Pinwheel Schedulers for Integer Distance Constraints

task sets with an original density less than 0.9,

Sr

³ can schedule about 90% of the tasks schedulable by

Sr

^b, and with original density less than 0.95,

Sr

¹⁰ can schedule about 92% of tasks schedulable by

Sr

^b. For

Sr

¹⁰⁰ in Figure 3.6, it performs almost as good as

Sr

^b. In other words, trying any base larger than 100 will not improve the schedulable percentage signicantly at all.

32

4.5 5 5.5 6 6.5 7 7.5 8 8.5 9

1 1.1 1.2 1.3 1.4 1.5 1.6

r

density function

points checked in Sxg(g=2)

Figure 3.7: Density function of

Sr

^g for Example 2.1

For all these reasons, we design a new scheduler

Sx

^g base on

Sr

^g.

Sx

^g transforms the distance constraints of all tasks to multiples of integer

x

using any base

g

. The execution times and distance constraints used in

Sx

^g can be real numbers. Moreover, the optimal schedule can be found in polynomial time.

Sx

4] tests all the integers in (

c

¹

=

2

c

¹] (starting from the largest one) until it nds a schedulable one. It thus takes exponential time to nd the integer with the minimum density increase. However, as will be shown in Lemma 3.3, only a subset of these numbers needs to be tested to nd the optimal

x

with the minimum total density increase. Moreover, since we may use any base

g

to transform the distance constraints,

x

can only be in the range of

c

¹

=g < x

c

¹.

Lemma 3.3

If

k

^g_v^;1

< r < r

⁰

k

_gv, for some

v

, 1

v

u

, then ^g

T

⁽

^r

⁾

^>

^g

T

⁽

^r

^0).

The proof of Lemma 3.3 is given in Appendix B. Lemma 3.3 shows that, for each

v

, the density function ^g

T

in the domain (

k

^g_v^;1

k

_gv] is monotonically decreasing. Therefore, for each

v

, only the largest integer less than

k

_gvis tested when searching for the integer with the minimum total density increase. This can be done in polynomial time. Figure 3.7 shows the density function ^g

T

for Example 2.1. Only two integers, ^f5 9^g, need to be tested to

nd the optimal solution.

Sx

^g applies

Sr

^g to nd all

k

_gv's and ^g

T

⁽

^k

^gv)'s. Among the largest integers less than

k

_gv for each

v

,

Sx

^g transforms the distance constraints with respect to ^f

x

^g, where

x

is the integer with the minimum ^g

T

value. According to Lemma A.1 in Appendix A, the ^g

T

values can be easily computed.

Sx

^g works as follows.

Algorithm Sx

^g

Input: DCTS

T

=^f

T

i = (

e

i

c

i)^j 1

i

n

^g, where

c

i

c

j ⁸

i < j



g

is the specialization base.

Output:

x

, ^g

T

⁽

^x

^{), and}

T

⁰=^f(

e

i

b

i)^j 1

i

n

^g, where

b

i^j

b

j ⁸

i < j

.

1 Apply

Sr

^g to get all

k

_gv's and ^g

T

⁽

^k

^{gv)'s, for 1}

^v

^u

^.

2 Let (

x

^g1

x

^g2

::: x

^g_j) be the resulting sequence of (^b

k

^{g1c b}

k

^g2c

:::

^b

k

_gu^c)

3 with duplicates and

x

^g_i

k

^g0 =

c

¹

=g

removed

4

i

=

j



5

for v

=

u downto

1

do

6

if

(

x

^g_i =

k

_gv)

7 ^g

T

⁽

^x

^{gi) = g}

T

⁽

^k

^gv)

8

i

=

i

^;1

9

elseif

(

x

^g_i

> k

_gv)

10 ^g

T

⁽

^x

^{gi) = kv +1gxgi g}

T

⁽

^k

^gv+1) // See Lemma A.1, where



(



_x^gg

i

) = 0.

11

i

=

i

^;1

12

endif

13 nd

x

such that ^g

T

⁽

^x

^{) = minx2fxg1
xg2
:::
xgjgg}

T

⁽

^x

^)

14

for i

= 1

to n do b

i=

x

g

^{blogg(ci=x )c}

15 output

x

^g

T

⁽

^x

^{) and}

T

⁰

:

Theorem 3.5

The time complexity of specialization operation

Sx

^g is

O

(

n

log

n

).

Proof: It is easy to see that all steps in Algorithm

Sx

^g can be done in

O

(

n

) time, except step 1 (applying

Sr

^g), which can be done in

O

(

n

log

n

). ²

3.3.2 Optimal Scheduler for Integer Constraints

Similar to

Sr

^b,

Sx

^b tests all possible bases to nd the optimal one such that the total density increase after specialization is minimized.

Sx

^b works as follows.

