chapter 5 non linear regression prediction of dwelling...

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CHAPTER 5

NON LINEAR REGRESSION PREDICTION

OF DWELLING TIME

Prediction based resource allocation is an efficient tool for dealing

with resource heterogeneity. However, linear time series prediction methods

give less precise parameter estimates when compared with the real data.

Nonlinear regression model can produce good estimates of the unknown

parameters in the model with relatively small data sets. Usually

numerical optimization algorithms are applied to determine the best-fitting

parameters. In this chapter, nonlinear regression based prediction method is used

to predict the future dwelling time of the resources for effective scheduling.

5.1 NON LINEAR PREDICTION MODEL

Non linear regression based prediction techniques are emerging as

powerful tool to predict the future patterns of behavior underlying hiding

dynamical systems which may avoid analysis using other traditional

techniques. In fact, nonlinear time-series analysis has been one of the major

areas of research in time-series analysis for more than two decades. Beyond

linear domain, there are many nonlinear forms to be explored.

5.1.1 Need for Nonlinear Regression Models

The goal of determining a regression is to obtain an equation from

which one can predict one variable based upon another variable. Linear time

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series models can be used for short range predictions. Although many

scientific and engineering processes can be described well using linear

models, or other relatively simple types of models, there are many other

processes that are inherently nonlinear. Linear models do not describe

processes that asymptote very well because for all linear functions the

function value can't increase or decrease at a declining rate as the explanatory

variables go to the extremes. There are many types of nonlinear models, on

the other hand, that describe the asymptotic behavior of a process well. Like

the asymptotic behavior of some processes, other features of physical

processes can often be expressed more easily using nonlinear models than

with simpler model types. The biggest advantage of nonlinear regression over

many other techniques is the broad range of functions that can be fit

(Clements et al, 2004). To obtain accurate future dwelling time in our

scheduling algorithm the nonlinear regression model is considered.

These models are generally more appropriate than linear models for

accurately describing dynamics of the series and for making multistep-ahead

forecasts. On the other hand, regression analysis is a statistical tool for the

investigation of relationships between variables.

5.1.2 Estimation of Regression Parameters

Estimation of the parameters of a nonlinear regression model is

usually carried out by the method of least squares or the method of maximum

likelihood as in linear regression models. Unlike linear regression, it is

usually not possible to find analytical expressions for the least squares and

maximum likelihood estimators for nonlinear regression models. Instead,

numerical search procedures are used for parameter estimation, which

requires intensive computations. In general, there is no closed-form

expression for the best-fitting parameters, as there is in linear regression. The

data are fitted by a method of successive approximations. Non-linear

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regression is an iterative procedure in which the number of iterations depends

on how quickly the parameters converge. The choice of initial starting values

is very important with the non-linear method because a poor choice may

result in slow convergence, convergence to a local minimum, or even

divergence. The optimization of nonlinear regression model parameters is

usually carried out by natural heuristic algorithms like GA, PSO, SA and TS

or by machine learning algorithms like ANN and CBR. None of the algorithm

for optimizing general nonlinear functions exists that will always find the

global optimum for a general nonlinear minimization problem in a reasonable

amount of time. Since no single optimization technique is invariably superior

to others, a variety of optimization techniques that work well in various

circumstances. Genetic algorithms are often applied as an approach to

solve global optimization problems.

5.2 GENETIC ALGORITHM (GA) BASED OPTIMIZATION

Genetic Algorithm (GA) is an evolutionary technique for large

space search (Fogel 1994, Xhafa et al, 2005). The general procedure of GA

search is given in Figure 5.1.

Yes

No

Figure 5.1 Genetic algorithm procedure

Start Initialization

ofpopulation

Valuation Fitness Value

Reproduction

Solution found? Stop

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5.2.1 Generation of Initial Population

A population is a set of chromosomes and each represents a

possible solution, which is a mapping sequence between tasks and machines.

The evolution usually starts from a population of randomly generated

individuals and is an iterative process, with the population in each iteration is

called a generation. The population size depends on the nature of the

problem, but typically contains several hundreds or thousands of possible

solutions.

