Omahen, K.

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Purdue University Purdue e-Pubs Department of Computer Science Technical Reports Department of Computer Science 1975 Some Insights Into Deterministic Scheduling Problems Involving Multiple Resources and Preemption K. Omahen Report Number: 75-155 Omahen, K., "Some Insights Into Deterministic Scheduling Problems Involving Multiple Resources and Preemption" (1975). Department of Computer Science Technical Reports. Paper 102. https://docs.lib.purdue.edu/cstech/102 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information. Some InsIghts Into Deterministic Scheduling ProblemsInvolvIng Multiple Resource. & PreemptionK. OmahenComputer ScIence DepartmentPurdue UniversityWest lafayette, IN 47907CSD-TR 155July 1975Keywords: deterministIc scheduling, preemptive-resume rules,multiple-resource scheduling, minimizing completion timeAbstractThis paper considers deterministic scheduling problems for systemshaving a nUmber of different resource types and an arbJtrary number ofunits of each resource. General results are obtained for preemptive-resume scheduling rules under the condition of zero preemption costs.Schedule completion time 15 the objective function considered, and thepaper examines the relative effect of demand scheduling and precedenceconstraints on the minimum schedule completion time.2IntroductionWith the exception of multiple-server systems, few scheduling problemsarising from multiple-resource systems have been treated In the literature.This paper attempts to offer Insights Into one cldss of muJtlpJe-resourcescheduling problems and to provide some general results as a by-product. AformaJ description of the problem to be considered Is given below:Problem.We are given a system having the folloWing resource characterfstics:I • No. of Types of ResourcesR1 ... Amount" of Resource-Type-I In System, 1.1.2, ...•1.A set of Jobs as described below are to be scheduled so as to minimizeschedule completion time (I.e., the tIme to fInish all jobs):N a No. of Jobs to be scheduled.J = Symbol denoting the nth Job, n. 1.2, •.• ,N.n _Vn ... [rnl,rn2····,rnI] ... Vector describing the resource requl"rementsof J ; I.e., r 1 units of Resource-Type-I, r 2 units of Resource-n n - nType-2. etc.Pn • Processing TIme for Jni this is the amount of time that theresources V are s' mu' taneous 1y requ Ired. If the reSOurce re-nqulrement V Is satisfIed. Job J progresses at unit rate.n nThe scheduling rule to be used Is assumed to be of the preemptive-resumetype with zero preemption costs, and the four cases to be considered are:Case 1. No Precedence ConstraJnts, Non-Demand Sche~ulln9Case 2. No Precedence Constraints, Demand SchedulingCase 3. Precedence Constraints, Non-Demand SchedulingCase 4. Precedence Constraints. Demand SchedulingProblems examined In the past have assumed that the scheduling decisioninvolves choosing the next job to be scheduled; such a viewpoint Is areasonable one to take when developing heuristic rules but Ignores thefundamental nature of the problem. This author believes that the followingalternative view Is more appropriate;Viewpoint. Scheduling rules for multiple-resource systems shouJd, ata schedUling epoch, be concerned with the choosing of some combinationof Jobs whose characteristics allow for simultaneous processing.3In order to demonstrate the utility of this approach, additional notationmust be introduced:,M• Number of distinct feasible Job combinationsC • N x I column vector denoting a feasible combination of Jobs,m"" a subset of the set of Jobs {InII ~ n ~ N},XmlXm2where XmnIf I n Is Included In em'otherwise.For a job combination em to be "feaslble," we require thatNEJ=Ir • 1,2, .•• ,1.That Is, the total resources required of all Jobs In the combinationmust not exceed the amount actually present In system.If we pursue the vtewpoJnt that the scheduling rule should be concerned withchoosing combinations of Jobs for processing, we may denote a schedule S as5 = [(Il'tll.(i2,t2), •.. ,(IK,tKl]where each pair (Ik,tk) '5 lnterpreted to be a combination schedulingInterval In which feasible combination with Index 'k Is to be servicedfor t k units of time.The above notation specifies the first through Kth combination schedulingintervals which constitutes the schedule.Some InsightsObserve that It Is Immediately possible to make a statement concerning,the mInimum schp.dule completion time for the situation In which there areneither precedence constraints on the order In which jobs must be processednor any requirement for demand processing.Lemma 1. The minimum schedule completion time TIfor Case I (Preemptlve-Resume Scheduling, No Precedence Constraints, Non-Demand Scheduling)is the solution of the fOllowing linear Programming (lP) problem:n=J,2" ..• Nx *t '" pmn m nTr .. MinMr""'IMr'"'"Subject to the N equality constraintswhere tma .