Summary

36 density increase and percentage of schedulability with those of using

Sx

(Here,

Sx

stands for

Sx

^g and

g

= 2 instead of the original

Sx

in 4]). From our simulation result, we nd that the performance of

HSx

is the same as

Sx

^b. In the following, we therefore compare the performance of

Sx

,

Sx

³ (

Sx

^g using only bases 2 and 3),

Sx

¹⁰ (

Sx

^g using bases less or equal to 10),

Sx

¹⁰⁰ (

Sx

^g using bases less or equal to 100) and

Sx

^b.

Figure 3.8 shows the dierences of the total densities after task sets are specialized by

Sx

and by

Sx

^b. Figure 3.8(a) shows the average dierence for the task sets which are and Figure 3.8(b) shows the average dierence for all task sets schedulable by

Sx

^b. Figure 3.8(a) has decreasing values when the total utilization for the task set is greater than the schedulability bound of

Sr

. In other words, for those schedulable cases, the density dierences between the two get smaller as the system utilization increases. However, we

nd the improvement of using

Sx

^b for the task sets with higher original utilization is not as substantial as

Sr

^b. This is due to the requirement for the distance constraints to be integers after specialization.

Figure 3.9(a) shows the percentage of schedulable tasks using

Sx

and using

Sx

^b. Among those schedulable by

Sx

^b, Figure 3.9(b) shows the percentage of those also schedulable by

Sx

.

Figures 3.10, 3.11 and 3.12 show similar comparisons as in Figure 3.9 except we compare

Sx

^bwith

Sx

³,

Sx

¹⁰ and

Sx

¹⁰⁰ respectively. We can see for the task sets with an original density less than 90%,

Sx

³ can schedule about 90% of the tasks schedulable by

Sx

^b, and with original density less than 95%,

Sx

¹⁰ can schedule about 90% of tasks schedulable by

Sx

^b. For

Sx

¹⁰⁰ in Figure 3.12, it performs almost as good as

Sx

^b.

65 70 75 80 85 90 95 100 0

0.005 0.01 0.015 0.02 0.025 0.03 0.035

utilization

density difference after specialization, Sx - Sxb

2 tasks 3 tasks 4 tasks 5 tasks 6 tasks 7 tasks

(a)task sets schedulable by

Sx

65 70 75 80 85 90 95 100

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

utilization

density difference after specialization, Sx - Sxb

2 tasks 3 tasks 4 tasks 5 tasks 6 tasks 7 tasks

(b)task sets schedulable by

Sx

^b

Figure 3.8: Comparison of density after specialization between

Sx

and

Sx

^b

65 70 75 80 85 90 95 100

0 2 4 6 8 10 12 14 16 18 20

utilization

difference in schedulable percentages, Sxb - Sx

2 tasks 3 tasks 4 tasks 5 tasks 6 tasks 7 tasks

(a)

65 70 75 80 85 90 95 100

0 10 20 30 40 50 60 70 80 90 100

utilization

percentage of schedulable task sets, Sx / Sxb

2 tasks 3 tasks 4 tasks 5 tasks 6 tasks 7 tasks

(b)

Figure 3.9: Comparison of schedulable tasks between

Sx

and

Sx

^b

38

65 70 75 80 85 90 95 100

0 2 4 6 8 10 12 14 16 18 20

utilization

difference in schedulable percentages, Sx(<=3) - Sx

2 tasks 3 tasks 4 tasks 5 tasks 6 tasks 7 tasks

(a)

65 70 75 80 85 90 95 100

0 10 20 30 40 50 60 70 80 90 100

utilization

percentage of schedulable task sets, Sx(<=3) / Sxb

2 tasks 3 tasks 4 tasks 5 tasks 6 tasks 7 tasks

(b)

Figure 3.10: Comparison of schedulable tasks between

Sx

,

Sx

³ and

Sx

^b

65 70 75 80 85 90 95 100

0 2 4 6 8 10 12 14 16 18 20

utilization

difference in schedulable percentages, Sx(<=10) - Sx

2 tasks 3 tasks 4 tasks 5 tasks 6 tasks 7 tasks

(a)

65 70 75 80 85 90 95 100

0 10 20 30 40 50 60 70 80 90 100

utilization

percentage of schedulable task sets, Sx(<=10) / Sxb

2 tasks 3 tasks 4 tasks 5 tasks 6 tasks 7 tasks

(b)

Figure 3.11: Comparison of schedulable tasks between

Sx

,

Sx

¹⁰ and

Sx

^b

65 70 75 80 85 90 95 100 0

2 4 6 8 10 12 14 16 18 20

utilization

difference in schedulable percentages, Sx(<=100) - Sx

2 tasks 3 tasks 4 tasks 5 tasks 6 tasks 7 tasks

(a)

