Quantitative Evaluation of Embedded Systems

1. Periodic schedules are linear programs2. Latency analysis of a periodic source3. Latency analysis of a sporadic source4. Latency analysis of a bursty source

Running a Periodic Regime

• 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

Periodic Schedule = Linear Program

AS B

C1ms 2ms

x3 y

x1

x2

3ms

µ

• 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

Quantitative Evaluation of Embedded Systems

1. Periodic schedules and linear programs2. Latency analysis of a periodic source3. Latency analysis of a sporadic source4. Latency analysis of a bursty source

What is latency?

Tim

e (s

)

Tokens

u(n)y(n)sup1n

Latency

Throughput

u(n)

y(n)

What is latency?

Tim

e (s

)

Tokens

u(n)δ)y(nsup1n

Latency

Throughput

u(n)

y(n)

Latency analysis for a periodic source

And a periodic schedule:

Given a source:1)μ(nu(n)

We inductively derive the following latency bound:

𝒙 (𝟏 )−𝒖 (𝟏 )=𝟎−𝟎≤𝐀 𝒙𝝁𝐦𝐚𝐱𝑩𝟎

Latency analysis for a periodic source

We derive the following latency bound:

And a periodic schedule:

Given a source:1)μ(nu(n)

Latency analysis for a periodic source

We derive the following latency bound:

And a periodic schedule:

Given a source:1)μ(nu(n)

Theorem: the latency of a dataflow graph for a source with

period and periodic schedule is smaller than:

Latency analysis for a sporadic source

And a periodic schedule:

Given a source: We inductively derive the following latency bound:

𝒙 (𝟏 )−𝒖 (𝟏 )≤𝟎−𝟎≤𝐀 𝒙𝝁𝐦𝐚𝐱𝑩𝟎

Theorem (monotonicity 2):Larger inter-arrival times in the source

will not worsen the latency.

Latency analysis for a bursty source

And a periodic schedule:

Given a source: As an exercise, derive that:

𝒙 (𝒏 )−𝒖 (𝒏 )≤𝑨 𝒙𝝁𝒎𝒂𝒙 𝑩𝟎+(𝜷−𝟏)𝝁Theorem: the latency of a dataflow graph for a sporadic bursty

source with period burst size ,and periodic schedule is smaller than: