Quantitative Evaluation of
Embedded Systems
1.
2.
3.
4.
Periodic schedules are linear programs
Latency analysis of a periodic source
Latency analysis of a sporadic source
Latency analysis of a bursty source
• Determine the MCM and choose a period μ ≥ MCM
• For each actor a initialize a start-time Ta := 0
• Repeat for each arc a—i—b :
Tb := Tb max (Ta + Ea – i μ)
until there are no more changes
Here, i denotes the number of initial tokens on an arc,
and Ea is the execution time of an actor a
x1
S
µ
A
x3
B
2ms
1ms
x2
C
y
• Choose a period μ ≥ MCM
• Initialize a start-time Ta := 0
• Repeat for each arc a—i—b :
Tb := Tb max (Ta + Ea – i μ)
until there are no more changes
3ms
𝟎 ≤ 𝒙𝝁 ≤ 𝐀𝒙𝝁 𝐦𝐚𝐱 𝑩𝟎 ≤ 𝝁 + 𝒙𝝁
Quantitative Evaluation of
Embedded Systems
1.
2.
3.
4.
Periodic schedules and linear programs
Latency analysis of a periodic source
Latency analysis of a sporadic source
Latency analysis of a bursty source
Time (s)
Throughput
y(n)
u(n)
Latency  supy(n)  u(n)
n 1
Tokens
Time (s)
Throughput
y(n)
u(n)
Latency  supy(n  δ)  u(n)
n 1
Tokens
Given a source:
u(n)  (n  1)μ
And a periodic schedule:
𝟎 ≤ 𝒙𝝁 ≤ 𝐀𝒙𝝁 𝐦𝐚𝐱 𝑩𝟎 ≤ 𝝁 + 𝒙𝝁
We inductively derive the following latency bound:
𝒙 𝟏 − 𝒖 𝟏 = 𝟎 − 𝟎 ≤ 𝐀𝒙𝝁 𝐦𝐚𝐱 𝑩𝟎
𝒙 𝒏+𝟏 −𝒖 𝒏+𝟏
= 𝑨𝒙 𝒏 𝒎𝒂𝒙 𝑩𝒖(𝒏) − 𝒖 𝒏 + 𝟏
= 𝑨𝒙 𝒏 𝒎𝒂𝒙 𝑩𝒖(𝒏) − 𝒖 𝒏 − 𝝁
= 𝑨(𝒙 𝒏 − 𝒖 𝒏 ) 𝒎𝒂𝒙 𝑩𝟎 − 𝝁
≤ 𝑨 𝑨𝒙𝝁 𝒎𝒂𝒙 𝑩𝟎 𝒎𝒂𝒙 𝑩𝟎 − 𝝁
≤ 𝑨 𝒙𝝁 + 𝝁 𝒎𝒂𝒙 𝑩𝟎 − 𝝁
≤ 𝑨 𝒙𝝁 + 𝝁 𝒎𝒂𝒙 𝑩𝝁 − 𝝁
