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Computer scheduling algorithms: Past, present and future

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Efficient scheduling techniques of computing resources are essential for achieving satisfactory performance for users as computer systems and their applications become more complex. In this paper, we survey research on scheduling algorithms, review previous classifications of scheduling problems and present a broader classification scheme. Using a uniform terminology for scheduling strategies and the new classification scheme, previous work on scheduling strategies is reviewed and trends in scheduling research are identified. Finally, a methodology for developing scheduling strategies is presented.
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COMPUTER SCHEDULING ALGORITHMS:
PAST, PRESENT, AND FUTURE
K. M. Baumgartner and B. W. Wah
ABSTRACT
Efficient scheduling techniques of computing
resources are essential for achieving satisfactory per-
formance for users as computer systems and their
applications become more complex. In this paper, we
survey research on scheduling algorithms, review pre-
vious classifications of scheduling problems, and
present a broader classification scheme. Using a uni-
form terminology for scheduling strategies and the new
classification scheme, previous work on scheduling
strategies is reviewed, and trends in scheduling
research are identified. Finally, a methodology for
developing scheduling strategies is presented.
1 INTRODUCTION
Early computer systems were centralized due to
the cost of replicating hardware and additional staffing.
As hardware costs dropped, it became possible for
smaller organizations to own computer systems. Con-
sequently, several computer installations could be
present on a college or industrial campus, and local
area networks (LANs) evolved to allow communica-
tion among the computer installations. The resulting
collection of resources and the communications
medium are Distributed Computer Systems (DCSs).
This trend is even more prevalent now as networks of
personal computers and workstations such as Sun,
Apollo, and MicroVAX are common in the work place.
The difference between workstations and personal
computers is the order of development of communica-
tion facilities. Personal computers were designed as
independent computers, and later networking capabil-
K. M. Baumgartner is with the Digital Equipment
Corporation, Maynard, MA 01754-2571. B. W. Wah is with
the Coordinated Science Laboratory, University of Illinois,
Urbana, IL 61801-3082.
This research was supported by the National Aeronautics and
Space Administration under grants NCC 2-481 and NAG 1-
613, and the National Science Foundation under grant MIP
88-10584.
First Workshop on Parallel Processing, National Tsing Hua
University, Taiwan, December 1990.
ity was added (this is similar to communication among
mainframe computers). The communication among
workstations was developed simultaneously with the
computing power, allowing communication services to
be more easily integrated. Now computing power is
literally distributed from desktop to desktop.
There are three distinguished characteristics of
local area networks: they are comprised of auto-
nomous systems so that control is not limited to one
location; there is a physical distribution of resources
(typically on the order of one kilometer); and the speed
of communication ranges from approximately one to
twenty megabits per second. Enslow [Ens78] more
formally specifies five requirements for a DCS: a mul-
tiplicity of general purpose resources (physical and
logical), a physical distribution of these resources, a
high level operating system to integrate control, system
transparency so that services may be requested by
name, and operation of the resources characterized by
cooperative autonomy.
The networks which connect computers and
workstations allow communication, but they also have
the capability to allow efficient sharing of resources.
Since the demands for computing power are continu-
ally increasing, the network can be used for scheduling
tasks during time when it is otherwise idle. DCSs can
provide a cost-effective solution to increase the com-
puting power available to a single computer userif jobs
can be scheduled to exploit potential parallelism. Liv-
ney and Melman [LiM82] have shown that in a system
of n independent processors modeled as M/M/1 sys-
tems [Kle75], the condition where a job is waiting for
service at one processor while another processor is idle
occurs 70% of the time for traffic intensities (the ratio
of the arrival rate to the service rate) ranging from 0.5
to 0.8. This idle-while-waiting condition indicates the
possibility of reducing the average job delay. With a
global scheduling strategy for a DCS, the occurrence of
the idle-while-waiting condition can be reduced and
consequently the overall performance of the DCS can
be improved.
1.1. Definition of the Scheduling Problem
The scheduling problem is shown in Figure 1. It
is a mapping of a set of jobs or tasks to a set of proces-
sors (or machines). The job characteristics (processing
time and precedence constraints), machine environ-
ment (number of processors, interconnection, power of
processors), and performance objectives are the input
to the scheduler, and the mapping, or schedule, is the
output. There are five components of the problem: the
events, the environment, the requirements, the
scheduler, and the schedule. The scheduler allocates
resources to events.
SCHEDULER
EVENTS
ENVIRONMENT
SCHEDULE
Figure 1. The Scheduling Problem.
Events are the smallest indivisible schedulable
entity. The environment refers to all characteristics
and capabilities of the surroundings that will impact the
schedule such as physical capabilities of the processors
and the communication mechanism. Requirements are
also input to the scheduler and may range from a real
time deadline to the requirement of determining if an
improvement in performance is possible. The
scheduler takes the events, environment, and require-
ments as input, and produces a schedule. In the pre-
cisely specified case, the schedule will be a set of
ordered pairs of events and times. Frequently it is not
possible to precisely state a schedule, in which case the
schedule is specified by a set of rules. These rules will
specify dynamic correcting actions when the distribu-
tion of events in the environment is such that the
correcting action will result in an improvement in some
system performance parameter.
