A Task-Type-Based Algorithm for the Energy-Aware Profit Maximizing Scheduling Problem in Heterogeneous Computing Systems

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by  Weidong Li

A Task-Type-Based Algorithm for the Energy-Aware Profit Maximizing Scheduling
Problem in Heterogeneous computing systems
Weidong Li , Xi Liu, Xuejie Zhang, Xiaobo Cai
Yunnan University
Kunming, China
Email: {weidong, xliu, xjzhang, xbcai}@ynu.edu.cn
5
1
0
Abstract—In this paper, we design an efficient algorithm B. Contributions and Outline
2
for the energy-aware profit maximizing scheduling problem,
This paper presents an accurate mathematical model for
n where the high performance computing system administrator
a istomaximizetheprofitperunittime.Therunningtimeofthe the problem proposed in [1]. A polynomial-time algorithm
J proposedalgorithmisdependingonthenumberof tasktypes, isthendevelopedtofindafeasiblesolutionfortheproposed

2 whiletherunningtimeof thepreviousalgorithm isdepending model.
2 onthenumberoftasks.Moreover,weprovethattheworst-case The contributions of this paper are:
performance ratio is close to 2, which maybe the best result.
1) An accurate mathematical model;
] Simulation experiments show that the proposed algorithm is
C more accurate than the previous method. 2) A task-type-based algorithm to find a more accurate
feasible solution, whose running time is independent of the
D
Keywords-highperformancecomputing;scheduling;resource
number of tasks;
. allocation; approximation algorithm; bag-of-tasks
s 3) The worst-case performance ratio.
c
The remainder of this paper is organized as follows.
[
I. INTRODUCTION Thenextsectionproposestheaccuratemathematicalmodel.
1
SectionIIIpresentsthetask-type-basedalgorithmandproves
v
A. Background and Motivation the worst-case performance ratio. Section IV gives the
4
1 experimental results. The last section discusses the useful
In high-performancecomputing(HPC) systems, itis well
4 extensions to the proposed model and lists ideas for future
5 known that when the performance is increased, the power work.
0 consumptionis increased,aswell asthe electricity costsfor
. the operators are increased. Recently, the high cost of the II. THE MATHEMATICALMODEL
1
0 HPC systems has lead to research that designs an efficient Asin [1],a usersubmitsa bag-of-taskstoprocess,where
5 resource allocation algorithm to reduce the required energy eachtaskisindivisibleandindependentofalltheothertasks.
1 consumption[1].Bycombiningtheenergyandperformance Thecost to theorganizationforprocessinga bag-of-tasksis
:
v objectives into a single profit objective, Tarplee et al. [1] the cost of electricity. The organization or service provider
i introduced a novel monetary-based model for HPC where should maximum the profit per bag, which is equal to the
X
there is a financial distinction between the service provider price minus the cost. However, the bag-of-tasks can take
r
a andtheusers.InHPCsystems,therearetwoimportantfacts: a considerable amount of time to compute when trying to
(a) The HPC systems are often composed of differenttypes increase the profit by reducing electricity costs. Thus, it is
of machines;(b)There are a largenumberof tasks butonly more reasonable for an organization to maximize the profit
smallnumberoftasktypes.Bysolvingalinearprogramand per unit time.
rounding carefully, they [1] designed an efficient algorithm Formally, assume that there are T task types and M
to find a feasible schedule. machine types. Let T be the set of tasks of type i and
i
In [1], a lower boundon the finishingtimes ofa machine T be the number of tasks in T . Similarly, let M be
i i j
type is used to replace makespan, which is defined as the set of machines of type j and M be the number of
j
the maximum finishing time of all machines. Therefore, machines in M . Denote by x the number of tasks of
j ij
the proposed mathematical model is inaccurate. For the type i assigned to a machine of type j, where x is the
ij
proposed algorithm [1], in the roundingprocess, the energy primary decision variable in the optimization problem. As
consumption maybe increased, which can be avoided by the definitionsfrequentlyused in schedulingalgorithms[1],
using a different method. Moreover, the running time is let ETC be a T ×M matrix where ETC is the estimated
ij
depending on the number of tasks, which can be improved, time to compute for a task i on a machine j. Similarly, let
too. Most importantly, the worst-case performance ratio of APCbeaT×M matrixwhereAPC istheaveragepower
ij
the proposed algorithm [1] is not given. consumption for a task i on a machine j.
