Salam Ayad Hussein*
| Emad Issa Abdul Kareem
© 2024 The authors. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
OPEN ACCESS
A major aspect of distributed systems is task scheduling. How significant is it to adequately allocate tasks to each one of the computer's processors for the purpose of achieving better performance? The indicated approaches attempt to reduce costs and improve performance while maximizing the utilization of the central processing unit (CPU) in such a situation. The main flaw in such methods is that they require a lot of scheduling and, as a result, a lot of time. To address such a flaw, a modified CPU job-scheduling method was suggested, which used the Round Robin technique (R.R.) and dual synchronized time-slices to reduce jobs' waiting time to be executed. Therefore, the present study proves that the suggested approach has optimization efficiency with an average percentage of (12.4%–62.07%), according to an original R.R. algorithm.
operating system, task scheduling, round robin (R.R.), time quantum (T.Q), optimization techniques
Task scheduling is an important field in the operating system. Task scheduling can be divided into two types: dynamic and static. Schedules are created throughout runtime in dynamic scheduling, while no knowledge regarding the task is available until it arrives [1].
It develops schedules prior to running time and has no ability to change them in static scheduling. Comparatively, all tasks should be identified ahead of time. Put differently, a static task scheduling algorithm schedules a group of tasks on processors with known communication and processing properties to improve a few performance metrics like complexity, Makespan, and CPU utilization [1]. The hybrid optimization scheduling approach is the focus of this study. The problem of timetabling was studied for many years, and various solutions were suggested. Since the 1960s, researchers have examined adequate as well as heuristic solution methods for the university and school timetabling problem [2]. When looking at the solution approaches' history, it's indicated that many solutions were suggested for such an issue. The operations research literature has been thoroughly indicated in the construction of university schedules over the last three decades [3].
In light of the prolonged execution time identified as a pivotal challenge in this undertaking, the primary objective is to ascertain an optimal sequence for the execution of jobs. The proposed methodology will consider both the time at which jobs are initiated and the time required for their completion. By balancing the entrance and execution times, the proposed approach aims to reduce the overall waiting time for jobs.
The proposed technique is based on the concept of dual synchronisation, which differs from the traditional approach of single synchronisation. In contrast to the traditional (R.R.) method, which does not take the length of the jobs into account, this technique considers the length of the jobs to be a key factor in the scheduling process. According to the proposed approach, the expected advantages Combine single and dual synchronisation for short and long jobs respectively. This approach will facilitate the completion of lengthy jobs without impacting the execution time of shorter jobs [4].
Thus, the objective of this paper will be to minimize execution time to achieve better usage of processors. Where, our contributions will be through creating a new, hybrid scheduling technique (by taking features from basic operating system scheduling techniques) and re-arrange it in a way that ensures better performance and the highest score in its fitness function [5].
Hussein et al. [6] demonstrated that the suggested hybrid planning framework is more effective and exacerbates the current planning gap. It aids in the study of the operating system and the use of a multi-programming environment. Creating a single priority activity policy resulted in identical scalability testing. The value of mixing complicated real-time and soft-time jobs in the system is also assessed.
Suppiah et al. [7] presented a novel queue-based multi-level CPU scheduling system. The suggested method assigns CPU service time and dynamically decides the time quantity (TQ) between different queues. The fuzzy toolbox is used to build the programming method utilized in the Math Lab application.
In the multi-level feedback queue programming technique, Jain and Jain [8], conducted a simulation study known as Markov chain analysis to examine the influence of wait state on overall system efficiency and performance. This study also underlines the fact that the comparison analysis is calculated in an arithmetic model with shifting values and d.
Maktum et al. [9] proposed utilizing a genetic algorithm to develop near-optimal solutions to the CPU scheduling problem. They developed a simple scheduling algorithm for uni-processor scheduling based on their evolutionary technique and compared it to SJF and FCFS scheduling to reduce average wait times.
Jawad [10] developed a neural fuzzy programming technique for the application, as well as a modified CPU scheduling algorithm, in order to increase reaction time while minimizing average time and response time. It accomplishes this by combining established schedules, such as SJF preemptive scheduling and Nero fuzzy scheduling.
Saroj et al. [11] offered a strategy that was simulated and implemented in C++ programming. This solution addresses a number of difficulties, including longer average wait times, rotation times, infinite hunger blocking, and hands-on execution. With two types of split times, the proposed timetable solved numerous difficulties and was simple to implement.
Ahmed and Brifcani [12] to learn advanced features, they employed a multi-layer artificial neural network, similar to deep neural networks. Deep learning was performed using the essential characteristics derived from the data.
Elmougy et al. [13] SRDQ is a revolutionary hybrid task scheduling technique that combines Shortest Job First (SJF) and Round Robin (RR) schedulers while taking the quantity of a dynamically changeable job into consideration. The suggested algorithms are primarily based on two essential keys: the first contains a large number of dynamic jobs to balance the waiting time between short and long tasks, and the second divides the ready queue into two sub-queues, Q1 for short tasks and Q2 for lengthy tasks. and another for quick jobs. the lengths. With two Q1 tasks and one Q2 task, task assignment to Q1 or Q2 resources is mutual.
Sai et al. [14] presented a hybrid approach to solve the scheduling problem. This approach combines the shortest and longest job scheduling techniques and indicates if it is suitable for both unit-processor and multiprocessor systems depending on the results. This technique can lead to efficient solutions when additional characteristics such as waiting time and execution time are taken into account.
Himthani et al. [15] suggested a multitasking scheduling approach that enhances system throughput and decreases context switching cost. Because it includes the benefits of both systems, the suggested strategy is a hybrid of Round Robin and Shortest Remaining Time First. The quantum time update is determined by the slice bit and the remaining time of the packet in this technique.
Processor scheduling can be defined as Allocating CPU time slots to processes procedure. In a multi-processing system, which can be regarded as a dynamic operation that grows more challenging [16, 17].