5.2.2 Fitness Evaluation

A fitness function value quantifies the optimality of a solution. The

value is used to rank a particular solution against all the other solutions. A

fitness value is assigned to each solution depending on how close it is actually

to the optimal solution of the problem.

5.2.3 Generation of New Population

The generation of new population consists of three process namely,

selection, crossover and mutation.

Selection : In selection process, the two parent chromosomes

from a population are selected according to their fitness (better

the fitness, bigger the chance to be selected).

Crossover : Crossover is a genetic operator that combines

two parent chromosomes to produce a new chromosome

(offspring). The idea behind cross over is that the new

chromosome may be better than both of the parents if it takes

the best characteristics from each of the parents. Crossover

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occurs during evolution according to a user-definable

crossover probability.

Mutation: Mutation is the process of replacing certain genes

or sub-sequences with new gene values to the current

population. The replacing of certain genes is done using

mutation probability. Mutation is an important part of the

genetic search which helps to prevent the population from

stagnating at any local optima.

5.2.4 Termination

Finally, the chromosomes from this modified population are

evaluated again. This completes one iteration of the GA. The GA stops when

a predefined number of evolutions is reached or all chromosomes converge to

the same mapping.

For its simplicity, GA is the most popular nature’s heuristic used in

algorithms for optimization problems. The main property that makes these

genetic representations convenient is that their parts are easily aligned due to

their fixed size, which facilitates simple crossover operations. Variable length

representations may also be used, but crossover implementation is more

complex in this case. Once the genetic representation and the fitness function

are defined, a GA proceeds to initialize a population of solutions and then to

improve it through repetitive application of the mutation, crossover and

selection operators.

5.3 RELATED WORKS

Recently, Genetic Algorithms (GAs) have been widely and

successfully applied to various optimization problems in Grid computing

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environment. GAs are well suited to the models with varying resolutions and

structures since they can search non-linear solution spaces without requiring

any gradient information or a priori knowledge about model characteristics.

Effective task scheduling in grid requires to model the available resources on

grid nodes and computation requests of tasks, determine the current load of

the system, and predict the task execution time. Many researchers have

successfully applied global optimization methods, such as the GA, rather than

local gradient-based optimization techniques (Collin & Takeshiyamada, 1998)

to Grid computing. GA is probabilistic meta-heuristic methods inspired by the

‘survival of the fittest’ principle of the neo-Darwinian theory of evolution

(Fogel 1994, Grefenstette 1986). Wu et al (2007) proposed a real-valued GA

to optimize the parameters of Support Vector Machine (SVM) for predicting

bankruptcy. Xue & Xue (2010) presented an optimization model for the task

scheduling problem and develop a Hybrid Clonal Selection Genetic

Algorithm (HCSGA) to solve it effectively. Joshua & Vasudevan (2011)

developed the smart Genetic Algorithm with the fitness function based on

makespan & flow time to schedule the task in Grid computing. Maleki & Ali

(2011) proposed an genetic algorithm based task scheduling algorithm to

assign the tasks to the grid resources with goal of minimizing the total

makespan of the environment. Wang et al (1997) presented a genetic-

algorithm-based approach to dispatch and schedule subtasks within grid

environments. The simulation tests presented in Wang et al (1997) for small-

sized problems (e.g., a small number of subtasks and a small number of

machines), the genetic-algorithm-based approach can be found the optimal

solution for these types of problems.

Levitin & Dai (2008) proposed a genetic algorithm to distribute

execution blocks within grid resources. The aim of the algorithm is to find the

best match between execution blocks and grid resources in which the

reliability and/or expected performance of the task execution in the grid

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environment maximizes. Salehi et al (2013) proposed a hybrid algorithm

combining genetic algorithm and network gravity algorithm to solve

scheduling tasks in the grid computing. Wiessman et al (2004) proposed a

novel GA based algorithm which schedules a divisible data intensive

application. Martino & Marco (2002) presented a GA based scheduling

algorithm where the goal of super scheduling was to minimize the release

time of jobs.

