#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.

*Problem Statement:*
#grid(
  columns: (1fr,15fr,1fr),
  [],[
    Given $n$ tasks $t_i, i in [0,n-1]$, 
    and an array of $m$ time periods $A = [x_0, dots.h.c, x_(m-1)]$ 
    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.
  ],
    []
)

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 

$
S = cases(
  &s_i(tau_i) > 0 "if" A[tau_i] = t_i,
  &s_(j eq.not i) < 0 "else",
)
$