Since tasks are indivisible in most cases, the x tasks of where MS = max 1 T x ETC is a lower
ij LB j Mj Pi=1 ij ij
type i may not be allocated equally to the M machines of bound on the makespan obtained by allowing tasks to be
j
typej. Foreverymachinej ∈M , letx bethe number divided among all machines. However, the approximation
k j ijk
of tasks of type i assigned to machine j . Clearly, x = of this method would be bad when the objective value is
k ij
x . The finishing time of a machine j ∈ M , close to 0 or little tasks of type i with large ETC are
Pk:k∈Mj ijk k j ij
denoted by F , is given by assigned to machines of type j. A similar phenomenon is
jk
also observed by Tarplee et al. [2].
T
Toovercometheobstaclementionedabove,wewillusea
Fjk =XxijkETCij. (1) different method. We replace MS(x) with a constant MS,
i=1
andthenobtainanapproximateintegerlinearprogram(ILP)
Thus, the maximum finishing time of all machines (i.e.,
for (4). By rounding the optimal fraction solution for the
makespan), denoted by MS(x), is given by
relaxationofILPbasedontheclassicroundingalgorithmfor
MS(x)=max max F . (2) thegeneralizedassignmentproblem[3],weobtainafeasible
jk
j k:jk∈Mj solutionfor(4).Itisdesiredtopointoutthat,inourmethod,
Inthispaper,forconvenience,machinesareturnedoffwhen the tasks of type i such that ETC > MS will not be
ij
notuse, whichmeansthatthe energyconsumedby thebag- assignedtomachinesoftypej,whichistoavoidincreasing
of-tasks is given by: the makespantoo much when roundingthe optimalfraction
solution.
M T
E(x)=XXxijAPCijETCij. (3) Let LB be the optimal makespan by ignoring the energy
consumption, and UB be the makespan of the feasible
j=1i=1
schedule by assigning each task to the machine with min-
Letpbethepricecustomerpaysandcbethecostperunit
imum average power consumption APC . For any given
ij
ofelectricalenergy.Theprofitthattheorganizationreceives constant ǫ > 0, Clearly, the makespan MS(x∗) of the
byexecutingabag-of-tasksisp−cE(x).TheEnergy-Aware optimalsolutionx∗for(4)liesin[LB(1+ǫ)t,LB(1+ǫ)t+1],
Profit Maximizing Scheduling(EAPMS) Problem definedin
for some t ∈ {1,2,...,⌈log UB/LB⌉}. By trying
[1] attempting to maximize the profit per unit time can (1+ǫ)
all possible values, we will find a feasible makespan MS
be formulated as the following nonlinear integer program such that MS(x∗) ∈ [MS/(1 + ǫ),MS], where MS =
(NLIP):
LB(1 + ǫ)t for some t. For convenience, from now on,
p−cE(x) assume that MS is a known constant satisfying
Maximizex MS(x)
MS(x∗)≤MS ≤(1+ǫ)MS(x∗). (5)
M M
subject to ∀i X X xijk =Xxij =Ti; (4) ForaconstantMS,asin[1],ouralgorithmisdecomposed
j=1k:jk∈Mj j=1 intotwo phases.Thisfirstphaseroundsthefractionoptimal
∀j Fjk ≤MS(x),for each jk ∈Mj; solutiontoobtainaschedulewherethenumbersxij oftasks
∀i,j x ∈Z ,for each j ∈M . of type i assigned to machines of type j are given. The
ijk ≥0 k j
secondphaseassignstaskstoactualmachinestoproducethe
The objective of (4) is to maximize the profit per unit full task allocation x . The next two subsections describe
ijk
time, where x is the primary decision variable. The first the two phases of this recovery procedure in detail.
constraintensuresthatall tasks of differenttypesin the bag There are two main differences between Tarplee, Ma-
are assigned to some machine type. Because the objective ciejewski,andSiegel’s(TMS,forshort)method[1]andour
is to maximize the profit per unit time, which is equivalent task-type-based(TTB,forshort)method(depictedinFigure
to minimize makespan, the second constrain ensures that 1.): (1) The TMS method uses one fractional solution to
MS(x) is equal to the maximum finishing time of all roundwhileweusemultiplefractionalsolutionsandchoose
machines. the best one; (2) In the first phase, the energy consumption
may increase in Tarplee et al.’s method while it will not
III. A TASK-TYPE-BASED ALGORITHM
increase in our method.