In order to fully investigate the problem of
scheduling, it is beneficial to consider a larger perspec-
tive as shown in Figure 2. The starting point is a prob-
lem. The problem can be deterministic,non-
deterministic, or undecidable. A deterministic problem
is clearly defined and tractable in terms of both time
and space. Traditionally, deterministic scheduling
problems refer to those scheduling problems that have
all information required for generating a schedule
specified a priori. This excludes any data dependent
tasks (i.e. decision points). An example would be
matrix multiplication. A non-deterministic problem
may be clearly defined, but too complex to find an
exact solution with finite resources, or it may be clearly
defined, but require too much space to find an exact
solution with finite resources, or it may require too
much time and space. Alternatively, a non-
deterministic problem has inputs which are not exactly
specified. For example a scheduling problem with
resource requirements specified as distributions is a
deterministic scheduling problem. There is no algo-
rithm to solve an undecidable problem even with
infinite processing and storage resources. An undecid-
able problem cannot be solved directly, nor can it be
represented directly as a deterministic problem. Only a
deterministic problem can be solved, so the non-
deterministic problem is solved using a deterministic
algorithm that approximates using heuristics. The
algorithm is the specified in a language which is in
turn given a machine representation.
PROBLEM ALGORITHM
LANGUAGE
MACHINE
REPRESENTATION
Figure 2. Mapping from Problem to Machine
Representation.
During the 1960s and early 1970s, a great deal of
attention was focused on scheduling problems that
emerged in a manufacturing environment [CoM67].
These problems were predominantly deterministic as
the arrival time of jobs requiring service, and the dura-
tion of service were exactly known. Scheduling prob-
lems that emerge in computer systems are non-
deterministic because exact information about resource
requirements is rarely available. Deterministic and
non-deterministic scheduling problems are discussed
further in Section 2.
Scheduling resources on distributed systems has
two aspects: intracomputer scheduling and intercom-
puter scheduling. Intracomputer scheduling concerns
scheduling within a computer (local scheduling) while
intercomputer concerns scheduling tasks among com-
puters (global scheduling). Intracomputer scheduling
occurs at many levels within the processor such as
through the memory hierarchy, at the device and func-
tional unit level, and also scheduling processes. When
a computer is composed of multiple processors,
scheduling among them is another level. Intercom-
puter scheduling is a level above processor scheduling,
and involves communication among independent com-
puters. The levels of scheduling are shown in Figure 3.
At each level, scheduling is a mapping of events to the
environment.
Multiple Computer
Multiprocessor
Device/Functional Unit
Memory Hierarchy
Figure 3. Levels of Scheduling.
1.2. Organization
This paper is organized as follows. In Section 2
previous classifications of scheduling problems are
reviewed. These previous classifications are for deter-
ministic scheduling problems, which are problems with
all resource requirements specified exactly. A new
classification of scheduling problems which incor-
porates both deterministic and stochastic scheduling
problems is presented next in Section 3. Using the new
classification, trends in research on scheduling prob-
lems are investigated. Terminology for scheduling
strategies is also introduced in that section. Section 4
reviews previous work on scheduling. A methodology
for developing scheduling strategies is discussed in
Section 5, and Section 6 contains concluding remarks.
2. PREVIOUS CLASSIFICATIONS OF
SCHEDULING PROBLEMS
Before discussing classifications of scheduling
problems, it is useful to state the reasons why a
classification is needed, and to identify the desirable
characteristics of a classification. The goal of develop-
ing a classification is to increase and organize overall
knowledge about a class of problems. This goal is
realized through two steps: specification of a problem,
and showing relationship among problems. A
classification implies developing a taxonomy, which
addresses both the above steps since a taxonomy
implies a specification by categorization. Categoriza-
tion in turn shows relationships since problems with
attributes in the same category will be related. Thus
the taxonomy helps organize knowledge about a class
of problems.
There are at least four attributes of a
classification that are desirable. First is to identify the
significant characteristics of the problem since these
will contribute to an efficient solution. Next is to
clearly show the relationship among problems. This is
beneficial because the solution to one problem may
indicate the solution to a closely related problem. Con-
versely if a problem has no solution, that may indicate
a related problem also has no solution. Expandability
and contractability of a classification are useful
because they allow important features of the problem
to be focused on and unimportant details of the prob-
lem can be eliminated to reduce the complexity of the
problem representation. Finally, it is desirable for the
classification to separate the problem specification
from the solution. The separation allows a clearer com-
parison of scheduling strategies for a given problem,
and avoids confusing a strategy and a problem.
Three classifications of scheduling problems are
discussed in this section. First is Conway and Miller’s
classification [CoM67], a four parameter scheme
where the categories are: A: the job arrival process
which for static arrivals indicates the number, and for
dynamic arrivals indicates the probability distribution
of the time between arrivals, B: the number of
machines in the shop, C: the flow pattern in the shop,
D: objective function. This classification has two of
the desirable characteristics stated earlier: it identifies
significant problem features, and separates the problem
from the solution, but lacks expandability and contrac-
tability. Much of the problem is unspecified such as
whether the machines are homogeneous or heterogene-
ous, if preemption is allowed, and details about the
kinds of tasks to be scheduled. There is no capacity for
specifying stochastic problems with this classification.
