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add first example of polyfit

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grizzly 2024-07-18 22:11:07 -04:00
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scheduling/scheduling.typ
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@ -8,29 +8,30 @@
The principle of #acr("EBS") is to use a function to determine which task is allowed to communicate at anypoint in time.
This principle provide a mathematically proovable way for each task to determine when to communicate.
This principle provide a mathematically proovable way for each task to determine when to communicate without collision.
*Problem Statement:*
#grid(
columns: (1fr,15fr,1fr),
[],[
Given $n$ tasks $t_i, i in [0,n-1]$,
and an array of $m$ time periods $A = [x_0, dots.h.c, x_(m-1)]$
and a schedule array of $m$ time periods $A = [x_0, dots.h.c, x_(m-1)]$
where each element $x_i in [t_0,dots.h.c, t_(n-1)]$ is the task allowed to communicate at time $tau_i$,
provide a set of $n$ scheduling functions $S=(s_0,dots.h.c,s_(n-1))$ such that,
for each time period $tau_i$, only $s_i(tau_i)$ associated with the task $t_i$ defined at by $A[i]$
validates a pre-defined _enable_ condition.
provide a scheduling functions $s$ such that,
for each time period $tau_i$, $s(tau_i) = A[tau_i]$.
],
[]
)
The _enable_ condition is any condition defined on the value of a scheduling function evaluated at a regular time.
For example, a simple _enable_ condition can be the positivity of the value.
In this case, the definition of the scheduling functions set is
= Polynomials
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
$
S = cases(
&s_i(tau_i) > 0 "if" A[tau_i] = t_i,
&s_(j eq.not i) < 0 "else",
A = [0,1,2,3,0,0,1,2,0,1,2,3,0,3,0,2,3,2,0,1]
$
#figure(
image("images/polyfit.svg", width:100%),
caption: "Scheduling function defined as a polynomial fit of the scheduling array."
)
$