The feasibility of evolutionary algorithm-based scheduling such as

GA for realistic scientific workloads is demonstrated by systems like Mars

(Bose et al 2004). Mars is a meta-scheduling framework for scheduling tasks

across multiple resources in a grid. Golconda & Ozguner (2004) studied and

compared five heuristic based algorithms and they found that GA produced

the best utility for users in comparison to heuristics such as Min-Min. Xhafa

et al (2005) presented Genetic Algorithms (GAs) based schedulers for

efficiently allocating jobs to resources in a Grid system in order to minimize

the makespan and flowtime. The above related works expose that GA method

will be suitable for any type of optimization problem. In this research work a

non linear prediction model is developed and the parameters will be estimated

using GA optimization.

5.4 NON LINEAR PREDICTION MODEL for DARA (NLPM-

DARA)

The dwelling time of the sporadic resource for each occurrence will

be varied randomly. The previous prediction model does not consider the

random dwelling time occurrence. To address this random dwelling time

problem, a non linear prediction model is developed to enhance the prediction

accuracy.

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5.4.1 Developing a non linear regression model

In general for a nonlinear regression model based on the occurrence

of the curve nature it is developed. The resource dwelling time occurrence is

similar to s-type curve and hence a commonly used logistic regression model

(Batts & Watts, 1988, Gallant, 1975) is modified as follows,

1

01

2

0

exp 1 1( )

1 exp 1

N

i iki

j j N

i iki

tt

t(5.1)

where )( jt = future dwelling time prediction model function,

jt = represents the dwelling time of

thj resource,

it = dwelling time of thi instant and

j , ik are the non linear model parameters which are to be

optimized

N = Past dwelling time history size (Maximum history size

taken is 10)

5.4.2 Optimization of nonlinear parameters

The non linear regression model is optimized by selecting the bestj

and bestN

bestbest1,,,

10values to make it fit to the historical data. The

procedures that are involved in the optimization process are described below.

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5.4.2.1 Initialize Population Pool

Genetic algorithm begins with a set of random solutions

(represented by chromosomes) called population pool. The population pool of

random solutions or chromosomes ( j , ik values) are generated and it is

denoted as

1,,1,0; pl NlX

where, pN is the pool size, in which each chromosome is of length 1N .

The chromosome length 1N indicates the number of genes i.e.

number of random weights to be optimized are 1N , j and 110 ,,, N .

j 0 1 2 3 4 5 6 7 8 9

The previous 10 occurrence (history) of dwelling time is taken as

chromosome length. Each gene value of every chromosome is an arbitrary

number to be generated within the interval [0, 10].

5.4.2.2 Fitness Evaluation

The chromosomes are then evaluated using the equation. The

fitness value is denoted as lf

12

,0

2

( ( ) )Rl N

j III jj

ft d

(5.2)

where, jIIId , is the actual dwelling time of the thj resource.

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NR = total number of resources

The difference between the actual dwelling time and the predicted

dwelling time of jth resource is squared for NR resources. For maximum fitness

value the prediction error will be minimum. For every resource in population

pool, the fitness value is evaluated. The objective of fitness evaluation is to

maximize the fitness value.

5.4.2.3 Selection

The selection operator then probabilistically populates the next

generation of chromosomes such that chromosomes with high fitness are

more likely to be selected. The technique to select the chromosomes, are

based on rank selection method. Here the rank is the maximum fitness value.

The operator selects two individuals at random and compares their fitness.

Only the individual with the highest fitness is propagated to the next

generation. If they have equal fitness the individual to be inserted is chosen at

random.

5.4.2.4 Crossover

In the crossover process two solutions (chromosomes) are selected

and exchanges the gene values with a probability of pC . The crossover

operation exchanges pCN. genes between two parent chromosomes and

produce 2/pN children chromosomes 12/,,1,0; poffq NqX .

5.4.2.5 Mutation

Perform uniform random mutation operation with a mutation

probability pM . In the mutation technique, a uniform random integer is

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generated and replaced in pMN. random positions of offqX and new

qX is

produced.

These crossover and mutation operators are applied to all members

of the new population. Often in addition to these operators, the best member

of the original population is copied into the new population unchanged, a

strategy called elitism.

5.4.2.6 Termination

The resultant newqX and the selection pool chromosomes are placed

in the population pool and the process is repeated until the termination

criteria. In this case, the termination criterion is set as reaching a maximum

number of repetitions of process. Once the maximum number of process

repetition is happened, the process is terminated and the chromosome (can be

represented as bestj

and bestN

bestbest1,,,

10), which has maximum fitness, in

the population pool is extracted.