Graham and Lawler classified deterministic
problems with requirements of optimal solutions. The
classification uses three sets of attributes: job charac-
teristics (α), machine environment (β), and objective
function (γ). These sets of attributes are subdivided in
to numerous components. In effect, Graham and
Lawler’s classification combined Conway and Miller’s
B and C categories into one field specifying the
machine environment, and provided a more detailed
specification for the job specification and machine
environment. The α/β/γclassification allows precise
specification of problems. In fact, by enumerating the
possible values of the categories, a finite set of prob-
lems can be named. Identification of significant
features, grouping sets of related characteristics, and
separating the problem from the solution are also
advantages of this classification. This classification is
especially good at showing the relationship among
problems using reducibility. As an example, the redu-
cibility for task structure is shown in the attribute task
precedence. Scheduling tasks governed by general
precedence constraints is more difficult than schedul-
ing tasks governed by tree-like precedence which in
turn is more difficult than scheduling independent
tasks. Solution to a more difficult problem implies the
solution of the easier problem. Unfortunately, this
classification does not include representations for sto-
chastic scheduling problems or representations for
problems with non-optimal objective function require-
ments. Since each attribute has values that are expli-
citly specified, a uniform representation of stochastic
problems using this representation would require a
large expansion of the representation. The set of sto-
chastic problems is extremely large, and an expanded
version of the α/β/γrepresentation would be unwieldy.
Gonzalez uses nine classification categories: the
number of processors, task duration (either equal or
unequal), precedence graph structure, task interrupti-
bility, processor idleness (whether this is allowed or
not), job periodicity, presence or absence of deadlines,
resource limited schedules, and homogeneous versus
heterogeneous processors. While this classification
identifies nine key features, there is no grouping of
related characteristic, no relationship among problems
shown, and the classification is not easily expandable.
Also, this classification is not designed for problem
specification.
3. A NEW CLASSIFICATION FOR SCHEDUL-
ING PROBLEMS
In this section, a new classification of scheduling
problems is presented. Additionally, terminology for
describing scheduling strategies is also described.
3.1 The ESR Classification Scheme
Table 1. The ESR Classification.
Category Attribute Values
Event Relation to independent
other events precedence
communication
Availability static
(arrivals) periodic
stochastic
Resource deterministic
Requirements stochastic
Environ- Number 1, k, n
ment Classes homogeneous
of Resources resources (1 class)
heterogeneous
resources
(> 1 class)
Physical speed
characteristics main memory size
special capabilities,
etc.
Paths Connecting topology
resources
Communication none
overhead deterministic
stochastic
Communication flow
mechanism broadcast
message passing
other
Require- Performance any solution
ments deadlines (real time)
good (improved)
solution
optimal solution
In the process of developing a new classification,
it was beneficial to use some of the methods of the pre-
vious classifications, specifically the grouping tech-
niques. Consequently, the classification uses three
groups corresponding to the input components of the
scheduling problem noted in Section 1. These three
categories: the events, the environment (or surround-
ings), and the requirements, comprise the new ESR
classification scheme. These correspond in part to the
categories of the α/β/γclassification. There is some
rearrangement, and this classification is not as explicit.
The categories and the attributes of the ESR
classification are summarized in Table 1.
The first category, E, is the event. The three
attributes of events considered are their relationship
with other events, arrival patterns, and the available
information about the resource requirements of each
event. Events may be independent in which case the
execution time is not constrained by other events.
Conversely, events may have precedence constraints
where an event must be executed before its successors.
Another relationship among events is communication,
where two events exchange information during or
between their execution. This relationship is especially
important when communication costs are considered.
Availability indicates whether events, or jobs, all
arrive simultaneously prior to execution, whether they
arrive periodically, or whether their arrival is governed
by a stochastic distribution. The first case is of simul-
taneous arrival is also called static arrivals. Periodic
arrivals are similar to static arrivals in the sense that
they can be considered to be a repetition of static
arrivals if arrivals are sufficiently far apart to allow
execution of the previous set of jobs to complete before
the next set arrives. Stochastic arrivals have interar-
rival times that are governed by a probability distribu-
tion.
The final attribute of events is the resource
requirements which may be completely specified as
constants and so be deterministic, or may be governed
by a probability distribution. Clearly these three attri-
butes are important in the scheduling process. The
available information about resource requirements is
especially important, however frequently in multipro-
cessor systems there is limited information available.
The second category, S, is the environment or
surroundings. Any environmental characteristics that
will impact a schedule such as the number of
resources, the number of classes of resources, the phy-
sical characteristics of each class, the topology of the
communications medium, the communication delay,
and the communication mechanism are included in this
category. The number of resources can be one, imply-
ing scheduling on one machine, a specific number such
as 2 or 3 (specified as k), or an arbitrary number, n.
The resources may be homogeneous (all are identical)
or homogeneous (all are unique or multiple classes of
processors). The physical characteristics of these
classes can also be indicated if they are significant.
