Multiprocessor Scheduling of Dependent Tasks to Minimize Makespan and Reliability Cost Using NSGA-II

International Journal in Foundations of Computer Science & Technology (IJFCST), Vol.4, No.2, March 2014
DOI:10.5121/ijfcst.2014.4204 27
Multiprocessor Scheduling of Dependent Tasks to
Minimize Makespan and Reliability Cost Using
NSGA-II
M.Rathna Devi and A.Anju
A.P/Department of IT, KCG College of Technology, Chennai, India
ABSTRACT
Algorithms developed for scheduling applications on heterogeneous multiprocessor system focus on a
single objective such as execution time, cost or total data transmission time. However, if more than one
objective (e.g. execution cost and time, which may be in conflict) are considered, then the problem becomes
more challenging. This project is proposed to develop a multiobjective scheduling algorithm using
Evolutionary techniques for scheduling a set of dependent tasks on available resources in a multiprocessor
environment which will minimize the makespan and reliability cost. A Non-dominated sorting Genetic
Algorithm-II procedure has been developed to get the pareto- optimal solutions. NSGA-II is a Elitist
Evolutionary algorithm, and it takes the initial parental solution without any changes, in all iteration to
eliminate the problem of loss of some pareto-optimal solutions.NSGA-II uses crowding distance concept to
create a diversity of the solutions.
KEYWORDS
Quality of Service, NSGA-II, elitism, makespan, reliability cost.
1. INTRODUCTION
Generally, heterogeneous computing (HC) environments use a distributed collection of different
high-performance machines, connected with high-speed links, for solving collections of
computationally intensive problems that have different computational requirements [1,
2].Scheduling is the main problem that arises in heterogeneous computing environments.
Scheduling means efficiently allocating a group of tasks coming into the system to the available
resources in the system. However, finding optimal schedules in an HC environment is an NP-
complete problem [3, 4]. In the HC environment considered, the tasks are assumed to be
dependent, i.e., communications between the tasks are needed. The execution time of each task in
each machine is already defined for the system. So static scheduling is done here. Static
scheduling is utilized in different types of analysis and environments. The static scheduling is
used for designing a system. Static scheduling is also used for post-mortem analyses i.e., to
evaluate the performance of a dynamic scheduler, to check how effectively the system is using
the resources available. Static scheduling is also used for simulation studies to know about the
efficiency of hardware,while the system runs. In future, high-powered computational grids [5]
will also be use static scheduling mapping techniques to distribute resources and computational

28
power. Hence the wide applicability of static scheduling makes it an important area for ongoing
research.
In the proposed work, each machine will execute only a single task at a particular time and no
task will interrupt other task while executing,(i.e) no dependency among the tasks. The proposed
work is done for non-preemptive tasks.The scheduling algorithms can be rated based on different
parameters like makespan, flowtime, communication cost, reliability cost and makespan. This
paper concentrates to minimize makespan and reliability cost. Makespan is the time taken for a
HC system to finish the last task. Reliability is defined as the system should not fail during the
time of executing the tasks. This scheduling problem proposed here is a multi-objective problem
with the objective to minimize the makespan and reliability cost of the system.
In Genetic Algorithm, the parental characteristics are changed after the first generation by
mutation. So, the time taken for obtaining sub-optimal solution is more. To solve this problem,
elitism concept is used. Using elitism concept a new population is constructed to allow some of
the better solutions from the current generation to carry over to the next, unaltered. Elitism is used
to eliminate the problem of loss of good solutions by keeping the elite population. Crowding
distance technique of NSGA-II is used to get diversity of the solution. In this paper, Non-
dominated Genetic Algorithm (NSGA-II) is used to select the best optimal schedule by
considering both the objectives and to get sub-optimal solution in minimum time.
The remainder of this paper is organized as follows. Section 2 provides existing methodologies.
Section 3 defines the computational environment parameters. Description of NSGA-II based
procedure is presented in Section 4. Section 5 presents the results for different problems
simulated and finally, Section 6 concludes with finishing remarks and future work.
2. EXISTING WORK
Job scheduling is a tedious work in multiprocessor system than in a single processor system.
Scheduling problem in multiprocessor system is always NP-hard. [6]. Now-a-days, more number
of genetic algorithm (GA) are proposed. Mitra and Ramanathan proposed a GA for scheduling of
non-pre-emptive tasks with precedence [7]. Lin and Yang developed a hybrid GA, where various
operators are applied at a different stage of the lifetime, for scheduling non-pre-emptive tasks in a
multiprocessor environment [8]. Monnier presented a GA implementation to solve a scheduling
problem for real-time non-pre-emptive tasks [9]. However, these algorithms have only one
objective such as minimizing cost, completion time or total tardiness. Oh and Wu presented a
multi-objective GA for scheduling non-preemptive tasks in a soft real-time system with
multiprocessors [10]. However, this algorithm did not refer to conflict between objectives, the so
called Pareto optimum, and assume that the performance of all processors is the same. Theys et
al. proposed a GA for static scheduling in a heterogeneous system [11]. Page and Naughton
presented a GA for dynamic scheduling algorithm on a heterogeneous system [12]. Dhodhi
presented a novel encoding method of GA for task scheduling on a heterogeneous system [13].
But, all these algorithms are designed for normal tasks without time constraints. Sandeep Jain and
Shweta Makkar presented the scheduling of dependent tasks using Genetic
Algorithm[14].kamaljit and Amit presented a heuristic based Genetic Algorithm to minimize
execution time and throughput of dependent tasks[15]. Amanpreet Kaur1 and Prabhjot Kaur
implemented mapping heuristic Genetic Algorithm to minimize Makespan and Schedule length