The obtained best weights are substituted in Equation (5.1) to

derive the final regression model as follows

1

01

2

0

exp 1 1( )

1 exp 1j

ik

Nbest

i ikibest

j Nbest

ii

tt

t (5.3)

The final regression model is able to determine the future dwelling

time resources. The non linear prediction model (NLPM) output is given to

the DARA for resource allocation process.

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5.4.3 Fuzzy based Resource Allocation

The future dwelling time of the resources from NLPM is given as

the input to the fuzzy scheduler as depicted in Figure 3.3. The three input

variables to the DARA technique are i) priority of job ii) requirement time

which is demanded by the jobs iii) future dwelling time of resource

respectively. The fuzzy inference system is inferred from the input vector

based on a set of rules. Then the output of inference system is compared with

the different threshold score values; if it is above a threshold score, the

corresponding job will be allotted to the resource immediately and stores the

details for the next scheduling.

5.5 RESULTS AND DISCUSSION

The NLPM-DARA technique is evaluated by using MATLAB

simulation. The performance of non linear prediction using GA optimization

is evaluated through MATLAB, then the resources allocated.

5.5.1 Non Linear Prediction Model Analysis

The Genetic Algorithm optimization is performed using the

parameters which is given in Table 5.1.

Table 5.1 GA Parameters for Simulation

Parameters Specifications Population Size 20Selection RankCross Over Probability 0.5Cross Over Type Single Point Mutation Probability 0.05Max. No. of Generations 50

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The non linear prediction model is evaluated interms of %

prediction accuracy. The prediction accuracy of linear prediction model and

non linear prediction model for sporadic and semi permanent type resources

are shown in Figure 5.2 (a) and (b) respectively. The history size ‘N’ of

values 1, 2,3,…,10 are used to evaluate the errors and accuracy.

1 2 3 4 5 6 7 8 9 1076

78

80

82

84

86

88

90

History Size "N"

Sporadic Resources

LPM-DARANLPM-DARA

(a)

1 2 3 4 5 6 7 8 9 1085

86

87

88

89

90

91

92

History Size "N"

Semi Permanent Resources

LPM-DARANLPM-DARA

(b)

Figure 5.2 Comparison of prediction accuracy of LPM-DARA and NLPM-DARA techniques (a) Sporadic resources (b) Semi permanent resources

125

Figure 5.2 illustrates the prediction accuracy of NLPM-DARA with

comparison to LPM-DARA techniques. It is observed that, the non linear

prediction model accuracy is improved by atleast 5% when compared to

linear prediction model. The maximum accuracy for sporadic resources is

90% and for semi permanent resources it is 92%. This is due to the GA

iterative computation for finding the best fitness value.

5.5.2 Resource Allocation Analysis

Resource dwelling time prediction can improve performance of

scheduling and this section describes the influence of predictions on grid

scheduling. A quantitative analysis is made based on dwelling time based

fuzzy scheduling with the same five datasets generated in the chapter 3. The

scheduling quality is evaluated according to average job makespan, resource

utilization and the failure rate. Makespan is calculated for each job as the time

from submission to completion and this value is averaged across all jobs.

Resource time utilization indicates the ratio of number of resources allocated

to total number of resources. In DARA technique, the fuzzy threshold

filtering has been performed in two locations. One is at the point of evaluating

the fuzzy score that deals with the sporadic resource type and the other is at

the point of evaluating the fuzzy score that deals with semi-permanent

resource type. The threshold values (Sth-III and Sth-II) are varied from 0.3 to 0.7

keeping the job and resource scenarios for five different datasets as mentioned

in chapter 3. The performance metrics for all the datasets under prediction

NLPM-DARA model for different threshold values are tabulated in Tables

5.2 to 5.6.

