Online Scheduling with Makespan Minimization: State of the Art Results, Research Challenges and Open Problems
aa r X i v : . [ c s . O S ] J a n Online Scheduling with Makespan Minimization: Stateof the Art Results, Research Challenges and OpenProblems
Debasis Dwibedy a, , Rakesh Mohanty a a Department of Computer Science and Engineering, Veer Surendra Sai University ofTechnology, Burla 768018, Odisha, India
Abstract
Online scheduling has been a well studied and challenging research problem overthe last five decades since the pioneering work of Graham with immense prac-tical significance in various applications such as interactive parallel processing,routing in communication networks, distributed data management, client-servercommunications, traffic management in transportation, industrial manufactur-ing and production. In this problem, a sequence of jobs is received one by onein order by the scheduler for scheduling over a number of machines. On arrivalof a job, the scheduler assigns the job irrevocably to a machine before the avail-ability of the next job with an objective to minimize the completion time of thescheduled jobs.This paper highlights the state of the art contributions for online schedulingof a sequence of independent jobs on identical and uniform related machineswith a special focus on preemptive and non-preemptive processing formats byconsidering makespan minimization as the optimality criterion. We present thefundamental aspects of online scheduling from a beginner’s perspective alongwith a background of general scheduling framework. Important competitiveanalysis results obtained by well-known deterministic and randomized onlinescheduling algorithms in the literature are presented along with research chal-lenges and open problems. Two of the emerging recent trends such as resourceaugmentation and semi-online scheduling are discussed as a motivation for fu-ture research work.
Keywords:
Competitive Analysis, Makespan, Online Algorithm, OnlineScheduling, Parallel Machines, Semi-online Scheduling
Email addresses: [email protected] (Debasis Dwibedy), [email protected] (Rakesh Mohanty)
Preprint submitted to Computer Science Review January 15, 2020 . Introduction
Scheduling is a quintessential phenomenon in our daily life. Everyday weschedule meetings, set deadlines for projects, organize work periods, schedulegames, manage time table for the lectures of various courses, allot rooms andplan maintenance operations. Each of these activities consists of a finite numberof jobs. Our main objective is to finish the jobs in minimum possible time bydesigning an efficient schedule. When we have complete knowledge of all jobsat the outset and we make a schedule based on a sequence of available jobson multiple machines, it is referred to as
Offline Scheduling [1-4]. However, inpractical applications, all jobs are not known at the beginning. Mostly, jobsare available on the fly and upon receiving a job, the scheduler is constrainedto assign the job irrevocably to one of the machines with no information onthe future incoming jobs. Such a scheduling is known as
Online Scheduling [5]. Online scheduling has been well studied in the literature based on jobcharacteristics, machine models, availability pattern of the jobs and optimalitycriteria [6].
Motivation . Offline m -machine( m ≥
2) scheduling for makespan minimizationobjective was shown to be
N P − Complete by a polynomial time reductionfrom the well-known Partition problem in [7-10]. Further, in online scheduling,the unavailability of all jobs at the beginning poses a non-trivial challenge indesigning and analyzing the scheduling algorithms. However, online algorithm [11, 12] provides a framework for processing a sequence of inputs which are givenin an online fashion and the performance of such an algorithm is measuredwidely by competitive analysis method [13]. For a basic understanding, wepresent briefly about online algorithm and competitive analysis as follows.
Online algorithms receive a sequence of inputs one by one and processeach input irrevocably upon its availability to produce a partial output prior tothe arrival of the next input. The partial output is produced by considering cur-rent and past inputs. Let us consider a sequence of inputs J = h J , J , .........J n i of finite size n . At any time step t , the input sequence < J , J ...J t − , J t > is received by an online algorithm, where 1 ≤ t ≤ n . Each input J t is pro-cessed as soon as it is received with no information on future input sequence < J t +1 , J t +2 , ..., J n − , J n > . Therefore, partial outputs O , O , ..., O n − are pro-duced in an incremental way on the fly before the final output O n is obtained.In contrast, an offline algorithm receives the whole input sequence at the begin-ning and processes them simultaneously to produce the output O . Competitive analysis method [8] measures the worst-case performance of anyonline algorithm
ALG by its competitive ratio (CR), defined as the smallest pos-itive integer k ( ≥ J = { J , J , ..., J n } , the following inequality holds C ALG ≤ k · C OP T ,where C ALG is the makespan obtained by online algorithm
ALG for any se-quence of J and C OP T is the optimum makespan incurred by the optimal of-2ine algorithm
OP T for J . The Upper Bound (UB) on the CR obtained by ALG guarantees the maximum value of CR , for which the inequality holds for all le-gal sequences of J . The Lower Bound (LB) on the CR of any online problem X ensures that there exists an instance of J such that ALG incurs makespan C ALG ≥ b · C OP T , where b is referred to as LB for X . The performance of ALG is considered to be optimal or tight for any online problem X , when the UB onthe CR achieved by ALG for X is also shown to be the LB on the CR of X .Sometimes, the performance of ALG is referred to as tight if C ALG = k · C OP T for all valid input sequences of X . The objective of an online algorithm is toobtain a CR closer to the LB of the online problem or to obtain a CR nearerto 1. We formally state the online scheduling problem with inputs, output, objec-tive and constraints as follows. • Inputs.
A list M =( M , M , ....M m ) of m ( ≥
2) machines, a sequence J = < J , J , .....J n − , J n > of n ( >> m ) jobs are given, where each J i is characterized by its processing time p i , where 1 ≤ i ≤ n . • Output.
Generation of a schedule, where completion time of the job thatfinishes last in the schedule i.e. makespan( C max ) is the output parameter. • Objective.
Minimization of the C max . • Constraints.
Jobs along with their processing times are revealed to thescheduler one by one in order; a newly available job must be scheduledirrevocably before the arrival of the next job; a job can not be forcefullypreempted while it is in execution, but the preemptive variant of onlinescheduling supports the preemption of an ongoing job prior to its comple-tion.
Online Scheduling has been intrinsically and widely used in modern scientificand technical computations. In many real life scenarios, online scheduling hasbeen found either as an independent problem or a segment of a larger problem.Major application domains of online scheduling are mentioned in Table 1.We now briefly discuss about some important applications as follows.
An interactive multi-processing system receives jobs on the fly. Upon re-ceiving a job, it has to immediately respond by allocating resource(s) such asmemory or processing unit(s) to the job with no knowledge of the future jobs.In practice, an interactive multi-processing system may be the operating systemrunning on a parallel processing enabled computer, router of the communicationnetworks, web server, robot navigation and motion planning system.3 able 1: Major Applications of Online Scheduling
Domain(S) Overview of Applications
Computers[6, 12, 14] Interactive Parallel Processing, High Per-formance Computing, Computer basedSimulation and Modeling, Robotics, Dis-tributed Data ManagementNetworks[15] Routing, Client Request ManagementProduction and Manufacturing[16-17] Production control System, ManufacturingProcess ManagementTransportation[18-19] Traffic control and signaling
Orders from clients arrive on the fly to a production system. The resourcessuch as human beings, machinery equipment(s) and manufacturing unit(s) haveto be allocated immediately upon receiving each client order with no knowledgeon the future orders. Online arrival of the orders have high impact on therenting and purchasing of the high cost machines in the manufacturing units.
It is not known in advance the number of vehicles running on the road andpassing through the traffic squares at any instance of time. For an effectivetransportation system, online scheduling can be very useful for managing trafficsignals in various squares of a street or city.