(non-negatlve) amount of time that the )E.th combinatIon,Cm, Is to be serviced In total to obtain the mInimumschedule COmpletion time;Pn a proCessing time for Job In;Xmna element of column matrIx CmIndicating Whether Job Inis present 10 the COmbination.!r~, The schedule completion time Is the sum of the amount of timespent In prOCessing each Possible combination of Jobs, and the conStraintsreflect the reqUirement that, for each Job, the amount of time spentIn combinations In Which the Job "progresses" must equal the proCesSingtime request of that Job. 0""" ,."., '" ''''''' ~ ,.-, ,.. """ """ "'''''',., ''''''. ">, ,., " ..... 0»,••• , " •• ,."'".,. ,••••Jfor which tmhas a non-Zero value WOuld be acceptable, Well-knOWn lPresults also Imply that no more than J of the combinations need be assignednon-zero Values and that the Set of J column vectors, A, will be linearlyindependent, whereA • iC It> 0),m mIt should be ObvIOUs that ImpoSing demand schedulIng and/or precedenceconstraints on the Set of Jobs Can neVer result In a smaller schedulecompletion than WOuld be obtained Using lemma I. However, there might existSituations In Which the minimum schedUle cOmpletion time might be Increased,due to demand scheduling and/or precedence conStraints, and therefore thesefactors will be next examined.for mUltiple-server sYstems, demand schedul'ng has been taken to meanthat no processor W/I' remain Idle If there Is some Job Which might beserviced. for mUltiple-resource sYStems thIs defln'tlon Is not adequatebecause ambigUities arise When there are many types of resOUrces in sYStem.5If we consider a scheduJlng epoch to be the instant at which a differentcombinatIon of Jobs goes Into service, a" more suitable definition of demandscheduJ Ing may be given In terms of a "max lma l comblnatlon."Maximal CombInation (when no precedence constraints): For a given setof Jobs In a multIple-resource system, each with resource-requirementsand (remaining) processing time requests, a (feasible) combination C.. IIs said to be a maximal comblnatl9" If there does not exist another(feasible) combinatIon CJ such that C1 Is properly contained In c)(i.e., such that every Job in C, Is also contained In ej ).Having defIned a maxima) combination, demand ~chedullng may be better describedas shown belOn!.Demand Scheduling:(for the remainingAt each scheduling epoch, a maximal combinationJobs In system) Is chosen for processing.The next result to be presented will apply to Case 2 (I.e., withdemand scheduling Imposed on Case I); this result demonstrates that theminImum schedule completion time T2 will always be the same as that possiblefor Case 1 where demand scheduling Is not required.lemma 2. The minimum schedule completIon time T2 for Case 2 (Preemptive-Resume Rule, No Precedence Constraints, Demand SchedulIng) will alwaysbe equal to the value TI of lemma 1.Proof. Given a schedule of minimum length 1 1 given by lemma I, aschedule which meets the requirements of demand scheduling may alwaysbe constructed using the algorIthm given In Appendix 1. I:JDemand schedulIng has the effect of Imposing constraints on the order Inwhich combinations may be processed because, whenever a combination CI Isproperly contained In some combination CJ, combination C, may be chosenfor servicing only if It Is impossfble to form combInation CJfrom the re-maining uncompleted Jobs In system.Job precedence constraints specify a partla) ordering on the set ofjobs, whereI .; nimplies that the ith Job must be processed to completion before the servicingI\!6of the ~th Job can commence. The effect of an arbItrary set of Job prece-dence constraints Is illustrated in the following lemma:Lemma 3. The minimum schedule completion time T3for Case 3 (Pre-emptive-Resume Rule. Precedence Constraints. Non-Demand Scheduling)will always be greater than or equal to the value T) of Lemma 1.Proof. Consider the effect of the Job precedence constraints on thepossible schedules that ,might be generated for Case 3.(a) We mLlst ff rst eliminate from consideration any Job comblna-t I on em such that J. E C , J E C • and J I < Infor I rI- n.I m n m(b) For the remaining job combinations. the job precedence re-lations have the following effect:(I) A combinatIon precedence relation between Ckand emwill be denoted as Ck < em If each job precedencerelation between a job In Ck and one In CmIs of theform J j < I n, where J I E Ck and I nE Cm.(II) If J, < I n and JJ < JR. for some J 1 ,JR. E Ckand JJ,JnE em'then combinations Ck and Cm may not be present In thesarre schedule. That Is,t k > 0 Implies tm.. 0andt m > 0 Implies t k• 0where t k and t m are combination processing times.