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4. The CTS has two Boolean variables s and ap and one clock xp . The values of s correspond to the two states in the figure. The CTS 30 J. Fisher et al. ; s s xp =1; ; xp : =0 Fig. 4. CTS for one rate is allowed to move from s to s when the clock x is in the range [1 − , 1), for some . When the CTS moves from s to s, it resets the clock back to 0. The variable ap is the scheduling variable that this CTS sets; it changes when the system moves from s to s, and does not change when the system moves from s to s.

Pi = p(t) (x), qij = αm (x) if j is the number assigned to state x + v m , and qji = αm (x − v m ) if j is the number assigned to state x − v m . 3 Uniformization A general solution of Equation (6) is [15,25,45] ∞ p(t) = p(0) · P (t) = p(0) · exp(Qt) = p(0) · k=0 (Qt)k . k! (8) 38 W. Sandmann and V. Wolf The infinite sum includes matrix exponentials and its computation is numerically unstable as Q contains strictly positive and negative entries leading to severe round-off errors [36]. As already mentioned in the introduction, different methods exist to compute the vector p(t) .

26 J. Fisher et al. 2 allows all possible interleavings of processes within a macro-step. We prove that we can construct a new scheduler that preserves system macro behavior (macro-step level behavior) but allows fewer interleavings. Let P = P11 ba · · · ba Pn1 be the bounded asynchronous composition of P1 , P2 , . . 4 We first formally define a macro-step g of P as a sequence of states g : s = s0 , s1 , . . , sm [bk ] holds for all 0 ≤ k ≤ n, – sm is the only final state in g. A macro-step induces a total and a partial order over the processes of P.

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7th International Conference on Geometry and Applications by Bohm J.

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