSIGNMENT POLICY WLAPS) WLAPS() is paraeterized by a constant (, ] It assues all processors are running at the sae speed At any tie t, WLAPS shares the processors aong all active jobs proportional to their adjusted eights at tie t, ie, each job j is assigned a fraction of the processors (j,t) a(t) DEFINITION 3 (SPEED SCALING POLICY AJW) At any tie t, AJW sets the speed of each processor to s(t) = Q( a(t) ) Intuitively, the speed s(t) ensures that the total rate of energy usage equals a (t), or s(t) = T if P (T ) < a (t) 3 SINGLE PROCESSOR RESULTS In this section, e propose a siple job rejection policy RWE (Reject When eighted flo Equals penalty) Cobining RWE ith WLAPS, e can obtain an O()-copetitive algorith for iniizing eighted flo plus penalty on a fixed-speed processor Belo e sho a ore general result on cobining RWE ith WLAPS and AJW to obtain an O()-copetitive algorith for iniizing eighted flo plus penalty plus energy under the dynaic speed scaling odel RWE is defined as follos DEFINITION 4 (JOB REJECTION POLICY RWE) RWE rejects a job j if it is not copleted by tie r(j) + c(j) (j) Algorith RAW Let (, ] be a constant RAW assues a (+ϵ)-speed processor here ϵ > can be any real RAW assigns jobs to the processor by WLAPS, ie, sharing the processor aong all active jobs proportional to their adjusted eights RAW scales the processor speed to ( + ϵ) Q( a(t)) (so the rate of energy usage is a(t)) RAW rejects jobs according to RWE We call the algorith RAW to stand for RWE-AJW-WLAPS Our ain result is the folloing 248

4 THEOREM Let = ϵ RAW is ( + ϵ)-speed 36( + 2(+ϵ) ϵ )2 -copetitive for iniizing eighted flo plus penalty plus energy To prove Theore, let OFF be the offline algorith that iniizes the cost under the condition that OFF scales the processor speed by AJW, and OFF rejects a job j at tie r(j) if OPT rejects j It is knon that the cost of OFF is at ost tice of OPT hen they do not reject jobs [7] Belo e sho that this relationship reains valid even if they reject jobs LEMMA 2 The cost of OFF is at ost tice the cost of OPT PROOF For each job j, recall that OFF rejects j hen j arrives if OPT rejects j Then, OFF schedules the reaining jobs by the algorith LLB [7], hich uses AJW to scale the processor speed [7] shos that the total eighted flo tie plus energy usage of LLB is at ost tice that of OPT Since the total penalty of OFF is the sae as that of OPT, the cost of OFF is at ost tice that of OPT Hence, it suffices to analyze RAW against OFF, hich ould incur an extra factor of to in the copetitive ratio At any tie t, let q a(j, t) be the reaining ork of j in RAW and let q o(j, t) be that in OFF Note that q a (j, t) (resp, q o (j, t)) becoes once RAW (resp, OFF) rejects j We are interested in the progress of the jobs that are not rejected by OFF DEFINITION 5 (LAGGING WORK) At any tie t, for any job j, the aount of lagging ork of j is defined as { if OFF rejects j at r(j) x(j, t) = ax{q a(j, t) q o(j, t), } otherise Intuitively, x(j, t) is the aount of useful ork that RAW is lagging behind OFF In particular, if j is already rejected by OFF, then all ork reaining in RAW is not useful At any tie t, let R(t) be the set of jobs being processed by RAW Recall that the total adjusted eight of jobs in R(t) is a(t), here a(t) is the total eight of all active jobs Let L(t) R(t) be those jobs in R(t) such that x(j, t) > We call L(t) the lagging jobs We denote ϕ(t) = j L(t) (j, t) Then Theore follos fro the folloing to leas LEMMA 3 (LOWER BOUND OF OFF) ( a (t) ϕ(t)) W o + C o, here W o and C o are the total eighted flo and the total rejection penalty of OFF, respectively LEMMA 4 (UPPER BOUND OF RAW) a (t) ((a(t) ϕ(t)) + o(t)), here o(t) is the total eight of active jobs in OFF at tie t Before proving Leas 3 and 4, e sho ho to use these to leas to prove Theore PROOF (Proof of Theore ) Recall that that < /2 By Leas 3 and 4, the total eighted flo tie of RAW is at ost (W o + C o + o (t)) = (2( + ϵ ))2 (( + )W o + C o) 6( + ϵ )2 (W o + C o) Note that for each job j rejected by RWE, the eighted flo tie incurred is c(j) (j) = c(j) (j) Hence, the total penalty of RAW is at ost its total eighted flo tie Furtherore, by running AJW, RAW has energy usage at ost its eighted flo tie In suary, the cost of RAW is at ost 8( + ϵ )2 ties the cost of OFF The latter is at ost tice the cost of OPT (by Lea 2) Theore follos PROOF (Proof of Lea 3) Consider any tie t Note that a (t) ϕ(t) is the total eight of jobs in R(t) \ L(t), hich are