A. Overview
B. b-Matching-Based Rounding
Note that (4) is a nonlinear integer program, which can
not be solved optimally in polynomial time. To obtain an Note thatif ETCij >MS,the tasksoftypei cannotbe
approximate solution of (4), one possible way is to convert assigned to the machines of type j in the optimal solution,
(4) to an equivalentlinear program (LP), and then to round by the definition of MS. This implies that xijk = xij =
the optimal fraction solution of LP to a feasible solution 0, if i,j,k satisfy that ETCij > MS and jk ∈ Mj. As
for (4). In [1], the authors obtained a linear program using mentioned in [1], 1 T x ETC is a lower bound on
Mj Pi=1 ij ij
variable substitution r ←1/MS and z ←x /MS , MS. Since
LB ij ij LB
Construct a weighted bipartite graph B(x) = (U,V,E;w),
where U = {u ,...,u } represent the set of task types.
n n 1 T
a a
p p The other node set V ={v |j =1,...,M,s =1,...,k }
s s js j
ke ke consistsofmachine-typenodes,wherek =⌈ T x′ ⌉and
ma ma j Pi=1 ij
k nodes v , s=1,...,k , correspond to machine type j,
j js j
for j =1,...,M.
energy energy
(a) TMS method (b) TTB method Asin [3],theedgesin E ofthebipartitegraphB(x) will
correspond to task-machine pairs (i,j), such that x′ > 0.
ij
Figure 1. Comparing the main ideas of two algorithms To construct the edges incident to the nodes in V corre-
sponding to machine type j, sort the task types in order
MS isconstantclosetoMS(x∗),wecansubstituteMS for of nonincreasing estimated times to compute ETCij. For
MS(x) in (4). Since p,MS,c are constants, the objective simplicity, assume that
maximizing (p −cE(x))/MS = p/MS −cE(x)/MS is
equivalentto minimizingE(x). Thus, we obtain an approx- ETC1j ≥ETC2j ≥...≥ETCTj. (7)
imateequivalentintegerprogrammingformulaforNLIP(4):
If T x′ ≤ 1, then k = 1, which implies that there
Pi=1 ij j
M T is only one node vj1 ∈V correspondingto machine type j.
Minimizex E(x)=XXxijAPCijETCij Foreachx′ij >0,include(vj1,ui)∈E.Otherwise,findthe
j=1i=1 minimum index i such that i1 x′ ≥ 1. Let E contain
1 Pi=1 ij
M those edges (v ,u )∈E, i=1,...,i , for which x >0.
j1 i 1 ij
subject to ∀i Xxij =Ti; Foreach s=2,...,kj−1, findthe minimumindexis such
j=1 (6) that is x′ ≥ s. Let E contain those edges (v ,u ),
Pi=1 ij js i
1 T i = i +1,...,i , for which x′ > 0. If is x′ > s,
∀j Mj Xi=1xijETCij ≤MS; t(hven ,sau−ls1o)∈puEt,eidg=esi(vj,s++1,1u,i.s.).,i∈jT,Efo.rFwinhaiPlclhyi,=x1p′ut>ije0d.ges
x ∈Z ,for each i,j; jkj i kj−1 ij
ij ≥0 For each edge (v ,u ) ∈ E, let the weight of edge
js i
xij =0,if ETCij >MS. (vjs,ui) be w(vjs,ui)=APCijETCij. For each task-type
nodeu ∈U,letthecapacityofu beb = M x′ ,where
Theorem 1. Any optimal solution x∗ for NLIP (4) is a i i i Pj=1 ij
b is an integer as M x′ = M x − M ⌊x ⌋ =
feasible solution for (6). i Pj=1 ij Pj=1 ij Pj=1 ij
T − M ⌊x ⌋ is an integer. From the construction of the
i Pj=1 ij
Replacing the constraint x ∈ Z with x > 0, we bipartitegraphB(x),itiseasytoverifythatthereareatmost
obtain the relaxation of (6), wijhich is≥a0linear pirjogram and T nodes in U and at most PMj=1kj ≤MT nodes in V. As
canbesolvedinpolynomialtime.NotingthatthereareTM there are T +M nontrivial constraints in (6), the number
variables and T + M nontrivial constraints, both are less of positive variablesin x is at most T+M, followingfrom
thepropertyoflinearprogramming.Combiningthefactthat
than that in the linear program (10) in [1]. By modifying
Shmoys & Tardos’s rounding method in [3], which is to there are one or two corresponding edges in E for each
find a minimum-weight matching of an auxiliary bipartite x′ij >0,thereare atmost2(T+M)edgesin E. Therefore,
graph B(x), we can convert a feasible solution x for the the minimum-costb-matchingBM, that exactly matches bi
relaxationof(6)toafeasiblesolutionxˆfor(6).Animportant times of the task-type node ui in E(x), can be found by
observation is that xˆ satisfies MS(xˆ)≤2MS and E(xˆ)= using the method in [4], whose running time is polynomial
E(x)≤E(x∗). in T and M.
