Final Thoughts

84 while still keep fairness or user satisfaction among programs. For example, for the CPU intensive tasks, they need larger time slices for I/O intensive tasks, the time slice can be smaller. Using our TOPS's, there is no context switching for real-time tasks until a timer event is raised. The time slice is between a task is run or resumed until it is preempted or nishes. Moreover, the size of time slice can also be changed dynamically for non-real-time tasks.

Mixed Scheduling Policies:

We can also reserve CPU bandwidth for dierent scheduling policies so that mixed scheduling policy is possible. For individual policies, the schedulability is tested using the allocated utilization. For example, if the

EDF

scheduling policy is allocated with execution time 1 and distance constraint 2, the tasks using

EDF

are schedulable if the utilization is less than 0.5. Each task can only execute in the time reserved for its policy. The overall schedulability test can be done by testing individual schedula-bility of each policy with its allocated utilization. Its proof of correctness is also an interesting topic.

real-time system. No matter how simple or complicated, the truly real-real-time system is a system we can dene and it will guarantee how "real-time" we want. We believe "time-driven" is an important technique to truly real-time. Therefore, we also believe that the time-driven pinwheel scheduling is a step in the right direction.

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The Correctness of Sr

^g

In this appendix we show that to nd an

r

in

Sr

^gso that ^g

T

⁽

^r

^{) = g}

T

^{= mincg1<r c1g}

T

⁽

^r

^),

we only need to compute ^g

T

⁽

^k

^{gv), for all}

^k

^gv in the special base of

T

.

Let



_gr =

T

i^j

c

i =

r g

^j for some integer

j

^0
i.e. the

r

-multiple subset of

T

on base

g

. It is obvious that for any

r

, there exists an

r

⁰,

c

¹

=g < r

⁰

c

¹, such that



_gr =



_r^g0. W.l.o.g.,

r

can be limited in (

c

¹

=g c

¹]. It is easy to see that



_gr ⁶= ^ (hence,



(



_gr) ⁶= 0) if and only if

r

=

k

_gv, for 1

v

u

. Recall that when

C

^!fg_r^g

B

, every

c

i in

C

is specialized to a number

b

i=

r g

^blogg(ci=r)c. If

r

=

l

^g_i (=

c

i

=g

^{dlogg(ci=c1)e}), we can derive

b

i=

c

i. That is, if

r

=

l

^g_i,

c

i will be specialized to itself, and hence, the density of

T

i will not increase after specialization. Therefore, if

r

=

l

^g_i =

k

_gv, for some

v

,



_gr is the set of tasks whose density will not increase after

C

^!fg_r^g

B

. To achieve the best schedulability, there must be at least one task whose density will not increase after specialization. So,

r

can be chosen only from the special base ^f

k

^g1

k

^2g

::: k

_gu^gof

T

(see Lemma A.1).

Suppose

C

^!fg_r^g

B

. For convenience of discussion, let

k

^0g =

c

¹

=g

. For a task

T

i ²



_k^ggv

(i.e.,

c

i =

k

_gv

g

^j, for some 1

v

u

and

j

^ 0), there are two possible ^c_bⁱi values due to the relationship between

k

_gv and

r

:

(a)

If

r > k

_gv, since

r=g < k

_gv

< r

, we have

b

i =

r g

^j;1 and ^c_bⁱⁱ = ^gk_r^gv.

(b)

If

r

k

_gv, since

r

k

_gv

< gr

, we have

b

i =

r g

^j and ^c_bⁱi = ^k_r^gv.

From (a) and (b), we know that, if

k

_v^g;1

< r

k

_gv, for some

v

, 1

v

u

, then

^g

T

⁽

^r

^{) =v}

;1

X

i⁼¹

gk

^g_i

r 

⁽



_k^gg

i

) +^Xu

i⁼v

k

^g_i

r 

⁽



_k^gg

i

)

:

(A.1)

92

In the above, a summation with the higher limit smaller than the lower limit is dened to have a value of 0. For example, if

v

= 1 in Equation (A.1), ^g

T

⁽

^r

^{) =Pui=1 krgi}

^

⁽

^

^kggi).

Moreover, we can prove the following lemma.

Lemma A.1

_k^g If

k

_v^g;1

r < k

_gv, for some

v

, 1

v

u

, and

r

⁶=

k

^0g, then ^g

T

⁽

^r

^{) =}

rv^g

T

⁽

^k

^gv);(

^g

^;1)

^

⁽

^

^gr).