This survey provides a comprehensive overview on the state of the art con-tributions for
Online Scheduling problem. In particular, the survey focuses ononline m -machine scheduling with respect to machine models such as identicaland uniform related; job characteristics such as preemptive and non-preemptive; optimality criterion such as makespan(please, see section 2 to understand ba-sic terminologies and notations related to the scope of our study). The surveypresents critical ideas, novel techniques along with important results obtained byseminal randomized and deterministic online scheduling algorithms to developbasic understanding from a beginner’s perspective without discussing much de-tail on the proof techniques. To make the survey less exhaustive, some of thewell studied areas such as online scheduling in real time systems, flow shop envi-ronment, unrelated machine, scheduling under machine availability or eligibilityconstraint, flow time objective and energy efficient scheduling are not covered. According to our knowledge, the existing literature lacks an exhaustiveoverview of the state of the art contributions of deterministic and randomizedonline scheduling algorithms. We present a summary of well-known surveys[6, 20-31] on online scheduling in Table 2. This survey studies the fundamen-4 able 2: Well-known Related Surveys on Online Scheduling
Year Author(s) Scope/Main Contributions α | β | γ . Overview of determinis-tic strategies for scheduling in various machinemodels such as single, uniform related and unre-lated machines, flow and open shop scheduling.1989 Cheng et al. [21] Job shop scheduling with deadline assignment.1998 Sgall [22] Scheduling strategies for variants of onlinescheduling by considering release time, preemp-tion and precedence constraints, speeds of ma-chines and shop scheduling.1998 Chen et al. [23] Methodologies, complexity analysis in non-preemptive and preemptive online scheduling forunit length jobs. Study of shop scheduling,family scheduling, resource constraint schedul-ing and scheduling with communication delays.2004 Pruhs et al. [24] Flow time minimization in single and parallelmachines.2006 Brucker [25] Shop scheduling, scheduling with deadline,batch scheduling and multi-purpose machinescheduling.2008 Leung et al. [26] Offline and online scheduling with processing setrestrictions in non-preemptive, preemptive set-tings for makespan minimization.2009 Albers [27] Energy efficient online scheduling. Specific re-sults in load balancing, makespan and flow timeminimization.2012 Sgall [28] Open problems in online scheduling for the max-imization of the throughput.2013 Lee et al. [29] Online m -machine scheduling under machine el-igibility constraint.2018 Epstein [30] Semi-online scheduling for makespan minimiza-tion in identical and uniform related machines.2019 Beaumont et al. [31] Scheduling in a parallel computing system withheterogeneous resources. Design of a frameworkto compare the performances of related algo-rithms based on achieved makespans and timecomplexities.5al aspects of online computation by a historical chronological overview on theseminal contributions for preemptive and non-preemptive online scheduling inidentical and uniform related machines. Fourteen well-known online schedulingalgorithms are presented. Deterministic and randomized online scheduling algo-rithms for makespan minimization are chronologically described for developinga basic understanding. Major research issues, open problems and two of theemerging recent trends are briefly presented as a motivation for future researchwork.
2. Preliminaries
We present basic terminologies, notations and definitions which are used inour survey in Table 3. Based on the scope of our literature survey, we makea classification of the online scheduling problem by three parameters such asparallel machine models, job characteristics and optimality criterion. We nowbriefly discuss about the parameters as follows.
In parallel machine system, all threads of a job execute simultaneously in aset of machines µ such that µ ⊆ { M , .........M m } . Here, we consider parallelprocessing of multiple jobs, by assuming that each job consists of a single threadof execution. We consider parallel machine models such as identical and uniformrelated machines in our survey, which are presented as follows. • Identical Machines : Here, all machines have equal speeds of processing ajob J i such that p ij = p i , ∀ M j , ≤ j ≤ m and p i ≥ • Uniform Related Machines or Uniform Machines : Here, the speeds of themachines may differ from one another. For a uniform machine M j withspeed S j , the execution time of job J i is p ij = p i S j . Job characteristics describe about the nature of the jobs and various con-straints related to job scheduling. We consider the following job characteristics. • Preemption.
It allows splitting of a job into pieces, where each piece isexecuted on same or different machines in non-overlapping time intervals. • Non-preemption.
It ensures that once a job J i with processing time p i begins to execute on machine M j at time t , then J i continues the executionon M j till time t + p i with no interruption. • Precedence Relation.
It defines dependencies among the jobs by the partialorder ’ ≺ ’ rule on the set of jobs [5]. A partial order can be defined ontwo jobs J i and J k as J i ≺ J k , which means execution of J k never startsbefore the completion of J i . The dependencies among different jobs can beillustrated with a precedence graph G ( p, ≺ ), where each vertex represents6 able 3: Basic Terminologies Notations and Definitions Terms Notations Definitions/Descriptions
Job[1] J i Program under execution which consists ofa finite number of instructions. A job is alsoreferred to as a collection of at least onesmallest indivisible sub task called thread .We use terms job and task in the samesense.Processing Time[1-3] p ij Total time of execution of a job J i on amachine M j . For identical machines pro-cessing time of J i is denoted by p i .Release Time[2, 25] r i Time at which any job J i becomes availablefor processing.Completion Time[3, 25] c i The time at which any job J i finishes itsexecutionDeadline[3, 25] d i Time by which job J i must be completed.Machine[1] M j An automated system capable of process-ing some tasks by following a set of rules.We use terms machine and processor in thesame sense.Load [11] l j Sum of processing times of the jobs thathave been assigned to machine M j .Speed [2, 3] S j The number of instructions processed bythe machine in unit timeSpeed Ratio s The ratio between the speeds of two ma-chines. For 2-machines with speeds 1 and S respectively. We have speed ratio: s = S = S p . A directed arc between twovertices in G ( p, < ) i.e J i → J k represents J i ≺ J k , where J i is referredto as predecessor of J k . If there exists a cycle in the precedence graph,then scheduling is not possible for the jobs. When there is no precedencerelation defined on the jobs, then they are said to be independent . We consider makespan ( C max ) as the optimal criterion. Makespan is de-fined as completion time of the job that finishes last in a schedule. Formally, C max = max ≤ i ≤ n c i . The objective is to obtain minimum C max . Anotherway of interpreting makespan is in terms of load balancing( L max ) . Here, themakespan is defined as the largest load incurred on any machine M j . Formally, L max = max ≤ j ≤ m l j . Here, the objective is to obtain minimum L max .
3. Well-Known Online Scheduling Algorithms
We present ten deterministic and four randomized online scheduling algo-rithms for identical and uniform related machines as follows.
Deterministic algorithms[11-12, 32] obtain same output while processing agiven input a number of times by following each time the same sequence of steps.Here, the output and running time depend on input(s) only. We now presentten well-known deterministic online scheduling algorithms as follows. • Algorithm List Scheduling(LS) was proposed by Graham [5] for non-preemptive online scheduling on identical machines. Algorithm LS assignsan incoming job J i to the machine M j , which is least loaded among themachines. • Algorithm Largest Processing Time(LPT) was proposed by Graham[33] for non-preemptive off-line scheduling on identical machines.
LPT first orders a list of jobs in non-increasing sizes. Then, it assigns job oneby one from the ordered list to the machine, which has least load aftereach assignment of a job.
LPT is an offline algorithm, however it hasbeen used significantly as an intermediate step in many online schedulingalgorithms. • Algorithm Refined List Scheduling(RLS) was proposed by Galambosand Woeginger [34] for non-preemptive online scheduling on m -identicalmachines. Upon the arrival of any job(say J i ), algorithm RLS first ordersthe machines in non-decreasing loads. J i is assigned to M if l ≤ αl m and J i is assigned to M if l + p i > αl m , where 0 ≤ α ≤ .
33. There after,
RLS assigns next incoming jobs ( J i +1 ) to M till the following inequalityholds: l β ≤ p i +1 ≤ β ( l m ), where 1 ≤ β ≤ .
25. Then, the machines arere-ordered again. 8
Algorithm ASSIGN-2:
Aspnes et al. [15] proposed the algorithm
ASSIGN-2 for non-preemptive online scheduling of uniform related ma-chines.