(IIi) If there are no precedence relations between any pair ofdifferent jobs, one In combInation CIand the second Incombination em' there is no combination precedence rela-tfon defined between Ciand Cm.The meanIng of a combinatIon precedence relation, CI< Cm' Is that ascheduJe containing combination scheduling intervals for both Cland Cmmust be such' that the IntervaJ for Cfprecedes the one for Cmo Theproof Is compJeted as follows. A well-known LP result Is that thesolution to the minimization problem of Lemma lies at an extreme pointfor which at most J combinations are assigned non-zero combination7processing times. SItuations (a) and (b - II) descrIbed above Causeone or more of the extreme points to be excluded due to the presenceof the Job precedence constraints, and It follows that 13 ~ Tt " [)We complete the coverage of the four cases by constderSng the effectof having both precedence constraInts and a requIrement for demand process-ing (Case ~). The approach taken here Is sImIlar to that for Lemma 2 exceptthat It wilt be shown that. gIven a solution to the Case 3 sltuation t onemay always construct a demand schedule having t~e same completion time.We again Interpret demand schedUlIng to mean that a max Sma 1 combination(for the remaining Jobs in system) is chosen at each scheduling epoch. How-ever, the presence of precedence constraints causes the definition of maximalcombInation to be modified:Maximal Comblnatlo~ (when precedence constraints): Given 8 set A ofjobs £Jnl having non-zero (rema1nlng) processing times, we consideronly those comblnatfons which can be formed from set S, a subset of Ahaving the followtng properties:B a {J IJ E A and a J =" (J < J and J E A))-n nm m n mThat Is (in the context of the Jobs In A), we consider only thosejobs for which either no precedence relations are defined or whichno Job precedes. A (feasible) combInation Cr Is said to be a maximalcombination If there does not exist any other (feasible) combinationCJsuch that Cj is properly contaIned In CJr where both Cj and CJ are,formed from Jobs In B on1y.Lemma %. The minimum schedule completion time T% for Case % (Pre-emptlYe-Resume Rule, Precedence Constraints, Demand Scheduling) will,always be the same as the minimum schedule completion time 13 forCase ).Proof •. Given a mInimum length schedule for Case ), a schedule forCase 4 may always be constructed uslng the same algortthm given rnAppendIx 1. Further details of the algorithm as It applIes to thecase where both precedence constraints and demand schedUling arepresent Is given In AppendIx 2.BThe preceding lenrnas wll J allow the following surrrnary to be gi Ven forthe relative effects of Job precedence constraints and demand schedulingon minimum schedule completIon time for preemptrve-resume rules for multl-resource systems.Theorem. For a multiple-resource system with a set of N Jobs{Jnll ~ n ~ N} to be 5ched~led, where each Job I n has an associatedresource requirement Vn and processing time Po' let 1, denote theminimum schedule completion time possible under each alternatlve-i1isted below:(I) Preemptive-Resume Rule, No Precedence Constraints, Non-Demand Scheduling(2) Preemptive-Resume Rule, No Precedence Constrarnts, DemandScheduJ ing(3) Preemptfve-Resume Rule, Precedence ConstraInts, Non-DemandSchedu1 Iog(4) Preemptive-Resume Rule, Precedence Constraints, DemandSchedulingThe relative values of T, are given by the following relations:T, • 12 ; T I .::.. T3; T2 .::.. 13 ;T, ~ 14 ; T2 ~ 14 ; T) • T~.Proof. Inmedlately follows from lemmas 1-4. cNo mention has been made concerning the effect of non-preemptiveschedulfng; this has been avoided because non-preemptive disciplines con-siderably complicate matters. Non-preemptive schedulJng requires that onebecome aware of the aJlowable transitions between Job combfnatlons, andthe descrIption of such transitions 1s somewhat unwleJdy for the generalcase. It Is Interesting to note that no anomalies have been observed Inthe above theorem; this suggests that anomalies such as those observedfor multiple-server systems are due to the Interaction between the con-straJnts Imposed by nonpreemptfve schedulIng and those 'mposed by precedenceconstraJnts and demand scheduling.9SummaryThis paper considered the deterministic scheduling problem for amultiple-resource system when preemptive-resume rules may be employed.The suggestion was made that the proper viewpoint to be taken for sucha system Is that the scheduling rule shouJd be concerned with choosingthe next combination of Jobs to he serviced rather than with the seJec-tlon of the next job for processing. Under the condItion that schedulecompletion time Is taken to be the objective function to he minimized,the followIng statements may be made when preemptive-resume rules areemployed:Job precedence constrafnts may cause an Increase in the minimumschedule completion time over that possible if the constraintswere not present.Demand scheduling never causes the minimum sc~edule completiontime to be greater than that possIble If demand scheduling werenot required.The problem of minimIzIng