29
ratio[16].But these algorithms are designed without taking the initial population into
consideration in all generations.
3. SYSTEM MODEL
The system is represented by Directed Acyclic Graph G= T, E where T represents a set of N tasks
to be executed in the system and E is the directed edges represents the data communication
between two tasks. Let ti,j be the execution time of task ti on machine mj, l ≤ j ≤ p. It is assumed
that the expected execution time ti,j,E l ≤ I ≤ n and 1 ≤ j ≤ p, is known. Let ek,l ε E indicate
communication from task tk to tl, where task tk (tl) is said to be an immediate
predecessor(successor)task of task tl (tk).Associated with directed edge ek,lε E is the volume of
data in terms of bytes ,which is denoted by dk,l, that will be transmitted from task tk to task tl upon
completion of task tk.
Fig 1. Directed Acyclic Graph
The scheduling problem for soft real-time tasks is formulated under the following assumptions:
computation time of each task is known. The problem is multi-objective job scheduling
considering, static scheduling of dependent jobs in heterogeneous environment, with the objective
of completion time and reliability cost. Static scheduling means the tasks in the system are
assigned to a particular machine before starting the system. So that there is no movement of task
from one machine to another during the compilation time.
4. PROPOSED SOLUTION
Traditional optimization techniques and search do not take care more on problem domains and are
not robust for multi-objective optimization problems. Previously Genetic Algorithm is used for
static scheduling [17] in heterogeneous multiprocessor system. But GA could not produce a better
pareto-optimal solution if more than one objective is considered. To overcome the above
problem, it is proposed to use NSGA-II [18] in solving the problem.
4.1 NSGA-II
Initialize the population Pt.
Create the child population Qt from the previous population Pt.
Combine the two populations Qt and Pt to form Rt.

