The Time Restricted Problem deals with some learners who have time constraints for some reasons. In this example, our goal is to meet their requirements within this time constraint. We map the TLCS (Time Limited Candidate Selector) algorithm [28] to distribute time over the learning path. We find time constraints particularly important because time factors can play an important role in exam preparation.
4.7.1 Assumptions
We share the given time (requested by the learner) to the required courses (after we have applied the minimum weight branching algorithm to find his learning sequence). Two important parameters in this problem
are “learning curves＂ and “pre-requests＂ (which we discuss further in section 4.7.2 and section 4.7.4 respectively). In the Time Restricted Problem, we are given the following parameters:
z the time constraints z the difficulty levels z the learning curve and z the course relations
4.7.2 Learning curves
The concept of learning curve is that when learning a course, if we spend more time, the score should be higher, we use figure 23 to show this concept. The X-axis is time and the y-axis is performance the learner achieves. 100% is the highest possible score.
Figure 23: Learning curve
4.7.3 Example
Suppose we have eight courses (Table 5) depicted in the directed graph shown in Figure 24. Each course has an examination date. The examination dates for each course are listed in the table. We assume that the learning curve of each course is one month (meaning that the learner must spend one or more months studying the course in order to pass). The learner starts with calculus (course1), and the given time constraint is the examination due date. Our goal is to reach (pass exams) course 8 given the time constraints distributed across the learning path and the starting time is in January 1.
Table 5: The courseware unit Course1 Calculus(exam in April)
Course2 Linear Algebra(exam in March)
Course3 Discrete Mathematics(exam in April) Course4 Differential Equation(exam in June) Course5 Probability Theory(Exam in June) Course6 Vector Analysis(exam in July) Course7 Concrete Mathematics(exam in July) Course8 Pattern Recognition(exam in November)
Figure 24: Relations of the courseware unit
First, we apply the Minimum weight branching algorithm to the directed graph (in Figure 24) to get learning path in Figure 25。
Figure 25: Find the branching
Our result is four learning paths:
z path 1:Course1ÆCourse4ÆCourse7ÆCourse8 z Path 2:Course2ÆCourse4ÆCourse7ÆCourse8 z Path 3:Course3ÆCourse6ÆCourse8
z Path 4:Course5ÆCourse8
Because the learner begins at course 1 (calculus), his learning path is path 1(Course1ÆCourse4ÆCourse7ÆCourse8). We map the TLCS algorithm to sequence the four courses.
1. Sequence the courses in list E.
2. If no courses in list E are late, stop, otherwise, identify the first late course, k.
3. Identify the latest position in the list in which the course k would not be late. Place course k in that position if it does not make any of the courses before k late, otherwise place it in late List L. Revise the sequence in list E and return to step 2.
According to TLCS algorithm, in step 1, the sequence in E is path 1(course1->course4->course7->course8). In step 2, we check if there is any late course (e.g. course 1 exam is in April and we need 1 month to study, therefore the time is enough). There in no course that is late.
At this point, we distribute our time constraint to these four courses.
First we distribute the whole of January to course 1 (we assume the learning curve requires one month at least) and test in April. We then share 1 month (whole of February) to course 4 and test in June, and then share 1 month (whole of March) to course 7 and test in July, and finally, share 1 month (whole of April) to course 8 and test in November. Under this arrangement, the learner can prepare his course sufficiently before tests.
4.7.4 Pre-requests
In time restriction problem, we discussed time sharing problem.
However, in some cases, we might encounter “pre-request＂ courses. The concept of a “pre-request＂ course states that when a learner completes a particular course, the learning curve will change. Courses that apply
this concept are called “pre-request＂ courses.
Figure 26 shows four learning curves, we note that after completing a pre-request course, the learning curve changes from learning curve 1 to learning curve 2 and so on.
Figure 26: Score percentage Example
Suppose the learning curve is typically 2 months for each course and the pre-request course is course 3. After finishing course 3, learning curve for each course becomes 1 month under the “pre-request＂ concept.
Suppose the learner begins at course 1 and the time constraint is 7 months. The directed graph for this is depicted in Figure 27. We approach this problem using two methods.
Figure 27: course relation
Method 1:
In this method, under normal conditions, we attempt to find the learning path.
First, we use the minimum weight branching algorithm to obtain the learning path in figure 28.