ASSIGN-2 works in phases by guessing the cost of
OPT at eachphase. Upon receiving the first job J , the cost of OPT is initializedto the load incurred by J on the fastest machine. The cost of OPT remains same till the completion of a phase. Each incoming job J i is as-signed to the machine M j , which has slowest speed among all machinesand l j + p i ≤ OP T ). Subsequently, the load of M j is updated andthe machines are ordered in non-decreasing speeds. The algorithm ends aphase, when it does not find an appropriate M j . A new phase is startedby initializing the loads of all machines to 0 and by doubling the previousvalue of OPT . Then, J i is assigned to the slowest machine M j for which l j + p i ≤ OP T ). • Algorithm Chen Vliet Woeginger(CVW) was proposed by Chen et.al. [35] for preemptive online scheduling of m -identical machines. Origi-nally, there was no particular name given to the algorithm. We name thealgorithm as CVW by extracting the first letter of the author’s names. Foreach incoming job J i , algorithm CVW computes the cost of
OPT , whichis max { m P ni =1 p i , max ≤ i ≤ n p i } . Then, job J i is assigned to any M j ,where 1 ≤ j ≤ m , if the new load of machine M j is at most ( m m m m − ( m − m )times of OPT . Otherwise, sequence the machines in non-decreasing loadsand assign ( m m m m − ( m − m ) OP T - l portion of job J i to machine M and therest part of job J i to machine M . • m-Machine Algorithm was proposed by Bartal et al. [36] for non-preemptive online scheduling of m -identical machines. The algorithm firstorders the machines in non-decreasing loads such that M has least loadand M m has highest load. When the first job J i is available, it is assignedto M and the machines are ordered in non-decreasing loads. Then, logi-cally group m -machines in to two, where first σm machines constitute firstgroup and rest m − σm machines form the second group. Each incomingjob J i +1 is assigned to M σm +1 th machine, if l σm +1 after the assignmentof J i +1 is at most 1 .
985 times of the average load incurred on first σm machines. Otherwise, J i +1 is assigned to M j , which has minimum loadover all machines. Note that, after each assignment of a job, the machinesare ordered in non-decreasing sequence of their loads. In the former case,machines from M σm +1 to M m are to be ordered and in the later case allmachines from M to M m are to be ordered. • Algorithm Compare Height to Average of Shorter Machines(CHASM) was proposed by Karger et al. [37] for non-preemptive onlinescheduling in identical machines.
CHASM always aims to maintain a lightload on the first k machines by keeping the next m − k machines with heavyload. Algorithm CHASM works in the following way: upon the arrivalof a new job J i , it first orders m -machines in non-decreasing sequence oftheir loads and assigns J i to the ( k + 1) st least loaded machine M k +1 if9 k +1 + p i ≤ . k P kj =1 l j ), where 1 ≤ k < m . Otherwise, assign J i tothe most lightly loaded machine M . • Algorithm H was proposed by Wen and Du [38] for preemptive onlinescheduling of 2-uniform related machines. Algorithm H works in the fol-lowing way: let the current loads of machines M and M be l and l respectively. Upon receiving a new job J i , Algorithm H first splits J i into two parts. The largest part of J i which is of size at most α.C i − OP T − l is assigned to the fastest machine M , where S ≥
1. The remaining partof J i , which is of size at most l − l is assigned to the machine M , where S = 1. Here, C i − OP T is the cost of
OPT before the assignment of J i . Wehave α = (1+ S )1+ S + S . • Algorithm M was proposed by Albers [39] for non-preemptive onlinescheduling on m -identical machines. Algorithm M always maintains m -machines, which are numbered in non-decreasing order of their loads afterthe assignment of each incoming job J i such that M be the machine withminimum load and M m be the machine with maximum load. A new job J i is assigned to machine M if one of the following inequalities holds.(a) L l ≤ α ( L h ) (b) λ m > l i − m and λ m > a. L l + L h m . Otherwise, J i is assigned to machine M k +1 . Here, we have the values of a = 1 . b = 0 . m , k = ⌊ m ⌋ and α = ( a − k − b ( a − m − k ) . If J i is scheduled on M ,then we have L l = P ki =1 l i and L h = P mi = k +1 l i . If J i is scheduled on M k +1 , then we have λ m = max ≤ j ≤ m l j and l i − m is the load of mostloaded machine M m before the assignment of J i . • Algorithm MR was proposed by Fleischer and Wahl [40] for non-preemptiveonline scheduling of m -identical machines. Algorithm MR schedules a se-quence of jobs in the following way: initially, sort all machines in non-decreasing order of their loads such that M and M m are the least loadedand most loaded machines respectively. When ever a new job J i arrives,assign J i to the machine M k +1 if λ > α ( l k +1 ) and l k +1 + p i ≤ a ( L iavg ).Otherwise, schedule J i on machine M . After each assignment of a newjob, the load of the corresponding machine is updated and all machinesare re-sorted. Here, we have k ≈ ⌊ . m ⌋ + 1, α ≈ . λ = k P ki =1 ( l i ),which is the average load of k least loaded machines before the assignmentof J i and L iavg be the average load incurred on m -machines after assigning J i . A randomized algorithm [41] flips coin while processing a given input. Thealgorithm produces a different output or follows a different order of executionsteps at each run. Here, the output and running time depend on the input(s) andrandom bits. We now present some well-known randomized online schedulingalgorithms as follows. 10
Algorithm Rand-2 was proposed by Bartal et al. [36] for non-preemptiveonline scheduling on 2-identical machines. Algorithm
Rand-2 maintainstwo logical schedules for each incoming job J i . In the first schedule, J i isassigned to the least loaded machine and in the second schedule J i is allo-cated on the most loaded machine. Overall expected discrepancies E and E are computed for both the schedules respectively, where discrepancy isthe difference in loads of the two machines at any instance of time. Thena value for x is chosen, where 0 ≤ x ≤ x ( E )+ (1 − x ) E ≤ L ,where L is the total load incurred by jobs J , J , ...J i − and J i . If such avalue of x exists, then J i is actually scheduled on the least loaded machinewith probability x and J i is assigned to the most loaded machine withprobability 1 − x . If there exists no such a value of x , then schedule job J i explicitly on the least loaded machine. • Algorithm Linear Invariant(LI) was proposed by Seiden [42] for non-preemptive online scheduling on m -identical machines, where 2 < m ≤
7. Algorithm LI schedules each incoming job J t on machine M withprobability p and on machine M with probability 1 − p , where p = l m − α ( l ) l m − α ( l ) − ( l m − α ( l )) . Here, l and l m are the loads of current least loadedmachine M and most loaded machine M m respectively, when all jobs J i ,where 1 ≤ i ≤ t are scheduled each time on the second least loaded machine M . We have l and l m as the loads of current M and M m respectively,when all J i are assigned each time to the least loaded machine M . Wehave values of α equals to 1.80, 2.04, 2.12, 2.11 and 2.10 for m =3,4,5,6,7respectively. It is observed that after each assignment of a job, machinesare re-indexed in non-decreasing order of their loads. • Algorithm BIAS was proposed by Epstein et al. [43] for non-preemptiveonline scheduling on 2-uniform-related machines, where speed of machine M is S = 1 and speed of machine M is S ≥
1. Algorithm
BIAS schedules an incoming job J i on the fastest machine M with probability S , otherwise assigns J i to the slowest machine M . • Algorithm RAND was proposed by Albers [44] for non-preemptive on-line scheduling on m -identical machines. Algorithm RAND is basically acombination of two deterministic algorithms
ALG and ALG . Any inputjob stream σ is scheduled by algorithm ALG with probability P = andby algorithm ALG with probability 1 − P . Algorithm
ALG : Machines are always indexed in the non-decreasing or-der of their loads. A new job J i is scheduled on machine M k +1 if theschedule is critical and l i − ,k +1 + p i ≤ b ( L i m ). Otherwise, J i is assignedto the least loaded machine M . Here, l ix,j be the load of the machine M ix,j , which is the j th current least loaded machine after scheduling jobsfrom J to J i by algorithm ALG x , where x ∈ { , } . L i be the totalprocessing time incurred by the jobs from J to J i . A schedule becomes critical if µ ix > α x ( L ix ) , k x + 1, where µ ix = k x P k x j =1 l ix,j . The values11f other parameters are initially set as follows: b = 1 . k = ⌈ m ⌉ , α = 1 − k −⌊ . m ⌋ − . k . Algorithm
ALG : Upon receiving a new job J i , algorithm ALG firstruns algorithm ALG . If the schedule obtained after assigning J i by algo-rithm ALG is balanced, meaning the machines are now equally loaded,then algorithm ALG sets the value of γ to max { b ′ L i m , b β − λ i m } , where λ i = P mj = k +1 l i ,j . Otherwise, the value of γ is set to b ′ L i m . Now, algo-rithm ALG assigns J i to the machine M k +1 if the schedule is criticaland l i − ,k +1 + p i ≤ γ . Otherwise, J i is scheduled on the least loaded ma-chine M . Initialize the values of other parameters as follows: b = 2, b ′ = 1 . k = ⌈ . m ⌉ , α = 0 .
