**International Journal of Engineering Technology and Computer Research (IJETCR) **
Available Online at www.ijetcr.org

**Volume 5; Issue 5; September-October: 2017; Page No. 92-99 **
**Journal Approved by UGC **

**Optimization of the Mixed Model Processor Scheduling**

**Dr. M. RAJA SEKAR **

Professor, CSE Department, VNRVJIET, Hyderabad, India.

*rajasekarm9@gmail.com *

Received14Aug. 2017; Accepted 14Sep. 2017

**Abstract **

Optimization of the mixed model processor scheduling (OMMPS) is nothing but allocating the processor time to
the number of process using mixed model of processor scheduling (MMPS) and minimizing the average
turnaround time of process. Usually round robin scheduling and priority scheduling are taken into account with
multilevel queue scheduling and and new scheduling concept is designed in which round robin scheduling
consists of shortest job first. In the first level queue dead line scheduling is implemented. All the processes,
which is ready to execute, must enter in to the ready queue. After entering in to the ready queue normally FIFO
scheduling is used in round robin scheduling. Instead of FIFO we have also used priority scheduling in which the
highest priority is given to the shortest processes and shortest are entering into the queue first. The nicety of
this paper is that we have considered dead line scheduling in parallel with round robin (RR) scheduling in order
to provide the execution of real time processes also. But we are not away from the concept of round robin
scheduling. In round robin scheduling the hard part is deciding what the length of the time slice could be. This
hard part is deciding what the length of the time slice could be. This hard part can be overcome by assuming the
processes follows normal distribution with mean µ and variance σ^{2 }and the time slice of each process is seal of
σ. Optimization of the mixed model processor scheduling consists of two queues namely q1 and q2. Where q1 is
meant for real time processes and q2 consists of remaining processes in which shortest process are executed
using normal dead line scheduling. Round robin scheduling is implemented for q2. The mentioned model applied
for randomly various set of processes and shown that their average turnaround time is reduced comparatively
with the usual pattern of round-robin scheduling and average waiting time can also be reduced.

**1. INTRODUCTION **

In optimization of the mixed model processor scheduling discipline, we have implemented the complete scheduling to classify the workload according to its characteristic to maintain separate queues serviced by the different schedulers [1, 2, and 3] which is the multi level queue scheduling. In this mixed model we have taken two queues. First queue q1 is used to have real time /deadline jobs. The jobs in q1 are executed using deadline scheduling discipline then second queue q2 is used to load the remaining sorted jobs [4, 5 and 6]. The job, which has shortest execution time, is entering into the q2 first and the process continues unless and until all the jobs loaded in to q2. When q2 is ready, the jobs in q2 are executed using round robin scheduling discipline. We have mixed round robin (RR) scheduling discipline with shortest job first scheduling discipline. So as to minimize the average turn around time as well as the average waiting time of the process [7, 8 and 9].

Each job will have its own CPU burst time. This time is nothing but the total amount of time that the process is in CPU to complete its execution. Common performance measures and optimization criteria to increase the system performance are processor utilization, throughput, turn around time, waiting time and response time [10, 11 and 12]. Process utilization is average fraction of time during which the process is busy [13, 14 and 15]. Throughput is the amount of work complete in a unit of time. The way to express the throughput is by means of the number of user jobs executed in a unit time [16, 17 and 18].

Turn around time of process or job is defined the time span between the process is submitted and until it is completed. Turn around time is sum of the execution time plus waiting time. Waiting time is the penalty imposed for sharing resources with others, in other words waiting time is the time that a process spends waiting for resource allocation due to contentions with others in a multiprogramming system. Response time is the time that elapses from a

movement of the last character of a common line launching a program or transaction is entered until the first result appears on the terminal[19, 20].

First come first served is simplest scheduling discipline. The work load is given in the order of arrival, without any preemption. FCFS scheduling discipline is quite straight forward. The performance of FCFS is very poor because of its excess average waiting time with no preemption; the resources utilization may be quite low[21, 22].

Round robin scheduling is for better scheduling to achieve good and relatively evenly distributed terminal response time. The performance of round robin scheduling is very sensitive to choice of the time slice. For this reason, in our model we have considered that if the processes follow normal distribution then the time slice can be taken as standard deviation of the normal distribution. In this paper we have clearly shown that the average turn around time and average waiting time of mixed model processor scheduling is less than the round robin scheduling [23, 24, 25, 26, 27 and 28].

**2.ROUND –ROBIN SCHEDULING **

Round robin is one of the oldest, simplest, fairest and most widely used scheduling algorithms, designed especially for time-sharing systems. A small unit of time, called times lice or quantum, is defined. All runnable processes are kept in circular queue. The CPU scheduler goes around this queue, allocating the CPU to each process for a time interval of one quantum. New processes are added to the tail of the queue.

The CPU scheduler picks the first process from the queue, sets a timer to interrupt after one quantum, and dispatches the process.

