10.3. Least laxity first (LLF) is a real-time scheduling algorithm for periodic tasks. Slack time, or laxity, is the amount of time between when a task would complete if it started now and its next deadline. This is the size of the available scheduling window. Laxity can be expressed as
Laxity = (deadline time) €“ (current time) €“ (processor time needed)
LLF selects the task with the minimum laxity to execute next. If two or more tasks have the same minimum laxity value, they are serviced on a FCFS basis.

a. Suppose a task currently has a laxity of t. By how long may the scheduler delay starting this task and still meet its deadline?
b. Suppose a task currently has a laxity of 0. What does this mean?
c. What does it mean if a task has negative laxity?
d. Consider a set of three periodic tasks with the execution profiles of Table 10.9a. Develop scheduling diagrams similar to those of Figure 10.5 for this set of tasks that compare rate monotonic, earliest-deadline first, and LLF. Assume preemption may occur at 5-ms intervals. Comment on the results.

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Process Arrival Time Execution Time Ending Deadline 20 40 10 A(2) 20 10 50 B(2) 50 100 15 50 C(2) 50 15 100 Process Arrival Time Execution Time Starting Deadline 100 30 60 20 20 40 20 20 70