the jobs j being processed by RAW ith x(j, t) = Note that x(j, t) = only if j is rejected by OFF at r(j) or q a (j, t) q o (j, t) at tie t Let C be the set of jobs rejected by OFF Let C(t) = C R(t) \ L(t) be those jobs in C that are still active in RAW at tie t and let N(t) R(t) \ L(t) be those jobs j not rejected by OFF but q a(j, t) q o(j, t) at tie t Then, ( a (t) ϕ(t)) = (j)+ (j) j C(t) j N(t) Note that for each job j, j reains in RAW for at ost c(j) units (j) of tie and incurs a eighted flo tie of at ost c(j) Hence, j C(t) (j) j C c(j) = Co For each job j in N(t), j is not copleted by OFF at tie t Hence, j N(t) (j) o (t) W o, here o (t) is the total eight of jobs in OFF at tie t PROOF (Proof of Lea 4) We use a potential function analysis Let j, j 2,, j na(t) be the active jobs in RAW at tie t arranged in increasing order of arrival ties We denote h(j i ) = i k= (ji) as the total eight of active jobs arrived no later than j i Recall that x(j, t) is the lagging ork of j at tie t Then, e define n a (t) Φ(t) = γ f(h(j i)) x(j i, t), i= here γ = and f(h) = h/q(h) Note that f(h) is nondecreasing We ill sho that Φ satisfies the folloing three con- (+ϵ) ditions Boundary condition: Φ = before the first job arrives and after all jobs are finished Discrete event condition: Φ does not increase at job arrival, copletion or rejection Running condition: At any other tie t, a (t) + dφ(t) (( a (t) ϕ(t)) + o (t)) Then, by integrating these conditions over tie, the lea follos The boundary condition is true as n a (t) = before any job is released and after all jobs are finished When a job j arrives, no atter hether OFF rejects j, x(j, t) =, so Φ does not increase When j is copleted by OFF, x(j, t) and Φ do not change When j is copleted or rejected by RAW, the ter corresponding to j disappears and other ters ay only decrease, so Φ does not increase Thus, the discrete event condition is true For the running condition, Lea 5 belo shos that dφ ( )ϕ(t) Note that 2 2 ax{ a (t), o (t)} ax{ a(t), o(t)} is no greater than ax{a(t), o(t)} 2a(t), and ϕ(t) a(t) a (t) Therefore, a(t) + dφ(t) a (t) + 2 ax{ a (t), o (t)} ( )ϕ(t) a (t) + ax{ a(t), o (t)} 2 a (t) ϕ(t) + a (t) [ ax{ a (t), o (t)} ϕ(t)] (( a(t) ϕ(t)) + o(t)) Since Q is concave, for any λ [, ] and h, ( λ)q() + λq(h) Q(( λ) + λh), ie, λq(h) Q(λh) Hence, f(λh) = λh h = f(h) Q(λh) Q(h) 249

5 LEMMA 5 At any tie t ithout discrete events, dφ 2 ax{ a(t), o(t)} ( )ϕ(t) PROOF We consider dφ as a cobined effect due to the action of RAW and OFF Let dφ a and dφ o be the rate of change of Φ due to RAW and OFF, respectively Then dφ = dφ a + dφ o For each active job j in RAW, if j L(t), x(j, t) > and h(j) ( ) a (t) RAW processes j at a speed ( + ϵ) (j,t) a(t) Q( a (t)) Hence, for each ter γ f(h(j)) x(j, t) here j L(t), the ter is decreasing at a rate at least γ γ h(j) Q(h(j)) ( + ϵ) (j, t) Q(a(t)) a(t) ( ) a (t) Q(( ) a (t)) ( + ϵ) (j, t) a (t) Q( a(t)) γ ( + ϵ) (j, t) = (j, t) here the first inequality coes fro that f(h) = h/q(h) is nondecreasing ith h, and the second inequality coes fro that Q is increasing For each job j / L(t), the ter γ f(h(j)) x(j, t) is non-increasing Hence, dφa For OFF, the orst case (in hich dφo j L(t) (j, t) = is the largest) occurs hen it processes the job ith axiu h(j), hich equals a (t), using all its speed s o = Q( o (t)) Then dφo γ f( a (t)) s o = γ a(t) Q( Q( a (t)) o(t)) If a (t) o (t), Q( a (t)) Q( o (t)) and dφ o γ a (t) Otherise, a (t) < o (t) Since f is nondecreasing, f( a (t)) f( o (t)) Hence, o dφ γf( o (t)) s o = γ o (t) Cobining the to cases, e have dφ o γ ax{ a(t), o(t)} Notice that = ϵ and = 2 2(+ϵ) +ϵ Hence, γ ax{ a (t), o (t)} = ax{ (+ϵ) a(t), o (t)} = ax{ a (t), o (t)} and the lea follos 2 Rearks on fixed speed setting Since the fixed speed setting is a special case of the arbitrary poer function setting (here energy usage of any algorith is zero), the previous result iediately iplies that RAW is 36( + ϵ )2 -copetitive using a ( + ϵ)- speed processor Yet e can tighten the analysis to achieve a better copetitive ratio In particular, e can directly copare RAW and OPT and sho leas siilar to Lea 3 and 4, hich can upper bound the total eighted flo of RAW to be at ost 6( + ϵ )2 ties the cost of OPT The total rejection penalty of RAW is at ost its eighted flo The folloing theore follos THEOREM 6 For the fixed speed setting, RAW is ( + ϵ)-speed 2( + ϵ )2 -copetitive for iniizing eighted flo plus penalty 4 MULTI-PROCESSOR RESULTS We consider