Proof: From Equation (A.1), we have, for

k

^g_v^;1

< r < k

_gv,

^g

T

⁽

^r

^{) = v}

;1

X

i⁼¹

gk

^g_i

r 

⁽



_k^gg

i

) +^Xu

i⁼v

k

^g_i

r 

⁽



_k^gg

i

)

=

k

_gv

r

v^X;1 i⁼¹

gk

^g_i

k

gv



(



_k^gg

i

) +^Xu

i⁼v

k

^g_i

k

gv



(



_k^gg

i

)

!

=

k

_gv

r

^g

T

⁽

^k

^gv)

Note that (

g

^;1)



(



_gr) = 0, for

k

^g_v^;1

< r < k

_gv. For

r

=

k

^g_v^;1, we have

^g

T

⁽

^k

^{gv;1) = v}

;2

X

i⁼¹

gk

_i^g

k

^g_v^;1



(



_k^gg

i

) + ^Xu

i⁼v^;1

k

^g_i

k

^g_v^;1



(



_k^gg

i

)

= ^vX;1

i⁼¹

gk

_i^g

k

^g_v^;1



(



_k^gg

i

) +^Xu

i⁼v

k

_i^g

k

_v^g;1



(



_k^gg

i

)^;(

g

^;1)



(



_k^gg

v ;1

)

=

k

_gv

k

^g_v^;1

v^X;1 i⁼¹

gk

^g_i

k

gv



(



_k^gg

i

) +^Xu

i⁼v

k

_i^g

k

gv



(



_k^gg

i

)

!

;(

g

^;1)



(



_k^gg

v ;1

)

=

k

_gv

k

^g_v^;1g

T

⁽

^k

^gv);(

^g

^;1)

^

⁽

^

^{kggv ;1)}

2

Recall that



(



_gr) ⁶= 0 if and only if

r

=

k

_gv, for 1

v

u

. Therefore, Lemma A.1 implies that ^g

T

⁽

^r

⁾

^>

^g

T

⁽

^k

^{gv), for all}

^r

^,

^k

^gv;1

< r < k

_gv. Thus, to nd an

r

in

Sr

^g so that

^g

T

⁽

^r

^{) = g}

T

^{= minc1=g<r c1g}

T

⁽

^r

), we only need to compute ^g

T

⁽

^k

^{gv), for all}

^k

^{gv in the}

special base of

T

.

Proofs of the Lemmas

Lemma B.1 k

_v^g+1g

T

⁽

^k

^gv+1);

^k

^gvg

T

⁽

^k

^{gv) = (}

^g

^;1)

^k

^gv

^

⁽

^

^{kgvg), for 1}

^{v < u}

^.

Proof: From Lemma A.1, for 1

v < u

, ^g

T

⁽

^k

^{gv) = kgv +1kgv g}

T

⁽

^k

^vg+1);(

^g

^;1)

^

⁽

^

^kgvg).

Therefore,

k

^g_v^+1g

T

⁽

^k

^vg+1);

^k

^gvg

T

⁽

^k

^{gv) = (}

^g

^;1)

^k

^gv

^

⁽

^

^{kggv). 2}

Lemma B.2

Let

T

²

U

(

d

) such that, for all

T

^{0 2}

U

(

d

), ^g

T

^0
g

T

^{= g
, and let}

f

k

^g1

k

^2g

::: k

_gu^gbe the special base of

T

based on

g

. It must be true that ^g

T

⁽

^k

^{gi) = g
, for}

all

k

^g_i, 1

i

u

.

Proof: The lemma is trivially true for

u

= 1. We prove the lemma for

u >

1 by contra-diction. Suppose to the contrary that there exists a

p

, 1

p

u

, ^g

T

⁽

^k

^gp)

^>

^{g
. We can}

show that there exists a task set

T

⁰ with the same total density and the same special base as

T

, but ^g

T

⁰

^>

^g
. And this contradicts the denition of

T

. To show this, there are two cases to be considered:

Case 1:

1

< p

u

.

Let's change the densities of



_k^ggp and



_k^gg

p;1

in

T

by changing the execution times of some of the tasks to create another task set

T

⁰.

(a) 

_k^gp^g is changed so that



(



_k^ggp) is increased by



= min^f(^g

T

⁽

^k

^{gp) ;g
)}

⁼

⁽

^g

^{;1) (1;}

"

)



(



_k^gg

p;1

)^g, and

(b) 

_k^gg

p;1

is changed so that



(



_k^gg

p;1

) is decreased by



,

where

"

is a small positive number. Note that



(

T

) =



(

T

⁰), hence

T

^{0 2}

U

(

d

). From Equation (A.1), we know that for

k

^g_v^;1

< r

k

_gv and 1

v

u

, ^g

T

⁽

^r

^{) =Pvi=1;1 gkrgi}

^

⁽

^

^kgig)+

94

Pui⁼vk^gi r



(



_k^gg

i

). From denitions of ^g
,

k

_gp and



, we know that for

k

^g0

< r

k

_gu, ^g

T

⁽

^r

^)

^g
, for 1

p

u

,

k

_gp

> k

^g_p^;1, and ^g

T

⁽

^k

^gp);g
(

^g

^;1)



. Therefore, we have:

(i)

if

r

k

^g_p^;1, then ^g

T

⁰⁽

^r

^);g
g

T

⁰⁽

^r

^);g

T

⁽

^r

^{) =}



⁽

^k

^gp;

^k

^gp;1)

^{=r >}

^{0, i.e., g}

T

⁰⁽

^r

⁾

^>

^g


(ii)

if

r

=

k

_gp, then ^g

T

⁰⁽

^k

^{gp);g
= (g}

T

⁰⁽

^k

^gp);g

T

⁽

^k

^{gp)) + (g}

T

⁽

^k

^{gp) ;g
)  (1;}

gk

^g_p^;1

=k

_gp)



+ (

g

^;1)

 >

0, i.e., ^g

T

⁰⁽

^r

⁾

^>

^g


(iii)

if

r > k

_gp, then ^g

T

⁰⁽

^r

^);g
g

T

⁰⁽

^r

^);g

T

⁽

^r

^{) =}

^g

⁽

^k

^gp;

^k

^pg;1)

^{=r >}

^{0, i.e., g}

T

⁰⁽

^r

⁾

^>

^g
.

So, ^g

T

⁰

^>

^g

T

, which contradicts the denition of

T

in the lemma.

Case 2: p

= 1.

In this case, we change the densities of



_k^gg

1

and



_k^ggu to create task set

T

⁰.

(a) 

_k^gg

1

is changed so that



(



_k^gg

1

) is increased by



= min^f(^g

T

⁽

^k

^{1g) ;g
)}

⁼

⁽

^g

^{;1) (1;}

"

)



(



_k^ggu)^g, and

(b) 

_k^ggu is changed so that



(



_k^gu^g) is decreased by



.

Again, we have



(

T

) =



(

T

⁰), hence

T

^{0 2}

U

(

d

), and from denition,

k

^g1

> c

¹

=g

=

k

_gu

=g

. Therefore, we have:

(i)

if

r

=

k

^g1, then ^g

T

⁰⁽

^k

^{1g);g
= (g}

T

⁰⁽

^k

^g1);g

T

⁽

^k

^g1))+(g

T

⁽

^k

^g1);g
)(1;

^k

^gu

^=k

^1g)



⁺

(

g

^;1)

 >

0, i.e., ^g

T

⁰⁽

^r

⁾

^>

^g


(ii)

if

r > k

^g1, then ^g

T

⁰⁽

^r

^);g
g

T

⁰⁽

^r

^);g

T

⁽

^r

^{) =}



⁽

^bk

^g1;

^k

^gu)

^{=r >}

^{0, i.e., g}

T

⁰⁽

^r

⁾

^>

^g
.

So, again, ^g

T

⁰

^>

^g

T

, which contradicts the denition of

T

in the lemma. ²

Lemma B.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).

Proof: From denition of



(



_gr), if

r

⁶=

k

_gv, for some

v

,



(



_gr) = 0. Thus, from Lemma A.1, if

k

_v^g;1

< r < k

_gv, for any

v

, 1

v

u

, ^g

T

⁽

^r

^{) = krgvg}

T

⁽

^k

^gv).

