Summary of Results

Our research is based on a pinwheel scheduler

Sr

. For distance-constrained task sets with real-number distance constraints and execution times, the scheduler

Sr

transforms the distance constraints into harmonic numbers with a base of 2. We have designed several pinwheel scheduling algorithms based on

Sr

for a single node and multiple nodes. We also studied how to practically use the pinwheel scheduling algorithms in a practical system environment. The pinwheel solutions can be used not only in the distance-constrained model but also the periodic task model. By taking advantage of the harmonic-period property in pinwheel scheduling, the eective schedule can be generated and stored in polynomial time and space. Therefore, a exible time-driven scheduling is possible and the schedule optimization can be done on-line. This provides a new scheme for the real-time system design.

In Chapter 3, we present several single-node pinwheel schedulers. We rst study the

Sr

^g scheduler , which is a version of

Sr

using any base

g

, and its schedulability condition.

Then, the

Sr

^b scheduler is derived by trying all possible bases using

Sr

^g. The schedulability condition of

Sr

^b is presented and its optimality in the class of schedulers using the single-number reduction technique is proved. Since the time complexity of

Sr

^b is exponential to the number of tasks, we also study the performance of

Sr

^b by simulation and compare it with a near-optimal heuristic algorithm

HSr

. We then present pinwheel schedulers

Sx

^g

and

Sx

^b, for handling systems with integer distance constraints. The optimality of

Sx

^b

is also proved and its performance studied. We also compare

Sx

^b with a near-optimal heuristic algorithm

HSx

, which transforms the distance constraints into integer multiples.

The various pinwheel schedulers studied in this dissertation are summarized in Table 6.1.

Figure 6.1 shows the the relationship among

Sx

,

Sr

,

Sx

^b, and

Sr

^b in term of the set of all distance-constrained task sets schedulable by each one of them. The proofs of the relationship can be found in Appendix D. In practice,

HSr

or

HSx

should be used to do the pinwheel transformation because it is near-optimal and can be done in polynomial time.

Table 6.1: Various pinwheel schedulers

distance constraint

g

= 2

g

^2 optimal

g

near-optimal heuristic

r

is an integer

Sx Sx

^g

Sx

^b

HSx

r

is a real number

Sr Sr

^g

Sr

^b

HSr

a = {(1,2),(1,3)}

b = {(1,2.1),(2.1,6)}

c = {(1,1.9),(1,4)}

d = {(1,2),(2.2,6.1)}

e = {(1,1.9),(2.1,6)}

f = {(1,2),(1.1,3)}

: Jitterless

C_X: the set of all DCTS’s schedulable by Scheduler x

C_Sr^b

C_Sx C_Sx^b a.

b.

e.

c. d.

f.

C_Sr

Figure 6.1: The relationships among

C

^Sx

C

^Sr

C

^Sxb and

C

^Srb.

80 We present the multiple-node pinwheel scheduling in Chapter 4 for the real-time sys-tems with multiple processor nodes and end-to-end timing constraints. To design real-time systems with end-to-end timing constraints, we need to schedule and coordinate tasks on dierent nodes. We extend

Sr

to the

DSr

algorithm so that the distance constraints on all nodes are transformed at the same time into harmonic values. After the schedules for indi-vidual nodes are produced, we provide algorithms to adjust the phases (relative start time) between each pair of consecutive nodes so that the overall end-to-end delay is minimized or reduced. Using the distributed pinwheel approach, schedules on dierent nodes are better synchronized and the end-to-end delays are more predictable. Our work provides a useful and complete design methodology to many practical real-time systems that are statically structured and well-dened.

In Chapter 5, we study how to apply the pinwheel scheduling algorithms in a time-driven system. We rst design a pinwheel schedule constructor

SSr

to generate the eective pinwheel schedule (the state time and nish time of each task) in polynomial time. Base on the eective schedule we can generate the complete pinwheel schedule (with preemptions) o-line (or even on-line) so that the time-driven approach can be applied. To eciently and economically schedule the tasks on-line, we design several time-driven on-line pinwheel schedulers (TOPS's) with dierent complexities according to dierent system requirements.

We also address dierent implementation issues and how to modify the TOPS's to solve them.

In summary, we have investigated the pinwheel scheduling problems in a single node and multiple nodes. We show that the pinwheel schedulers can achieve both good predictabil-ity and schedulabilpredictabil-ity. The comparisons of dierent schedulers in periodic and distance-constrained task model is shown in Table 6.2 21].

EDF

has been shown to be optimal in terms of the schedulability condition. However, due to its dynamic scheduling nature, its behavior is not as predictable as the other two schedulers. When systems are heavily overloaded, systems become unstable and most of the tasks will have insucient time to execute and thus miss their deadlines.

EDF

also has high jitters since the schedules are

Table 6.2: Comparison of Schedulers

Scheduler

EDF RM Sr

^b

schedulability condition 1

n

(2^1=n;1)

n

(2^1=n;1)

jitter high low^y none

end-to-end delay long short shortest

resource reservation dicult easy^y easy worst case blocking time large large small transient overload unstable stable stable

yfor high-priority tasks

highly dynamic with no control for regularity.

RM

and

Sr

^b have the same schedulability condition. However, assuming the execution time is constant, pinwheel schedules are jitter-less 32] and can be generated o-line (or even on-line) in polynomial time 19]. Therefore it provides the best predictability. In other words, it is easy to predict when some specic resources may be needed by a task using

Sr

^b while

RM

can only predict resource usages for high-priority tasks. Therefore, resource reservation is easier to do in

Sr

^b than in

RM

. In multiple-node systems, pinwheel schedules in dierent nodes can be properly aligned to provide shorter end-to-end delays 19]. In comparison, the end-to-end delay for tasks using

RM

may be more dicult to adjust 42]. In practice, since a pinwheel eective schedule can be generated in polynomial time, the schedule optimization can be done in a more dynamic fashion and the scheduling can be exible and adaptive to the system environment change.

Therefore, we believe our work creates a new scheme for a better real-time system design.

在文檔中 Real-Time Systems Scheduling Using the Distance-Constrained Model (頁 91-94)