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Table 5.2 Performance metrics for Dataset-I

S.No Sth-III Sth-IIUtilization

(in %) Failure rate

(in %) Makespan

(in sec)

1 0.3 0.3 77.41 22.59 70.068

2 0.3 0.4 71.62 28.38 55.418

3 0.3 0.5 82.53 17.47 61.9

4 0.3 0.6 76.86 23.14 57.102

5 0.3 0.7 81.12 18.88 58.078

6 0.4 0.3 80.73 19.27 55.696

7 0.4 0.4 73.89 26.11 59.188

8 0.4 0.5 80.35 19.65 62.802

9 0.4 0.6 79.34 20.66 65.386

10 0.4 0.7 80.65 19.35 57.078

11 0.5 0.3 79.84 20.16 65.556

12 0.5 0.4 79.63 20.37 56.758

13 0.5 0.5 81.5 18.5 58.214 0.5 0.6 78.89 21.11 52.5

15 0.5 0.7 78.63 21.37 59.904

16 0.6 0.3 77.56 22.44 59.74

17 0.6 0.4 82.21 17.79 65.67

18 0.6 0.5 77.19 22.81 53.272

19 0.6 0.6 80.1 19.9 61.3

20 0.6 0.7 76.6 23.4 64.67

21 0.7 0.3 76.54 23.46 59.904

22 0.7 0.4 73.56 26.44 56.496

23 0.7 0.5 78.88 21.12 71.902

24 0.7 0.6 76.22 23.78 48.314

25 0.7 0.7 79.13 20.87 65.556

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Table 5.3 Performance metrics for Dataset-II


(in %) Failure rate

(in %) Makespan

(in sec) 1 0.3 0.3 70.76 29.24 143.9482 0.3 0.4 69.56 30.44 111.4963 0.3 0.5 73.12 26.88 155.744 0.3 0.6 70.13 29.87 151.745 0.3 0.7 69.11 30.89 102.3486 0.4 0.3 70.63 29.37 106.3367 0.4 0.4 68.78 31.22 93.068 0.4 0.5 70.41 29.59 142.029 0.4 0.6 72.21 27.79 130.66410 0.4 0.7 70.16 29.84 106.0211 0.5 0.3 67.94 32.06 119.39212 0.5 0.4 71.59 28.41 149.37613 0.5 0.5 73.76 26.24 105.20814 0.5 0.6 68.07 31.93 119.96415 0.5 0.7 71.01 28.99 138.59616 0.6 0.3 70.94 29.06 109.21217 0.6 0.4 72.88 27.12 101.00418 0.6 0.5 74.42 25.58 92.13219 0.6 0.6 67.63 32.37 144.42420 0.6 0.7 70.51 29.49 150.05621 0.7 0.3 69.52 30.48 122.9822 0.7 0.4 71.35 28.65 92.26423 0.7 0.5 70.24 29.76 121.23624 0.7 0.6 70.14 29.86 138.9425 0.7 0.7 70.16 29.84 96.692

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Table 5.4 Performance metrics for Dataset-III


(in %) Failure

rate (in %)Makespan

(in sec)

1 0.3 0.3 74.89 25.11 87.948

2 0.3 0.4 79.7 20.3 95.348

3 0.3 0.5 75.23 24.77 94.51

4 0.3 0.6 78.76 21.24 93.184

5 0.3 0.7 70.44 29.56 131.008

6 0.4 0.3 73.25 26.75 159.816

7 0.4 0.4 74.96 25.04 95.448

8 0.4 0.5 72.63 27.37 109.064

9 0.4 0.6 73.38 26.62 148.548

10 0.4 0.7 77.03 22.97 104.628

11 0.5 0.3 71.96 28.04 159.86

12 0.5 0.4 76.43 23.57 95.924

13 0.5 0.5 78.23 21.77 101.22

14 0.5 0.6 75.14 24.86 106.544

15 0.5 0.7 74.89 25.11 128.608

16 0.6 0.3 71.43 28.57 123.34

17 0.6 0.4 74.01 25.99 128.768

18 0.6 0.5 73.23 26.77 147.788

19 0.6 0.6 76.17 23.83 93.624

20 0.6 0.7 72.87 27.13 105.036

21 0.7 0.3 75.53 24.47 103.172

22 0.7 0.4 72.17 27.83 86.416

23 0.7 0.5 77.25 22.75 149.836

24 0.7 0.6 71.19 28.81 99.308

25 0.7 0.7 74.17 25.83 129.752

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Table 5.5 Performance metrics for Dataset-IV