Figure 28 The learning paths are as follow:
path 1: Course1ÆCourse4ÆCourse7ÆCourse8 Path 2: Course2ÆCourse4ÆCourse7ÆCourse8 Path 3: Course3ÆCourse6ÆCourse8
Path 4: Course5ÆCourse8
Since the learner starts from course 1, we use path 1 and distribute the 7 months to these courses (Course1ÆCourse4ÆCourse7ÆCourse8), noting that we need a minimum of 2 months per course:
z course 1:2 months z course 4:2 months z course 7:2 months z course 8: 2 months
We note that the total is 8 months, thus, over our time constraint limitation of 7 months. We find that, under normal conditions, this method does not achieve our goal.
In the next method, we attempt to find the learning path in the directed graph that contains “pre-request＂ courses.
Method 2:
We suppose that course 3 is a pre-request course. If the learner starts from course one, the next logical course will be the pre-request course (i.e. course 3). From there on, we continue to follow the learning path in Figure 28 to course 8. Thus, our path is (course 1 -> course 3 -> course 6 -> course 8). We show the time distribution below:
z course 1: 2 months z course 3: 2 months
z course 6:1 month (it becomes one our because of pre-request course 3)
z course 8: 1 month
In this method, our total months spent is 6 months – achieving our goal within the time constraint of 7 months.
The results are shown below in figure 29.
Figure 29: Result learning path
Having demonstrated the above two-methods, we can conclude that if a directed graph contains a pre-request course, method one is not suitable, in other words, using our original algorithm. So, we modify our algorithm in method 2 (by changing the learning path to suit the pre-request course)
to suit the pre-request course.
Summary
In this chapter, we discussed our model and how it maps the logical view to the physical view. First, we obtained a directed graph containing difficulty levels, time constraints, capability indicators, course relations and later, learning curves. Using the minimum weight branching algorithm, we were able to find the learning path of the user that took these parameters into account and modeled them in the physical environment.
We demonstrated the applicability of this approach through several examples.
In chapter 5 we will demonstrate the applicability of our model into some physical networks such as the ARPA network, the pacific basin network and so forth.
Chapter 5
Illustrative examples and simulation results
In this chapter, we use some examples to see the course access reliability from the logical to the physical view. We demonstrate the applicability of this in four different physical networks: The ARPA network, The Pacific Basin Network, the TANet network and the three-Cube Network. In this chapter, we use the logical view that depicted in the figure 24 as an example and applied it into the physical network mentioned above. Figure 25 is redrawn as Figure 30, and map the courseware unit into the physical network, compute the difficulties of each learning path and their successful probability in the physical network.
Figure 30: Learning paths
All the possible Paths for each course in Figure 30:
Path for course 1: Course1ÆCourse4ÆCourse7ÆCourse8 Path for course 2: Course2ÆCourse4ÆCourse7ÆCourse8 Path for course 3: Course3ÆCourse6ÆCourse8
Path for course 4: Course7ÆCourse8 Path for course 5: Course5ÆCourse8 Path for course 6: Course6ÆCourse8 Path for course 7: Course7ÆCourse8
ARPA network:
Consider the physical network topology in figure 31 which consists of 21 nodes, the user is in node 11; there are 26 links, and other data storage locations.
Figure 31: ARPA network
The Allocation tree of ARPA network is depicted below:
Figure 32: Allocation of ARPA network
We wish to determine the probability that a user can finish a course successfully in the ARPA network environment.
From figure 32, suppose we run simulations from all courses, for example,
simulation 1 starts at course one, simulation 2 starts at course two and so on. Our results are as follows:
Simulation 1
z Simulation 1 needs to take another 3 courses (c4, c7 and c8 in figure 32 and Figure 30)
z the difficulty is 9(3+3+3)
z Its allocation site is 9, 12,14,7 in APRA network (figure 31).
Simulation 2
z Simulation 2 needs to take another 3 courses (c4,c7,c8 in figure 32 and Figure 30)
z the difficulty is 11(5+3+3),
z Its allocation site is site 9, 12,14,7 in APRA network (figure 31).
Simulation 3
z Simulation 3 needs to take another 2 courses (c 6,c8 in figure 32 and Figure 30)
z and the difficulty is 4(1+3),
z Its allocation site is site 9, 12, 14 in APRA network (figure 31).