Interconnection among resources is called the path
among them. There are infinite possibilities for this
category including bus, ring, multistage networks, and
arbitrary point-to-point connections. Communication
overhead may be zero in which case events flow
among resources at no cost, or may have a constant
(deterministic) or varying (stochastic) cost. Lastly, the
communication mechanism may be a flow of events,
broadcast of information, message passing, or take
some other form. The scheduler uses information
about the surroundings to make predictions about the
relative costs of different schedules.
The final category is the requirements, R, of the
schedule. In the simplest case the requirements are for
any solution or schedule. Other cases include decreas-
ing response time (or another performance metric)
resulting from a previous schedule, meeting a fixed
deadline, and finding the schedule that uses the
minimum number of processors, and so on.
A key point about this classification is that it is
high level, meaning that attributes and their values can
be specified in general terms or in more detail. When a
scheduling problem is specified using this
classification, the values of the attributes are expanded
if they reflect a characteristic that will be used in creat-
ing the schedule. Unlike the deterministic scheduling
problems considered with the α/β/γclassification, sto-
chastic scheduling problems are not enumerable; it is
impossible to list every problem. With this
classification, it is possible to specifically note those
values the scheduler uses, and leave less important
parameters expressed in more general terms. Problem
specification is indicated by the three categories using
the following notation:
E: { }S: { }R: { }
The sets correspond with each category, and include as
much detail as is desired. If the sets are empty, it is
presumed that there is no useful information available
about that category. For example, deterministic prob-
lems can be specified as
E:
requirements
deterministic resource
static or periodic arrivals
event precedence
arbitrary acyclic
S:
overhead
communication
deterministic
heterogeneous resources
n processors
R: parameter
performance
optimize some .
Stochastic scheduling problems, in contrast, can be
specified as
E:
requirements
stochastic resource
stochastic arrivals
event precedence
arbitrary acyclic
S:
overhead
communication
stochastic
resources
heterogeneous
n processors
R: response time
reduce .
3.2 Scheduling Terminology
There is a considerable conflict in the literature
regarding terminology used to describe attributes of
scheduling strategies. This section discusses terminol-
ogy for classifying scheduling strategies. First, some
previous classifications of scheduling strategies are
reviewed. Using the results of these studies, terminol-
ogy used in this paper is described.
Wang and Morris developed a classification of
scheduling algorithms [WaM85]. The criteria for
classification is whether the strategy is source initiated
or sink initiated meaning whether overloaded resources
look to alleviate their load, or lightly loaded resources
actively pursue more work. Additionally, the level of
information dependency is a factor. Information
dependency refers to the level at which a resource has
information about the current state (workload) of other
resources.
The terminology used in global scheduling (or
load balancing) [ChK79, NiH81, EaL86, Ezz86,
WaJ83, BaW85] is varied and conflicting. Some
features commonly discussed (using different names)
are whether the scheduling intelligence is centralized
or distributed, whether the rule basis is static meaning
independent of the current state of the system, or
dynamic meaning decisions are state dependent. This
characteristic is also referred to as being deterministic
versus probabilistic, or adaptive versus non-adaptive
(adaptability also refers to an different attribute dis-
cussed below). These features are useful for compar-
ing scheduling strategies. The comparison has indi-
cated that these characteristics are related to the poten-
tial of an algorithm, as will be discussed in Section 4.
The informal classification used for load balanc-
ing problems above was formalized by Casevant and
Kuhl in an attempt to unify the diverse notation used
[CaK88]. Their classification was designed for distri-
buted computing systems, and consists of two parts: a
hierarchical classification and a flat classification. The
hierarchical classification is used to show where some
characteristics are exclusive. The flat classification
gives definitions of attributes that are not exclusive.
Several observations about this taxonomy are:
(a) Adaptability refers to long term algorithm state-
dependency rather than short term state depen-
dency for rule basis. Adaptability is available
with both a static and dynamic rule basis since the
algorithm can be static (or dynamic) for one time
interval, and then change for the next time inter-
val.
(b) Load balancing and optimality are considered
strategy characteristics rather than problem
requirements (elements of the set R).
(c) One time reassignment and dynamic reassignment
(which correspond to preemption in the one pro-
cessor case) are considered strategy characteris-
tics rather than environment capabilities (elements
of the set S).
(d) Bidding in the flat portion of the classification and
cooperation in the hierarchical portion of the
classification are not distinct.
The terminology summarized in Table 2 is not
exclusively adopted from any of these sources for the
following reasons. Wang and Morris’ classification
focuses on only two aspects of the strategy (initiation
location and information dependency), so it not exten-
sive enough. Casevant and Kuhl’s taxonomy is not
used exclusively because there is overlap between their
classification of solutions and the classification of
problems described in Section 2.3.
The first characteristic specified is the level of
scheduling as described in Section 1. The rule basis is
included next and may be static or dynamic. Note this
is different that the static versus dynamic sets used to
describe event arrivals. Next is the location of control.
If control has a degree of distribution, the controlling
processors can negotiate to make scheduling decisions,
or function independently. Initiation and adaptability
are the two final characteristics. The tradeoffs of these
different characteristics are discussed in the next sec-
tion.
The relationship among the six characteristics is
shown in Figure 4. Each path through the relationship
graph indicates a set of attributes a schedule may have.