(in %) Failure rate

(in %) Makespan

(in sec)

1 0.3 0.3 81.8 18.2 131.411

2 0.3 0.4 75.45 24.55 112.505

3 0.3 0.5 79.03 20.97 92.888

4 0.3 0.6 78.19 21.81 97.481

5 0.3 0.7 68.46 31.54 117.947

6 0.4 0.3 76.86 23.14 123.515

7 0.4 0.4 75.29 24.71 97.622

8 0.4 0.5 80.17 19.83 98.519

9 0.4 0.6 74.59 25.41 85.961

10 0.4 0.7 69.93 30.07 113.117

11 0.5 0.3 73.46 26.54 108.782

12 0.5 0.4 72.6 27.4 130.316

13 0.5 0.5 80.7 19.3 118.2

14 0.5 0.6 70.26 29.74 92.786

15 0.5 0.7 72.38 27.62 80.618

16 0.6 0.3 78.34 21.66 134.39

17 0.6 0.4 68.19 31.81 101.381

18 0.6 0.5 67.18 32.82 131.195

19 0.6 0.6 71.69 28.31 86.243

20 0.6 0.7 65.56 34.44 83.099

21 0.7 0.3 74.24 25.76 83.003

22 0.7 0.4 63.86 36.14 116.453

23 0.7 0.5 79.67 20.33 84.653

24 0.7 0.6 80.19 19.81 107.093

25 0.7 0.7 70.11 29.89 83.999

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Table 5.6 Performance metrics for Dataset-V


(in %) Failure rate

(in %) Makespan

(in sec)

1 0.3 0.3 76.07 23.93 165.265

2 0.3 0.4 80.65 19.35 152.76

3 0.3 0.5 85.84 14.16 120.755

4 0.3 0.6 72.67 27.33 172.755

5 0.3 0.7 75.69 24.31 162.665

6 0.4 0.3 71.99 28.01 107.67

7 0.4 0.4 73.13 26.87 102.525

8 0.4 0.5 86.77 13.23 110.165

9 0.4 0.6 74.38 25.62 156.525

10 0.4 0.7 72.21 27.79 120.565

11 0.5 0.3 79.09 20.91 102.655

12 0.5 0.4 74.36 25.64 150.76

13 0.5 0.5 79.31 20.69 120.41

14 0.5 0.6 78.37 21.63 101.275

15 0.5 0.7 85.01 14.99 162.165

16 0.6 0.3 75.09 24.91 151.675

17 0.6 0.4 70.13 29.87 120.565

18 0.6 0.5 82.34 17.66 116.01

19 0.6 0.6 76.96 23.04 150.175

20 0.6 0.7 73.91 26.09 107.755

21 0.7 0.3 83.24 16.76 160.025

22 0.7 0.4 71.63 28.37 133.26

23 0.7 0.5 81.19 18.81 162.395

24 0.7 0.6 79.29 20.71 136.095

25 0.7 0.7 76.13 23.87 173.37

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Tables 5.2 to 5.6 shows for all possible Sth-III and Sth-II, threshold

values and its corresponding utilization, failure and makespan for all the five

scenario datasets. Table 5.7 shows the observed the maximum resource

utilization achieved in the sporadic and semi permanent thresholds for five

different datasets. Similarly, Table 5.8 shows the observed the minimum

makespan achieved in the sporadic and semi permanent thresholds for five

different datasets.

Table 5.7 Thresholds for maximum resource utilization

Sth-III Sth-II Max. Utilization (%)

Dataset-I 0.3 0.5 82.53

Dataset-II 0.6 0.5 74.42

Dataset-III 0.3 0.4 79.7

Dataset-IV 0.3 0.3 81.8

Dataset-V 0.4 0.5 86.77

Table 5.8 Thresholds for minimum Makespan

Sth-III Sth-II Min. Makespan(Sec)

Dataset-I 0.7 0.6 48.314

Dataset-II 0.6 0.5 92.132

Dataset-III 0.7 0.4 86.416

Dataset-IV 0.5 0.7 80.618

Dataset-V 0.5 0.6 101.275

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From Table 5.7 and 5.8, it can be seen that, none of the threshold

combination values are repeated for maximum utilization and minimum

makespan.

