schedj

scheduling/scheduling.typ

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+#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",
+)
+$