Some paths and the corresponding combination of sets
of attributes are not possible. For example is is not
possible to have a centralized strategy with coopera-
tion. Centralized implies one decision point. Static
Table 2. Revised Classification of Scheduling Strategies.
Terminology
Characteristic Values Explanation
Level of
Scheduling Intra-
resource
Inter-
resource
Refers to scheduling
within a node or resource
as opposed to among
nodes or resources. There
may be multiple sub-
levels of intra-resource
scheduling. Also called
global versus local.
Rule Basis Static
Dynamic Refers to the flexibility of
schedule rules to react to
the current state of the
system. A static schedule
bases rules on unchanging
system characteristics. A
dynamic schedule bases
rules on the current state
of the system. Also called
state-dependency.
Location of
Control Distributed
Hierarchical
Centralized
Describes where the
responsibility for
scheduling decisions lies.
This applies primarily to
dynamic timing since
static timing implies a
centralized decision.
Hybrids are also possible.
Cooperation Negotiated
Independent Describes the interaction
among locations of control
(distributed or hierarchical
only). This applies to
distributed or hierarchical
control since centralized
does not have separate
modules to cooperate.
Initiation Source
Sink
Both
Which processor initiates
job movement, the
overloaded processor
(source initiated) or the
underloaded processor
(sink initiated).
Adaptability Adaptable
Non-
adaptable
Refers to flexibility of the
algorithm, and whether the
algorithm changes
execution based on the
arrivals to the system.
strategies are considered centralized since although
each participating strategy may make different deci-
sions, the choice was determined a priori by one
scheduling intelligence. In Section 4, this terminology
is used to discuss the results obtained in previous work
on scheduling problems.
Previous classifications of scheduling problems
are limited to deterministic scheduling problems. The
ESR classification includes both deterministic and sto-
chastic problems, and specifies problems using three
sets of characteristics, events, environment (or sur-
roundings), and requirements. The ESR scheme
specifies problems in a flexible manner, so that attri-
butes of the three sets can be indicated with varying
degrees of completeness allowing attention to be
focused on important features of the problem, and less
important details to be left in general terms.
4 STATE OF THE ART IN SCHEDULING
RESEARCH
In this section, trends in scheduling problems
that research is focused upon are identified. This infor-
mation indicates the current importance of scheduling
problems on DCSs. The trends also show the progres-
sions of understanding of scheduling problems. Table
3 chronologically lists studies on scheduling problems.
Problems are uniformly specified in this table. Three
categories indicating trends are communication over-
head, resource requirements, arrival characteristics.
Figure 5 shows a graphical representation of the trends.
The progression is towards the general problem of
scheduling tasks on distributed computing systems.
This is not unexpected; the earlier work is on the more
specific problems associated with deterministic
scheduling, and later work focuses on the more general
problems associated with stochastic scheduling prob-
lems.
The ESR classification scheme can be used to
specify a problem, and to make comparisons among
similar problems. It becomes difficult to show rela-
tionships among problems using all the attributes for a
wide range of problems because it is possible for a
given problem to be less general than another problem
for one attribute, while being more general for another
attribute. Because of this, the results of previous stu-
dies are shown by grouping problems by the charac-
teristic that dominated the trends shown in the previous
section which is whether processing requirements are
specified stochastically or deterministically.
Level of
Scheduling
Rule
Basis
Location
of Control
Cooperation
Initiation
Apaptability
intra-processor interprocessor
static dynamic
centralized hierarchical distributed
negotiated independent
source both
non-adaptive
sink
adaptive
Figure 4. Relationship Among Strategy Characteristics
Sto77
Stb78
Sto78
Ras79
ChH80
LoL81
Mal82
Wil83
BaH85
JeB85
CoM67
GoR72
RaC72
Bak74
LiH77
Gon77
GrL77
ApI78
Gon79
Weg80
EnH80
KrH80
Jak80
Efe82
Mon82
Leu82
Bal83
ChA82
Pin83
OuS80
WaH82
LiM82
HwC82
WaJ83
WaP83
MeT84
KeL84
BaW85
TaT85
NiH85
NiX85
Sta85
Ezz86
EaL86
JuW86
Alo86
BaW87
ChK79
NiH81
WaM85
Resource
Requirements
Static/Periodic Arrival
StochasticDeterministic EVENTS
Stochastic
Deterministic/
Stochastic
None
ENVIRONMENT
Communications
Overhead
Figure 5. Trends in Scheduling Research
4.1. Problems with Deterministic Resource
Requirements
A great deal of research was done for scheduling
tasks with exact a priori knowledge of execution
requirements during the 1960s and 1970s. This work
has been described in several books [CoM67, Bak74,
Cof76] and survey papers [Gon77, GrL77]. The prob-
lems are represented as follows.
E:
requirements
deterministic resource
static arrival
precedence
arbitrary acyclic
S:
by flow
communication
flow shop paths
open shop,
resources
heterogeneous
n processors
R:
approximate
nearoptimal
optimal
Note that this specification includes less general prob-
lems (i.e. events that are independent tasks). When
specifying a set of problems with the ESR
classification, a more general problem encompasses
less general problems. This is a class of problems that
frequently occur in a manufacturing environment and
are scheduling at a inter-resource level. Scheduling
strategies for this type of problem have been discussed
extensively [CoM67, GrL77]. Solutions are optimal,
near optimal, or approximate. Schedules may be expli-
cit time, event pairs or static rules such as schedule the
shortest job first. Scheduling intelligence in this case is
centralized.
