Math Problem Statement
Use the decreasing time algorithm to create a priority list for the digraph below, then schedule the project with two processors.
T1 (9)T2 (2)T3 (12)T4 (7)T5 (5)T6 (6)T7 (5)T8 (5)T9 (8)End[Graphs generated by this script: setBorder(60,30,60,30);initPicture(0,3,0,2); fontsize=14;fontstyle='none';marker="arrow";line([0,2],[1,1]);line([0,1],[1,1]);line([0,1],[1,0]);line([0,0],[1,1]);line([1,2],[2,2]);line([1,2],[2,1]);line([1,1],[2,2]);line([1,1],[2,1]);line([1,1],[2,0]);line([1,0],[2,1]);line([1,0],[2,0]);line([2,2],[3,1]);line([2,1],[3,1]);line([2,0],[3,1]);fontbackground='white';fontfill='blue';text([0,2],'T1 (9)','aboveleft');dot([0,2]);fontfill='blue';text([0,1],'T2 (2)','belowleft');dot([0,1]);fontfill='blue';text([0,0],'T3 (12)','belowleft');dot([0,0]);fontfill='blue';text([1,2],'T4 (7)','aboveleft');dot([1,2]);fontfill='blue';text([1,1],'T5 (5)','belowleft');dot([1,1]);fontfill='blue';text([1,0],'T6 (6)','belowleft');dot([1,0]);fontfill='blue';text([2,2],'T7 (5)','aboveright');dot([2,2]);fontfill='blue';text([2,1],'T8 (5)','belowright');dot([2,1]);fontfill='blue';text([2,0],'T9 (8)','belowright');dot([2,0]);fontfill='blue';text([3,1],'End','belowright');dot([3,1]);]
Task 5 is done by Select an answer Processor 1 Processor 2 Correct starting at time
Task 6 is done by Select an answer Processor 1 Processor 2 Correct starting at time
The finishing time for the schedule is
Solution
To solve this problem using the decreasing time algorithm with two processors, we will follow these steps:
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Create the Priority List Using Decreasing Time Algorithm:
- First, we list all tasks in descending order based on their duration.
- Then, we respect the task dependencies (as indicated by the graph) when scheduling.
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Assign Tasks to Processors and Calculate Starting and Finishing Times:
- We assign tasks to processors based on their availability and the priority list.
Step 1: Create the Priority List
The tasks are as follows:
- T3 (12 units)
- T1 (9 units)
- T9 (8 units)
- T4 (7 units)
- T6 (6 units)
- T5 (5 units)
- T7 (5 units)
- T8 (5 units)
- T2 (2 units)
Priority List in Decreasing Order:
- T3 (12)
- T1 (9)
- T9 (8)
- T4 (7)
- T6 (6)
- T5 (5)
- T7 (5)
- T8 (5)
- T2 (2)
Step 2: Assign Tasks to Processors
-
T3 (12 units):
- This is the highest priority task, so it is assigned to Processor 1 starting at time 0.
-
T1 (9 units):
- This task is assigned to Processor 2 starting at time 0.
-
T9 (8 units):
- Processor 1 will be available after T3 completes at time 12, so T9 can start on Processor 1 at time 12.
-
T4 (7 units):
- Processor 2 will be available after T1 completes at time 9, so T4 can start on Processor 2 at time 9.
-
T6 (6 units):
- Processor 2 will be available after T4 completes at time 16, so T6 can start on Processor 2 at time 16.
-
T5 (5 units):
- Processor 1 will be available after T9 completes at time 20, so T5 can start on Processor 1 at time 20.
-
T7 (5 units):
- Processor 2 will be available after T6 completes at time 22, so T7 can start on Processor 2 at time 22.
-
T8 (5 units):
- Processor 1 will be available after T5 completes at time 25, so T8 can start on Processor 1 at time 25.
-
T2 (2 units):
- Processor 2 will be available after T7 completes at time 27, so T2 can start on Processor 2 at time 27.
Summary of Scheduling
-
Processor 1:
- T3: Start at 0, End at 12
- T9: Start at 12, End at 20
- T5: Start at 20, End at 25
- T8: Start at 25, End at 30
-
Processor 2:
- T1: Start at 0, End at 9
- T4: Start at 9, End at 16
- T6: Start at 16, End at 22
- T7: Start at 22, End at 27
- T2: Start at 27, End at 29
Answers to the Specific Questions:
- Task 5 is done by: Processor 1, starting at time 20.
- Task 6 is done by: Processor 2, starting at time 16.
- The finishing time for the schedule is: 30 units.
Would you like more details on how this was calculated, or do you have any further questions?
Here are 8 related questions to expand your understanding:
- How would the schedule change if we added a third processor?
- What are some advantages of using the decreasing time algorithm in scheduling?
- How do dependencies between tasks impact the final schedule?
- Can this method be adapted for real-time task scheduling?
- What happens if two tasks with the same duration but different dependencies are next in line for scheduling?
- How does task preemption affect the scheduling strategy?
- What are potential challenges in scheduling when tasks have variable processing times?
- How would the schedule be affected if task T3 had a higher dependency?
Tip: In scheduling problems, consider both task duration and dependencies to ensure that the final schedule is efficient and meets all constraints.
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Math Problem Analysis
Mathematical Concepts
Scheduling Algorithms
Task Dependency Graphs
Processor Allocation
Formulas
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Theorems
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Suitable Grade Level
Advanced Undergraduate