Note that the running time of Shmoys & Tardos’s round- The modified Shmoys& Tardos’s roundingmethodalgo-
ing method [3] is dependenton the number of tasks, which rithm to constructa schedule xij from a feasible solution x
is very large in reality [1]. To reduce the running time, of the relaxation of (6) is summarized as follows.
we will replace minimum-weight matching by minimum- ALGORITHMA
weight b-matching [4] to design an algorithm whose run- Step1.FormthebipartitegraphB(x)withweightsonits
ning time is dependent on the number of task types. For edges as described above.
completeness, we present the modified Shmoys & Tardos’s Step 2. Use the method in [4] to find a minimum-weight
rounding method in [3] as follows. Here, for simplicity, we (integer) b-matching BM that exactly matches bi times of
onlyshowhowtoconstructthebipartitegraphB(x)andthe the task-type node ui in B(x).
edgeweights,ignoringthefractionsolutionofthematching. Step 3. For each edge (v ,u ) ∈ BM, assign a task of
js i
Given a feasible solution x for the relaxation of (6), let type i on a machine of type j, which implies that xˆ =
ij
x′ = x − ⌊x ⌋, for i = 1,...,T and j = 1,...,M. ⌊x ⌋+|{(v ,u )|(v ,u )∈BM}|, for every i,j.
ij ij ij ij js i js i
Theorem 2. [3] The schedule xˆ obtained by ALGORITHM Obviously, there are at most Mj unassigned tasks of type
A hasmakespanatmost2MS,andtheenergyconsumption i, which can be assigned using LPT algorithm. It is easy
is at most solution E(x∗). to verify that our method is equivalent to the LPT al-
gorithm in [1]. However, the running time is reduced to
C. Task-Type-Based Local Assignment
O( M (T logT +TM )), not depending on the number
Recall that a feasible schedule is to assign every indivis- Pj=1 j
oftasks,whichisalwaysahugenumberintheHPCsystem.
ible task to a specific machine. The solution xˆ obtained
ij ALGORITHMBshowsthepseudo-codeforassigningtasks
in the last subsection is to assign xˆ tasks of type i to
ij to machines for each type.
machines of type j. To obtain a feasible schedule, we need
to schedule the tasks already assigned to each machine
ALGORITHMB Assign tasks to machines for each type.
type to specific machines within that group. In a group of
machinesof type j, ETC and APC are only dependent
ij ij 1: For j =1 to M do
on the task type i. Thus, the total energy consumed by
2: RelabeltheindicessuchthatETC ≥···≥ETC ;
machines of type j is T xˆ APC ETC , which is a 1j Tj
Pi=1 ij ij ij 3: For i=1 to T do
constant. Therefore,we only need to schedule tasks to min-
4: For each machine j ∈M do
imize makespan, which is equivalent to the multiprocessor k j
5: Assign Ni (defined in (9)) tasks of type i to
scheduling problem [5]. Tarplee et al. [1] use the common k
it, if there are unassigned tasks;
longest processing time (LPT) algorithm to assign tasks to
6: End for
machines for each machine type, where the T xˆ tasks
Pi=1 ij 7: Use LPT algorithm to assign the remaining tasks
are sorted in descending order by execution time, and each
of type i (at most M );
task is assigned to the machine that will complete earliest. j
8: End for
As shown in [1], the effect of the sub-optimality of LPT
9: End for
algorithmontheoverallperformanceofthesystemsconsider
is insignificant, as the number of tasks is large generally.