scheduling on > processors in the speed scaling odel We give an algorith MultiRAW hich is 2( + ϵ)- speed 2(+ ϵ )2 (log +2)-copetitive for iniizing eighted flo plus penalty plus energy Note that MultiRAW follos the job rejection policy RWE given in Section 3 The fixed speed odel is treated as a special case; e can odify the result of MultiRAW to get an (+ϵ)-speed 2(+ ϵ )2 -copetitive algorith for eighted flo plus penalty For siplicity, e first assue that the nuber of processors is a poer of to We ill explain ho to reove this assuption later Our algorith is built on MultiLAPS proposed by [6], in hich they consider uneighted jobs that cannot be rejected and the poer ϕ(t) ) /α Each group runs independently and processes the n a (t) jobs ith the latest arrival ties by sharing the processors ithin the group equally aong the jobs Note that each of these jobs is duplicated into log copies and processed siultaneously by the log groups Since the jobs are checkpointable, at any tie, the processing rate of a job is the axiu processing rate aong all the copies of the job in the log groups Algorith MultiRAW To handle eighted jobs, rejection penalty, and arbitrary poer functions, e extend the algorith MultiLAPS as follos function is P (s) = s α ithout axiu speed bound Their objective is to iniize the total flo tie plus energy usage Algorith MultiLAPS [6] Let (, ] be a constant and n a (t) be the nuber of active jobs in MultiLAPS at tie t MultiLAPS divides the processors into log groups such that for l =,, log, the l-th group consists of l = 2 l processors each running at speed 2( na(t) l For the l-th group, e set the speed of each processor to s l = 2(+ϵ)Q( a(t) l ); recall that Q(y) = in{p (y), T }, and a (t) is the total eight of the active jobs at tie t We share the processors ithin a group proportional to the adjusted eight of the jobs Hence, a job j is processed by (j,t) a (t) l processors in the l-th group (Job rejection policy RWE) We reject a job j if it is not finished by tie r(j) + c(j) (j) Our ain result is the folloing theore Sections 4 to 45 are devoted to proving this theore THEOREM 7 Let ϵ > be any real and = ϵ MultiRAW 2(+ϵ) is 2(+ϵ)-speed 2(+ ϵ )2 (log +2)-copetitive for iniizing eighted flo plus penalty plus energy 4 Restricting input instances and offline algorith Canonical instances We first sho that it suffices to consider soe specific input instances Recall that jobs ay have varying parallelizability and the parallelizability of a phase is given by a speedup function Γ We call Γ parallel up to σ processors if Γ(y) = y for all y σ and Γ(y) = σ for y > σ We call an instance canonical if each job phase is parallel up to σ processors for soe σ [, ], here σ ay be different for different phases LEMMA 8 For any input instance I, e can transfor I into a canonical instance such that the cost of MultiRAW does not change and the cost of OPT ay only decrease Canonical instances ere first introduced in [6,4], and Lea 8 can be proven in a siilar ay as in [6, 4] In the folloing, e consider canonical instances only Restricted offline algorith To prove Theore 7, e need to copare MultiRAW against the optial offline algorith OPT Without loss of generality, OPT rejects jobs at their release tie It is soeties easier to copare MultiRAW against an offline algorith that rejects the sae set of jobs as OPT but orks on a single processor ith axiu speed T Let OFF be such an offline algorith that iniizes the eighted flo alone under the condition that the single processor alays runs at speed Q( o(t) ), here o (t) is the total eight of active jobs in OFF Note that OFF does not have energy concern in its objective LEMMA 9 The eighted flo of OFF is at ost the eighted flo plus energy of OPT 25

6 To prove Lea 9, e generalize the algorith LLB (Latest Lag Behind) given in [7] to transfor OPT (hich iniizes eighted flo plus energy of the jobs not rejected on processors) to an offline algorith LLB on a single processor ith axiu speed T that at any tie t, follos the speed Q( b(t) ), here b (t) is the total eight of active jobs in LLB A ne feature of LLB is the ability to handle the case that OPT can run ultiple jobs by tie sharing at the sae tie Algorith LLB Consider any tie t Let n b (t) be the nuber of active jobs in LLB For any job j, let p b (j, t) and p o (j, t) be the ork done on j up to tie t by LLB and OPT, respectively Furtherore, let d(j, t) = p o (j, t) p b (j, t) We say a job j is lagging at tie t if d(j, t) > Let