The major scheduling aims can be summarized as [18, 19]:
1. Being fair between processes.
2. Maximizing the number of completed processes for each unit time (throughput).
3. Avoiding the indefinite postponement regarding any process (Starvation Free).
4. Minimizing overhead, i.e., minimizing the wasted time in scheduling.
5. Balancing resource usage, i.e., using all resources all the time.
6. Enforcing a priority scheme for allowing certain processes to get more CPU time.
7. Degrading under heavy loads.
With regard to significance, there are numerous scheduling types which were delivered as follows [20, 21]:
3.1 First in first out scheduling (FIFO)
Processes are dispatched based on the arrival time on the Ready Queue in such a type. As soon as the process has access to the CPU, it will run to the point where it’s finished. Due to the fact that FIFO scheduling isn't non-preemptive, it might be used in a single programming environment. It can’t be utilized as a master scheme in multi-programming yet, only as part of its [22, 23].
Disadvantages [24, 25]:
a. Less execution time process endures, i.e., the waiting time is often excessively long.
b. Favours CPU-bound processes over I/O-bound processes.
c. The first process will get the CPU first; remaining processes can use the CPU exclusively after the existing process has achieved its execution.
3.2 Round robin scheduling
Round-robin scheduling has been considered comparable to FCFS scheduling, with the exception that the CPU bursts are given time quantum limits (T.Q.). In addition, a timer has been set to whatever value was set for time quantum in a case where a process is given the CPU. In the case where the process completes its burst prior to the expiration of the quantum timer, it’s swapped out of the CPU in the same way that the standard FCFS algorithm is. The process has been swapped out of the CPU and then moved to the back end of the ready queue in a case where the timer goes off first. The ready queue is kept as a circular queue, which means that as soon as all processes have had their turn, the scheduler will give the first process another turn, etc. [24, 26].
Disadvantages [18, 25]:
a. Setting the quantum too short increases the overhead and lowers the CPU efficiency, but setting it too long may cause a poor response to short processes.
b. The average waiting time under the RR policy is often long.
c. If the time quantum is very high, then RR degrades to FCFS.
3.3 Shortest job first (SJF)
It has been indicated that an extremely simple method solves the problem; actually, it's an idea that has been borrowed from operations research [C-54, PV-56] and utilized for job scheduling in computer systems. SJF is the name of this novel scheduling discipline, and it must be easy to remember since it accurately describes policy: it runs the shortest job first, after that the next shortest, etc. [27].
Disadvantages [13, 25]:
a. SJF may cause starvation if shorter processes keep coming. This problem is solved by aging.
b. It cannot be implemented at the level of short-term CPU scheduling.
3.4 Priority scheduling
Priority scheduling has been considered a more general version of SJF, where every one of the jobs is given a priority and the maximum priority job is served first. SJF prioritizes according to the inverse of the expected burst time; the smaller the expected burst, the higher the priority. Actually, priorities are conducted with the use of integers within a fixed range, yet there isn't consensus on whether high priorities should be conducted with small or large numbers. In this study, high priorities are represented by a low number, with 0 being the highest possible priority [28, 29].
Disadvantages [30, 31]:
a. If high-priority processes use up a lot of CPU time, lower-priority processes may starve and be postponed indefinitely. The situation where a process never gets scheduled to run is called starvation.
b. Another problem is deciding which process gets which priority level assigned to it.
The major effect of the previously mentioned disadvantages is that they might cause an oscillation in the optimization efficiency of these approaches.
The suggested approach has considered a finite number of jobs, each with a different execution time (length). Those jobs have been placed in the job pool from which they have been assigned to be implemented by an OS, as shown in Figure 1 below:
Figure 1. The proposed job's sequence
4.2 The proposed method flowchart
The suggested model will be concentrated on the job's execution time using a proposed scheduling technique via taking advantage of the R.R. technique, as in the following flowchart (Figure 2).
Figure 2. The proposed method flowchart
4.3 The proposed method steps
Basically, a number of jobs will be assigned (N) with various lengths. As in the R.R. technique, a time quantum (T.Q.) will be specified in order to be utilized to execute job slices from main jobs. Normally, the T.Q. should be commensurate with the length of the shortest job (T.Q. less than the shortest job length). The execution starts (based on the job's arrival sequence) by executing dual synchronized slices from both sides of each job.
For each round, three parameters will be considered as input: No. of jobs, T.Q, and length of jobs. Initially, two variables (right_side, left_side) need to be assigned. The right_side will be equal to the longest job, while the left_side needs to be assigned to be equal to the start state (initially equal to 0). For each execution round, these two variables in addition to the max_length variable will be updated using Eqs. (1)-(3) respectively (see Figure 3).
$ { Right\_\ Side }_i= { Right_\ Side }_{i-1}-T_{.} Q$ (1)
${ Left\_\ side }_i= { left_\ side }_{i-1}+T \cdot Q$ (2)
$\ { Max\_\ length }_i= { Max_\ length }_{i-1}-T . Q$ (3)
Figure 3. The proposed technique implementation
The results have been presented via quantitative evaluation. Three hundred random jobs have been used in this issue (see Figure 4 and appendix A). The randomness came from assigning a random time for the job to enter and a random time for the job to be executed. The evaluation of the suggested scheduling algorithm has been detected by finding the optimal ratio gained via the proposed algorithm. The suggested algorithm is an optimization algorithm that balances between the two above- noted criteria (i.e., the time of job entry and the time that the job will take to be executed). The results have been presented via quantitative evaluation. The optimization ratio of the proposed approach comparing with traditional R.R has been calculated via Eq. (4).
optimization_Ratio $=\left(\frac{{right_\ side }}{{ Job_\ lingth }}\right) \times 100$ (4)
In Eq. (4), the left-side variable has been ignored because it is a common factor for both approaches (i.e., traditional and proposed approaches), while the right-side factor corresponds to the proposed approach. Thus, the right-side factor contributes to the optimality of the proposed approach compared to traditional R.R.
