Exercise 9: Stabilization
Task 1: Recovering Reasonable Constraints
In this exercise, we want to get a better handle on the set of constraints from the lecture:
S(1)≥2
δ+
1−1 ϑ
T
(1) R−
ϑ ≥σh+ϑS(1) +d (2)
B2
ϑ > σh+R++T+ 2S(1) (3)
B1> σh+R+ (4)
B3> R++ (M −1)(T+S(1)) + (ϑ+ 1)S(M) +σh (5) B2≤R−
ϑ + (M−1)
T
ϑ − S(1)
+S(M) (6)
R+
ϑ ≥(ϑ+ 1)S(M) +σh (7)
2(S(1)− S(M))≥σh (8)
a) ChooseR− tight according to (2) andR+/ϑequal to the r.h.s. of (7) plusd, respec- tively. Show that for these choices (1) and (8) hold. (Hint: Use these equalities to expressS(1) in terms ofS(M) and other terms not involvingR+ orR−.)
b) Fix these choices and consider the remaining inequalities. Which of the terms on the r.h.s. of the inequalities are inO(σh+d)? (Hint: Recall thatϑ∈ O(1).)
c) In addition, suppose now that for α >1, we have that B3 =αB2=α2B1 and can choose B1 > 0 arbitrarily. For which values of α can you choose M so that the system of inequalities is satisfied?
Task 2: If it Were so Simple. . .
a) Modify the Srikanth-Toueg algorithm so that its pulses can be triggered by an exter- nal NEXT signal in the vein of Definition 9.7. (Hint: Add a second, smaller timeout to the transition fromreadyto propose.)
b) What choices ofB1,B2, andB3can this solution support?