65 70 75 80 85 90 95 100

0 10 20 30 40 50 60 70 80 90 100

utilization

percentage of schedulable task sets, Sx(<=100) / Sxb

2 tasks 3 tasks 4 tasks 5 tasks 6 tasks 7 tasks

(b)

Figure 3.12: Comparison of schedulable tasks between

Sx

,

Sx

¹⁰⁰ and

Sx

^b

distance constraints, a generalized pinwheel scheduler

Sr

^g is presented. We have also shown the schedulability condition of

Sr

^g. Moreover, we present an optimal pinwheel algorithm,

Sr

^b, among those using the single-number reduction technique. It has also been shown that using

Sr

^b, which tries all possible bases, a system has a better chance of nding a good schedule than using

Sr

. We have also designed a heuristic algorithm

HSr

and found that

HSr

and

Sr

^b has almost the same performance in our simulation study. Therefore, we believe that

HSr

is the best practical algorithm for pinwheel scheduling using the single-number reduction technique. For systems requiring integer distance constraints, pinwheel scheduler

Sx

^g which is derived from

Sr

^g can be used. We also present

Sx

^b, which tries all possible bases using

Sx

^g, and prove that

Sx

^b is the optimal pinwheel algorithm for integer distance constraints using the single-number reduction technique. A polynomial-time heuristic algorithm

HSx

is then designed and its performance compared to

Sx

^b.

Using optimal pinwheel schedulers like

Sr

^b or

Sx

^b, most tasks will be scheduled with a shorter distance constraint than required by the applications. Therefore, tasks will be executed more frequently than originally requested. This may appear to be a waste of system resources since some other tasks may become unschedulable. However,

Sr

^b or

Sx

^b

40 makes task executions more predictable. With the increased predictability, most tasks can run more eciently since less time will be wasted in dealing with system uncertainty 25, 26].

We believe that system utilization will not degrade using the pinwheel scheduling approach.

Multiple-Node Pinwheel Scheduling

Real-time applications running on distributed systems often have timing constraints on tasks running on several processors. To design real-time systems with end-to-end perfor-mance requirements, we need to have algorithms to schedule and to coordinate tasks on dierent processor nodes. In this chapter, an end-to-end scheduling approach based on the pinwheel scheduling model is presented for distributed real-time systems. We present how tasks on dierent nodes may be transformed to have periods consisting of only harmonic numbers. With harmonic periods, on each node, the start and nish times of each task are the same relative to its ready time in each period. Phase alignment algorithms are then applied to adjust the phases between schedules of neighboring nodes so that the overall end-to-end delay is minimized. Using the proposed pinwheel model, schedules on dierent nodes are better synchronized but suer less delays. For many practical real-time systems, this approach provides a more predictable performance and a shorter end-to-end delay.

Algorithms for allocating resources and scheduling tasks are important to the success of real-time systems with end-to-end performance requirements 1, 42, 44]. A good algo-rithm will improve the schedulability, dependability and scalability of systems. Previous work on the problem has studied dierent algorithms when systems are using the

priority-41

42 driven approach. In this chapter, an end-to-end scheduling approach based on the pinwheel scheduling algorithms 3, 15] is presented for distributed real-time systems. Previous pin-wheel algorithms are all designed for a single node. We extend one of the most powerful pinwheel algorithms,

Sr

(refer to Section 2.3.2), to the

DSr

algorithm so that all nodes are processed at the same time 19]. We then produce the single-node schedule for each node after it is processed by

DSr

and consists of only harmonic distance constraints. After the schedules for individual nodes are produced, we then provide algorithms to adjust the phases (relative start time) between these nodes so that the overall end-to-end delay for transactions is minimized or reduced.

Using the pinwheel approach, schedules on dierent nodes are better synchronized and the end-to-end delays are more predictable. We believe that our work provides a useful and complete design methodology to many practical real-time systems that are statically structured and well-dened, such as those in automated factory, automobile and avionics systems 25, 26].

The rest of this chapter is organized as follows. Section 4.1 reviews the basic end-to-end scheduling model. Section 4.2 presents the distributed pinwheel scheduling algorithm

DSr

. We discuss its complexity and the schedulability condition bound. In Section 4.3, we present the algorithms that can be used to reduce the end-to-end delays by adjusting the phases of pinwheel schedules on dierent nodes. Section 4.4 contains a summary of this chapter.

在文檔中 Real-Time Systems Scheduling Using the Distance-Constrained Model (頁 49-55)