From 85d4edf508fc9eed0e24a602feac9871f83b94ae Mon Sep 17 00:00:00 2001
From: grizzly <gd@crans.org>
Date: Fri, 19 Jul 2024 18:17:01 -0400
Subject: [PATCH] update

---
 scheduling/images/polyfit.svg | 436 +++++++++++++++++-----------------
 scheduling/scheduling.typ     |  28 ++-
 2 files changed, 244 insertions(+), 220 deletions(-)

diff --git a/scheduling/images/polyfit.svg b/scheduling/images/polyfit.svg
index 8035999..69880ae 100644
--- a/scheduling/images/polyfit.svg
+++ b/scheduling/images/polyfit.svg
@@ -6,7 +6,7 @@
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diff --git a/scheduling/scheduling.typ b/scheduling/scheduling.typ
index 52c8282..0e49ddd 100644
--- a/scheduling/scheduling.typ
+++ b/scheduling/scheduling.typ
@@ -26,7 +26,7 @@ This principle provide a mathematically proovable way for each task to determine
 = 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
+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]
 $
@@ -34,4 +34,28 @@ $
 #figure(
   image("images/polyfit.svg", width:100%),
   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.