As DCSs evolved, scheduling problems related
to computing environment received more attention.
The ESR specification of these problems is
E:
resource requirements
deterministic
static arrivals
precedence
arbitrary acyclic
S:
overhead
communication
deterministic
resources
heterogeneous
n processors
R:
approximate
nearoptimal
optimal
Some early scheduling problems for the mul-
tiprocessor environment considered scheduling tasks
with acyclic precedence requirements with no com-
munication among tasks [RaC72, GoR72]. These stu-
dies assumed that a reasonable estimate of resource
requirements could be obtained with a preprocessing
phase of a program. Results of these studies included a
method for determining the minimum computation
time, and a method to determine the minimum number
of processors to achieve the minimum computation
time [RaC72]. Also, a comparison of a centralized
versus a decentralized algorithm indicated that the
decentralized algorithm performed better.
Scheduling tasks with intertask communication
is a more difficult and a more realistic problem.
Improving performance requires limiting excessive
communication and evenly distributing the work load
among processors. If all the tasks are scheduled on one
processor, there is no communication cost, but there is
no benefit from a multiplicity of processors. If the
tasks are distributed such that processor utilization is
completely uniform, full advantage of the multiplicity
of processors is realized, however communications
costs may be so large that the benefits of concurrent
execution are eliminated causing processing to be more
expensive than with one processor. Since the goals of
limiting communication and balancing load are directly
conflicting, obtaining maximum performance is a
tradeoff.
Several studies were performed by Stone [Sto77,
StB78, Sto78, RaS79] in which jobs were represented
by graphs. The nodes represent tasks with execution
requirements, the edges are labeled with communica-
tion costs. The representation is actually serial since
the program will be executing on only one node at a
time. The communication is information that one
module sends to its successor on completion. An
assignment of tasks is specified by a cut that divides
the graph in to as many sections as there are proces-
sors. The cost of the assignment is equal to the sum of
the execution costs plus the sum of the communication
costs between tasks that are not assigned to the same
processor, that is the sum of the weights of the edges
on the cut. Consequently, the minimum cost assign-
ment corresponds to the minimum cutset. The time
requirements of this method may allow it to be used
dynamically for two or three processors, but not
efficiently for larger numbers of processors. A similar
mapping problem for larger systems was explored by
Bokhari [Bok81] in which communicating modules are
placed as much as possible on adjacent processors.
Other approaches to this problem include graph match-
ing [ChH80, RaS79], mathematical programming
[Weg80, ChH80], a branch and bound algorithm
[MaL82], and heuristics [ApI78, ChH80, KrH80,
LoL81, Efe82]. An optimal solution for the n proces-
sor case was found by Chou and Abraham [ChA82].
The solutions to these problems are for static sets
of jobs, and the scheduling strategies are static as well.
Their execution time is too long for them to be effec-
tive for dynamically arriving tasks. Their utility is then
for the planning phase of a system (prediction of
minimum or maximum execution time), or for real-
time systems where timing is critical.
4.2. Problems with Stochastic Resource Require-
ments
The studies reviewed in the previous subsection
used exact information about the execution and com-
munication requirements of the jobs being scheduled.
Frequently in scheduling problems on computer sys-
tems, this information is not known, and only proba-
bilistic information may be available. Further, dynami-
cally arriving sets of jobs are more common in com-
puter systems than static sets of jobs. This change in
available information changes the methods and expec-
tations of the scheduler. The problem is easier in some
cases because with exact a priori information, there
may be much information that it cannot be used
efficiently [Pin83]. The class of problems discussed in
this section have an ESR representation as follows:
E:
requirements
stochastic resource
dynamic arrivals
independent tasks
S:
point interconnection
bus or point-to-
homogeneous resources
n processors
R:
communication
minimize
distribute load
evenly
The following issues are important when
developing strategies for dynamic scheduling problems
with stochastic resource requirements: balanced load
versus minimized communication tradeoff, location of
control, the status information used for the scheduling
decisions, and the initiation point. The balanced load
versus communication tradeoff was discussed above.
Location of control is also a tradeoff. A central-
ized location of control may allow the the scheduling
strategy to be simpler. However, the decision point has
the potential of becoming a bottleneck and a critical
failure point. If a distributed decision is made, the
overhead of distributing status information can be so
high as to eliminate the benefits of scheduling. A com-
parison of centralized versus distributed strategies
using a trace driven simulation was performed by Zhou
[Zho86]. The results indicate that neither strategy is
always superior, and that the overhead for communi-
cating information is important for both. The third
consideration is what status information to use for
scheduling decisions. Several studies on this topic
have indicated that excessive status information is not
only unnecessary, but can be detrimental [LiM82,
WaM85, TaT85].