If the process is still running at the end of the quantum, the CPU is preempted and the process is added tail of the queue. If the process finishes before the end of the quantum, the process itself releases the CPU voluntarily. In either case, the CPU scheduler assigns the CPU to the next process in the ready queue. Every time a process is granted the CPU, a context switch occurs, which adds overhead to the process execution time. When other processes are waiting for processor no process can run for more than one time slices. If the process need more time than allocated time slice, then process is preempted after one time slice and placed at the end of the ready queue for next allocation.

**Figure 1: Mixed model Processor scheduling **
The performance of round robin is dependent on the
choice of time slice. Usually this is in milliseconds.

Rotating priority basis, the processor time is effectively allocated to processes, hence it called Round Robin.

**3.TIME QUANTUM SIZES **

The size of the time quantum chosen influences how effective optimization of the mixed model processor scheduling will be. If the quantum is too large then the MMPS becomes too similar to SJF. If the quantum is very small, then there will be a lot of context switching. A general guideline is to make the time quantum a little larger that the time allowed for typical interaction. But it may not work well. In order to overcome this difficulty, we have considered that processes time follows normal distribution with mean

µ and σ^{2} variance in order to fix the time slice. Let us
consider the size of the time quantum is σ.

**4. THE NORMAL DISTRIBUTION **

This distribution is extremely important in stastical applications because of the central limit theorem, which states that, under very general assumptions, the mean of a sample of n mutually independent random variables, having distribution with finite mean and variance, is normally distributed in limit as n goes to infinity. The probability density function of normal distribution is

where µ and σ^{2} are mean and variance of the
distribution .

**4.1 . MIXED MODEL PROCESSOR **
**SCHEDULING **

The optimization of mixed model processor scheduling algorithm is designed specifically for time- sharing systems. This model is similar to round robin.

For mixed model processor scheduling the ready queue is treated as a SJF, but preemption is added Most systems involve parameters and variables, which are random variables due to uncertainties.

Probabilistic methods are powerful in modeling such systems. Stochastic programming deals with situations where uncertainty is present in the data or model. Usually in deterministic mathematical programming the data(coefficients) are known whereas in stochastic programming the data are unknown, instead we may have a probability distribution present, such models are also sometimes called chance-constrained linear programs, as the name implies, is mathematical(i.e. linear, integer, mixed-integer, non-linear ) programming but with a stochastic element present in the data. For example the modeling of an investment portfolio so as to meet random liabilities, modeling of strategic capacity investments, power systems, that is modeling the operation of electrical power supply systems. So as to meet customers demand for electricity etc.[1, 2, 3, 4, 5, 6 and 7].

This paper focuses mainly on LSFP with joint probabilistic constraints, in which some of the data is random with at least one deterministic constraint.

The objective of this paper is to maximize the ratio of two linear functions subjected to a set of probabilistic

and deterministic inequalities and non negative constraints on the variables. We shall introduce the elements of LSP with joint probability constraint techniques. Also we shall discuss the transformations of LSFP problems into non-linear programming problems and solving through SLP method and branch and bound technique [8, 9, 10 and 11].

The SLP method, also known as the cutting plane method, was originally presented by Cheney and Goldstein and Kelly [12,13,14 and 15]. The concept of solving a series of linear programming problem in order to obtain the solution of the original non-linear programming problem, is known as sequential linear programming. Each linear programming problem is generated by approximating non-linear objective and constraint functions using first order Taylor series.

Though the LSFP with joint probability constraints model is non-linear, after adopting the concepts of chance –constraints. The model become partially linear. In the sense that objective function turn turns to be linear. Our one of the interest was in converting non-linear constraints to linear constraints which was fulfilled by SLP method [16, 17, 18 and 19].

Once the solution is obtained by SLP method, the B and B techniques comes in to picture to obtain integer solutions. This method starts by considering the original solution with complete feasible region, which is called root node problem/solution. The lower bounding and upper bounding procedures are applied to the root problem. For that, algorithm is given in section 3.b. If the bound satisfies the constraints, then an integer solution obtained and the procedure terminates, else if the constraints not satisfied, the corresponding branch terminates or else procedure moves recursively. Among the set of all integer solutions, which is optimize the objective function is considered as optimal solution.

In this paper Linear Fractional Programming problem and linear stochastic programming problem have been discussed succeeding sections. The formulation of Linear stochastic fractional programming problem with joint probability constraints with the various cases of stochastic data have been given in next section[21,22].

**5. METHODOLOGY **

In real world situations a decision maker faces the problem of optimization ratio. For example, in a manufacturing company one may have to optimize output/cost, Profit/Cost etc. Optimizing the ratio of two linear functions subject to some constraints is

called Linear fractional programming problem(LFPP).

