Center. Unwemzty of Wisconsin-Mad~son. IBM. T.J.. Watson. Resea7ch. Center ...... a scheduling algorithm we call. LIST for the problem. NMRTS. This algorithm.
Scheduling
Parallelizable
John IBM
T.J.
Tasks
Turek
Watson
Yorktown
to Minimize
Walter
Research Heights,
Center
Unwemzty
NY
Compute.
Lisa
Fleischer
Science
operations
Research Ithaca,
L.
Resea7ch
Yorktown
Department
Wolf
Watson
Heights,
Jason
Tiwari
Watson
Yorktown
Department
T.J.
ReseaTch He%ghts,
Glasgow
Ismel
NY
Institute
of Technology
Hazfa,
for
Philip
Schwiegelshohn
Unzverstty Information Dortmund,
of Dovtmund Technology
Center
NY
Technton
Center
NY Uwe
Institute
T.J.
Joel IBM
Time
WI
Prasoon IBM
Unwerstty
Response
Ludwig
of Wisconsin-Mad~son Madmen,
Cornell
Average
IBM
T.J.
Systems
Israel
S. Yu
Watson
Yorktown
Research Hetghts,
Center
NY
Germany
1
Abstract
Introduction
Consider A parallelizable (or malleable) task is one which can be run on an arbitrary number of processors, with a task execution time that depends on the number of processors allotted to it. Consider a system of M independent parallelizable tasks which are to be scheduled without preemption on a parallel computer consisting of P identical processors. For each task, the execution time is a known function of the number of processors allotted to it. The goal is to find (1) for each task i, an allotment of processors ~,, and (2) overall, a non-preemptive schedule assigning the tasks to the processors which minimizes the average response time of the tasks. Equivalently, we can minimize the flow tnne, which is the sum of the completion times of each of the tasks.
a multiprocessor
cessors dent
and
a task
tasks
i c
to
be
{1, . . . . M},
processors
be
allotted
a task
are
without to
required
i are
later
completion
think
of the
and
although
explicitly tasks
malleable.
i, and
the
total
for
all
number
number
t.The
finding
the
given
average
measure
by ~ = in
active
problem
the
In
minimizes
r,
(61,
other
we are concerned
that
task
refer
the
or i,
>0.
that
cannot
of a (We
of task
. . . . BM).
sense
processors
of processors.
a schedule
each time
allotment of
We
o~ paralleksm
to be
a pToa task
paralleltzable
for
degree
a
along
to
contiguous.
a starting
legal
within
along
allotted
consist,
and
informally
place
stretches
be
same
stretches
as either
will ,&
is required
t the
to
to @ as the
we denote
time
The
schedule
refer
schedule
tzme
required
allotment
sometimes
~;
the
i at some
can
h (~,)
processors
interchangeably
A
processor
height
and
allotted
at task
One
to
unison
task
i as taking
width the
in
complete ).
of pro-
allotted
processors
the
7, + t,(~, of task
whose
&
i, of
taskezecutzon number
task
start
then
whose
cessor not
will
ttme
a tmn.e axis are
200
They
is, the to
task
number
processors
that
pro-
indepen-
each
its
of the
of the
That required
that
that
execute
execution
axis,
and
All
to
rectangle
to such
SPAA 94-6194 Cape May, N.J, USA 0 1994 ACM 0-89791-671 -9/94/0006..$3.50
all
tzrne r,.
malleable
Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association of Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission.
it.
M
an arbitrary
function
preemption.
task
star-tmg
In this paper we tackle the problem of finding a schedule with minimum average response time in the special case where each task in the system has sublinear speedup. This natural restriction on the task execution time means simply that the efficiency of a task decreases or remains constant as the number of processors allotted to it increases. The scheduling problem, with sublinear speedups has been shown to be ,f P-complete in the strong sense. We therefore focus on finding a polynomial time algorithm whose solution comes within a fixed multiplicative constant of optimal. In particular, we give an algorithm which finds a schedule having a response time that is within 2 times that of the optimal schedule and which runs in 0( M(it42 + P)) time.
to
of
Assume
. . . . P}
E {1,
of P identical
consisting
allotted
tmne t~(L?; ) is a known cessors
T
scheduled. can
~;
consisting
system
at
) A any
exceed
words,
with
is that
average
response
by
response
in computer
time
is an important
performance.
Note
and the
standard
completion
of
times
of all
pletion makespan
count
(as is the
equally,
Modulo
be removed
problem,
not
case in the
problem).
obviously the
tasks
time
the
without
just
the
last
well-studied factor
affecting
we are attempting
com-
say t~. In other
minimum
&,
which
the
solution
ters
can to
to minimize
of the
words,
task
We refer
to this
response
time
case
the
system,
allotments
rather
problem
an execution
parame-
of the
(for
NMRTS
because time
part
as NMRTS
scheduling).
of MRTS,
are fixed
than
problem.
non-malleable
is indeed
we can
define
for
a special
each
task
i
function
(1)
i=l This
quantity
is typically
For
simplicity,
we shall
tion
to the
terms by
minimum
of the
the
to this
time
of the
[Cof7 of
time
6].
The
a solu-
problem
function
Equation (for
time
paper
response
in
as MRTS
jlow
this
objective
formula
problem
the in
average
value
flow
called speak
problem
Turek,
in
NMRTS
the
1. We
malleable
shall
refer
The
rest
time
tion
2 we briefly
response
of
present
sake
of simplicity
time
of the
number
this
ti(~~)
is not
time
actually
~,
allotted
a restriction:
t,(~;)
time
assume
gives
function
that
the
i is a non-increasing
of processors
function
cution
we shall
of task
to
exe-
function
to it.
However,
task
execution
Any
rise
defined
the
a new
a surrogate
task
the
three
MRTS-SS.
exe-
which
by
less
optimal the
number
of
remaining
We
processors
processors
do
simply
the
the
make
work,
and
In this
paper
we focus
of the
means
tasks
that
task
i
is
allotted
the
our
number
umrk
attention
function
~;.
function
Inother
in which
Formally,
this
=~itt(~i) for
az(/3i)
a non-decreasing
processors
on systems speedup. of
words,
the
foralll
of
< i < M,
task
the bound
possible
to
tasks
in
Section
to
be 5.
used
optimal
one.
ponents
of
as input bound
initial the
(Geometrically,
of course,
striction the
on the
efficiency
as the refer
task
of
number to this
which
work
malleable
execution
a task
restricted
decreases
for
MRTS
or
of
re-
simply
remains
2
Previous
remark
zncreaszng
function
time
is minimized
each
task,
work. not
that
in passing
and
Of
of /3, for trivially
ordering
course,
typically
if,
each
the
such
We
The
corresponding
minimum
special
and
a, (~,)
the
strong
the
strong
optimal paper for
case
task
i,
then
the
in order
has
to
of increasing assumption
is
been
studied
malleable
and
sense.
Thus
sense
as well.
flow
time
we present
the
problem
and
which
The
current
runs
Let
X; for
will
to
refer
in 0(M(M2 makes
entire have
is ;\ ’P-complete
in
and
the
system
which + P))
Shachnai in
denote
having
by
be .\-’P-complete
task
algorithm
MRTS-SS
paper
studied
to
MRTS-SS
solution an
was shown
cost T.
finds
flow
time
of the In
use of a new
We
com-
order
also
of
outline
a schedule
with
by
{’Ti + t.(~, )},
in
a schedule XT