30
Rt = Pt U Qt
Find the all non-dominated fronts Fi of Rt.
Initiate the new population Pt+1 = 0 and the counter=1
While Pt+1 + Fi ≤ Npop, do: Pt+1Pt+1UFi, i i+
Arrange the last front Fi in descending order using crowding distance and choose the first (Npop –
Pt+1) elements of Fi.
Use Genetic operators such as selection, crossover and mutation to obtain the new child
population Qt+1 size Nobj.
4.1.1 Initialization
Initialize the population Pt using equality and inequality constraints. After initialization step, it
creates chid population Qt from the currently existing population Pt and then combines both
populations to form Rt. Where Rt is define as:[19]
Rt = PtUQt
Before schedule creation the height of all tasks are found out. Then using permutations different
possibilities of tasks in each height is found out. Then the tasks are allocated to the processor in
order of this permutation combination results. A schedule would be illegal if a task is scheduled
to be executed before its ancestor. Suppose that T and T‘ are tasks assigned to the same processor,
and that T’ is an ancestor of T. By the definition of height, we have height (T’) height (T). If we
order the tasks in ascending order of height, then T‘ will be executed before T, and the schedule
will be legal.
4.1.1.1 Algorithm
This algorithm randomly generates a schedule of the task graph TG for a multiprocessor system
with p processors.
GSl. [Initialize.] Find height ht’ for all task in task graph.
GS2. [Find the tasks based on their height and keep separately] Divide the tasks in task graph into
different groups, G(h) (G(h) represents a group of tasks with height h), based on the value of ht’.
GS3.[Task permutation] Task within the particular height is get permutated to get different
possibilities.
GS4. [Allocation to processors] From the different possibilities, allocate the jobs to processors p1
to pn. Each time start the allocation from subsequent processors.
By repeatedly applying the algorithm for all the possibilities, the initial population of search
nodes can be generated.
4.1.2 Non-Dominated Sorting
After the initialization, the population is sorted based on non-domination. Each solution assigned
a fitness value according to its non-dominated level, where level one is considered to be the best
level. The solution at the level one did not dominate by any of other solution. Whereas solutions
at other level dominated by least one solution. Perform the non dominated sorting to the initial
population and identify the different rank to each population: rank1, rank2, rank3....etc. Rank

International Journal in Foundations of Computer Science Technology (IJFCST), Vol.4, No.2, March 2014
31
provides the value of non-dominated rank across each solution. All the solutions have sorted
according to its non-dominated rank.
for each (pεP)
for each (qεP)
if (pq) then
Sp = SpU{q}
else if (q p) then
np = np+1
end
end
if (np= = 0) then
F1 = F1U{p}
end
end
While ( Fi#Ф)
Q = Ф
for each (pεFi)
for each(qεSp)
nq = nq-1
If( nq = 0) then
Q = Q U {q}
end
end
end
i = i+1
Fi = Q
End
4.1.2.1 Crowding Distance
To have difference in chromosomes, we have to find out crowding distance. Following algorithm
is The crowding distance is calculated by the below algorithm.
l = |I|
set I[i]distance = 0
end
for each objective m
I = sort(I,m)
I[1]distance = I[l]distance = α
end
for (I = 2 to (l-1))
I[i]distance=I[i}distance+I(k+1)m-I(k-1)m/fmax
m-fmin
m
end
I(k).m

32
4.1.3 Selection
After the chromosomes are sorted based on non-domination and with crowding distance found
out, the selection is done using a crowded-comparison-operator (n) and best solution is selected.
All solution is having two criterias.
1. A Non-domination rank (ri) in population
2. A local Crowding distance (I[i]distance)
i n j
if (ri rj) or
if (ri = rj)
and
(I[i]distance I[j]distance)
4.1.4 Crossover
The crossover operators play the most important part in evolutionary algorithm. This is done by
selecting chromosomes from the parental generation and interchanging their genes, new
chromosomes are obtained. By this, we can obtain better quality offspring that will enter the next
generation and enable the search to be done on new regions of solution space not searched yet.
There are many types of crossover operators in the evolutionary computation, which depends on
the chromosome representation.
4.1.4.1 Algorithm
This algorithm performs the crossover operation on two strings (A and B) and generates two new
strings.[19]
C1. [Select crossover sites.] Randomly generate a number, c, between 0 and the maximum height
of the task graph.
C2. [Loop for every processor.] For each processor Pi in string A and siring B, do-step C3.
C3. [Find the crossover points.] Find the final job Tji in machine Pi that has height c, and Tki is the
task following Tji. That is, c =height' (Tji) height' ( Tki ) and height' (Tji) are the same for all i.
C4. [Loop for every processor.] For each processor Pi in string A and string B, do step CS.
C5. [Crossover.] Using the crossover sites selected in step C3, exchange the bottom halves of
strings A and B for each processor Pi.
Although the crossover operation is powerful, it is random in nature and may eliminate the
optimal solution. Typically, its application is controlled by a crossover probability whose value is
determined experimentally. Furthermore, we can always preserve the best solution found by
including it in the next generation.
4.1.5 Mutation
Mutation is a GA operation that modifies one or more genes from their current value.This might
result in completely new gene values being added to the gene pool. With this new gene values, it
is able to arrive at better solution than the previous one. Mutation operator helps to prevent the
result stopping at any local optima. Mutation is done based on mutation probability. This
probability indicates how many chromosomes has to undergo mutation operation. This probability

