#import "@preview/acrostiche:0.3.2": *

#init-acronyms((
  "EBS": ("Equation-Based Scheduling",),
))

#align(center)[#text(size:2em)[Equation-Based Scheduling]]


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 without collision.

*Problem Statement:*
#grid(
  columns: (1fr,15fr,1fr),
  [],[
    Given $n$ tasks $t_i, i in [0,n-1]$, 
    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 scheduling functions $s$ such that, 
    for each time period $tau_i$, $s(tau_i) = A[tau_i]$.
  ],
    []
)

= 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
$
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."
)