Simulation 4
z simulation 4 needs to take another 2 course(c 7,c8 in figure 32 and Figure 30)
z Its allocation site is site 9, 12 in APRA network (figure 31).
Simulation 6
z simulation 6 needs to take another 1 course(c 8, in figure 32 and Figure 30)
z and the difficulty is 3,
z Its allocation site is site 9, 12(figure 31).
Simulation 7
z simulation 7 : it need to take another 1 course(c 8, in figure 32 and Figure 30)
z and the difficulty is 3,
z Its allocation site is site 9, 12 in APRA network (figure 31).
Table 6 shows the results (Success ratio) of applying conditional Multi-access Probability to the physical APRA network from course 1 to course 8.
Table 6: Simulation result of APRA network
Course C1 C2 C3 C4
Success ratio
0.95 0.9301 0.9116684 0.8967487
Course C5 C6 C7 C8
Success ratio
0.88456843 0.8164889 0.756879874 0.7148863
Pacific Basin network：
Consider the network topology in figure 33 which consists of 19 nodes, the user is in node 7; there are 26 links, and other data storage locations.
I
Figure 33: Pacific Basin network
In this example, we need to allocate 8 media. The total access reliability uses conditional Multi-access Probability. The tree of the physical network is displayed below in Figure 34
Figure 34: Allocation tree for Pacific Basin network
We wish to determine the probability that a user can finish a course successfully in the Pacific Basin network environment.
From figure 34, suppose we run simulations from all courses, for example, simulation 1 starts at course one, simulation 2 starts at course two and so on. Our results are as follows:
Simulation 1
z Simulation 1 needs to take another 3 courses (c4, c7 and c8 in figure 34 and Figure 30)
z the difficulty is 9(3+3+3)
z Its allocation site is 6,8,9,3 in the physical network (figure 34).
Simulation 2
z Simulation 2 needs to take another 3 courses (c4,c7,c8 in figure 34 and Figure 30)
z the difficulty is 11(5+3+3),
z Its allocation site is site 6,8,9,3 in the physical network (figure 34).
Simulation 3
z Simulation 3 needs to take another 2 courses (c 6,c8 in figure 34 and Figure 30)
z and the difficulty is 4(1+3),
z Its allocation site is site 6, 8, 9 in the physical network (figure 34).
Simulation 4
z simulation 4 needs to take another 2 course(c 7,c8 in figure 34 and Figure 30)
z and the difficulty is 2,
z Its allocation site is site 6, 8 in the physical network (figure 34).
Simulation 6
z simulation 6 needs to take another 1 course(c 8, in figure 34 and Figure 30)
z and the difficulty is 3,
z Its allocation site is site 6, 8(figure 34).
Simulation 7
z simulation 7 : it need to take another 1 course(c 8, in figure 34 and Figure 30)
z and the difficulty is 3,
z Its allocation site is site 6, 8 in the physical network (figure 34).
Table 7 shows the results (Success ratio) of applying conditional Multi-access Probability to the physical Pacific Basin network from course 1 to course 8.
Table 7: simulation result of Pacific Basin network
Course C1 C2 C3 C4
Success ratio
0.90 0.83474767 0.801546323 0.79654851
Course C5 C6 C7 C8
Success ratio
0.754684132 0.73546454 0.731544842 0.71488512
TANet topology:
Consider the physical network topology in figure 35 which consists of 9 nodes, the user is in node 4; there are 14 links, and other data storage locations.
Figure 35: TANet topology
Figure 36: Allocation of TANet
We wish to determine the probability that a user can finish a course successfully in the TANet network environment.
From figure 36, suppose we run simulations from all courses, for example, simulation 1 starts at course one, simulation 2 starts at course two and so on. Our results are as follows:
Simulation 1
z Simulation 1 needs to take another 3 courses (c4, c7 and c8 in figure 36 and Figure 30)
z the difficulty is 9(3+3+3)
z Its allocation site is 2,1,5,6 in the physical network (figure 36).
Simulation 2
z Simulation 2 needs to take another 3 courses (c4,c7,c8 in figure 36 and Figure 30)
z the difficulty is 11(5+3+3),
z Its allocation site is site 2,1,5,6 in the physical network (figure 36).