The jobs' optimization rate will be divided into three periods: period 1 includes jobs with an optimization rate in the range (0% to 33%); period 2 includes jobs with an optimization rate in the range (33% to 66%); and period 3 includes jobs with an optimization rate in the range (66% to 100%).
Figure 4. Optimization ratio for each job of the 300 jobs
The benefit of the proposed approach is that it does not affect the execution times of the shortest jobs, as shown in Figure 4, where jobs with an execution time that is less than half that of the longest job within the synchronized jobs group would take their own execution time. The jobs' optimization rate will improve as soon as their execution times increase. Thus, it is evident that the rate of optimization and job execution time are directly correlated. additionally, it was discovered that a job group will achieve the highest optimization rate when it is larger than 40 seconds and includes equal values, or at the very least, nearly equivalent values.
Finally, it should be mentioned that the system overhead will never increase because both long and short jobs are executed at each clock cycle at least from the left side (unlike the standard R.R. that only at most executes one side of long and short jobs at each clock cycle). And for fairness and scientific integrity, it should be mentioned that the mathematical complexity might increase in this approach due to the ongoing updating in job's length measurement from the start until the end of the execution.
The standard round robin and the proposed method have been compared quantitatively using a sample of 400 randomly chosen jobs with random different lengths (which makes the proposed approach applicable at any different scenario). In this experiment, the attributes were: job length, start state, right-side and left side. In addition to T.Q which is less than the shortest job. When These attributes had been generated randomly. The quantitative results of the experiment are shown in Tables 1 and 2. In comparison to the conventional Round Robin, the proposed method has been able to minify the long job Time of execution to around (12.4%–62.7%), as shown in Figure 5.
Figure 5. Quantitative comparison between traditional round robin and the proposed method
The suggested method has the priority of the execution of jobs according to entering time as well as the strategies for taking the long execution jobs into account. It combines the benefits of the round-robin method with dual synchronized time slices. In light of this, there was a noticeable decrease in the average waiting time by (12.4% to 62.07%). This proves that the proposed method is successful in reducing job execution time. This led us to the conclusion that the proposed method accomplished the primary study objective for which it was designed.
(1) In order to shorten the time required to complete jobs, it is suggested that the classic round robin be coupled with the best aspects of spatial or temporal notions.
(2) For real time system, we recommend reducing the number of jobs or using a parallel processing to achieve the optimal results from the proposed method.
(3) In cloud computing, the proposed method can be a useful solution for task scheduling in virtual machines queues or in the request schedule manager after fog computing layer.
(4) It is recommended that the technical complexity should be measured between classic R.R and this approach to find the system overhead.
This appendix contains a table (Table 1) of the results of Three hundred random jobs have been used. The optimization ratio of the proposed approach comparing with traditional R.R has been calculated via Eq. (4).