y(j) be the latest tie hen j has becoe lagging; if j is non-lagging, let y(j) = t We denote the active jobs in LLB at tie t as j, j 2,, j nb (t), arranged in increasing order of y(j i), ie, y(j ) y(j 2) y(j nb (t)) Let l be the nuber of lagging jobs ( l n b (t)) Let s o be the total speed of the processors in OPT at tie t Let s x s o be the total speed OPT assigns to all jobs j ith d(j, t) = LLB sets its speed s b = Q( b(t) ), and focuses on the job j a defined to be j l if l >, and j nb (t) otherise Details are as follos: If OPT is not running any job j i ith d(j i, t) =, then LLB runs j a ith speed s b If OPT is processing soe jobs j i ith d(j i, t) =, but s x s b (ie, LLB has enough speed to prevent all of the fro becoing lagging), then LLB follos OPT s speed on each of those jobs and runs j a ith the reaining speed s b s x Otherise (s b < s x), LLB follos OPT s speed for each of those jobs j i, in descending order of the index i, until there is soe job j hich LLB cannot follo OPT s total speed on j LLB then assigns all of its reaining speed to j Roughly speaking, LLB attepts to catch up ith the progress of OPT; it gives priority to the job that has becoe lagging ost recently, hile trying to avoid creating ore ne lagging jobs We can sho that the eighted flo of LLB is at ost the eighted flo plus energy of OPT Its proof is given in Appendix B Since OFF iniizes eighted flo, Lea 9 follos 42 Loer bound on processing of MultiRAW At any tie t, consider any active job j in MultiRAW We define σ(j, t) such that the next available ork of j belongs to a phase that is parallel up to σ(j, t) processors We call σ(j, t) the saturated nuber of j at tie t Intuitively, any processor allocated to j beyond σ(j, t) is asted and cannot further increase the processing rate At any tie t, let R(t) be the set of jobs j ith positive adjusted eight, ie, (j, t) > Note that R(t) is the set of jobs being processed by MultiRAW and the processing rate of a job j R(t) is the axiu of its processing rate in the log groups We classify the ork of j into to types as follos We label the ork as unsaturated if it is processed by no ore processors than its saturation nuber in all groups; and label the ork as saturated otherise We call a job j R(t) saturated at tie t if MultiRAW is processing its saturated ork, and call j unsaturated otherise Intuitively, by running j on log groups, MultiRAW guarantees at least one group has a sufficient processing rate on j LEMMA At any tie t, consider any job j R(t) If j is unsaturated, the processing rate on j is at least ( + ϵ) (j,t) a (t) Q( a(t) ) If j is saturated, the processing rate on j is at least ( + ϵ) σ(j, t) Q( (j,t) ) σ(j,t) PROOF If j is unsaturated, consider the processing rate of j by the first group, hich has /2 processors each running at speed 2( + ϵ)q( a(t) ) The job j receives (j,t) /2 a fraction of the processors Since j is unsaturated, the processing rate on j equals (t) /2 (j,t) 2( + ϵ)q( a(t) a ) Since Q is non-decreasing and (t) /2 a(t) a(t), e obtain the desired bound /2 If j is saturated, e let the k-th group be the group such that (j,t) k σ(j, t) < 2 a(t) k (j,t), here a(t) k is the nuber of processors in the group Note that k ust exist since σ(j, t) = log (j,t) log a The processing rate on j by this group (t) equals (j,t) k 2( + ϵ)q( a(t) a(t) k ) ( + ϵ)σ(j, t)q( a(t) k ) Since a(t) k (j,t) (j,t), e obtain the desired bound σ(j,t) σ(j,t) 43 Bounding eighted flo of MultiRAW To bound the eighted flo of MultiRAW, e identify a subset of jobs such that the eighted flo of MultiRAW is at ost a constant ties the total eighted flo of these jobs in MultiRAW Then, e upper bound this cost by the eighted flo of OFF and the cost of OPT in Section 44, and the desired result follos No, e copare MultiRAW against OFF For any job j and tie t, let q a (j, t) and q o (j, t) be the reaining aount of unsaturated ork of j in MultiRAW and OFF, respectively In particular, q a(j, t) (resp, q o(j, t)) is zero if j has been rejected by MultiRAW (resp, OFF) by tie t Recall that OFF only rejects j at r(j) We define the aount of lagging ork of a job j, denoted x(j, t), in the sae ay as in Definition 5 in Section 3 Consider any tie t It is useful to have a detailed breakdon of the set R(t) of jobs being processed by MultiRAW We divide R(t) into to sets S(t) and U(t) containing the saturated and unsaturated jobs, respectively We further divide U(t) into L(t) and N(t) hich contain jobs ith x(j, t) > and