Therefore, according to Lemma A.1, if

k

^g_v^;1

< r < r

⁰

k

_gv, for some

v

, 1

v

u

, then

^g

T

⁽

^r

⁾

^>

^g

T

⁽

^r

^{0). 2}

De
nitions and Algorithms of SLINE Operations

In this appendix, we present the detailed denitions and algorithms of SLINE operations.

h

j is the relative phase on nodes

N

j and

N

j⁺¹ as dened in Section 4.3.1. Each SLINE

W

corresponds to a serrated line function

W

(

h

j) dened in X-domain 0

w

). In our algorithms, we repeatedly reduce the domain size of a serrated line but keep the minimum value of drop point. To keep track of the original

h

j value, each point in our representation has two domains: the current domain and the original domain.

The current domain is the X-domain of the serrated line function

W

represents. The original domain is the original X-domain of each point before any CUT operation. All points used in our discussion are in the current domain unless we specify explicitly. Sometimes, we use

t

as a shorthand of

h

j.

De
nition: An SLINE

W

=^f

s w

(

x y

) ^f

d

¹

d

²

:::

^ggwhere

1.

s

is the slope, which is the slope of the line segments of the serrated line it represents

2.

w

is the width, or the current domain size

3. (

x y

) is the origin, where

x

is in the original domain

4. The last eld is a set of drop points. Each drop point

d

i has three elds, (

p a p

⁰)

p

is the X-coordinate of the drop point in the current domain,

a

is the drop amount of the value at the drop point, and

p

⁰ is the X-coordinate of the drop point in the original domain. The drop points are sorted in increasing order on

p

.

96

5.

W

(

t

) returns the value of

W

at

t

.

6.

W

⁰(

t

) returns the X-coordinate in the original domain of

W

at

t

.

2

Given a

W

(

t

) and a

W

⁰(

t

), we can dene a

W

. Suppose

F

¹

::: F

u are delay functions of

t

with period

w

. From the denition of delay function in Equation (4.3), they are serrated line functions and have slope equal to 1.

De
nition: Three operations are dened for SLINE: SUM, MAX and CUT.

1. SUM^f

W

^g=

W

and MAX^f

W

^g=

W

.

2. SUM^f

W F

¹

::: F

u^g is an SLINE,

S

, where

S

(

t

) =

W

(

t

) +

F

¹(

W

⁰(

t

))+

F

²(

W

⁰(

t

)) +

:::

+

F

u(

W

⁰(

t

)) and

S

⁰(

t

) =

W

⁰(

t

),⁸

t

²0

w

).

3. MAX^f

W F

¹

::: F

u^g is an SLINE,

S

, where

S

(

t

) = max^f

W

(

t

)

F

¹(

W

⁰(

t

))

F

²(

W

⁰(

t

))

::: F

u(

W

⁰(

t

))^gand

S

⁰(

t

) =

W

⁰(

t

), ⁸

t

²0

w

).

4. Let

u

=

w=E

k,

E

k

w

!, and

V

E^k(

t

) is a function of

t

to a set of

u

values, where

V

E^k(

t

) =

f

t

^0j

t

⁰=

t

+ (

v

^;1)

E

k 1

v

u

^{g 8}

t

²0

E

k)

CUT^f

W E

k^g is an SLINE,

S

, where

S

(

t

) = min^f

W

(

v

) ^j

v

²

V

E^k(

t

)^g and

S

⁰(

t

) = min^f

W

⁰(

v

)^j

W

(

v

) =

S

(

t

)^{g 8}

t

²0

E

k).

2

Suppose

S

is an SLINE with

v

drop points and width

E

k, and there are

u

periodic delay functions,

F

¹

::: F

u, with the same period

E

k. Because the period of

F

i is

E

k for every

F

i, there is a

p

i ²0

E

k) s.t.

F

i(

p

i) = 0. W.l.o.g. suppose the

p

i's are in increasing order.

From the denition of SLINE,

d

iare sorted on

d

i

:p

in increasing order. For convenience, to insert, delete or append drop points on an SLINE

S

will not update the original indices of the drop points unless we specify "

reindex S

". During the operations of SLINE, we may generate drop points with the same

p

value. Before we return

S

, we can reduce these

98 drop points to one drop point, called

compact

. For example, if there are two drop points

d

i and

d

j,

i < j

and

d

i

:p

=

d

j

:p

. After compacting, only

d

i will be kept while

d

i

:a

changes to

d

i

:a

+

d

j

:a

and

d

i

:p

⁰changes to min^f

d

i

:p

⁰

d

j

:p

^0g.

在文檔中 Real-Time Systems Scheduling Using the Distance-Constrained Model (頁 97-111)