Several static scheduling strategies have been
proposed. Proportional branching is a static, sink ini-
tiated strategy where jobs are routed to processors with
a probability determined by the relative power of the
processors [ChK79]. Ni and Hwang found optimal
static sink initiated strategies for single and multiple
job classes again with centralized control [NiH81]. An
optimal static source initiated strategy was found by
Tantawi and Towsley for scheduling jobs that are
modeled as independent tasks [TaT85].
Dynamic strategies have more potential than
static strategies because they can react to changes in
the system state. Chow and Kohler proposed three
dynamic centralized strategies with a job dispatcher
(sink initiated) and found the one that maximizes
throughput gives the best performance.
Distributed dynamic strategies are more complex
than scheduling strategies using centralized control
because of the added task of coordinating independent
actions. Typically distributed and adaptive strategies
involve negotiation among participating processors.
This negotiation involves communication of status
information, and the selection of processors involved in
the scheduling decision. The simplest method is to
maintain a centralized table with load information, and
processors can consult the table before sending jobs for
remote execution [HwC82]. This method has the simi-
lar problems as the centralized dispatcher in the sense
of reliability and bottleneck. Another technique is to
have each processor broadcast its load and keep track
of the loads of the other processors in effect duplicat-
ing the table at each processor in the system [LiM82,
Zho86]. Alternatively, only significant load changes
can be broadcast resulting in a decrease in network
overhead [LiM82, Zho86]. Other methods of exchange
involve nearest neighbors [Wil83, KeL84]. Stankovic
has proposed a strategy that includes a bidding phase
where negotiation takes place [RaS84, Sta85]. Some
significant results of these previous studies are as fol-
lows.
(a) Load balancing is beneficial since load imbalance
occurs frequently in a system with ten or more
processors.
(b) Excessive state information to make a load
balancing decision is not necessary and may be
detrimental.
(c) Sink initiated strategies have the potential for
improved performance over source initiated.
(d) Dynamic strategies have greater potential that
non-adaptive strategies.
(e) Centralized strategies may create reliability and
bottleneck problems
(f) Scheduling communication should not interfere
with regular message transfer.
(g) There is a mismatch between the capabilities of
the network and the communications required for
scheduling operations.
5. A METHODOLOGY FOR DEVELOPING
SCHEDULING STRATEGIES
There are three components of a scheduling stra-
tegy where improvement is possible: representation,
procedure, and evaluation. Representation involves the
identification of features significant to the problem.
The procedure is the steps performed in the strategy.
Finally, the evaluation is the measurement of parame-
ters associated with performance. Evaluation may then
involve measurement of system parameters for use in
procedure decisions, and measurement of system
parameters for performance evaluation.
The methodology for developing a scheduling
strategy uses the following three assumptions: the sys-
tem in question is a distributed computer system as dis-
cussed in Section 1, the arrival rate is stochastic, and
the jobs resource requirements are expressed stochasti-
cally.
The overall flow of the methodology is shown in
Figure 6. There are eight steps, and each is made con-
sidering the key results noted in Section 4.3. The first
step is to develop a problem representation for the
three components environment, events and require-
ments. In this step, environmental attributes of the sys-
tem are identified which may aid in the scheduling pro-
cess are identified. Underlying communication net-
Determine
representation
for problem
using ESR
classification
Find set of
states that
can be altered
to improve
system perf.
Determine s
to states
that cause
improvement
Identify and
develop support
tools to
detect states
and perform s
Develop proc.
(or sequence)
detecting the
states, and
performing s
Implement
strategy
Alter
evaluation
criteria
Evaluate
efficiency
of sequence
Figure 6. Methodology for Developing Scheduling
Strategies
work capabilities such as broadcast or multicast capa-
bility and bandwidth are important at this point as they
may be helpful during later steps.
The procedure specification is next and consists
of identifying system states that will allow improve-
ment. Such states are those in which the idle-while-
waiting condition arises, or those states which make
the idle-while-waiting condition more likely. The next
step is to determine feasible rearrangements of events
or s that redistributed the jobs to correct the condi-
tion. Clearly the redistribution must use the environ-
ment capabilities indicated when the problem represen-
tation was specified. Consequently, it may be neces-
sary at this point to iterate to the first step and redefine
the representation to include more features to aid in
identifying the states where improvement is possible,
and the redistribution to effect the improvement.
The next step in procedure development is to
determine the support tools for state detection and
implementing the s. It is important to assure that
these operations can be performed efficiently. If this in
not the case, iteration may again be necessary. Candi-
dates for change will be both the representation where
it may be possible to identify additional capabilities
that can be utilized, and determining new states and s.
It may be necessary to adjust expectations because net-
work limitations preclude complex state detection and
rearrangement. Conversely, care should be taken to
assure that network capabilities are fully exploited.
With tools developed to identify states and s,
the final step in the procedure development is to deter-
mine the overall sequence of operations. During the
formulation of this sequence other system responsibili-
ties need to be considered. The sequence development
may require adjustment to each of the four steps men-
tioned previously.
After the sequence has been generated, perfor-
mance evaluation is necessary. This consists of two
steps, determining the evaluation criteria (i.e. delay or
throughput), and measuring the performance. Obvi-
ously, these steps must consider the set R. Also impor-
tant is assuring the overhead of the scheduling strategy
does not impede other system missions.