= 𝑨𝒙𝝁 𝒎𝒂𝒙 𝑩𝟎
Given a source:
u(n)  (n  1)μ
And a periodic schedule:
𝟎 ≤ 𝒙𝝁 ≤ 𝐀𝒙𝝁 𝐦𝐚𝐱 𝑩𝟎 ≤ 𝝁 + 𝒙𝝁
We derive the following latency bound:
𝒙 𝒏+𝜹 −𝒖 𝒏
= 𝑨𝜹 𝒙 𝒏 𝒎𝒂𝒙𝟎<𝒊≤𝜹 𝑨𝜹−𝒊 𝑩𝒖(𝒏) − 𝒖 𝒏
= 𝑨𝜹 𝒙 𝒏 − 𝒖 𝒏 𝒎𝒂𝒙𝟎<𝒊≤𝜹 𝑨𝜹−𝒊 𝑩𝟎
≤ 𝑨𝜹 (𝑨𝒙𝝁 𝒎𝒂𝒙 𝑩𝟎) 𝒎𝒂𝒙𝟎<𝒊≤𝜹 𝑨𝜹−𝒊 𝑩𝟎
= 𝑨𝜹+𝟏 𝒙𝝁 𝒎𝒂𝒙𝟎≤𝒊≤𝜹 𝑨𝜹−𝒊 𝑩𝟎
≤ 𝑨𝒙𝝁 𝒎𝒂𝒙 𝑩𝟎 + 𝜹𝝁
Given a source:
u(n)  (n  1)μ
We derive the following latency bound:
𝒚 𝒏+𝜹 −𝒖 𝒏
Theorem:
= 𝑪𝒙 𝒏 + 𝜹 𝒎𝒂𝒙 𝑫𝒖(𝒏) − 𝒖(𝒏)
𝟎 ≤ 𝒙𝝁the
≤ 𝐀𝒙latency
𝝁+a
𝒙𝝁 dataflow graph for a source with
𝝁 𝐦𝐚𝐱 𝑩𝟎 ≤ of
≤ 𝑪 𝑨𝒙𝝁 𝒎𝒂𝒙 𝑩𝟎 + 𝜹𝝁 𝒎𝒂𝒙 𝑫𝟎
period 𝝁 and periodic schedule 𝒙𝝁 is smaller than:
𝑪 𝑨𝒙𝝁 𝒎𝒂𝒙 𝑩𝟎 + 𝜹𝝁 𝒎𝒂𝒙 𝑫𝟎
And a periodic schedule:
Given a source:
𝒖 𝟏 ≥𝟎
𝒖 𝒏+𝟏 ≥𝒖 𝒏 +𝝁
We inductively derive the following latency bound:
𝒙 𝟏 − 𝒖 𝟏 ≤ 𝟎 − 𝟎 ≤ 𝐀𝒙𝝁 𝐦𝐚𝐱 𝑩𝟎
𝒙
𝒏 + 𝟏 − 𝒖 𝒏 + 𝟏2):
(monotonicity
= 𝑨𝒙 𝒏 𝒎𝒂𝒙 𝑩𝒖(𝒏) − 𝒖 𝒏 + 𝟏
𝟎 ≤ 𝒙𝝁 ≤ 𝐀𝒙𝝁 𝐦𝐚𝐱
𝑩𝟎 ≤ 𝝁inter-arrival
+ 𝒙𝝁
Larger
in𝑩𝒖(𝒏)
the source
≤ 𝑨𝒙times
𝒏 𝒎𝒂𝒙
−𝒖 𝒏 −𝝁
= 𝑨(𝒙 the
𝒏 − latency.
𝒖 𝒏 ) 𝒎𝒂𝒙 𝑩𝟎 − 𝝁
will not worsen
≤ 𝑨 𝑨𝒙𝝁 𝒎𝒂𝒙 𝑩𝟎 𝒎𝒂𝒙 𝑩𝟎 − 𝝁
≤ 𝑨 𝒙𝝁 + 𝝁 𝒎𝒂𝒙 𝑩𝝁 − 𝝁
= 𝑨𝒙𝝁 𝒎𝒂𝒙 𝑩𝟎
Theorem
And a periodic schedule:
Given a source:
𝒖 𝟏 ≥𝟎
𝒖 𝒏 + 𝜷 ≥ 𝒖 𝒏 + 𝜷𝝁
As an exercise, derive that:
𝒙Theorem:
𝒏 − 𝒖 𝒏 ≤ 𝑨𝒙𝝁 𝒎𝒂𝒙 𝑩𝟎 + (𝜷 − 𝟏)𝝁
the latency of a dataflow graph for a sporadic bursty
And a periodic schedule:
source
with
period
𝝁
burst
size
𝜷,
𝟎 ≤ 𝒙𝝁 ≤ 𝐀𝒙𝝁 𝐦𝐚𝐱 𝑩𝟎 ≤ 𝝁 + 𝒙𝝁
and periodic schedule 𝒙𝝁 is smaller than:
𝑪 𝑨𝒙𝝁 𝒎𝒂𝒙 𝑩𝟎 + 𝜹𝝁 𝒎𝒂𝒙 𝑫𝟎 + 𝜷𝝁