Table 1. Optimization ratio for each job of the 300 jobs
|
NO. |
Job Length |
Leftt_slice |
Right_slice |
Optimization Rate |
|
1 |
99 |
98 |
1 |
1.01010101 |
|
2 |
62 |
61 |
1 |
1.612903226 |
|
3 |
62 |
61 |
1 |
1.612903226 |
|
4 |
99 |
95 |
4 |
4.04040404 |
|
5 |
49 |
47 |
2 |
4.081632653 |
|
6 |
49 |
47 |
2 |
4.081632653 |
|
7 |
49 |
47 |
2 |
4.081632653 |
|
8 |
49 |
47 |
2 |
4.081632653 |
|
9 |
93 |
88 |
5 |
5.376344086 |
|
10 |
93 |
88 |
5 |
5.376344086 |
|
11 |
101 |
95 |
6 |
5.940594059 |
|
12 |
101 |
95 |
6 |
5.940594059 |
|
13 |
101 |
95 |
6 |
5.940594059 |
|
14 |
71 |
65 |
6 |
8.450704225 |
|
15 |
98 |
89 |
9 |
9.183673469 |
|
16 |
98 |
89 |
9 |
9.183673469 |
|
17 |
102 |
92 |
10 |
9.803921569 |
|
18 |
102 |
92 |
10 |
9.803921569 |
|
19 |
20 |
18 |
2 |
10 |
|
20 |
73 |
65 |
8 |
10.95890411 |
|
21 |
91 |
81 |
10 |
10.98901099 |
|
22 |
62 |
55 |
7 |
11.29032258 |
|
23 |
82 |
72 |
10 |
12.19512195 |
|
24 |
92 |
80 |
12 |
13.04347826 |
|
25 |
105 |
91 |
14 |
13.33333333 |
|
26 |
102 |
88 |
14 |
13.7254902 |
|
27 |
102 |
88 |
14 |
13.7254902 |
|
28 |
87 |
74 |
13 |
14.94252874 |
|
29 |
87 |
74 |
13 |
14.94252874 |
|
30 |
116 |
98 |
18 |
15.51724138 |
|
31 |
109 |
92 |
17 |
15.59633028 |
|
32 |
109 |
92 |
17 |
15.59633028 |
|
33 |
107 |
90 |
17 |
15.88785047 |
|
34 |
68 |
57 |
11 |
16.17647059 |
|
35 |
114 |
95 |
19 |
16.66666667 |
|
36 |
114 |
95 |
19 |
16.66666667 |
|
37 |
114 |
95 |
19 |
16.66666667 |
|
38 |
70 |
57 |
13 |
18.57142857 |
|
39 |
118 |
96 |
22 |
18.6440678 |
|
40 |
118 |
96 |
22 |
18.6440678 |
|
41 |
125 |
100 |
25 |
20 |
|
42 |
114 |
91 |
23 |
20.1754386 |
|
43 |
110 |
87 |
23 |
20.90909091 |
|
44 |
100 |
78 |
22 |
22 |
|
45 |
100 |
78 |
22 |
22 |
|
46 |
118 |
92 |
26 |
22.03389831 |
|
47 |
118 |
92 |
26 |
22.03389831 |
|
48 |
117 |
91 |
26 |
22.22222222 |
|
49 |
117 |
91 |
26 |
22.22222222 |
|
50 |
84 |
65 |
19 |
22.61904762 |
|
51 |
17 |
13 |
4 |
23.52941176 |
|
52 |
85 |
65 |
20 |
23.52941176 |
|
53 |
132 |
100 |
32 |
24.24242424 |
|
54 |
132 |
100 |
32 |
24.24242424 |
|
55 |
98 |
74 |
24 |
24.48979592 |
|
56 |
98 |
74 |
24 |
24.48979592 |
|
57 |
128 |
96 |
32 |
25 |
|
58 |
128 |
96 |
32 |
25 |
|
59 |
16 |
12 |
4 |
25 |
|
60 |
127 |
95 |
32 |
25.19685039 |
|
61 |
135 |
100 |
35 |
25.92592593 |
|
62 |
135 |
100 |
35 |
25.92592593 |
|
63 |
119 |
88 |
31 |
26.05042017 |
|
64 |
99 |
73 |
26 |
26.26262626 |
|
65 |
55 |
40 |
15 |
27.27272727 |
|
66 |
138 |
100 |
38 |
27.53623188 |
|
67 |
138 |
100 |
38 |
27.53623188 |
|
68 |
105 |
76 |
29 |
27.61904762 |
|
69 |
65 |
47 |
18 |
27.69230769 |
|
70 |
65 |
47 |
18 |
27.69230769 |
|
71 |
65 |
47 |
18 |
27.69230769 |
|
72 |
65 |
47 |
18 |
27.69230769 |
|
73 |
136 |
96 |
40 |
29.41176471 |
|
74 |
136 |
96 |
40 |
29.41176471 |
|
75 |
143 |
100 |
43 |
30.06993007 |
|
76 |
143 |
100 |
43 |
30.06993007 |
|
77 |
113 |
79 |
34 |
30.08849558 |
|
78 |
113 |
79 |
34 |
30.08849558 |
|
79 |
136 |
95 |
41 |
30.14705882 |
|
80 |
136 |
95 |
41 |
30.14705882 |
|
81 |
136 |
95 |
41 |
30.14705882 |
|
82 |
79 |
55 |
24 |
30.37974684 |
|
83 |
49 |
34 |
15 |
30.6122449 |
|
84 |
111 |
77 |
34 |
30.63063063 |
|
85 |
111 |
77 |
34 |
30.63063063 |
|
86 |
111 |
77 |
34 |
30.63063063 |
|
87 |
111 |
77 |
34 |
30.63063063 |
|
88 |
101 |
70 |
31 |
30.69306931 |
|
89 |
65 |
45 |
20 |
30.76923077 |
|
90 |
131 |
90 |
41 |
31.29770992 |
|
91 |
131 |
90 |
41 |
31.29770992 |
|
92 |
134 |
92 |
42 |
31.34328358 |
|
93 |
134 |
92 |
42 |
31.34328358 |
|
94 |
136 |
93 |
43 |
31.61764706 |
|
95 |
136 |
93 |
43 |
31.61764706 |
|
96 |
147 |
100 |
47 |
31.97278912 |
|
97 |
84 |
57 |
27 |
32.14285714 |
|
98 |
146 |
99 |
47 |
32.19178082 |
|
99 |
146 |
99 |
47 |