34

Algorithm Sx

^b

Input: DCTS

T

=^f

T

i = (

e

i

c

i)^j 1

i

n

^g, where

c

i

c

j ⁸

i < j

. Output:

r

, ^g

T

⁽

^r

^{), and}

T

⁰=^f(

e

i

b

i)^j 1

i

n

^g, where

b

i^j

b

j ⁸

i < j

.

1

foreach

base

g

^2f2 3 5 6 7

:::

^b

c

n

=c

^1cg

do

^f

2 apply

Sx

^g using the base

g



3

if

(^g

T

⁽

^r

) is the minimum ever found),

g

=

g



4 g

5 apply

Sx

^g using the base

g



6 output

g

x

^g

T

⁽

^x

^{) and}

T

⁰

:

The following theorem proves the optimality of

Sx

^b.

Theorem 3.6

Given a distance-constrained task set

T

, if

Sx

^b can not
nd a schedule of integer distance constraints for

T

then no scheduler based on the pinwheel-problem with single-number reduction can
nd a schedule of integer distance constraints.

Proof: We prove this theorem by contradiction. Assume some pinwheel scheduler nds a schedule for distance constraint multiset

C

= ^f

c

i^g using single-number reduction with respect to^f

x

^gwhile

Sx

^bcan not nd one. W.l.o.g., suppose the algorithm transforms

C

into a set of distance constraints consist of solely multiples

B

=^f

b

i^j

b

i =

xg

^j for some integer

j

^g, for an integer

x

²(

c

¹

=g c

¹] and an integer base

g

such that

b

i

c

i and

gb

i

> c

i.

Now we can use

Sx

^g with respect to ^f

x

^g to transform

C

into

B

⁰=^f

b

⁰_i^g, so that

b

⁰_i

c

i

and

gb

⁰_i

> c

i. From the denitions of

B

and

B

⁰, we know that both consist of solely multiples. Therefore,

B

=

B

⁰, i.e.

B

is the transformed multiset for

C

using

Sx

^g with respect to^f

x

^g.

Since

Sx

^bsearches through all bases (including

g

),

B

⁰will be found using

Sx

^b. Therefore,

Sx

^b can nd a schedule for

C

. This contradicts our assumption. ² According to Theorem 3.4, the schedulability condition of

Sr

^b is the same as

Sr

. In 4], the schedulability condition for

Sx

is shown to be at least 0

:

65. That is, if the utilization of a task set

T

is less than 0

:

65,

Sx

is guaranteed to nd a schedule meeting the distance

constraints for

T

. However, the bound is not tight and the proof is very complicated. We conjecture that if there is a tight bound for

Sx

, it is also the bound for

Sx

^b. Since

Sx

^b

is a special case of

Sr

^b, the schedulability condition for

Sx

^b cannot be better than the schedulability condition for

Sr

^b, which is

n

(2^1=n;1) for a system of

n

tasks, or ln2 = 0

:

69 when

n

is large. On the other hand,

Sx

^b is a general case of

Sx

, so the schedulability condition for

Sx

^b is at least as high as the schedulability condition for

Sx

. Therefore, the bound of the schedulability condition for

Sx

^b is somewhere in 0

:

65 ln2]. At this time, we do not have the exact tight bound yet. Even though we don't have simple schedulability condition for

Sx

^g, the schedulability still can be checked in polynomial time by running

Sx

^g.

3.3.3 Heuristic Scheduler HSx

Similar to

Sr

^b,

Sx

^b is of exponential-time complexity. We have designed a near-optimal heuristic scheduler

HSx

for

Sx

^b, so that the schedulability of most task sets can be checked in polynomial time.

HSx

applies

HSr

to get the sub-optimal base

g

⁰ for task set

T

, then applies

Sx

^g using the base

g

⁰ to schedule the task set. If ^g

T

⁰⁽

^x

^)
1,

^x

is the integer with the minimum total density increase using base

g

⁰, the task set

T

is schedulable.

HSx

works as follows.

Algorithm HSx

Input: DCTS

T

=^f

T

i = (

e

i

c

i)^j 1

i

n

^g, where ⁸

i < j c

i

c

j. Output:

x

, ^g

T

⁰⁽

^x

^{), and}

T

⁰ =^f(

e

i

b

i)^j 1

i

n

^g, where ⁸

i < j b

i^j

b

j.

1 apply

HSr

to nd the suboptimal base

g

⁰

2 apply

Sx

^g using the base

g

⁰

3 output

g

⁰

x

^g

T

⁰⁽

^x

^{) and}

T

⁰

:

The time complexity of

HSx

is

O

(

n

²log(

c

n

=c

¹)) as

HSr

. Simulation has been conducted to compare the performance of

Sx

^b and

HSx

. In the simulation, the number of tasks ranges from 2 to 20 and the original density ranges from 65% to 100%. We randomly generated 10000 task sets (uniformly distributed) for each simulation and compared the

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.

在文檔中 Real-Time Systems Scheduling Using the Distance-Constrained Model (頁 44-49)