At the conclusion of the preliminary study, the
strategy can be implemented. This may require itera-
tion through all of the five steps of the methodology.
Again, performance measurement will be necessary.
6. CONCLUDING REMARKS
Previous classifications of scheduling problems
are limited to deterministic scheduling problems. In
this paper, we have presented the ESR classification
which includes both deterministic and stochastic prob-
lems. The ESR scheme specifies problems in a flexible
manner, so that attributes of the three sets can be indi-
cated with varying degrees of completeness allowing
attention to be focused on important features of the
problem, and less important details to be left in general
terms. Using the attributes of the ESR classification,
previous research on scheduling problems has been
discussed and trends in scheduling problems that have
been studied have been identified. Additionally, termi-
nology for discussing scheduling strategies has been
defined. Finally, a methodology for developing
scheduling strategies has been described. It uses the
the ESR representation of the problem to focus key
aspects that contribute to an efficient solution.
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Table 3. Summary of Scheduling Problems Studied.
Event Environment
Resource Communication
Relationship Availability Requirements Number Resources Overhead
Reference
I P C Static Periodic Stoch Determ Stoch 1 k n Homog Heter None Some
[CoM67] X X X X X X X X X X
[GoR72] X X X X X X X X X
[RaC72] X X X X X X X X X X X
[Bak74] X X X X X X X X X X
[Sto77] X X X X X X X X X X X X
[LiH77] X X X X X X X X X X
[GrL77] X X X X X X X X X X X
[Gon77] X X X X X X X X X X
[StB78] X X X X X X X X X X X
[Sto78] X X X X X X X X X X X
[ApI78] X X X X X X X
[Gon79] X X X X X X X X
[ChK79] X X X X X X X X X
[RaS79] X X X X X X X X X X X
[OuS80] X X X X X X X X X X
[Weg80] X X X X X X X X X
[EnH80] X X X X X X X
[ChH80] X X X X X X X X X X X
[KrH80] X X X X X X X X X
[JaK80] X X X X X X X X
[NiH81] X X X X X X X X X X
[LoL81] X X X X X X X X X X
[Efe82] X X X X X X X X X
[WaH82] X X X X X X X X X
[LiM82] X X X X X X X X X
[HwC82] X X X X X X X X X
[ChA82] X X X X X X X X
[Mon82] X X X X X X X X
[Leu82] X X X X X X X
[MaL82] X X X X X X X
[BaL83] X X X X X X
[Wil83] X X X X X X X X
[Pin83] X X X X X X X X X
[WaP83] X X X X X X X X
[WaJ83] X X X X X X X X
[MeT84] X X X X X X X X X X X X
[KeL84] X X X X X X X X
[RaS84] X X X X X X X X X X X
[BaW85] X X X X X X X X
[BaH85] X X X X X X X
[JeB85] X X X X X X X
[TaT85] X X X X X X X X X
[WaM85] X X X X X X X X X
[WaJ85] X X X X X X X X
[NiH85] X X X X X X X
[NiX85] X X X X X X X X
[Sta85] X X X X X X X X
[JuW86] X X X X X X X X
[Ezz86] X X X X X X X X
[EaL86] X X X X X X X X
[Alo86] X X X X X X X X
[BaW87] X X X X X X X X
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Chapter
In Parts I and II a number of stylized and (supposedly) elegant mathematical models are discussed in detail. The deterministic models have led to a number of simple priority rules as well as to many algorithmic techniques and heuristic procedures. The stochastic models have provided some insight into the robustness of the priority rules. The results for the stochastic models have led to the conclusion that the more randomness there is in a system, the less advisable it is to use very sophisticated optimization techniques. Or, equivalently, the more randomness the system is subject to, the simpler the scheduling rules ought to be.
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This book on scheduling covers theoretical models as well as scheduling problems in the real world. Michael Pinedo includes a CD-ROM that contains slide-shows from industry and movies dealing with implementations of scheduling systems. © 2008 Springer Science+Business Media, LLC All rights reserved.
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In multiprocessor systems, an efficient scheduling of a parallel program onto the processors that minimizes the entire execution time is vital for achieving a high performance. The problem of multiprocessor scheduling can be stated as finding a schedule for a general task graph to be executed on a multiprocessor system so that the schedule length can be minimize. This scheduling problem is known to be NP-Hard. In multiprocessor scheduling problem, a given program is to be scheduled in a given multiprocessor system such that the program's execution time is minimized. The objective is makespan minimization, i.e. we want the last job to complete as early as possible. The tasks scheduling problem is a key factor for a parallel multiprocessor system to gain better performance. A task can be partitioned into a group of subtasks and represented as a DAG (Directed Acyclic Graph), so the problem can be stated as finding a schedule for a DAG to be executed in a parallel multiprocessor system so that the schedule can e minimized. This helps to reduce processing time and increase processor utilization. Genetic algorithm (GA) is one of the widely used technique for this optimization. But there are some shortcomings which can be reduced by using GA with another optimization technique, such as simulated annealing (SA). This combination of GA and SA is called memetic algorithms. This paper proposes a new algorithm by using this memetic algorithm technique.
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