32.19178082 |
|
100 |
136 |
91 |
45 |
33.08823529 |
|
101 |
114 |
76 |
38 |
33.33333333 |
|
102 |
108 |
72 |
36 |
33.33333333 |
|
103 |
133 |
88 |
45 |
33.83458647 |
|
104 |
134 |
88 |
46 |
34.32835821 |
|
105 |
134 |
88 |
46 |
34.32835821 |
|
106 |
122 |
80 |
42 |
34.42622951 |
|
107 |
110 |
72 |
38 |
34.54545455 |
|
108 |
153 |
100 |
53 |
34.64052288 |
|
109 |
153 |
100 |
53 |
34.64052288 |
|
110 |
118 |
77 |
41 |
34.74576271 |
|
111 |
138 |
90 |
48 |
34.7826087 |
|
112 |
138 |
90 |
48 |
34.7826087 |
|
113 |
94 |
61 |
33 |
35.10638298 |
|
114 |
94 |
61 |
33 |
35.10638298 |
|
115 |
144 |
93 |
51 |
35.41666667 |
|
116 |
152 |
98 |
54 |
35.52631579 |
|
117 |
127 |
81 |
46 |
36.22047244 |
|
118 |
102 |
65 |
37 |
36.2745098 |
|
119 |
154 |
98 |
56 |
36.36363636 |
|
120 |
154 |
98 |
56 |
36.36363636 |
|
121 |
154 |
98 |
56 |
36.36363636 |
|
122 |
154 |
98 |
56 |
36.36363636 |
|
123 |
74 |
47 |
27 |
36.48648649 |
|
124 |
74 |
47 |
27 |
36.48648649 |
|
125 |
74 |
47 |
27 |
36.48648649 |
|
126 |
74 |
47 |
27 |
36.48648649 |
|
127 |
156 |
99 |
57 |
36.53846154 |
|
128 |
156 |
99 |
57 |
36.53846154 |
|
129 |
139 |
87 |
52 |
37.41007194 |
|
130 |
55 |
34 |
21 |
38.18181818 |
|
131 |
122 |
74 |
48 |
39.3442623 |
|
132 |
122 |
74 |
48 |
39.3442623 |
|
133 |
139 |
84 |
55 |
39.56834532 |
|
134 |
139 |
84 |
55 |
39.56834532 |
|
135 |
139 |
84 |
55 |
39.56834532 |
|
136 |
139 |
84 |
55 |
39.56834532 |
|
137 |
139 |
84 |
55 |
39.56834532 |
|
138 |
131 |
79 |
52 |
39.69465649 |
|
139 |
131 |
79 |
52 |
39.69465649 |
|
140 |
150 |
90 |
60 |
40 |
|
141 |
154 |
92 |
62 |
40.25974026 |
|
142 |
154 |
92 |
62 |
40.25974026 |
|
143 |
154 |
92 |
62 |
40.25974026 |
|
144 |
154 |
92 |
62 |
40.25974026 |
|
145 |
152 |
90 |
62 |
40.78947368 |
|
146 |
114 |
67 |
47 |
41.22807018 |
|
147 |
143 |
84 |
59 |
41.25874126 |
|
148 |
143 |
84 |
59 |
41.25874126 |
|
149 |
143 |
84 |
59 |
41.25874126 |
|
150 |
143 |
84 |
59 |
41.25874126 |
|
151 |
162 |
95 |
67 |
41.35802469 |
|
152 |
156 |
91 |
65 |
41.66666667 |
|
153 |
156 |
91 |
65 |
41.66666667 |
|
154 |
155 |
90 |
65 |
41.93548387 |
|
155 |
160 |
92 |
68 |
42.5 |
|
156 |
160 |
92 |
68 |
42.5 |
|
157 |
71 |
40 |
31 |
43.66197183 |
|
158 |
174 |
98 |
76 |
43.67816092 |
|
159 |
174 |
98 |
76 |
43.67816092 |
|
160 |
174 |
98 |
76 |
43.67816092 |
|
161 |
174 |
98 |
76 |
43.67816092 |
|
162 |
73 |
41 |
32 |
43.83561644 |
|
163 |
125 |
70 |
55 |
44 |
|
164 |
140 |
78 |
62 |
44.28571429 |
|
165 |
140 |
78 |
62 |
44.28571429 |
|
166 |
167 |
93 |
74 |
44.31137725 |
|
167 |
167 |
93 |
74 |
44.31137725 |
|
168 |
178 |
99 |
79 |
44.38202247 |
|
169 |
178 |
99 |
79 |
44.38202247 |
|
170 |
132 |
73 |
59 |
44.6969697 |
|
171 |
85 |
47 |
38 |
44.70588235 |
|
172 |
85 |
47 |
38 |
44.70588235 |
|
173 |
85 |
47 |
38 |
44.70588235 |
|
174 |
85 |
47 |
38 |
44.70588235 |
|
175 |
125 |
69 |
56 |
44.8 |
|
176 |
182 |
100 |
82 |
45.05494505 |
|
177 |
141 |
77 |
64 |
45.39007092 |
|
178 |
141 |
77 |
64 |
45.39007092 |
|
179 |
141 |
77 |
64 |
45.39007092 |
|
180 |
141 |
77 |
64 |
45.39007092 |
|
181 |
171 |
93 |
78 |
45.61403509 |
|
182 |
72 |
39 |
33 |
45.83333333 |
|
183 |
72 |
39 |
33 |
45.83333333 |
|
184 |
165 |
89 |
76 |
46.06060606 |
|
185 |
165 |
89 |
76 |
46.06060606 |
|
186 |
173 |
93 |
80 |
46.24277457 |
|
187 |
151 |
81 |
70 |
46.35761589 |
|
188 |
102 |
54 |
48 |
47.05882353 |
|
189 |
172 |
91 |
81 |
47.09302326 |
|
190 |
172 |
91 |
81 |
47.09302326 |
|
191 |
123 |
65 |
58 |
47.15447154 |
|
192 |
172 |
90 |
82 |
47.6744186 |
|
193 |
140 |
73 |
67 |
47.85714286 |
|
194 |
75 |
39 |
36 |
48 |
|
195 |
75 |
39 |
36 |
48 |
|
196 |
149 |
77 |
72 |
48.32214765 |
|
197 |
105 |
54 |
51 |
48.57142857 |
|
198 |