Using the notation, the LFPP is stated as follows,

where T is an mxn matrix, b is an mx1 vector, c, d and x are nx1 vectors respectively and α, β are scalars. Let F be the set of feasible solutions of this problem.

Using the transformation Y = Tx where t is sacalar, have replaced the problem by the following two problems:

The denominator (d′X+β)>0 ∀ x∈F, then the problem is infeasible whereas if (d′X+β)<0, ∀ x∈F then the problem is infeasible. The denominator (d′X+β) for all practical cases, cannot have two problems, therefore, ultimately amounts to solving only the problem.

**5.1 Linear Model **

The chance –constrained stochastic programming was first studied by Jonson[2].The mathematical formulation can be stated as follows

Optimize f(X)= c’X (3) Subject to:

1-pi is the least probability with which i^{th} constraint is
satisfied. Assume that the random variables are
normally distributed with known parameters.

In real life situation, there are many applications of stochastic programming.

**5.2 Fractional Programming Problem **

Let us consider the mathematical form of LSFP as follows: Maximise

Thus the deterministic fractional non-linear programming problem equivalent to the given fractional stochastic linear programming is

Using the transformation Y=Tx, the above model reduces to non linear programming problem

Using the transformation Y=Tx, the above model reduces to non linear programming problem

**5. 3 Proposed Algorithm **

1. Start with an initial point Yint and a scalar and scalar t. The point Yint must be feasible.

2. Linearize the non-linear functions in constraints about the point Yj as

3. Formulate the approximation LSFP problem with joint probability constraints as

4. Solve the approximating LPP to obtain the solution vector Ynext and scalar tnext

5. Evaluate the original constraints at Ynext i.e Get gi(Ynext), i=1 to m.If gi(Ynext)<r for i=1 to m, where r is a prescribed small positive tolerence, all the original constraints can be assumed to have been satisfied.

Hence stop the procedure by taking Yopt=Ynext and Yopt=Tnext.

6. Let next=int+1. Goto step 4.

7. Solve the problem as if all the variables were real numbers i.e not integers. This solution is the upper bound of the LSFP problem.

8. Choose one variable at a time that has a non integer value, say xj and branch that variable to the next higher integer value for one problem and to the next lower integer value for the other. The real valued solution of variable j can be expressed as xj.

9. Now variable xj is an integer in either branch. Fix the integer of xj for the following steps of branch and bound. Select the branch which yield the maximum objective function with all constraints satisfied. Then repeat step2 on another variable.

10. Stop the particular branch if the solution is not satisfying the constraints of the original problem else stop the branch when all the desired integer variables are obtained, that is all the integer variables find integer values.

**6. RESULTS AND CONCLUSION **

A company manufactures n types of machines.

Further, their production requires m processes. The processing time tij are known to be independently distributed normal variables with estimated means uij

and standard deviations sij. We also know the average profit for each machine. The annual average fixed cost to manufacture the machine is α. The mean capital invested is Rs. Ci per machine of type i. Below table may provide us more details.

The combination of the SLP and the B and B technique takes advantage of an exact method. The SLP method quickly reaches a non-integer solution that is close to optimum and branch and bound technique guarantees the global optimum as well as integer solution. Since a good approximation is obtained by the SLP method, if does not take many branches for the B&B to reach the optimal solutions.

**Figure 1: Multi processor Scheduling **

**Figure 2: Graph in between number of product and **
**CMAX **

**Figure 3: LSFP Comparison **

**Figure 4: LSFP vs ILPP**

**Table 1: Data for manufacturing **

**Resources Machine types ** **Availability **

1 234 213 980 321 452

2 432 521 643 643 675

3 123 431 612 125 342

4 678 653 890 764 432

5 908 784 765 532 145

6 765 476 432 145 653

7 643 098 165 653 543

8 231 712 653 421 642

9 621 453 532 543 234

10 213 543 432 653 765

11 542 213 765 324 345

12 521 764 875 542 231

**Table 2: Process timings **

**Process ** **T1 ** **T2 ** **T3 ** **T4 ** **T5 **

P1 432 234 452 126 764

P2 543 654 786 234 564

P3 321 564 764 343 234

P4 654 125 675 432 765

P5 642 698 892 998 342

P6 453 678 436 654 621

P7 543 654 982 892 980

P8 543 432 564 643 987

P9 953 643 247 864 231

P10 982 126 875 321 421

P11 098 412 762 542 509

P12 412 864 986 721 986

P13 213 543 221 981 891

P14 986 213 653 981 965

P15 213 975 098 213 165

P16 981 543 128 098 254

In addition, the B&B technique generates many sets
of solutions. The competitive alternative provides
management with several options and flexibility. This
method can be applied to any optimization problem
whose objective function is to optimize the ratio of
two linear cost functions subject to constraints with
some stochastic variables. AB and B technique for
LSFP with joint probability constraints successfully
yield better results for the machine manufacturing
**problem which we have discussed. **

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