5.5.3 Comparative Analysis for Fixed Threshold

In order to evaluate LPM-DARA technique with the DARA as well

as with LPM-DARA the Sth-III and Sth-II values are kept fixed as 0.5. The

performance metrics for these three techniques are compared and shown in

Figure 5.3.

(a)

(b)

Figure 5.3 (Continued)

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(c)

Figure 5.3 Comparison of the DARA, LPM-DARA and NLPM-DARA techniques (a) Utilization (b) Makespan and (c) Failure rate

5.5.4 Comparative Analysis of Averaged Threshold Value

The performance metrics of Sth-III are averaged individually for the

clarity as like in the previous chapter for further analysis of the output of the

NLPM-DARA model. Based on the averaged value of each category of the

performance metrics are presented in Figure 5.4 (a), (b) and (c).

In general a considerable improvement has been made by the non-

linear prediction model (NLPM-DARA) when compared with LPM-DARA.

LPM-DARA reported the utilization rate between 42-75% (Figure 4.9(a)) for

all the datasets where as in NLPM-DARA it narrowed down to 70-80%

(Figure 5.4(a)). It is noteworthy to mention that, while increasing the fuzzy

threshold from 0.3 to 0.7 the % utilization does not produce marked

difference in NLPM-DARA (maximum 1.4%) for all the datasets, but in the

case of LPM-DARA it reported maximum of 6.2%. It reveals that, the

allocation of the jobs to sporadic resources are still very effective compared to

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LPM-DARA, where in higher fuzzy threshold values % utilization was

reduced considerably. As like in the case of LPM method, this method also

scored better efficiency in lower fuzzy threshold values.

In the case of makespan metrics of NLPM-DARA the structural

pattern of the graph was more or less retained as like LPM-DARA (Fig. 4.9c).

Overall the makespan was reduced by 30 seconds with the previous model.

This showed the effectiveness of the improved NLPM-DARA method.

Dataset I produced the better makespan and difference between the Dataset I

and all other Datasets are as like LPM-DARA (approximately 40 seconds).

0.3 0.4 0.5 0.6 0.740

45

50

55

60

65

70

75

80

Util

izat

ion

in%

STh-II

Dataset I Dataset II Dataset III Dataset IV Dataset V

(a)


135

0.3 0.4 0.5 0.6 0.7

60

70

80

90

100

110

120

130

140

150

160

Mak

espa

nin

sec

STh-II


(b)

0.3 0.4 0.5 0.6 0.70

5

10

15

20

25

30

35

40

45

50

Failu

rera

te

Sth-II


(c)

Figure 5.4 Compilation of performance metrics for (a) Resource utilization in %, (b) Makespan in seconds and (c) Failure rate in %.

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5.5.5 Comparison with other Techniques

To compare all the proposed DARA techniques with other

conventional techniques, the mean value of resource allocation, makespan

and failure rate are shown in Figure 5.5.

(a)

(b)


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(c)

Figure 5.5 Comparison with other algorithms (a) Mean Utilization (b)

Mean Makespan (c) Mean Failure Rate

From Figure 5.5 it is observed that the NLPM-DARA technique is outperformed when compared to all other techniques like FCFS, PSO, SA, DARA and LPM-DARA. This is due to the better prediction of dwelling time of the resources.

5.6 SUMMARY

In this paper, the previously proposed LPM-DARA was enhanced by replacing the time series prediction model by a nonlinear regression model. The nonlinear regression model was blindly initiated and then fine-tuned as per the dwelling time of the historical data. The model understood the historical dwelling time of the sporadic resources and hence the nature of the sporadic resources. This led the way to better resource allocation performance in terms of utilization, failure rate and makespan. The experimental validation under different fuzzy threshold had proved that the performance metrics achieved better results rather than the conventional resource allocation mechanisms including LPM-DARA and heuristic search based resource allocation mechanisms. Hence, the comparison in terms of reliability would be naturally a supporting end to NLPM-DARA, primarily compared against LPM-DARA.

chapter 5 non linear regression prediction of dwelling...

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