Simulation 3
z Simulation 3 needs to take another 2 courses (c 6,c8 in figure 36 and Figure 30)
z
z and the difficulty is 4(1+3),
z Its allocation site is site 2, 1, 5 in the physical network (figure 36).
Simulation 4
z simulation 4 needs to take another 2 course(c 7,c8 in figure 36 and Figure 30)
z Its allocation site is site 2, 1 in the physical network (figure 36).
Simulation 6
z simulation 6 needs to take another 1 course(c 8, in figure 36 and Figure 30)
z and the difficulty is 3,
z Its allocation site is site 2, 1(figure 36).
Simulation 7
z simulation 7 : it need to take another 1 course(c 8, in figure 36 and Figure 30)
z and the difficulty is 3,
z Its allocation site is site 2, 1 in the physical network (figure 36).
Table 8 shows the results (Success ratio) of applying conditional Multi-access Probability to the physical TANet network from course 1 to course 8.
Table 8: Simulation of TANet
Course C1 C2 C3 C4
Success ratio
0.90 0.85456346 0.822345432 0.79345435
Course C5 C6 C7 C8
Success ratio
0.78604895 0.73104895 0.729867474 0.71204576
3-Cube network:
Consider the physical network topology in figure 37 which consists of 8 nodes, the user is in node 1; there are 12 links, and other data storage locations.
Figure 37: 3-Cube network
Figure 38: Allocation of 3-cube network
We wish to determine the probability that a user can finish a course successfully in the 3-Cube network environment.
From figure 38, suppose we run simulations from all courses, for example, simulation 1 starts at course one, simulation 2 starts at course two and so on. Our results are as follows:
Simulation 1
z Simulation 1 needs to take another 3 courses (c4, c7 and c8 in figure 38 and Figure 30)
z the difficulty is 9(3+3+3)
z Its allocation site is 2,3,5,6 in the physical network (figure 38).
Simulation 2
z Simulation 2 needs to take another 3 courses (c4,c7,c8 in figure 38 and Figure 30)
z the difficulty is 11(5+3+3),
z Its allocation site is site 2,3,5,6 in the physical network (figure 38).
Simulation 3
z Simulation 3 needs to take another 2 courses (c 6,c8 in figure 38 and Figure 30)
z
z and the difficulty is 4(1+3),
z Its allocation site is site 2, 3, 5 in the physical network (figure 38).
Simulation 4
z simulation 4 needs to take another 2 course(c 7,c8 in figure 38 and Figure 30)
z Its allocation site is site 2, 3 in the physical network (figure 38).
Simulation 6
z simulation 6 needs to take another 1 course(c 8, in figure 38 and Figure 30)
z and the difficulty is 3,
z Its allocation site is site 2, 3(figure 38).
Simulation 7
z simulation 7 : it need to take another 1 course(c 8, in figure 38 and Figure 30)
z and the difficulty is 3,
z Its allocation site is site 2, 3 in the physical network (figure 38).
Table 9 shows the results (Success ratio) of applying conditional Multi-access Probability to the physical 3-cube network from course 1 to course 8.
Table 9: simulation of 3-cube network
Course C1 C2 C3 C4
Success ratio
0.90 0.87349589 0.850935043 0.85012092
Course C5 C6 C7 C8
Success ratio
0.780940885 0.83049928 0.782349023 0.77239490
Summary
In this chapter we demonstrated the applicability of our approach from the logical view to the physical view using real-life physical network examples such as the ARPA network, TANet, the pacific basin network and the 3-cube network. We used the data allocation scheme to distribute courseware unit on the physical network and CMP to calculate the success ratio and reliability of the physical network environment.
Chapter 6
Conclusion and future work
Here we summarize the issue we have mentioned in chapter 1. In the beginning, we mentioned mapping the minimum-branching learning model to e-learning for wireless environments, formulating a course planning algorithm, and enhancing this model with the addition of a time constraint.
First, we mapped Multi-access Probability and moving patterns to e-learning. Second, we used branching to solve course planning problem.
Third, we used the time restricted algorithm to solve time restricted problem and enhance the model. Limitations for this research are that statistic data and physical network parameters must be ready before these methods can be used to solve them.
Future Work
In pre-request courses, there are many types of functions we can use, the condition we use in our example may not be suitable for real life, in future, and we may use more realistic functions to match the experiment.
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