However, this leads to another problem, that the running
time of LPT algorithm will increase dramatically when the
D. Performance Analysis
numberoftasksgrowsrapidly.NotethatintheHPCsystem,
In summary, for each t ∈ {1,...,⌈log UB/LB⌉},
the number of types of tasks is always much less than that (1+ǫ)
of tasks. For example, in the simulations of [1], there are let MS = LB(1 + ǫ)t. Then, use ALGORITHM A and
30 task types, yet there are 11,000 tasks. An important ALGORITHM B to find a feasible solution for (4). Among
thesesolutions(atmost⌈log UB/LB⌉),choosetheone
observation is that we do not need to assign one task at (1+ǫ)
with maximum profit per unit time. It is easy to verify that
a time when assign the tasks of same type.
thetotalrunningtimeisindependentofthenumberoftasks.
Each group of machines of type j is processed indepen-
Foramaximizationproblem,ifalgorithmAcanproducea
dently.The task typesare sortedin descendingorderby ex-
feasiblesolutionwiththeobjectivevalueatleastOPT/ρfor
ecution time ETC , which can be done within O(T logT)
ij
anyinstance,whereOPT denotestheoptimalvalue,thenρ
time. Without loss of generality, assume ETC ≥ ··· ≥
1j
ETC . For each machine j ∈M , let Li be the current is called the worst-case performanceratio or approximation
Tj k j k
ratio.
load of machine j after assigning tasks of type i, where
k
Combining (5) and Theorem 2, the objective of the
the load of machine j is the total processing time of tasks
assigned to it. Initiallyk, L0 =0 for each j ∈M . Let AL schedule xˆ is at least
k k j i
be the average load of machines of type j after assigning p−cE(xˆ) p−cE(x∗) p−cE(x∗)
≥ ≥
the tasks of type i, where 2MS 2MS 2(1+ǫ)MS(x∗)
ALi = Pk:jk∈MjLik−1+ETCijxˆij. (8) ≥ 2+12ǫOPT.
M
j
It implies that the worst-case performance ratio of the
For k = 1, ...,M , assuming there are N unas-
j unassign proposed algorithm is 2+2ǫ, for any ǫ>0.
signed tasks, schedule min{N ,Ni} tasks of type i
unassign k
simultaneously to machine jk, where IV. EXPERIMENTALRESULTS
AL −Li−1 Simulation experiments were performed to compare the
Ni =max{⌊ i k ⌋,0}. (9)
k ETC quality of TMS and TTB methods. As in [1], the software
ij
was written in C++ and the LP solver used the simplex
If the load of a machine j is increased meaning Ni > 0,
k k method from COIN-OR CLP [6].
we have
Without loss of generality, assume that c = 1 for all the
AL −ETC <Li =Li−1+NiETC ≤AL . (10) experiments.As in [1], let E be the lower bound on the
i ij k k k ij i min
minimum energy consumed when ignoring makespan, and 100
p = γE , where γ = p/E is a parameter that will 90
min min
80
be used to affectthe price per bag. Clearly, when γ is large e
m
enough, the focus is to minimize the makespan [1]. Thus, ti 70 TTB method
we only consider the case that γ ∈[1,1.5]. nit 60
u 50 TMS method
For all the simulations, there are nine machine types per 40
and 40 machines of each type for a total of 360 machine,
fit 30
as in [1]. Our first experiment is based on a benchmark o
pr 20
[7] with nine machine types and five task types, where
10
the missing values are deleted. The workload consists of
0
12, 000 tasks divided among 5 task types. When γ is 5 10 15 20 25
varying, different solutions produced by the TMS and TTB Figure 3. 25 randomized experiments
methods are shown in Table 1. The table shows that every
solution produced by the TTB method is better than that Although experiments show that the solution produced
produced by the TMS method. Especially, when γ = 1, by the TTB algorithms is close to the optimal solution,
because the rounding method in the TMS method will this does not hold in a worst-case scenario. It is interesting
increasetheenergyconsumption,theTMSmethodproduces andchallengingto designa polynomial-timealgorithmwith
a solution with negative objective value, while the TTB worst-case performance ratio less than 2.
method produces the optimal solution.
ACKNOWLEDGEMENT
γ = 1 1.1 1.2 1.3 1.4 1.5
The work is supported in part by the National Natural
TMS -0.6 985.1 1998.8 3505.4 5491.4 7933.9 Science Foundation of China [Nos. 11301466, 61170222],
and the Natural Science Foundation of Yunnan Province of
TTB 0.0 986.1 2009.0 3529.8 5510.7 7986.0
China [No. 2014FB114].
Table 1. The solutions with γ varying from 1 to 1.5 REFERENCES
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