449 and β = 1 − ( b − k m . Notethat: after each assignment of a new job, the machines are re-numberedaccording to the non-decreasing order of their loads.
4. Historical Overview of Online Scheduling
The theory of sequencing and scheduling have been emerging as an interest-ing area of research over the past few decades. In early fifties, the main focus ofresearch was on offline single machine scheduling. After a decade, the curiositywas transferred to define potential advantages of multiprocessing systems. Asan outstanding outcome, the systems witnessed increase in throughput. Still,the quest was for designing application specific scheduling models and to ob-tain optimum schedule for processing of multiple jobs. This resulted in theemergence of a number of scheduling setups. One of such setups is the onlinescheduling , which basically deals with the online arrival of a sequence of jobs.Online Scheduling setup was first proposed and validated by Graham in 1966[5], however it has gained significant research interests after the introductionof competitive analysis in 1985 [13]. We now present an overview of the earlycontributions on online scheduling from year 1966 to 1984 as follows.Graham [5] initiated the study of non-preemptive online scheduling on m -identical machines in the objective to explore several multiprocessing timinganomalies. He considered a sequence of n ( ≥
2) jobs and ordered them througha static list , where the ordering of the jobs is decided upon receiving all jobs.Here, the jobs are ordered only once and the ordering remains same throughoutthe scheduling of all jobs. Graham proposed the famous LS algorithm, whichalways scans the list until an eligible job is found. A job J i is called eligible , if ex-ecution of J i has not been started and processing of all its predecessors have beencompleted. Algorithm LS adopts a greedy strategy by immediately schedulingthe first eligible job of the list to the lowest loaded machine prior to make an-other scan of the list. Algorithm LS achieves a performance ratio of 2 − m , whichis the ratio between the largest makespan obtained over all possible ordering ofthe jobs to the optimum makespan( C OP T ) , where C OP T = m · ( P ni =1 p i ). In [33],Graham proposed LPT algorithm, which orders a sequence of jobs in a dynamiclist , where the ordering of the jobs may change during the scheduling process.12lgorithm
LPT always places the eligible job with largest size among the avail-able jobs in the top of the list and schedules it by algorithm LS . Algorithm LPT achieves a performance ratio of 1 . − m . These two studies of Graham showeda new direction in scheduling, which is based on the arrival pattern of the jobs.The scenario of selecting the eligible job from the top of the list without lookingat the entire task list was later formed the concept of arrival of jobs one by one .Coffman and Graham [45] followed the work of Graham [5,33] and designed twonon-enumerative algorithms with worst case complexities of n and ( n ) re-spectively for makespan minimization of a job schedule. They considered a listof jobs, where all jobs have equal execution time. The main objective of bothalgorithms is to obtain a list A ∗ such that makespan of A ∗ is minimal over alllist A . A novel computational model was introduced, which partitions a givenproblem into two equal sized jobs. The applicability of the model was shown inpreemptive scheduling, where a job can be splitted and shared among multiplemachines.Sahni [46] studied scheduling of n -independent jobs on m -identical machines forminimizing the makespan. He designed an offline algorithm by dynamic pro-gramming approach, which has the worst case complexity of O ( min { n , nM } ),where M characterizes the cost of OPT . Furthermore, he proposed three approx-imation algorithms for scheduling problems such as single machine schedulingwith deadline, scheduling on m -identical machines for minimizing completiontime and scheduling on 2-identical machines to minimize weighted mean flowtime. Here, all proposed algorithms obtain costs which are not far than a value ǫ from the optimum cost, where 0 < ǫ <
1. The worst case complexities of bothalgorithms were proved to be O ( n ǫ ).Sahni and Cho [47] proposed a nearly online algorithm for preemptive schedul-ing of n -independent jobs on m -uniform related machines. They specified eachjob J i with release time r i . The worst case time complexity of the algorithmwas proved to be O ( m n + mnlogn ). The algorithm ensures at most O ( nm )preemption in executing all jobs. Here, they assumed that there are at most v distinct release times of the jobs and designed an algorithm which has v phases.At any phase k , where 1 ≤ k < v , a selected number of jobs are scheduledthrough a deterministic procedure. In fact, those jobs are chosen which havenon-zero remaining processing times and are available on or before time r k . Inthe last phase, the jobs are scheduled in the interval [ r v , d ], where d is assumedto be the common deadline for all jobs.Hariri and Potts [48] proposed a branch and bound algorithm to obtain a pro-cessing order of the jobs, which minimizes the sum of weighted completion timesin a single machine offline scheduling environment. They considered a list ofjobs, where each job is specified by its release time and weight. They com-puted earliest time of completion for each job. They obtained a lower boundby assuming Langragean relaxation for the release time constraints. Here,
Lan-gragean multipliers are chosen in such a way that the generated job sequenceyields optimum cost for the relaxed problem. Later, they provided a method toderive better release time constraints than the original ones to increase the lower13ound. Blazewicz et al.[49] studied parallel machine scheduling of a sequenceof unit size jobs under various resource constraints schemes.Initial two decades(1966-1985) were focused mostly on exploring variants of theGraham’s
List Scheduling setup in optimizing the makespan.
Greedy, Dynamicprogramming , branch and bound algorithmic design strategies were introducedfor the multiprocessor scheduling problem. A more systematic study on online m -machine scheduling was started after the development of competitive analy-sis . Hence forth, two of the major aspects of online algorithm design such asdeterministic and randomized strategies were studied extensively. We presentthe state of the art on deterministic and randomized online scheduling in thecoming sections.
5. Deterministic Online Scheduling: State of The Art
This section presents important results with insights and research chal-lenges in the design of deterministic online algorithms for preemptive and non-preemptive online scheduling with makespan minimization for identical anduniform- related machines.
Upper Bound Results.
Graham’s LS [5] algorithm was provably the firstdeterministic online algorithm for m -machine scheduling problem. The objec-tive of algorithm LS is to assign jobs to the machines such that at the endof scheduling, all machines incur nearly equal load. However, equal load shar-ing policy fails to obtain optimum makespan in all cases. One such case isthe availability of the largest job as the last job i.e. n th job of the sequencewhile the assignment of first n − LS obtains a makespan( C LS ) which isnearly twice of the value of the optimum makespan( C OP T ). Let us considera list of m − m + 1 jobs, where a sequence of m ( m −
1) jobs each of size size m unit is available at the end of thesequence. We now have C LS ≥ m −
1, where C OP T = ( m − m ) · mm = m . There-fore, we have C LS C OPT ≥ − m . An equal UB was shown by Graham to obtain(2 − m )-competitiveness of algorithm LS . It was a non-trivial challenge to designimproved competitive online algorithm with CR asymptotically lesser than 2.After a quest of over two and half decades, the first improvement over algorithm LS was presented by Galambos and Woeginger [34]. They proposed algorithm RLS and proved a UB of 2 − m - ǫ m , where ǫ m > m ≥
4. Bartal et al. [36]proposed the m-Machine Algorithm and achieved UB .
985 for m ≥
70. Thealgorithm assigns incoming jobs to the machines in such a way that there existsalways a set of machines with light load and rest machines with heavy load.So, whenever a job arrives, it can be assigned to the lightly loaded machine ofthe set of heavily loaded machines or to the smallest loaded machine among allmachines. The objective is to obtain a makespan which is smaller than twicethe value of the optimum makespan. Karger et al. [37] proposed the algorithm14
HASM and obtained an improved UB .
