FIXED-PRIORITY SERVERS - Hard Real-Time Computing Systems: Predictable Scheduling Algorithms and Applications - page 135

Substituting this value in U we obtain

K
K (1 − 1 /n ) −

1) K 1 /n +

U lub − U s = n − n =

nK 1 /n K 1 /n + K 1 /n = − − n =

n ( K 1 /n = − 1);

that is,

= U s + n ( K 1 /n

U lub − 1) . (5.10)

Now, noting that

C s
T s

T 1 − T s

2 T s

R s − 1

U s = = =

2

we have

R s =(2 U s +1) .

Thus, K can be rewritten as
K = 3
2 R s

=

+ 1
2

U s +2
2 U s +1 ,

and finally

U lub = U s + n U s +2
2 U s +1

1 .

1 /n

− (5.11)

Taking the limit as n →∞ , we find the worst-case bound as a function of U s to be

given by

U lub = U s +ln U s +2
2 U s +1

.

lim
n→∞

(5.12)

A plot of Equation (5.12) as a function of U s is shown in Figure 5.11. For comparison,
the RM bound is also reported in the plot. Notice that for U s < 0 . 4 the presence of
DS worsens the RM bound, whereas for U s > 0 . 4 the RM bound is improved.

Deriving Equation (5.12) with respect to U s , we can find the absolute minimum value
of U lub :

2 U s +5 U s − ∂U lub

∂U s

=1+ (2 U s +1)
( U s +2)

(2 U s +1) 2( U s +2)

(2 U s +1) 2

− 1

( U s + 2)(2 U s +1) .

=

The value of U s that minimizes the above expression is

= √ 33

− 5

U s 0 . 186 ,

4

is U lub

so the minimum value of U lub 0 . 652.