update
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parent
52b5fdeb98
commit
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2 changed files with 244 additions and 220 deletions
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|
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@ -1020,7 +1020,7 @@ z
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Before Width: | Height: | Size: 35 KiB After Width: | Height: | Size: 35 KiB |
|
|
@ -26,7 +26,7 @@ This principle provide a mathematically proovable way for each task to determine
|
||||||
= Polynomials
|
= Polynomials
|
||||||
|
|
||||||
Contrary to popular beliefs, polynomials are not boring.
|
Contrary to popular beliefs, polynomials are not boring.
|
||||||
Let us consider a problem with four tasks $(t_0,t_1,t_2,t_3)$ and a schedule array of 20 time period
|
Let us consider a problem with four tasks $(t_0,t_1,t_2,t_3)$ and a schedule array of 20 time periods
|
||||||
$
|
$
|
||||||
A = [0,1,2,3,0,0,1,2,0,1,2,3,0,3,0,2,3,2,0,1]
|
A = [0,1,2,3,0,0,1,2,0,1,2,3,0,3,0,2,3,2,0,1]
|
||||||
$
|
$
|
||||||
|
|
@ -34,4 +34,28 @@ $
|
||||||
#figure(
|
#figure(
|
||||||
image("images/polyfit.svg", width:100%),
|
image("images/polyfit.svg", width:100%),
|
||||||
caption: "Scheduling function defined as a polynomial fit of the scheduling array."
|
caption: "Scheduling function defined as a polynomial fit of the scheduling array."
|
||||||
)
|
)<fig-polyfit>
|
||||||
|
|
||||||
|
== Clock Sensitivity
|
||||||
|
In order for every task to know when they can transmit, they all need to evaluate the scheduling function at the same times.
|
||||||
|
In the real world, that could be very difficult as the clock of each task can drift over time.
|
||||||
|
Some people suggested to use atomic clock chips as the RTC of clock so that, after a syncronisation phase, all timings across tasks would remain exact #footnote[Atomic clocks drift is estimated at one second per hunred million year (Wikipedia).].
|
||||||
|
However, I suspect that the person proposing this solution have atomic clock in their office drawer and don't think of us, mere students, that do not have 6k to invest in a fancy clock.
|
||||||
|
|
||||||
|
This raises the question; How sensitive is a scheduling function to clock imprecision?
|
||||||
|
After all, the function is exact on the start of the period but has no constraint during the period.
|
||||||
|
We define the clock sensitivity of a scheduling function $"cs"(s)$ as the maximum time delta around any period start without an incorrect task decision.
|
||||||
|
$
|
||||||
|
"cs"(s) = max_(delta t)(s(tau_i plus.minus delta t) = A[tau_i]; forall tau_i)
|
||||||
|
$
|
||||||
|
|
||||||
|
Let us consider again the example displayed in @fig-polyfit.
|
||||||
|
As the degree of the polynomial grows to perfectly fit the schedul, extreme variations appear during the periods.
|
||||||
|
These extreme variations induce high derivative of the function at the sampling time that makes the clock sensitivity very small.
|
||||||
|
|
||||||
|
== Problems with Polynomials
|
||||||
|
A single polynomial fitted to the schedul does not seem like a good approach as it require a very precise clock to obtain the correct values.
|
||||||
|
Moreover, storing the polynomial coefficient for the function requires at least as much memory as storing the schedule itself.
|
||||||
|
Finally, evaluating the function is more computation-intensive than looking up a value in a table.
|
||||||
|
|
||||||
|
= Regression, imprefect fit, and flatten scheduling array.
|
||||||
|
|
|
||||||
Loading…
Add table
Add a link
Reference in a new issue