945 for all m . Algorithm CHASM outperforms algorithm LS for m ≥
6. An improvement over
CHASM was proposedby Albers [39]. She designed the
Algorithm M and proved a UB of 1 .
923 fora general case of m -machine. Fleischer and Wahl [38] proposed the algorithm MR , which obtains a UB of 1 . m → ∞ . Algorithm MR is currently thebest deterministic pure online algorithm for non-preemptive online schedulingon m -identical machines for makespan minimization, where m ≥ Lower Bound Results.
Faigle et. al. [50] proved that LS is optimal onlinescheduling algorithm for m = 2 ,
3. The LB − m for 2-machine case was shownby considering online availability of a sequence of three jobs < J , J , J > with p =1, p =1 and p =2. Similarly, for m = 3, they showed LB − m by con-sidering online arrival of a sequence of seven jobs, where the jobs are of sizes(1 , , , , , ,
6) respectively. Further, they proved LB .
707 for m ≥ m + 1 jobs, where first m jobs are of size 1 unit each,next m jobs are of size 1 + √ √ LB .
837 for m ≥ m + 1 jobs, where first m jobs are of size x +1 unit each, second m jobs are of size xx +1 unit each, third m jobs are of size x unit each, next ⌊ m ⌋ jobs are of size y unit each, next ⌊ m ⌋ − z unit and last ( m + 3 − ⌊ m ⌋ − ⌊ m ⌋ ) jobs are of size 2 y unit each, where x, y, z are positive real values. Albers [39] obtained a better LB of 1 .
852 for m ≥
80. She considered a special class of job sequence, wherejobs are available in four rounds. In each round a specific number of equal sizejobs arrive. The main idea is to schedule the jobs round-wise such that themakespan incurred at each round is not more than 1 .
852 times of the value of C OP T . Gormley et al. [52] improved the LB to 1 .
853 for m ≥
80 by consideringan adversary strategy. The adversary strategy was presented as a game tree.Here, the game tree has two kinds of nodes, the adversary request nodes andthe online move nodes. Each adversary request node is a non-leaf node. Thechildren of adversary request nodes are online move nodes, which are generatedat each move as per the current request. Each online move node is either a leafor has a single adversary request node as a child.Rudin III [53] proved that no deterministic online algorithm can achieve a com-petitive ratio smaller than 1 .
88. Later, he along with Chandrasekaran [54]obtained LB √ m = 4. For the general case of m -machine, they showed LB √ − ǫ , where ǫ is a positive constant. They used the job master strategyto produce successive layers of jobs, where each layer contains m jobs. Thelayers are considered in such a way that if any two of the jobs in the same layerare scheduled on the same machine, then the makespan of the correspondingmachine will become at least √ − ǫ times of the value of the C OP T . Therefore,irrespective of the scheduling algorithm, by the completion of a series of layers,it can be known that either there has been assignment of one job to each ma-chine at each layer or a competitive ratio of at least √ − ǫ has already beenachieved. We now present the summary of the important competitive analysisresults achieved by deterministic online algorithms for non-preemptive schedul-15ng on identical machines in Table 4. Table 4: Summary of Important Competitive Analysis Results
Year Author(s) Result(s) Bound − ( m ), m =2 , − ( m − ǫ m ), m ≥ . m ≥ . m ≥ . m ≥ . m → ∞ UB.1989 Faigle et al. [50] 1 . m ≥ . m ≥ . m ≥
80 LB.2000 Gormley et al. [52] 1 . m ≥
80 LB.2001 Rudin III [53] 1 . m → ∞ LB.2003 Rudin III, Chandrasekaran [54] √ − ǫ , m = 4 LB. Research Challenges: • Minimizing or diminishing the gap between current best LB and UB of[1.853, 1.9201] on the CR . • Classification, characterization of input job sequences and ranking of on-line scheduling algorithms based on real world inputs. • The design of deterministic algorithms for online scheduling on m -identicalmachines have been witnessed various strategies such as greedy , input char-acterization , game tree , layering and job master . However, it will beinteresting to develop a unified deterministic strategy for scheduling anarbitrary sequence of large jobs. • Finding the exact competitiveness achievable by deterministic online al-gorithms.
Offline scheduling in non-identical machines was introduced in late seventiesby Hrowitz and Sahni [55]. However, the study of online scheduling in uni-form related machines was initiated in year 1993 by Aspnes et al. [15]. Theyproposed the deterministic algorithm
ASSIGN-2 and achieved a UB of 8. Al-gorithm ASSIGN-2 works on the idea of assigning each incoming job J i to thelowest speed machine(say M ) as long as l + p i ≤ C OP T (assuming l as theload of M before the assignment of J i ). As the value of C OP T is not known,algorithm
ASSIGN-2 follows a doubling strategy . Initially, a smaller value ischosen for C OP T and later the value of C OP T will be set to twice of its previousvalue, when the scheduling of job J i on M makes l + p i > C OP T . Bermanet al. [56] obtained UB .
82 for large m . Further, they proved LB s 2 .
28 and16 .
43 for m = 6 and m = 9 respectively. They proposed an algorithm, whichworks in phases. In each phase, a sequence of jobs are scheduled. The makespanobtained in each phase is represented through one of the nodes of a graph. Theyverified the achieved competitive ratios through a computer based search in thegraph.Ebenlendr and Sgall [57] proved LB .
56 for the setting, where the speeds ofthe machines and the processing times of the jobs are in a geometric sequence.They proposed a new lower bound inequality, which is based on the total num-ber of jobs scheduled and the number of jobs assigned per machine. The LB was derived as follows: the behavior of the algorithm is first examined in anyof the machines to obtain the UB on the number of jobs that can be scheduledon that machine. Basically, this UB is a ratio of number of jobs and machinespeed. They considered such UB s on every machines, which is at least 1 as ineach step of the algorithm only one job can be scheduled. Finally, they obtainedthe LB by assuming the common ratio of the geometric sequence to be 1.Jez et al. [58] obtained LB s 2 .
14 and 2 .
31 for m = 4 , LB s by analyzing the possible orders of scheduling the jobs onthese machines. They achieved better LB s of 2 .
34 and 2 .
46 for m = 6 , Table 5: Summary of Important Competitiveness Results
Year Author(s) Result(s)
Bound1993 Aspnes et al. [15] 8 UB2000 Berman et al. [56] 5 .
82 UB2000 Berman et al. [56] 2 . m = 6, 2 . m = 9 LB2012 Ebenlendr and Sgall [57] 2 .
56 LB2013 Jez et al. [58] 2 . m = 4, 2 . m = 5,2 . m = 9 LB Research Challenges: • Minimizing or diminishing the gap between current LB and UB of [2.56,5.82] on the CR . • Development of an alternative to doubling strategy for the improvementof the existing competitive bounds. • Design of efficient competitive online deterministic algorithms with a newparameter or function based on the size of the jobs.17
To obtain a tight bound for large m . Here, we survey important results obtained by deterministic algorithms forpreemptive online scheduling in identical and uniform related machines as fol-lows.
Identical Machines.
Less attention has been paid to the online preemptivescheduling on identical machines. To the best of our knowledge, the only deter-ministic online algorithm
CVW was proposed by Chen et al. [35] for makespanminimization in identical machines. They obtained LB m m m m − ( m − m for m ≥ .
58 for m → ∞ . The overall idea of the algorithm is to maintainthe load of the least loaded machine as small as possible so that whenever a largesize job arrive it can be assigned to the least loaded machine. The objective isto obtain a bound, which is asymptotically lesser than 2. Uniform Related Machines.
Wen and Du [59] achieved UB S .S S + S .S + S for m = 2, where S and S are the speeds of machine M and M respectively.Epstein [60] studied a special case, where the speeds of the machines are definedby the following inequality: S j − S j ≤ S J S j +1 , for j = 2 ......m −
1. He obtained an UB for each sequence of speeds as P mj =1 S i X (1 − S X ) i − − , where X = P mj =1 S i .Ebenlendr and Sgall [61] obtained UB C OP T . They showed that algorithm LS of Graham [5]achieves UB m and LB log m for preemptive online scheduling on m -uniform related machines. Recently, Ebenlendr et al. [62] obtained a UB ,which lies between the values 2 .
054 and 2 . Research Challenges: • Determination of competitive bounds i.e. UB for identical machines caseand LB for uniform related machines case. • Design of strategies that avoid or minimize idle periods of uniform relatedmachines while scheduling the jobs online. • How to guess the value of the optimum makespan( C OP T )? In general, asthe choices for C OP T go up, the number of cases in the analysis of anyonline algorithm grow exponentially. • Design of nearly optimum online algorithms with best competitive ratios.
6. Randomized Online Scheduling: State of the Art
This section is devoted to an overview of the state of the art contributionsin design of randomized algorithms for online scheduling with makespan min-imization. Important results achieved for non-preemptive online scheduling inidentical and uniform related machines are discussed followed by an overview ofthe seminal works in preemptive online scheduling.18 .1. Non-preemptive, Identical Machines
Design of randomized algorithms for online scheduling has received notableresearch interests after the seminal work of Bartal et al. [36]. They proposedthe algorithm
Rand-2 for non-preemptive online scheduling on 2-identical ma-chines. The objective of algorithm
RAND-2 is to maintain an expected loaddifference of ( ) L between two machines at any instance of time, where L isthe sum of processing time of the jobs that have already received. Algorithm Rand-2 achieves LB s of 1 .
33 and 1 . m = 2 , m -identical machines with an objective to minimize the makespan mustbe at least ( m m m m − ( m − m )-competitive for all m . The bound tends to ee − ≈ .
58 as m → ∞ . The overall idea of the algorithm is presented as follows:upon receiving a new job J i , the algorithm first computes m probabilities0 ≤ x , ..........x m ≤
1, where P mj =1 x j = 1. Then, schedule job J i on ma-chine M j with probability x j , j = 1 , .....m .Sgall [64] proved that any randomized algorithm A must be at least (1 + mm − ) m − )-competitive. He showed that the LB tends to 1 . m → ∞ . Hedefined an interesting ordering of the machines by considering last m jobs of anyjob sequence. The idea is to order the machines in such a way that the index ofthe i th machine remains unchanged after the assignment of each new job J i to it.Initially, when no job is scheduled on any of the machines, then a new job J i isscheduled arbitrarily on any one of the m -machines. The objective of such order-ing is to maintain the ratio of the loads of m machines as 1 : L : L : ... L m − ,where L = mm − . This implies C A ≥ L m − and C OP T = L + L + .... + L m − m = L m − m ( L − . Therefore, C A C OPT ≥ L m − · ( m · ( L − L m − ≥ mm − ) m − .Seiden [65] proposed the algorithm LI by generalizing the 2-machine algorithmof Bartal et al. [36]. Algorithm LI schedules an incoming job J i either on theleast loaded machine or the second least loaded machine with certain probabil-ity. The objective is to keep the load of one of the m machines as low as possibleto schedule on it the largest job that likely to arrive in future. Algorithm LI achieves competitive ratios of 1 . . . . . m = 3, 4, 5, 6, 7 respectively.Albers [44] envisioned a strategy for designing of randomized algorithms bycombining simple deterministic policies. She developed the algorithm RAND ,which is a combination of two deterministic algorithms
ALG and ALG . Sheobtained an upper bound of 1 .
916 for all m . Upon the arrival of a job, algo-rithm RAND invokes
ALG i , i ∈ { , } with probability , then schedules thewhole job sequence by the chosen algorithm. Algorithm RAND aims at main-taining two schedules at any instance of time unlike algorithm
RAND-2 andalgorithm LI , those maintain separate schedules upon each job arrival. Here,Albers proved that none of the known deterministic online strategies can beatthe performance of algorithm RAND .Tichy [66] obtained LB s of 1.425, 1.495 and 1.504 for machines m = 3 , , k th least loaded machine,where k is a positive integer. In [67], he showed the problem for three identicalmachines case. He considered a critical job sequence which is characterized bysome integer parameters and proved by contradiction that no randomized algo-rithm can obtain a lower bound less than or equal to 1 . Research Challenges:
Table 6: Summary of Important Competitiveness Results
Year Author(s) Result(s) Bound . m =2 and 1 . m =3 LB1994 Chen et al. [63] 1 . m → ∞ LB1997 Sgall [64] 1 .
582 LB2000 Seiden [65] 1 . m =3, 1 . m =4 1 . m =5, 1 . m =6, 1 . m =7 UB2002 Albers [44] 1 .
916 UB2002 Tichy [66] 1 . m = 3, 1 . m = 5, 1 . m = 6 LB2004 Tichy [67] 1 . m = 3 LB • Tighten the gap between the current best lower and upper bounds of1 .
582 and 1 .
916 respectively on the competitive ratio for general case of m machines. • Development of more refined strategies with stronger invariants to con-struct improved bounds for the problem. • Design of a unified strategy to schedule a sequence of very large size jobs. • Development of a new framework to measure the performance of any on-line scheduling algorithm based on the input characterization approach ofSeiden [63] for any job sequence and a general case of m machines. Indyk [68] initiated the study of randomized algorithms for non-preemptiveonline scheduling on uniform related machines. He obtained an upper bound of5 .
436 by extending the work of Aspnes et al. [15].Epstein et al. [43] proposed the algorithm
BIAS for online scheduling in twouniform related machines with speeds 1 and S ≥ . < S <
2. They showed thatrandomization does not improve the bounds for the speeds S ≥ . . C OP T is made along with the ini-tialization of variables c j , x j and r . Here, c j is referred to as the capacityof machine M j which is the amount of work machine M j can do under load C OP T . We have c j = C OP T ( S j ) for any M j , where S j is the speed of ma-chine M j . The variable r is initialized to p ( ). Upon receiving a job, first thevalue of C OP T is checked. The value of C OP T is updated if
OnlyFor ( S j , C OP T , σ ) > C OP T .Cap ( S ) for some S ∈ { S , S , ....S m } .(Here, σ is referred to as asequence of jobs and OnlyFor( S j , C OP T , σ ) defines the sum of sizes of thosejobs for which p i S j > C OP T for some j ∈ { j = 1 , , ...m } ). Then, the value of C OP T is updated to r ( C OP T ). The values of x j and c j are updated as follows: x j +1 = ( C OP T ) S j and c j +1 = x j +1 + C OP T . Here, the value of r is chosenuniformly at random from the interval [ − z, − z ]. For a negative z , we have r = r y +1 and for a positive z , we have r = r y with any integer variable y . Themain idea of the algorithm is to increase the value of C OP T at each phase bya factor r instead of doubling its value as soon as the current value of C OP T seems to be very small. Hence, the jobs arriving upfront can be scheduled onthe relatively faster machines and the jobs coming later in the sequence can bejudiciously assigned to the slower machines. Therefore, the loads of the fastermachines can be increased, which in turn minimizes the overall makespan of theschedule.Epstein and Sgall [70] obtained an improved lower bound of 2 by characterizingthe input parameters such as number of jobs, processing times of the jobs, num-ber of machines and their speeds. For an infinite number of machines, where amachine M j has speed S j = x j for any variable x ¡1, they obtained a lower boundof 1 + x . Here, they considered an infinite job sequence, where the processingtime p i = x i for any job J i such that the sum of processing times of all jobsis always x − x . For a general case of m machines, they obtained a simple lowerbound of x x m by considering m largest jobs each with processing time p i = x i and machines with speed S j = x j , where 0 < x < x (1 − x m )1 − x . Then, by judiciously modifying the values ofinput parameters, they obtained improved lower bounds of 1 .
33 and 1 . m = 2 ,
100 number of machines respectively. We now present the summary ofimportant competitive results in non-preemptive online scheduling on uniformrelated machines by randomized algorithms in Table 7.
Table 7: Summary of Important Competitiveness Results
Year Author(s) Result(s) Bound .
436 UB2000 Berman et al. [69] 4 .
311 UB2000 Berman et al. [69] 1 . . Research Challenges: Close or diminish the gap of [2, 4.311] for the current best lower and upperbounds on the competitive ratio. • Design of competitive randomized online algorithms with a new functionand parameter based on the speeds of the machines. • Can a randomized algorithm beat the performance of the current bestdeterministic strategy in obtaining the lower bound?
Here, we present an overview of the state of the art literature in design ofrandomized algorithms for preemptive online scheduling in identical and uni-form related machines as follows.
Identical Machines.
Seiden [65] proposed the first randomized algorithmfor preemptive online scheduling on identical machines by slightly modifyingthe original algorithm of Chen et al.[63]. Seiden considered the notion of jobsplitting as the preemptive characteristic of the scheduling algorithm. The algo-rithm schedules a sequence of jobs on m identical machines in the following way:whenever a job J i is received, a time slot is assigned for job J i on each machinestarting from the most loaded machine M m to the least loaded machine M .The time slot for J i in machine M m is defined as a function( f ) of the followingtwo parameters: sum of the processing times of all jobs(T) and largest process-ing time( P b ). Formally, the time slot in M m can be ( L m , f ( T, P b )]. The timeslot for job J i on rest of the machines( j < m ) are allocated in the following way:( L j , L j +1 ]. The objective is to keep k machines lightly loaded and m − k ma-chines heavily loaded. Here, the bounds on the competitive ratios were shownas a function of real constants and were verified through computer programswritten by tool Mathematica . However, a general bound for the algorithm wasnot derived.
Uniform Related Machines.
Ebenlendr and Sgall [61] obtained an upperbound of 2 .
71 for preemptive online scheduling on uniform related machines byusing the doubling strategy. They made a conjecture on the improvement overdoubling strategy that the initial value for optimal makespan( C OP T ) can beguessed by considering an exponential distribution.Epstein and Sgall [70] obtained lower bounds on the competitive ratio for theworst case combination of speeds for any fixed m . These bounds approach to 2when m → ∞ and all hold for randomized algorithms.
7. Recent Trends
Online scheduling poses a non-trivial research challenge for designing op-timal algorithms due to unavailability of complete input information at theoutset. Recent studies have guaranteed a performance improvement over theonline deterministic and randomized strategies by relaxing one or more stringentconstraints of the pure online scheduling setting. The relaxation includes avail-ability of additional computational power or extra piece of information(EPI) to22n online scheduling algorithm. This section highlights some of the relaxed vari-ants of the online scheduling model and reports important competitive analysisresults as follows.
Resource augmentation model was pioneered by Kalyansundaram and Pruhs[71]. Here, an online scheduling algorithm is given some additional resourcessuch as high speed machines(speed augmentation) or memory space (memoryaugmentation) as compared to its optimum offline(OPT) counterpart. We nowdiscuss on speed augmentation and memory augmentation as follows. This model gives as input a set of relatively high speed machines to anonline algorithm for scheduling a sequence of jobs and compare its performancewith
OPT that schedules jobs on relatively slower machines. For instance, anonline scheduling algorithm is given a set of machines with speeds S j ≥ ∀ M j ,where algorithm OPT operates machines with speed S j =1, ∀ M j . Berman andCoulston [72] studied preemptive online scheduling on a single machine withspeed augmentation for minimizing P ni =1 c i − r i , where r i is the release timeof job J i and c i is the completion time of J i . They considered that an onlinealgorithm is given a machine with speed u times faster than the machine givento the optimal offline algorithm. They proposed the algorithm Balance , whichalways schedules least executed job. They achieved a UB of uu − . Algorithm Balance was shown to be ( u )-competitive for u ≥
2. Lam and To [73] studiedpreemptive online scheduling with hard deadlines. They explored a trade-offbetween increment of speed and number of machines. They proved that anyonline algorithm that schedules jobs by earliest deadline(EDF) is optimal for m =2, if the algorithm is given machines with speeds 1 . EDF rule achieves optimalitywith (2 − xm + x ) times faster machines and x ≥ Buffer . A buffer B ( k ) is given, which is capable of keeping at most k jobs,where k ≥
1. Availability of B ( k ) allows an online algorithm either to keep anincoming job temporarily on the buffer or to schedule one of the available jobsdirectly on a machine. An online algorithm now can see at most k +1 jobs at anytime step prior to make a scheduling decision. Some of the interesting studiesfor online scheduling in identical machines with buffer of varying sizes are due to[77-79, 81]. In [82], the authors considered online hierarchical scheduling with B (1) in 2-identical machines, where the machines are of different capabilities inthe sense that machine M can process any job and machine M can processonly some designated jobs. An available job J i is given with its p i and g i , if g i =1, then J i can only be processed by machine M , if g i =2, then J i can be23rocessed by either of the machines. We now represent important competitiveanalysis results achieved for online scheduling in identical machines with bufferin Table 8. Table 8: Important Results for Online Scheduling with Buffer
Year, Author(s) Machine(s), B ( k ) Competitive Ratio(s) k ≥ .
33 Tight1997, Zhang [78] 2-identical, k =1 1 .
33 Tight2004, Dosa, He [79] 2-identical, k =1 1 .
25 Tight2012, Lan et al. [81] m -identical, k =(1 . m . k =1 1 . Parallel Schedules . Upon receiving a job J i , 1 ≤ i ≤ n , an online algorithmmakes two copies of J i and virtually schedules each of the copies by two inde-pendent procedures. Therefore, two parallel schedules are constructed at anytime step. After constructing two virtual schedules for the entire job sequence,one of the schedules is chosen that has incurred minimum C max for actual as-signment of all jobs. Here, an extra space is given to maintain solutions forparallel schedules. In 1997, Keller et al. [77] first studied non-preemptive onlinescheduling on 2-identical machine with parallel schedules and obtained a tight bound of 1 .
33. In 2012, Albers and Hellwig [83] investigated the general case of m -identical machine and achieved UB .
75. An open issue is to obtain a tight bound for m -identical machine setting. It will be interesting further to exploreuniform machine setting with parallel schedules. The study of online scheduling with
Extra Piece of Information(EPI) pio-neered the concept of semi-online scheduling. Kellerer et al. [77] first envisionedthat availability of additional information on future inputs is quite natural incontrast to the constraint of no information at all. For an instance, the numberof jobs that are going to be submitted to a multi-user time shared system isnot known a priori. However, the minimum and maximum time required toprocess each job can be known in advance by previous history. This revitalizedthe area of online scheduling to explore practically significant new
EPI s. Wenow report important results, achieved by some well-known semi-online policieswith a classification of
EPI s as follows.
An online algorithm is given a priori, the sum of the processing times of alljobs. Kellerer et al. [77] first introduced
Sum as an
EPI for non-preemptiveonline scheduling on 2-identical machine and achieved a tight bound of 1 .
33 onthe CR . Recent contributions in this setting are due to [80, 84, 86-92]. Importantresults achieved for online scheduling with known Sum is reported in Table 9.24 able 9: Important Results for Online Scheduling with Known Sum
Year, Author(s) Machine(s) Competitive Ratio(s) .
33 Tight1998, Gilrich et al. [84] m -identical 1 .
66 UB2004, Angelleli et al. [86] m -identical (1 . , . m → ∞ m -identical (1 . , .
6) LB and UB respectively, m ≥ . , . LB and UB respectively2008, Angelleli et al. [80] 2-uniform 1 .
33 Tight for s =1, (1+ s +1 ) Tight for s ≥ . .
369 UB2010, Angelleli et al. [90] 2-uniform 1 .
359 LB for s =1 . s Tight2015, Keller et al. [92] m -identical 1 .
585 Tight
An online algorithm is given with the value of the optimum makespan(Opt) for an online sequence of jobs prior to their scheduling and availability. Epstein[93] first considered
Opt as an
EPI for online scheduling on 2-uniform machineand obtained UB . Opt as an
EPI in bin stretching problem and achieved UB . Opt is reported in Table 10.
Table 10: Important Results for Online Scheduling with Known Opt
Year, Author(s) Machine(s) Competitive Ratio(s) m -identical 1 .
625 UB2003, Epstein [93] 2-uniform 1 .
414 UB2009, Ng et al. [89] 2-uniform 1 .
366 Tight2011, Dosa et al. [91] 2-uniform min { s , s s +5 , s +1 } LB for s ≥ s +1)4 s +5 LB for s ∈ [1 . , . { s +109 s +7 , s +1616 s +7 , s +73 s +10 } LB for s ∈ [1 . , . s +109 s +7 Tight for s =1 . s +12 Tight for1 . ≤ s ≤ . A job with largest processing time(
Max ) is known at the outset. He andZhang [97] introduced
Max as known
EPI in online scheduling on 2-identicalmachines and obtained a LB of 1 .
33. Further studies for online scheduling with25nown
Max are due to [98-101]. Important results achieved for online schedulingwith known Max is reported in Table 11.
Table 11: Important Results for Online Scheduling with Known Max
Year, Author(s) Machine(s) Competitive Ratio(s) .
33 Tight2002, Cai [98] m -identical 1 .
414 UB2004, He, Jiang [99] 2-uniform s +3 s +12 s +2 s +1 LB for preemptive schedul-ing2008, Wu et al. [100] m -identical 2 − m − Tight2013, Lee, Lim [101] m -identical 1 .
618 Tight for m =4 and 1 .
667 Tightfor m =5 Processing times of an online sequence of jobs are not known a priori. How-ever, maximum and minimum time required to process each job is given to anonline algorithm at the outset. For instance, it is given that size of each job J i ,(1 ≤ i ≤ n ) is in between ( p, rp ), where p > r ≥
1. He and Zhang [97]initiated study of online scheduling on 2-identical machine with
T GRP ( p, rp )and achieved LB .
33. Further advancements in this setting are the outcomes ofthe following contributions [99, 102]. Later, we shall report some of the results,where minimum processing time ( T GRP ( lb )) for each job or maximum process-ing time ( T GRP ( ub )) for each job were considered in combine with some other EPI s. Recently [103], online hierarchical scheduling on 2-uniform machine with
T GRP (1 , α ) has been studied and a LB of 1 + α has been shown, where α ≥ TGRP isreported in Table 12.
Table 12: Important Results for Online Scheduling with Known TGRP
Year, Author(s) Machine(s) Competitive Ratio(s) . { s + r/ rs/ s + rs/ , s + ss +1 } Tightfor s > √ s ≤ r < s − s with preemptive scheduling2005, He, Dosa [102] 3-identical 1 . r ∈ (2 , . r +22 r +3 UBfor r ∈ (2 . , α ) LB An online algorithm is given with the arrival pattern of a sequence of jobs.For an instance, it is known at the outset that the jobs arrive one by one with non-decreasing sizes (Decr) . Seiden et al. [104] introduced
Decr as a known26 PI for preemptive online scheduling on 2-identical machine and achieved a tight bound of 1 .
16. For m =3, LB .
18 was shown and for a general case of m -identical machine, LB .
36 was obtained. Recent works for online schedulingwith known
Decr can be found in [105, 106] and we report important results oncompetitive analysis in Table 13.
Table 13: Important Results for Online Scheduling with Known Decr
Year, Author(s) Machine(s) Competitive Ratio(s) m -identical 1 .
36 LB for m → ∞ , 1 .
16 Tightfor m =2, 1 .
18 LB for m =32005, Epstein, Favoholdt [105] 2-uniform 1 .
28 Tight2012, Cheng et al. [106] m -identical (1 . , .
25) Tight for m =3 and m > An online algorithm is given with more than one
EPI s at the outset. An-gelelli [85] initiated the study of online scheduling on 2-identical machine withcombined
EPI s. They considered an online algorithm has the prior knowledgeof minimum processing time for all jobs and sum of the sizes of the jobs. Heachieved a tight bound of 1 .
33 on the CR . Tan and He [107] considered twocombined EPI s. First, they considered
Sum , Max and obtained a LB of 1 . LB of 1 .
11 with known
Sum and
Decr . Recent con-tributions in this setting are due to [108-113]. Important results achieved in theliterature for online scheduling with combined
EPI s are reported in Table 14.
Here, the
EPI given to an online algorithm is not exact. For instance, thealgorithm knows a nearest value of actual
Sum but not the exact value. Tanand He [114] initiated the study of online scheduling in m -identical machineswith inexact EPI s. They considered independently inexact C OP T , Sum, Maxand obtained lower bound of 1 .
8. Open Problems(OP) • OP : Defining a new performance measure.
Can we come up with an alternate performance measure than competitiveanalysis for online scheduling algorithms? The reason is quite intense inthe sense that comparing the cost of an online scheduling algorithm withan actual lowest possible cost would be more realistic than comparing withthe lowest cost obtained by an unrealistic offline scheduling algorithm.Now, the non-trivial challenge is to define the actual lowest possible cost.27 able 14: Important Results for Online Scheduling with Combined EPIs
Year, Author(s) Machine(s),EPIs Competitive Ratio(s)
T GRP ( lb ) 1 .
33 Tight2002, Tan, He [107] 2-identical,Sum, Max 1 . T GRP ( ub ) 1 . ub ∈ (0 . , . ub Tight for ub ∈ (0 . , T GRP ( ub ) (1 + b +1 ) Tight for ub ∈ [ b , b +1) b (2 b +1) ], ( b − ) ub +0 . b +1 b )Tight for ub ∈ ( b − b ( b − , b − ] and b ≥ . , .
4) LB and UB respectively2007, Wu et al. [111] 3-identical,Sum, Max 1 .
33 Tight2012, Cao et al. [112] 2-identical,Opt, Max 1 . T GRP (1 , r ),Decr 1 .
16 Tight for 1 ≤ r < .
5, 1 .
16 LBfor r ≥ . • OP : Fairness criteria.
Is an online scheduling algorithm fair in sharing resources such as ma-chines and time? It is worth of considering fairness in online schedulingfor multi user systems, where the main concern is to share the resourcesfairly among the users. Here, a scheduling algorithm mainly focuses onoptimizing the objective for each user than the overall objective of thesystem. Now, the question is: how to analyze the performance of such online scheduling algorithms? • OP : Realistic Job Characteristics as new EPIs.
It is an open issue to explore the realistic job characteristics that can beknown in advance for the improvement of the existing bounds obtainedby online scheduling algorithms in multiprocessor systems. • OP : Generic unified online scheduling model.
Can we define an unified model for illustrating all variants of the onlinescheduling? It will be interesting to design a generic online algorithmbased on the model that can be applicable for all settings of the onlinescheduling problem. 28 OP : Characterization of input job sequences for practical ap-plications.
How to characterize input job sequences? Characterization of input se-quences will provide a mapping rule for the practitioner to implementvarious theoretical online scheduling strategies in real world applications.
9. Concluding Remarks
We presented the state of the art results for preemptive and non-preemptiveonline scheduling with makespan minimization. Important contributions on de-terministic and randomized online algorithms in parallel machine models suchas identical and uniform-related were discussed. The basic concepts of onlinealgorithm, optimum offline algorithm and competitive analysis were presentedfrom a beginner’s perspective. Well-known related previous surveys on onlinescheduling for the last five decades were summarized in a chronological way.Fourteen well-known online scheduling algorithms along with their competitiveanalysis results were presented.Two emerging research trends such as resource augmentation in online schedul-ing and semi-online scheduling with extra piece of information were also high-lighted. We explored non-trivial research challenges and open problems in oursurvey. We hope that our survey will help the naive researchers to gain a ba-sic and comprehensive understanding of the emerging area of online schedulingwith makespan minimization and inspire for future research.
Acknowledgment
This work is partially supported by Department of Computer Science and Engi-neering of Veer Surendra Sai University of Technology, Burla, Sambalpur, India.
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