x(j, t) =, respectively We call L(t) the lagging jobs and N(t) the nonlagging jobs Recall that the total adjusted eight of jobs in R(t) is a (t) Let ϕ(t) = j L(t) (j, t) Siilar to Lea 4 in the single processor setting, e try to bound the total eighted flo tie of MultiRAW by eighted flo tie incurred due to jobs in R(t) \ L(t) LEMMA (WEIGHTED FLOW OF MULTIRAW) a (t) (( a (t) ϕ(t)) + o (t)), here o (t) is the total eight of active jobs in OFF at tie t The proof is siilar to that of Lea 4 In particular, by Lea, the processing rate of each unsaturated job j in MultiRAW is at least (+ϵ) (j,t) a(t) a Q( ) On the other hand, the processing (t) rate of an unsaturated job j in OFF is at ost Q( o(t) ) Hence, by redefining the potential function Φ(t) to γ n a(t) i= f( h(j i) ) x(j i, t), e can prove a lea identical to Lea 5 (see Appendix A) Then e can prove Lea in the sae ay as Lea 4 44 Bounding non-lagging jobs and saturated jobs In the folloing, e sho that (a(t) ϕ(t)) can be bounded by the eighted flo of OFF and the cost of OPT Let C be set of jobs rejected by OPT, hich is also the set of jobs rejected by OFF Let C(t) = C (S(t) N(t)) Then, a (t) ϕ(t) = j C(t) (j, t)+ j N(t)\C(t) (j, t)+ j S(t)\C(t) (j, t) 25

7 We further break don the proof into to parts Let W, E and C be the eighted flo, energy and penalty of OPT, respectively Let W o be the eighted flo of OFF LEMMA 2 (j, t) + (j, t) W o + C j C(t) j N(t)\C(t) PROOF For each j C(t), j reains in MultiRAW for at ost c(j)/(j) units of tie, incurring a eighted flo of at ost c(j) Hence, j C(t) (j) j C c(j) = C For each j N(t) \ C(t), j is unfinished by OFF at tie t Hence, j N(t)\C(t) (j) o(t) = W o, here o(t) is the total eight of active jobs in OFF at tie t The lea follos by observing that (j, t) (j) for any job j at any tie t LEMMA 3 (j, t) W + E j S(t)\C(t) PROOF Consider any job j not rejected by OPT Let (j) be the union of all saturated ork in j We divide the saturated ork (j) into infinitesial pieces {x, x 2, } such that () ithin a piece of ork, the saturation nuber reains the sae; and (2) MultiRAW processes each piece continuously at a fixed rate, and so as OPT; and (3) there is no job arrival, rejection or copletion during the processing of a piece Each piece is infinitesial For any piece x (j), e let σ(x) be its saturation nuber Let s(x) and s (x) be the processing rate on x by MultiRAW and OPT, respectively Let (x) be the adjusted eight of j hen x is being processed by MultiRAW Let p(x) be the size of x Let S be the set of jobs that are not rejected by OPT Then, (j, t) = p(x) s(x) (x) j S(t)\C(t) j S x (j) Since x is a piece of saturated ork, by Lea, s(x) ( + ϵ)σ(x)q( (x) ) > σ(x)q( (x) ) Hence, p(x) σ(x) σ(x) s(x) (x) p(x) (x)/σ(x) p(x) (j)/σ(x) The last inequality coes Q( (x)/σ(x)) Q((j)/σ(x)) fro the fact that is increasing ith y (as Q is nonnegative and y Q(y) concave) and (x) (j) Consider the sae piece of ork x The eighted flo incurred by x in OPT is p(x) s (x) (j) OPT processes x at the rate s (x); the ost energy efficient ay is to use σ(x) processors each running at speed s (x) The rate of energy usage is σ(x) P σ(x) (s (x)/σ(x)) Considering the cost in OPT, e have W + E ( ) p(x) p(x) s (j) + (x) s (x) σ(x) P ( s (x) σ(x) ) j S x (j) It reains to sho (j) + σ(x) P ( s (x) ) (j)/σ(x) To s (x) s (x) σ(x) Q((j)/σ(x)) this end, e consider to cases If s (x) σ(x)q((j)/σ(x)), then (j) (j)/σ(x) P (y) ; otherise, e note that is increasing ith s (x) Q((j)/σ(x)) y y σ(x) s (x) P ( s (x) σ(x) ) = P (s (x)/σ(x)) s (x)/σ(x) = P (Q((j)/σ(x))) Q((j)/σ(x)) (j)/σ(x) Q((j)/σ(x)) The last equality follos fro the fact that σ(x)t s (x) > σ(x)q((j)/σ(x)) Therefore, T > Q((j)/σ(x)) and e have P (Q((j)/σ(x))) = (j)/σ(x) 45 Conclusion Proof of Theore 7 We are ready to prove Theore 7 PROOF (Proof of Theore 7) By Leas 2 and 3, e have ( a(t) ϕ(t)) W o + W + E + C Together ith Lea and that <, the eighted flo of MultiRAW is 2 at ost (( + )W o + W + E + C ) (5W o + W + E + C ), hich by Lea 9, is at ost 25 ties the cost of OPT The total energy usage of MultiRAW is at ost log ties its eighted flo tie The rejection penalty of MultiRAW is at ost its eighted flo tie Hence, MultiRAW is at ost ( 25 (log + 2))-copetitive against OPT Finally, putting = ϵ, e conclude that MultiRAW is ( + 2(+ϵ) ϵ )2 (log + 2)- copetitive So far, e have assued that is a poer of to If is not a poer of to, let = /2 and i = ( i j=, j)/2 here i is the nuber of processors in the i-th group We can sho that i 3 i+ Repeating the arguent of Lea, e can sho that each saturated job j has a processing rate at least 2 ( + ϵ) σ(j, t) Q( (j,t) ) The only consequence is that the 3 σ(j,t) right-hand-side of Lea 3 is eakened to 5(W + E ) The copetitive ratio of MultiRAW is increased by a factor of becoes 2( + ϵ )2 (log + 2) 3 25 and Rearks on fixed speed setting If all the processors have speed one and energy is not a concern, e can largely siplify the algorith by running a single group ith all the processors We assue that the online algorith is given ( + ϵ)-speed processors and e share the processors to the active jobs proportional to the adjusted eight (j, t) Then, siilar to Lea, e can sho that each unsaturated job is processed at a rate at least (j,t) a(t) ( + ϵ) and each saturated job is processed at a rate at least ( + ϵ)σ(j, t) We can check that Leas, 2 and 3 reain true, here E = Hence, the total eighted flo is at ost 25 ties the cost of OPT The rejection penalty of the online algorith is at ost its total eighted flo tie Note that = ϵ 2(+ϵ) Hence, e have the folloing theore THEOREM 4 For the fixed speed setting, there is a ( + ϵ)- speed 2( + ϵ )2 -copetitive algorith Note that for each job j, e only process one copy of j Hence, the results hold even if the jobs are non-checkpointable 5 REFERENCES [] S Albers Energy-efficient algoriths Counications of the ACM, 53(5):86 96, 2 [2] N Bansal, A Blu, S Chala, and K Dhadhere Scheduling for flo-tie ith adission control In Proc ESA, pages 43 54, 23 [3] N Bansal, H L Chan, and K Pruhs Speed scaling ith an arbitrary poer function In Proc SODA, pages 693 7, 29 [4] N Bansal and K Dhadhere Miniizing eighted flo tie ACM Transactions on Algoriths, 3(4):39, 27 [5] H L Chan, J Edonds, T W La, L K Lee, A Marchetti-Spaccaela, and K Pruhs Nonclairvoyant speed scaling for flo and energy Algorithica, 6(3):57 57, 2 [6] H L Chan, J Edonds, and K Pruhs Speed scaling of processes ith arbitrary speedup curves on a ultiprocessor In Proc SPAA, pages,

8 [7] S H Chan, T W La, and L K Lee Non-clairvoyant speed scaling for eighted flo tie In Proc ESA, pages 23 35, 2 [8] S H Chan, T W La, and L K Lee Scheduling for eighted flo tie and energy ith rejection penalty In Proc STACS, pages , 2 [9] J Edonds Scheduling in the dark Theor Coput Sci, 235():9 4, 2 [] J Edonds and K Pruhs Scalably scheduling processes ith arbitrary speedup curves In Proc SODA, pages , 29 [] B Kalyanasundara and K Pruhs Speed is as poerful as clairvoyance Journal of the ACM, 47(4):67 643, 2 [2] R Motani, S Phillips, and E Torng Nonclairvoyant scheduling Theor Coput Sci, 3():7 47, 994 [3] F Yao, A Deers, and S Shenker A scheduling odel for reduced CPU energy In Proc FOCS, pages , 995 [4] J Zhu, H L Chan and T W La Non-clairvoyant eighted flo tie scheduling on different ulti-processor odels To appear in Proc WAOA, 2 Appendix A: Lea 5 for Multi-processor Setting In this appendix, e consider > processors and prove Lea 5 for the ulti-processor setting Recall that j, j 2,, j na (t) are the active jobs in MultiRAW at tie t arranged in increasing order of (ji) As entioned in Section 4, Φ(t) = γ n a(t) i= f( h(j i) ) x(j i, t), here γ = and (+ϵ) f(h) = h/q(h) arrival ties, and h(j i) = i k= Lea 5 At any tie t ithout discrete events, ax{ a(t), o(t)} ( )ϕ(t) dφ 2 PROOF We consider dφ as a cobined effect due to the action of MultiRAW and OFF Let dφ a and dφ o be the rate of change of Φ due to MultiRAW and OFF, respectively Then dφ = dφ a + dφ o Note that OFF is an offline algorith on a single processor ith axiu speed T, hich at any tie t, follos the speed Q( o(t) ), here o(t) is the total eight of active jobs in LLB Consider the schedule of MultiRAW For each active job j in MultiRAW, if j L(t) R(t), j is an unsaturated job, x(j, t) > and h(j) ( ) a(t) By Lea, MultiRAW processes j at a rate ( + ϵ) (j,t) Q( a(t) a(t) ) Hence, for each ter γ f( h(j) ) x(j, t) here j L(t), the ter is decreasing at a rate at least γ γ h(j)/ ( + ϵ) (j,t) Q( a(t) ) Q(h(j)/) a (t) ( ) a(t)/ ( + ϵ) (j,t) a(t) Q(( ) a (t)/) a Q( ) (t) γ ( + ϵ) (j, t) = (j, t) here the first inequality coes fro that f(h) = h/q(h) is increasing ith h, and the second inequality coes fro that Q is increasing For each job j / L(t), the ter γ f( h(j) ) x(j, t) is nonincreasing Hence, dφ a j L(t) (j, t) = ϕ(t) The processing rate of an unsaturated job j in OFF is at ost its speed on the single processor, hich is s o = Q( o(t) ) Thus, the orst case (in hich dφ o is the largest) occurs hen OFF processes the job ith axiu h(j), hich equals a (t), using all its speed s o Then dφo γ f( a(t) ) s o = γ a(t)/ Q( a (t)/) Q( o (t)/) If a (t) o (t), Q( a (t)/) Q( o (t)/) and dφ o γ a (t) Otherise, a (t) < o (t), since f is nondecreasing, f( a(t) ) f( o(t) ) Hence, dφ o γf( o(t) )so = γ o (t) Cobining these to cases, e obtain that dφo γ ax{ a(t), o(t)} Finally, notice that = 2 Hence, γ ax{ a (t), o (t)} = (+ϵ) ax{ a (t), o (t)} and the lea follos = 2 ϵ and 2(+ϵ) +ϵ = ax{a(t), o(t)} Appendix B: Offline Schedule Transforation This appendix shos the folloing lea on the perforance of LLB that transfors OPT (hich iniizes eighted flo plus energy on processors) to an offline algorith LLB on a single processor ith axiu speed T, hich at any tie t, follos the speed Q( b(t) ), here b(t) is the total eight of active jobs in LLB LEMMA 5 The eighted flo of LLB is at ost the eighted flo plus energy of OPT Before proving Lea 5, e first restate the algorith LLB given in Section 4 Algorith LLB Consider any tie t Let n b (t) be the nuber of active jobs in LLB For any job j, let p b (j, t) and p o (j, t) be the ork done on j up to tie t by LLB and OPT, respectively Furtherore, let d(j, t) = p o (j, t) p b (j, t) We say a job j is lagging at tie t if d(j, t) > Let y(j) be the latest tie hen j has becoe lagging; if j is non-lagging, let y(j) = t We denote the active jobs in LLB at tie t as j, j 2,, j nb (t), arranged in increasing order of y(j i ), ie, y(j ) y(j 2 ) y(j nb (t)) Let l be the nuber of lagging jobs ( l n b (t)) Let s o be the total speed of the processors in OPT at tie t Let s x s o be the total speed OPT assigns to all jobs j ith d(j, t) = LLB sets its speed s b = Q( b(t) ), and focuses on the job j a defined to be j l if l >, and j nb (t) otherise Details are as follos: If OPT is not running any job j i ith d(j i, t) =, then LLB runs j a ith speed s b 2 If OPT is processing soe jobs j i ith d(j i, t) =, but s x s b (ie, LLB has enough speed to prevent all of the fro becoing lagging), then LLB follos OPT s speed on each of those jobs and runs j a ith the reaining speed s b s x 3 Otherise (s b < s x ), LLB follos OPT s speed for each of those jobs j i, in descending order of the index i, until there is soe job j hich LLB cannot follo OPT s total speed on j LLB then assigns all of its reaining speed to j Note that any job phase has a speed-up function Γ satisfying Γ(y) = y for y (, ] Therefore, LLB using a single processor ith axiu speed T can guarantee all its speed to be fully utilized to process the jobs and no speed ould be asted To prove Lea 5, e exploit potential functions Let F b be the total eighted flo of LLB, and let F o and E o be the total eighted flo and energy of OPT, respectively For any tie t, let F b (t) be the total eighted flo incurred up to tie t by LLB Define F o(t) and E o(t) siilarly We derive a potential function Φ(t) that satisfies the folloing three conditions: Boundary condition: Φ = before the first job arrives and after all jobs are finished Discrete event condition: Φ does not increase hen a job arrives, or is copleted by LLB or OPT, or hen a lagging job is changed to non-lagging or vice versa 253

9 Running condition: At any other tie t, df o (t) df b (t) + dφ(t) Integrating these conditions over tie, e can conclude that F b F o + E o ; Lea 5 follos Potential function Φ(t) Consider any tie t Recall that the active jobs in LLB are denoted as j j nb (t) Define the coefficient c i of j i to be i k= (j k) The potential function Φ(t) is defined as follos Φ(t) = n b (t) i= f(c i) ax(d(j i, t), ) here f(x) = P (P ( x )) Note that P is the first derivative of P Since P is convex, P is non-decreasing, hich together ith that P (x) is non-decreasing, iplies that P (P (x)) is also non-decreasing Therefore, f(x) = P (P ( x )) is a nondecreasing function of x The boundary condition is obvious No e check the discrete event condition Recall that l is the nuber of lagging jobs When a job j arrives at tie t, e have d(j, t) =, y(j) = t and the coefficients of all lagging jobs of LLB reain the sae, so Φ does not change When OPT copletes a job or LLB copletes a nonlagging job, Φ does not change When LLB copletes a lagging job or a lagging job changes to non-lagging at tie t, that job ust be j l and d(j l, t) The coefficients of other lagging jobs stay the sae, so Φ does not change When one or ore job(s) changes fro non-lagging to lagging, those jobs ust have the largest index aong all lagging jobs and their d(j, t) s are all The coefficients of other lagging jobs stay the sae, so Φ does not change It reains to check the running condition Consider any tie t ithout discrete events Recall that s b is the current speed of LLB, and s o is the current total processor speed of OPT Let o(t) be the total eight of active jobs in OPT Then, = b (t) and df o = o(t) Since P is convex, the energy usage of the processors in OPT is de o P ( s o ) If all active jobs in LLB are non-lagging, ie, d(j i, t) for all i n b (t), then these jobs are also active in OPT and hence o(t) b (t) In this case, Φ reains zero and thus dφ(t) = Then the running condition follos easily since df b(t) = b (t) o(t) df o(t) Henceforth, e assue at least one active job in LLB is lagging, ie, l > We define a real nuber such that b (t) = c l, hich is the total eight of lagging jobs Then the rate of change of Φ can be bounded easily (Lea 6) More interestingly, e can also sho that at any tie t, Q( b (t)) T (Lea 7) LEMMA 6 dφ(t) f( b (t)) ( s b + s o) PROOF We consider ho Φ changes in an infinitesial aount of tie (fro tie t to t + ) here no job arrives, or copletes, or changes fro lagging to non-lagging or vice versa Recall that s x is OPT s total speed on jobs j i ith d(j i, t) = Then, OPT s total speed on jobs j i ith d(j i, t) is s o s x In the definition of LLB, if Case 3 happens at t, soe jobs including job j becoe lagging, hich corresponds to a discrete event condition Right after t, j becoe the ne j a and LLB ill follo Case or Case 2 Thus, it suffices to consider Cases and 2 for the running condition Consider each job j i ith d(j i, t) = By Cases and 2 in the definition of LLB, either both LLB and OPT are not orking on j i, or they are running j i ith the sae speed (totaling to s x for all such j i s) Therefore, d(j i, t) reains and the processing of j i does not affect the value of Φ No consider other jobs j i ith d(j i, t) Recall that LLB is running j l ith the reaining speed s b s x Since LLB is using df b a single processor ith axiu speed T, j l is also processed by LLB at a rate of s b s x Consider the processing of OPT The orst case that causes the largest increase in dφ is that OPT is also using all its reaining speed s o s x to run j l It is because any job j k ith k > l is non-lagging and OPT cannot increase Φ by processing j k Also, OPT cannot increase Φ by processing a job that is not active in LLB In this case, d(j l, t) changes at a rate of at ost (s b s x)+(s o s x) = ( s b +s o) Therefore, Φ changes at rate at ost f(c l ) ( s b + s o) = f( b (t)) ( s b + s o) LEMMA 7 At any tie t, e have P ( b(t) ) T PROOF We ill prove b(t) P (T ); then the lea follos by taking P on both sides of the inequality Consider the current tie t If b(t) P (T ), then b(t) b(t) P (T ) If b (t) > P (T ), let t be the last tie before t here b(t) P (T ) Then the total eight of lagging jobs at t is at ost P (T ) At any tie after t, LLB s speed is T Since the total speed of OPT is also at ost T, by the definition of LLB, for each job arrived after t, LLB s progress is at least OPT s progress Hence, those ne jobs cannot be lagging Furtherore, Case 3 in the definition of LLB does not occur, and thus the total eight of lagging jobs cannot increase after t and reains at ost P (T ) at tie t In other ords, b (t) P (T ), ie, P (T ) b (t) We are no ready to prove the running condition, hich together ith the boundary and discrete event conditions, iplies Lea 5 LEMMA 8 At any tie t ithout discrete events, df o(t) dφ(t) df b (t) + PROOF Recall that j l is the lagging job ith the largest index, and j l+,, j na are non-lagging and ust also be active jobs in OPT The total eight of these jobs is b (t) b (t) = ( ) b (t), so o (t) ( ) b (t) Also recall that df b(t) = b (t) and df o(t) o(t) + P ( s o ) ( ) b(t) + P ( s o ) We note a fact (given in [3]) that for any real x, P (P (x)) s o P (x) P (P (x))+p (s o) x Lea 6, together ith this fact, gives df b (t) + dφ(t) b (t) + P (P ( b(t) )) ( s b + s o ) b (t) + ( s b )P (P ( b(t) )) + P (P ( b(t) ))( so b (t) + ( s b )P (P ( b(t) ))+ (P ( b(t) ) P (P ( b(t) so )) + P ( ) b(t) ) b (t) b (t) + P ( s o )+ P (P ( b(t) )) ( s b + P ( b(t) )) Since s b = Q( b(t) ), here Q(y) = in(p (y), T ), e have s b = in(p ( b(t) ), T ) By Lea 7, s b in(p ( b(t) ), P ( b(t) )) = P ( b(t) ) Thus, the last ter of the above inequality is at ost, and hence df b(t) dφ(t) ( ) b (t) + P ( s o ) df o(t) ) + 254