33
value should be very low. If it is set to high, the search will be a simple random search. The
mutation operation is summarized in the following algorithm: [21]
4.1.5.1 Algorithm
This algorithm performs the mutation operation on a string and generates a new string.[19]
MI. [Pick a task]. Randomly pick a task, Ti.
M2. [Match height.] Search the string for a task, Tj, with the same height.
M3. [Exchange tasks.] Create a new chromosome by interchanging the two jobs, Ti and Tj, in the
schedule.
Typically, the frequency of applying the mutation operator is controlled by a mutation probability
whose value is determined experimentally.
4.2 Completion Time
It is the time when the latest job will be finished. For a scheduling problem the value should be
minimized [20].
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In above equations, notations are de.ned as follows:
Indices:
i, j task index, i, j = 1, 2, . . . , N
m processor index, m = 1, 2, . . . , M.
Parameters:
G = (T ,E) : task graph
T = {τ1, τ 2, . . . , τ N}: a set of N tasks
E = {eij }, i, j = 1, 2, . . . , N, i ≠ j : directed edges
N : Number of jobs
M : Number of machines
ti : the ith task, i = 1, 2, . . . , N
eij : precedence relationship between task τ i and task τ j
eij=1,if there are precedence ip between task τ i and τ j.
eij=0 otherwise
cim :computation time of task τ i on mth processor
di : deadline of task τ i
pre*( τ i ): set of all predecessors of task τ i
suc*( τ i ): set of all successors of task τ i

34
pre(τ i ) : set of immediate predecessors of task τ i
suc(τ i ) : set of immediate successors of task τ i
tiE : earliest start time of task τ i
tiF : finish time of task τ i
tiF=min{tiS +cim,’,di}
where cim ‘= cim|xim
tSi : real start time of task τ i
4.3 Reliability Cost
Reliability is defined as, the system will not fail when a task is under execution. Consider a
heterogeneous system with M PEs, P={P1,P2…PM}, and a DAG containing N nodes,{u1,u2,...uN}.
Let ti(x) be the computation time of node ui for PE Pj. Let fj be the failure rate of PE Pj . Let gkb be
the failure rate of the communication link from Pk to Pb . Let wij be the volume of data that task ui
needs to send to task uj. dkb represents the delay to transmit a single length data from Pk to Pb . Let
Xij denotes a binary number whether task ti is assigned to Pj, for assigned, 0, for not assigned.
Reliability Cost as follows[21]:
Rij, represents the reliability cost for task ti to be scheduled on Processor Pj . Let pred(x’) be the
set which includes all ti ’s predecessors, Rij can be expressed as follows:
Thus, to maximize system reliability, the Reliability Cost is needed to be minimized. Reliability
means how the system is working without fault when some group of task is assigned to it.
Reliability cost says about how well the system when a group of tasks are assigned to it. When
reliability is minimized the reliability cost will be maximized.
5. TEST RESULTS
Numerical tests are performed with a randomly generated task graph. P-Method [22] is used for
the generation task graph. Directed Acyclic Graph is generated by random number concept using
P-method. Element aij of the matrix is equal to 1 if there is a predecessor relationship from τ i to

