FIXED-PRIORITY SERVERS - Hard Real-Time Computing Systems: Predictable Scheduling Algorithms and Applications - page 125

The resulting utilization is then

C s
T s

+ C 1
T 1

+ ... + C n
T n

U = =

U s + T 2 −

T 1

+ ... + T n −

T n− 1
T n− 1

+ 2 T s −

T n

= =

T 1 T n

+ 2 T s
T 1

T 1
T n −

U s + T 2
T 1

T n
T n− 1

= + ... + n.

Defining

⎧
⎨

R s = T 1 /T s
R i = T i +1 /T i
K =2 T s /T 1 =2 /R s

⎩

and noting that

T n
T 1

R 1 R 2 ...R n− 1 = ,

the utilization factor may be written as

n− 1

K
R 1 R 2 ...R n− 1 −

U = U s + R i + n.

i =1

Following the approach used for RM, we minimize U over R i , i =1 ,...,n − 1.

Hence,

∂U
∂R i

K
R i ( n− 1

j = i R j ) .
Thus, defining P = R 1 R 2 ...R n− 1 , U is minimum when
⎧
⎨

=1 −

R 1 P = K
R 2 P = K
...
R n− 1 P = K ;

⎩

that is, when all R i have the same value:

R 1 = R 2 = ... = R n− 1 = K 1 /n .

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);