159 |
81 |
78 |
49.05660377 |
|
199 |
89 |
45 |
44 |
49.43820225 |
|
200 |
93 |
47 |
46 |
49.46236559 |
|
201 |
93 |
47 |
46 |
49.46236559 |
|
202 |
186 |
93 |
92 |
49.46236559 |
|
203 |
93 |
47 |
46 |
49.46236559 |
|
204 |
93 |
47 |
46 |
49.46236559 |
|
205 |
107 |
54 |
53 |
49.53271028 |
|
206 |
109 |
55 |
54 |
49.5412844 |
|
207 |
129 |
65 |
64 |
49.6124031 |
|
208 |
137 |
69 |
68 |
49.6350365 |
|
209 |
139 |
70 |
69 |
49.64028777 |
|
210 |
149 |
75 |
74 |
49.66442953 |
|
211 |
157 |
79 |
78 |
49.68152866 |
|
212 |
157 |
79 |
78 |
49.68152866 |
|
213 |
161 |
81 |
80 |
49.68944099 |
|
214 |
173 |
87 |
86 |
49.71098266 |
|
215 |
175 |
88 |
87 |
49.71428571 |
|
216 |
175 |
88 |
87 |
49.71428571 |
|
217 |
177 |
89 |
88 |
49.71751412 |
|
218 |
177 |
89 |
88 |
49.71751412 |
|
219 |
179 |
90 |
89 |
49.72067039 |
|
220 |
183 |
92 |
91 |
49.72677596 |
|
221 |
183 |
93 |
91 |
49.72677596 |
|
222 |
183 |
92 |
91 |
49.72677596 |
|
223 |
185 |
93 |
92 |
49.72972973 |
|
224 |
191 |
96 |
95 |
49.7382199 |
|
225 |
191 |
96 |
95 |
49.7382199 |
|
226 |
195 |
98 |
97 |
49.74358974 |
|
227 |
195 |
98 |
97 |
49.74358974 |
|
228 |
195 |
98 |
97 |
49.74358974 |
|
229 |
195 |
98 |
97 |
49.74358974 |
|
230 |
195 |
98 |
97 |
49.74358974 |
|
231 |
199 |
100 |
99 |
49.74874372 |
|
232 |
199 |
100 |
99 |
49.74874372 |
|
233 |
199 |
100 |
99 |
49.74874372 |
|
234 |
199 |
100 |
99 |
49.74874372 |
|
235 |
199 |
100 |
99 |
49.74874372 |
|
236 |
168 |
84 |
84 |
50 |
|
237 |
168 |
84 |
84 |
50 |
|
238 |
134 |
67 |
67 |
50 |
|
239 |
94 |
47 |
47 |
50 |
|
240 |
168 |
84 |
84 |
50 |
|
241 |
148 |
74 |
74 |
50 |
|
242 |
38 |
19 |
19 |
50 |
|
243 |
134 |
67 |
67 |
50 |
|
244 |
168 |
84 |
84 |
50 |
|
245 |
168 |
84 |
84 |
50 |
|
246 |
94 |
47 |
47 |
50 |
|
247 |
154 |
77 |
77 |
50 |
|
248 |
154 |
77 |
77 |
50 |
|
249 |
94 |
47 |
47 |
50 |
|
250 |
168 |
84 |
84 |
50 |
|
251 |
154 |
77 |
77 |
50 |
|
252 |
190 |
95 |
95 |
50 |
|
253 |
154 |
77 |
77 |
50 |
|
254 |
182 |
91 |
91 |
50 |
|
255 |
148 |
74 |
74 |
50 |
|
256 |
38 |
19 |
19 |
50 |
|
257 |
134 |
67 |
67 |
50 |
|
258 |
180 |
90 |
90 |
50 |
|
259 |
184 |
92 |
92 |
50 |
|
260 |
180 |
90 |
90 |
50 |
|
261 |
78 |
39 |
39 |
50 |
|
262 |
152 |
76 |
76 |
50 |
|
263 |
196 |
98 |
98 |
50 |
|
264 |
198 |
99 |
99 |
50 |
|
265 |
190 |
95 |
95 |
50 |
|
266 |
186 |
93 |
93 |
50 |
|
267 |
156 |
78 |
78 |
50 |
|
268 |
162 |
81 |
81 |
50 |
|
269 |
122 |
61 |
61 |
50 |
|
270 |
94 |
47 |
47 |
50 |
|
271 |
168 |
84 |
84 |
50 |
|
272 |
154 |
77 |
77 |
50 |
|
273 |
190 |
95 |
95 |
50 |
|
274 |
168 |
84 |
84 |
50 |
|
275 |
154 |
77 |
77 |
50 |
|
276 |
190 |
95 |
95 |
50 |
|
277 |
182 |
91 |
91 |
50 |
|
278 |
38 |
19 |
19 |
50 |
|
279 |
180 |
90 |
90 |
50 |
|
280 |
144 |
72 |
72 |
50 |
|
281 |
176 |
88 |
88 |
50 |
|
282 |
184 |
92 |
92 |
50 |
|
283 |
146 |
73 |
73 |
50 |
|
284 |
78 |
39 |
39 |
50 |
|
285 |
180 |
90 |
90 |
50 |
|
286 |
184 |
92 |
92 |
50 |
|
287 |
198 |
99 |
99 |
50 |
|
288 |
160 |
80 |
80 |
50 |
|
289 |
156 |
78 |
78 |
50 |
|
290 |
130 |
65 |
65 |
50 |
|
291 |
122 |
61 |
61 |
50 |
|
292 |
68 |
34 |
34 |
50 |
|
293 |
114 |
57 |
57 |
50 |
|
294 |
80 |
40 |
40 |
50 |
|
295 |
182 |
91 |
91 |
50 |
|
296 |
82 |
41 |
41 |
50 |
|
297 |
143 |
84 |
84 |
58.74125874 |
|
298 |
130 |
77 |
77 |
59.23076923 |
|
299 |
148 |
92 |
92 |
62.16216216 |
|
300 |
138 |
92 |
91 |
65.94202899 |
This appendix contains a table (Table 2) of the results of four hundred random jobs have been used. The optimization ratio of the proposed approach comparing with traditional R.R has been calculated.
Table 2. Quantitative comparison between traditional round robin and the proposed method
|
X Values |
The Proposed Method |
Traditional R. R |
