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Quantitative Evaluation of
Embedded Systems
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Quantitative Evaluation of
Embedded Systems
1.
2.
3.
4.
Matrix equations - revisited
Monotonicity
Eigenvalues and throughput
Eigenvalues and the MCM
Quantitative Evaluation of
Embedded Systems
1.
2.
3.
4.
Matrix equations - revisited
Monotonicity
Eigenvalues and throughput
Eigenvalues and the MCM
x(n  1)  (0) x(n)
Cycles with a 0 execution time cause livelocks
y(n)  (0) x(n)
y
0 ms
But when logging events, this is mathematically okay...
y(n)  0
y
Theorem:
2ms of tokens on any cycle is constant!
1msThe number
x(n  1)  A x(n) max B u(n)
u
A
C
4ms
B
Therefore, every cycle must y(n)
containatC
least
onemax
token,D u(n)
x(n)
otherwise a deadlock occurs.
D
3ms
3
 0
2  
    0

  1 
 1
0  3 

   1
 1
 
2
1



 
0
 2
An
2 
 
1 

0 
 
3  
n times



 A   A
A0  Ι








Ax  α   Ax   α
AB C  ABC
Ax max y   Ax  max Ay 
AB  BA
x(n  1)  A x(n) max B u(n)
y(n)  C x(n) max D u(n)
C A n 1 x(1)


n 1 i
y(n)  max  max CA
B u(i)
 0 i n

D u(n)


Quantitative Evaluation of
Embedded Systems
1.
2.
3.
4.
Matrix equations - revisited
Monotonicity
Eigenvalues and throughput
Eigenvalues and the MCM
 x1   y1 
Define :  x 2    y 2  iff x1  y 1 and x 2  y 2 and x 3  y 3
   
 x3   y 3 
Theorem:
(max,+) matrix addition is a monotone operator
If x  y then Ax  Ay
x(n  1)  A x(n) max B u(n)
y(n)  C x(n) max D u(n)
C A n 1 x(1)


y(n)  max  max CA n 1i B u(i)
 0 i n

D u(n)


C A n 1 0

y(n) C A n1 0 n 1i
y(n)  max  max CA
B0
 0 i n

D 0


x1
u
A
x3
B
1ms
x2
 
 2  2 




y
x(n  1)   5
4
5  x(n) max  4  u(n)




2ms
 1 
  1  




max   u(n)
y(n)    1    x(n)
C
3ms
Time (s)
Theorem:
(max,+) matrix addition is a monotone
operator
y(n)  CAn 0
and as a consequence,
removing the input gives a best-case approximation of behavior.
Tokens
Cx
Quantitative Evaluation of
Embedded Systems
1.
2.
3.
4.
Matrix equations - revisited
Monotonicity
Eigenvalues and throughput
Eigenvalues and the MCM
In linear algebra...
a pair (x, λ) is an eigenpair for a matrix A if:
Ax  λ  x
with λ  x denoting elementwise multiplication
and assuming that x  0
In (max,+) algebra...
a pair (x, λ) is an eigenpair for a matrix A if:
Ax  λ  x
with λ  x denoting elementwise addition
and assuming that x  
In linear algebra, we find for any eigenpair (x, λ) that
A x  λ x
n
n
If (x, λ) is an eigenpair then so is (αx, λ), for any scalar α
And so, in (max,+) algebra we find for any eigenpair (x, λ)
A x  nλ  x
n
If (x, λ) is an eigenpair then so is (α+x, λ), for any shift α
x1
A
x3
B
2ms
1ms
x2
C
3ms
y
 2  2 


 5
4
5 


  1  


x(n  1)  A x(n) max B u(n)
y(n)  C x(n) max D u(n)
C A n 1 x(1)


y(n)  max  max CA n 1i B u(n  i)
 0 i n

D u(n)


y(n)  C A n 1 0
n 1
y(n)  C A x
 C((n  1)λ  x)
 nλ  (Cx  λ)
Time (s)
y(n)  nλ  Cx  λ
λ
Tokens
Cx  λ
Tokens
Theorem:
The best-case throughput is always smaller than 1/λ,
t  Cx
λ
1 λ eigenvalue of the associated
with λ the biggest
matrix.
y(t) 
λ
Cx  λ
Time (s)
Quantitative Evaluation of
Embedded Systems
1.
2.
3.
4.
Matrix equations - revisited
Monotonicity
Eigenvalues and throughput
Eigenvalues and the MCM
x1
A
x3
B
2ms
1ms
x2
C
3ms
y
 2  2 


x(n  1)   5
4
5  x(n)


  1  


y(n)    1    x(n)
x1
A
1ms
x2
x3
B
y
2
5

 

4
1

5 x  λ  x

 
2
x  
Theorem:
n l
mis
nthe

:
A
 xxof
some
 xi
Every eigenvalue
cycle
mean
cycle...
n
2ms
m i,k,l : A (i,k) 
k
λ(nxl)λ

nλ
x
C
3ms
n i,k
i(i,:k)x i  k
i i
l
 A (k,
k)  x k  lλ  x k
where Al(k,k) is the duration of some cycle with l tokens on it
hence λ = Al(k,k)/l is a cycle mean.
x1
A
x3
B
2ms
1ms
x2
C
3ms
y
Let λ=MCM and xi represent a critical token...
Now given matrix A of a dataflow graph,
let us construct the following two matrices:

A λ  maxA  λ   max A n  nλ
n
n0
n0
A λ  maxA  λ   (A  λ)A λ
n
n 1
In A λ an entry (i,j) represents
the max duration of any path
from i to j minus λ for each
token on that path.

x1
A
1ms
x2
x3
B
2ms
y
Since xi represents a critical token:
So for the ith column we find:
A 
λ
(i,i)
( A λ ) (  ,i )  ( max A λ ) (  ,i )  ( A λ ) (  ,i )
AndTheorem:
consequently:
λ
The maximal
ofAaλ )graph
A(Acycle
)(  ,i) mean
(A  λ)(
(  ,i)  λ
C maximum eigenvalue
is the
ofits
 ((A
λ)((max,+)
A λ )) matrix.
λ
(  ,i)
3ms
 (A λ )(  ,i)  λ
λ
In A an entry (i,j) represents
the max duration of any path
from i to j minus λ for each
token on that path.

 (A λ )(  ,i)  λ

So (A λ )(  ,i) , λ is an eigenpair of A
0
(max,+) matrix addition is a monotone operator
C A x(1)
Every eigenvalueis the cycle mean of some cycle...
 cycle mean is anneigenvalue!
1 i
But not every
y(n)  max  max CA
B u(n  i)
i n mean of a graph
The maximal
 0cycle
is the maximumeigenvalue
of its (max,+) matrix
D
u(n)

and 1/MCM is the maximal throughput.
1
Thus removing the input gives anbest-case
approximation of behavior.


See the book by [Baccelli, Cohen, Olsder and Quadrat] for more!
Like for the question “can the 1/MCM throughput actually be achieved?”
Simulate 6 firings
Give the (max,+)
matrix equations
𝑥 𝑛+1 =
𝑥 𝑛 𝑚𝑎𝑥
𝑢(𝑛)
Calculate the MCM
Determine a periodic schedule for arbitrary µ
Plot the latency for a period µ
Optimize the periodic schedule for µ = 15
Optimize the periodic schedule for arbitrary µ
Plot the delayed latency for a period µ
Plot the minimal delay for a period µ
Optimize the periodic schedule for arbitrary µ