schedule completion time when thereare neither job precedence relations nor the requirement fordemand processing is a conventional Linear Programming problemIn which one determines the amount of tIme that each Job combina-tion should be serviced.Bibliography1. Omahen, K. Capacity bounds for multi-resource queues. Tech. Rpt.CSD-TR 137, Computer Science Department, Purdue University, March 1975.2. Kafura. D. G. Analysis of Scheduling Algorithms for a Model of aMultiprocessor Computer System. Ph.D. Dfssertatlon. Computer ScienceDepartment, Purdue University, August 1974.Jobs {J !P (k-I) > O.. n nwe proceed to (c).Included in combination Ik but notJob Indices defined as follows:B • {n IJ E :: I' • J $ CI }n k n k10Appendix 1Suppose that we are given a schedule 51 for Case 1 having minImumcompletion time and that we wish to construct a schedule for Case 2.bymeans of an ~Igorlthm whIch, at the ~th step, Insures that the combinatIonchosen for the kth epoch meets the requirements of demand scheduling.Define:Pn(k) c remaining processIng. time for job Inafter the completionof the ~th comblnat~on scheduling Interval In schedule S2(k).5Z(k) = [(Il'tl),(iZ·tZ)·····(IK·tK)]- schedule which results after k steps of the algorithm givenbelow, where 52 (0) = 510 Note that K need not have the samevalue for all 5Z(k).The algorithm consists of repeating the following step for k = 1,2, ..• untIla demand schedule Is obtained.Step - k: Assume that the first k-) combination scheduling Intervalseach satisfy the requirements of demand scheduling. Examine the kthcombination In schedule S2(k-l). ik:(a) If i k Is a maximal combination for the set ofl.,::.n ~N}, S2(k) Is taken to be S2(k-l), andOtherwise, go to (b).(b) GIven C, Is properly contained In some maximal combination ell'k kwe may redlstrlb~te the remaining processIng times of those JobsLet B denote a set ofWe replace the kth combination scheduling Interval In S2(k-l),(ik,tk), wIth the interval (ik,tk), wheret k' = min[tk.mln [p (k-1)]l).nEB nFor each nEB, we systematically modIfy the remainder of theschedule S2(k-l) so as to subtract the portIon of the processing· '.II(b) Continued.times which are now Included In the kth Interval. The quantitytic. represents the amount of processing time for Job Inwhichmust be subtracted off from the remaining combination schedulingintervals. These remaining Intervals (Ik+1,tk+I), etc. must besequentially examined and modified until thIs amount of processingtime tic. for Job I n has been properly subtracted. Suppose that weare examining {I J ,tJ> for j > k and that wnrepresents the pro-cessing time for Job J which has yet to be subtracted (w an ntic. Initially). If I n e e lJ , we proceed to the next combinationInterval. Otherwise, one of the followrng actions takes place:(I) If t J ~ wn ' we replace (IJ'tJ> In the schedule withinterval (Il.t j ), where ell'" Ci - I n , Also, weJ Jdecrement quantIty wn by amount t j.(1) If t J > wn ' replace OJ'tJ) In the schedule with twocombInation schedulfng Intervals (Ij,wn),(lj.tj-wn)where elj = e lj - I n. Also, quantity wn Is set to zero.For a given value of n, additional combInation scheduling inter-vals are examined & modIfied until the amount of processfng timetk for job I n has been subtracted entIrely. ThIs process Js thenrepeated until all values of n Included In the set of Indices Bhave been treated. The reSUlting schedule Is then defined as S2(k}.{e} Update the remaining processing tImes; I.e.,P (k) a P (k-I) - X *tkfor n • 1,2, •.• ,Nn n mnwhere (Ik,tk) Is the kth combination scheduling Interval In 52(k)andm-I k•This process' 15 guaranteed to termInate In a fInite number of steps, andthe final value of 52 (k) Is a demand schedule of the same length as 51because the algorithm never causes the completion time to Increase and adecrease In completion time would contradict the assumption that 51 Is ofminimum length.12Appendix 2The algorithm described In Appendix 1 works equally wetl when we aregiven a schedule 53 for Case 3 haYI~g minimum completion time possible andwish to construct the correspondl~g demand schedule 54 for Case 4. Thekey to belIeving that the algorithm stili works properly Is to understandthe ImplicatIons of the definitIon of demand scheduling when precedenceconstraints are present. As mentioned In the main body of the paper, theJobs included In a maxImal combination are not preceded by any of theremaining Jobs In system. It Is this characteristic tha~ prevents anyconflicts In precedence constraints from arising when" the algorithm system-atically modifies the scheduJe.'.

Some Insights Into Deterministic Scheduling Problems Involving Multiple Resources and Preemption

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Some Insights Into Deterministic Scheduling Problems Involving Multiple Resources and Preemption

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