|
1 |
14 |
14 |
|
2 |
84 |
168 |
|
3 |
11 |
11 |
|
4 |
58 |
58 |
|
5 |
91 |
183 |
|
6 |
73 |
73 |
|
7 |
62 |
154 |
|
8 |
65 |
65 |
|
9 |
55 |
139 |
|
10 |
84 |
168 |
|
11 |
59 |
143 |
|
12 |
61 |
61 |
|
13 |
32 |
32 |
|
14 |
18 |
65 |
|
15 |
46 |
93 |
|
16 |
2 |
49 |
|
17 |
54 |
54 |
|
18 |
67 |
134 |
|
19 |
63 |
63 |
|
20 |
32 |
32 |
|
21 |
27 |
74 |
|
22 |
38 |
85 |
|
23 |
16 |
16 |
|
24 |
47 |
94 |
|
25 |
14 |
14 |
|
26 |
84 |
168 |
|
27 |
11 |
11 |
|
28 |
58 |
58 |
|
29 |
9 |
98 |
|
30 |
88 |
177 |
|
31 |
76 |
165 |
|
32 |
4 |
4 |
|
33 |
68 |
137 |
|
34 |
57 |
57 |
|
35 |
18 |
18 |
|
36 |
56 |
125 |
|
37 |
74 |
149 |
|
38 |
16 |
16 |
|
39 |
15 |
15 |
|
40 |
11 |
11 |
|
41 |
48 |
122 |
|
42 |
13 |
87 |
|
43 |
74 |
148 |
|
44 |
24 |
98 |
|
45 |
18 |
18 |
|
46 |
13 |
13 |
|
47 |
12 |
12 |
|
48 |
19 |
38 |
|
49 |
54 |
54 |
|
50 |
67 |
134 |
|
51 |
63 |
63 |
|
52 |
32 |
32 |
|
53 |
14 |
14 |
|
54 |
84 |
168 |
|
55 |
11 |
11 |
|
56 |
58 |
58 |
|
57 |
91 |
183 |
|
58 |
73 |
73 |
|
59 |
62 |
154 |
|
60 |
65 |
65 |
|
61 |
55 |
139 |
|
62 |
84 |
168 |
|
63 |
84 |
143 |
|
64 |
61 |
61 |
|
65 |
27 |
74 |
|
66 |
38 |
85 |
|
67 |
16 |
16 |
|
68 |
47 |
94 |
|
69 |
91 |
138 |
|
70 |
73 |
73 |
|
71 |
62 |
154 |
|
72 |
65 |
65 |
|
73 |
76 |
174 |
|
74 |
57 |
57 |
|
75 |
56 |
154 |
|
76 |
97 |
195 |
|
77 |
41 |
118 |
|
78 |
1 |
1 |
|
79 |
77 |
154 |
|
80 |
77 |
130 |
|
81 |
77 |
154 |
|
82 |
64 |
141 |
|
83 |
34 |
111 |
|
84 |
41 |
41 |
|
85 |
99 |
199 |
|
86 |
48 |
48 |
|
87 |
35 |
135 |
|
88 |
59 |
59 |
|
89 |
27 |
74 |
|
90 |
38 |
85 |
|
91 |
16 |
16 |
|
92 |
47 |
94 |
|
93 |
91 |
183 |
|
94 |
73 |
73 |
|
95 |
62 |
154 |
|
96 |
65 |
65 |
|
97 |
55 |
139 |
|
98 |
84 |
168 |
|
99 |
59 |
143 |
|
100 |
61 |
61 |
|
101 |
32 |
32 |
|
102 |
18 |
65 |
|
103 |
46 |
93 |
|
104 |
2 |
49 |
|
105 |
76 |
174 |
|
106 |
57 |
57 |
|
107 |
56 |
154 |
|
108 |
97 |
195 |
|
109 |
77 |
154 |
|
110 |
64 |
141 |
|
111 |
34 |
111 |
|
112 |
41 |
41 |
|
113 |
6 |
101 |
|
114 |
19 |
114 |
|
115 |
95 |
190 |
|
116 |
41 |
136 |
|
117 |
77 |
154 |
|
118 |
72 |
149 |
|
119 |
39 |
39 |
|
120 |
28 |
28 |
|
121 |
91 |
182 |
|
122 |
65 |
156 |
|
123 |
26 |
117 |
|
124 |
81 |
172 |
|
125 |
21 |
21 |
|
126 |
78 |
159 |
|
127 |
80 |
161 |
|
128 |
76 |
76 |
|
129 |
48 |
122 |
|
130 |
13 |
87 |
|
131 |
74 |
148 |
|
132 |
24 |
98 |
|
133 |
18 |
18 |
|
134 |
13 |
13 |
|
135 |
12 |
12 |
|
136 |
19 |
38 |
|
137 |
67 |
134 |
|
138 |
45 |
45 |
|
139 |
47 |
114 |
|
140 |
50 |
50 |
|
141 |
41 |
131 |
|
142 |
48 |
138 |
|
143 |
90 |
180 |
|
144 |
63 |
63 |
|
145 |
17 |
109 |
|
146 |
42 |
134 |
|
147 |
16 |
16 |
|
148 |
92 |
184 |
|
149 |
74 |
167 |
|
150 |
43 |
136 |
|
151 |
92 |
186 |
|
152 |
55 |
55 |
|
153 |
90 |
180 |
|
154 |
17 |
107 |
|
155 |
65 |
65 |
|
156 |
88 |
88 |
|
157 |
38 |
138 |
|
158 |
99 |
199 |
|
159 |
25 |
125 |
|
160 |
38 |
138 |
|
161 |
1 |
1 |
|
162 |
36 |
75 |
|
163 |
39 |
78 |
|
164 |
33 |
72 |
|
165 |
7 |
62 |
|
166 |
24 |
79 |
|
167 |
54 |
109 |
|
168 |
52 |
52 |
|
169 |
66 |
66 |
|
170 |
78 |
157 |
|
171 |
52 |
131 |
|
172 |
34 |
113 |
|
173 |
29 |
105 |
|
174 |
76 |
152 |
|
175 |
38 |
114 |
|
176 |
46 |
46 |
|
177 |
54 |
152 |
|
178 |
98 |
196 |
|
179 |
1 |
99 |
|
180 |
6 |
6 |
|
181 |
26 |
118 |
|
182 |
68 |
160 |
|
183 |
92 |
148 |
|
184 |
10 |
102 |
|
185 |
32 |
132 |
|
186 |
99 |
199 |
|
187 |
53 |
153 |
|
188 |
43 |
143 |
|
189 |
79 |
178 |
|
190 |
99 |
198 |
|
191 |
47 |
146 |
|
192 |
57 |
156 |
|
193 |
69 |
139 |
|
194 |
68 |
68 |
|
195 |
55 |
125 |
|
196 |
31 |
101 |
|
197 |
58 |
123 |
|
198 |
6 |
71 |
|
199 |
37 |
102 |
|
200 |
64 |
129 |
|
201 |
67 |
162 |
|
202 |
32 |
127 |
|
203 |
95 |
190 |
|
204 |
4 |
99 |
|
205 |