35
τ j ; otherwise, aij is equal to zero. An adjacency matrix is generated randomly with all its lower
triangular and diagonal elements equal to zero. The elements above the diagonal elements of the
adjacency matrix are verified by a Bernoulli process with parameterε , which represents a
success. When the Bernoulli trial is a success, then the element is assigned a value of 1, for a
failure the element is given a value of 0. With this method, a probability parameter of ε = 1
creates a totally sequential task graph, and ε = 0 creates an inherently parallel one. Values of ε
that lie in between these two extremes generally produce task graphs that possess intermediate
structures.
For the tasks’ computation time and delay, random numbers based on exponential distribution and
normal distribution is used as follows:
cEim = random value for exponential distribution with mean 5
cNim = random value for normal distribution with mean 5
rE = random value for exponential distribution with mean cEi
rN = random value for normal distribution with mean cNi
dEi = tEi + max{cEim
∀ m} + rE + Communication time
dNi = tEi + max{cNim ∀ m} + rN + Communication time
where cEim and cNim is the execution time of task ti on processor Pm based on exponential
distribution and normal distribution, respectively. dEi and dNi is the deadline of task ti based on
exponential distribution and normal distribution, respectively.
Data transmitted between the tasks are randomly generated between the values 1 and 10.Data
matrix is represented as n×n matrix, where ‘n’ represents the number of tasks in the task graph.
Data will be transmitted when there is a dependency between the tasks, otherwise in the matrix
the data is represented as zero. In other words the data matrix is based on adjacency matrix.
For calculating the reliability cost, the processor failure rate and the communication link failure
rate is necessary. The link failure rate is represented as a p×p matrix, where ‘p’ represents the
number of machines. The processor failure rate is a single dimensional matrix, and the number of
column is equal to the number of machines. Both processor failure rate and link failure rate is
generated randomly between the values 0.0000075 and 0.0000125.
The parameters of GA were set to 0.9 for crossover (pC, ), 0.1 for mutation (pM, ), and
population size (popSize) is taken as twice the number of jobs taken.

36
CASE I - 2 Machines and 10 Jobs
Fig. 2. Pareto optimal solutions for makespan for Normal and Exponential distribution(1 iteration)
The difference in the pareto-optimal solutions for computation time generated with the Normal
and Exponential distribution for single iterations. With NSGA-II both the objectives get
converged and both makespan and reliability cost get minimized.
Fig. 3. Pareto optimal solutions for makespan for Normal and Exponential distribution(5 iteration)
The difference in the pareto-optimal solutions for computation time generated with the Normal
and Exponential distribution for 5 iterations. When NSGA-II is used and number of iterations get
increased both the objectives get converged and both the makespan and reliability cost get
minimized.
Fig 4. Pareto optimal solution for iteration 1 and 5(Normal distribution for makespan)

37
The pareto -optimal solutions for Computation time generated with Normal distribution for
different number of iterations and it is observed that more number of pareto-optimal solutions is
obtained, when the number of iterations gets increased.
Fig 5. Pareto optimal solution for iteration 1 and 5(Exponential distribution for makespan)
The pareto optimal solutions for Computation time generated with Exponential distribution for
obtained, when the number of iterations get increased.
CASE II - 4 Machines and 50 Jobs
Fig 6. Pareto optimal solution for makespan for Normal distribution
The difference in the pareto optimal solutions for computation time generated with the Normal
distribution for different iterations. When the number of iterations get increased both the
objectives get converged and both the makespan and reliability cost get minimized.

38
Fig 7. Pareto optimal solution for iteration 1 and 5(Normal distribution for makespan)
The pareto-optimal solutions for Computation time generated with Normal distribution for
obtained, when the number of iterations get increased.
6. CONCLUSION
This paper gives solution for the scheduling problem in Multiprocessor system with multiple
objectives to minimize the makespan and reliability cost. From the above work it is cleared that
the deterministic methods are not efficient in solving multi-objective problems. The Non-
dominated sorting Genetic algorithm-II based approach is a random search method that can be
used to solve hard combinatorial optimization problems. The simulation was done for the
computation time generated with Normal and exponential distribution. From the simulation
results, it is found that the two objectives makespan and reliability cost is minimized successfully.
In future, study has to be done to select the correct parameter. It is planned to extend the proposed
work for, dynamic scheduling of both dependent and independent tasks considering different
objectives. It is planned to compare the simulated results with other scheduling algorithms also. It
can also be extended by considering the objective like maximizing the processor utilization.
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Authors
Rathna Devi received M.Tech degree in Information Technology from P.S.G College
of Technology Coimbatore, India. Since July 2013, she has been working as an
Assistant professor in KCG College of Technology, Chennai, India. Her research area
is on Grid Computing and Evolutionary Algorithms.
A.Anju received M.E degree in Computer Science Engineering from Sathyabama
University, Chennai, India. Since July 2013,she has been working as an Assistant
professor in KCG College of Technology, Chennai, India. Her research area is on Grid
Computing and Evolutionary Algorithms.

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