93 |
186 |
|
206 |
80 |
173 |
|
207 |
78 |
171 |
|
208 |
51 |
144 |
|
209 |
22 |
100 |
|
210 |
62 |
140 |
|
211 |
78 |
156 |
|
212 |
63 |
63 |
|
213 |
46 |
127 |
|
214 |
70 |
151 |
|
215 |
81 |
162 |
|
216 |
10 |
91 |
|
217 |
5 |
93 |
|
218 |
14 |
102 |
|
219 |
87 |
175 |
|
220 |
46 |
134 |
|
221 |
82 |
82 |
|
222 |
18 |
116 |
|
223 |
28 |
28 |
|
224 |
97 |
195 |
|
225 |
33 |
94 |
|
226 |
30 |
30 |
|
227 |
1 |
62 |
|
228 |
61 |
122 |
|
229 |
27 |
74 |
|
230 |
38 |
85 |
|
231 |
16 |
16 |
|
232 |
47 |
94 |
|
233 |
55 |
139 |
|
234 |
84 |
168 |
|
235 |
59 |
143 |
|
236 |
61 |
61 |
|
237 |
32 |
32 |
|
238 |
18 |
65 |
|
239 |
46 |
93 |
|
240 |
2 |
49 |
|
241 |
76 |
174 |
|
242 |
57 |
57 |
|
243 |
56 |
154 |
|
244 |
97 |
195 |
|
245 |
77 |
154 |
|
246 |
64 |
141 |
|
247 |
34 |
111 |
|
248 |
41 |
41 |
|
249 |
40 |
136 |
|
250 |
22 |
118 |
|
251 |
95 |
191 |
|
252 |
32 |
128 |
|
253 |
6 |
101 |
|
254 |
19 |
114 |
|
255 |
95 |
190 |
|
256 |
41 |
136 |
|
257 |
55 |
139 |
|
258 |
84 |
168 |
|
259 |
59 |
143 |
|
260 |
61 |
61 |
|
261 |
32 |
32 |
|
262 |
18 |
65 |
|
263 |
46 |
93 |
|
264 |
2 |
49 |
|
265 |
76 |
174 |
|
266 |
57 |
57 |
|
267 |
56 |
154 |
|
268 |
97 |
195 |
|
269 |
9 |
98 |
|
270 |
88 |
177 |
|
271 |
76 |
165 |
|
272 |
4 |
4 |
|
273 |
77 |
154 |
|
274 |
64 |
141 |
|
275 |
34 |
111 |
|
276 |
41 |
41 |
|
277 |
40 |
136 |
|
278 |
22 |
118 |
|
279 |
95 |
191 |
|
280 |
32 |
128 |
|
281 |
6 |
101 |
|
282 |
19 |
114 |
|
283 |
95 |
190 |
|
284 |
41 |
136 |
|
285 |
91 |
182 |
|
286 |
65 |
156 |
|
287 |
26 |
117 |
|
288 |
81 |
172 |
|
289 |
2 |
20 |
|
290 |
4 |
17 |
|
291 |
4 |
16 |
|
292 |
19 |
38 |
|
293 |
41 |
131 |
|
294 |
48 |
138 |
|
295 |
90 |
180 |
|
296 |
63 |
63 |
|
297 |
38 |
110 |
|
298 |
72 |
144 |
|
299 |
10 |
82 |
|
300 |
36 |
108 |
|
301 |
31 |
119 |
|
302 |
88 |
176 |
|
303 |
45 |
133 |
|
304 |
22 |
22 |
|
305 |
17 |
109 |
|
306 |
42 |
134 |
|
307 |
16 |
16 |
|
308 |
92 |
184 |
|
309 |
74 |
167 |
|
310 |
43 |
136 |
|
311 |
92 |
185 |
|
312 |
55 |
55 |
|
313 |
67 |
140 |
|
314 |
73 |
146 |
|
315 |
26 |
99 |
|
316 |
59 |
132 |
|
317 |
20 |
20 |
|
318 |
86 |
173 |
|
319 |
23 |
110 |
|
320 |
52 |
139 |
|
321 |
20 |
65 |
|
322 |
44 |
89 |
|
323 |
45 |
45 |
|
324 |
36 |
36 |
|
325 |
1 |
1 |
|
326 |
36 |
75 |
|
327 |
39 |
78 |
|
328 |
33 |
72 |
|
329 |
60 |
150 |
|
330 |
90 |
180 |
|
331 |
37 |
37 |
|
332 |
65 |
155 |
|
333 |
89 |
179 |
|
334 |
64 |
64 |
|
335 |
82 |
172 |
|
336 |
62 |
152 |
|
337 |
66 |
66 |
|
338 |
78 |
157 |
|
339 |
52 |
131 |
|
340 |
34 |
113 |
|
341 |
26 |
118 |
|
342 |
68 |
160 |
|
343 |
92 |
184 |
|
344 |
10 |
102 |
|
345 |
32 |
132 |
|
346 |
99 |
199 |
|
347 |
53 |
153 |
|
348 |
43 |
143 |
|
349 |
79 |
178 |
|
350 |
99 |
198 |
|
351 |
47 |
146 |
|
352 |
57 |
156 |
|
353 |
42 |
122 |
|
354 |
80 |
160 |
|
355 |
12 |
92 |
|
356 |
33 |
33 |
|
357 |
22 |
100 |
|
358 |
62 |
140 |
|
359 |
78 |
156 |
|
360 |
63 |
63 |
|
361 |
19 |
84 |
|
362 |
8 |
73 |
|
363 |
65 |
130 |
|
364 |
20 |
85 |
|
365 |
5 |
93 |
|
366 |
14 |
102 |
|
367 |
87 |
175 |
|
368 |
46 |
134 |
|
369 |
33 |
94 |
|
370 |
30 |
30 |
|
371 |
1 |
62 |
|
372 |
61 |
122 |
|
373 |
21 |
55 |
|
374 |
34 |
68 |
|
375 |
15 |
49 |
|
376 |
7 |
7 |
|
377 |
11 |
68 |
|
378 |
13 |
70 |
|
379 |
27 |
84 |
|
380 |
57 |
114 |
|
381 |
82 |
182 |
|
382 |
35 |
135 |
|
383 |
47 |
147 |
|
384 |
99 |
199 |
|
385 |
31 |
71 |
|
386 |
19 |
19 |
|
387 |
40 |
80 |
|
388 |
15 |
55 |
|
389 |
51 |
105 |
|
390 |
48 |
102 |
|
391 |
53 |
107 |
|
392 |
44 |
44 |
|
393 |
91 |
182 |
|
394 |
45 |
136 |
|
395 |
23 |
114 |
|
396 |
14 |
105 |
|
397 |
36 |
36 |
|
398 |
32 |
73 |
|
399 |
41 |
82 |
|
400 |
16 |
16 |
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