Cycle time of a P-time Event Graph with affine-interdependent residence durations

Abstract

In this paper, we widen the class of P-time Event graphs by introducing affine-interdependent residence durations. This new class is studied through a general algebraic model. Considering a periodic behavior, we provide conditions of existence of a trajectory and propose a technique allowing the determination of extremal solutions. We show that the cycle time is intrinsic to this new model: it depends on the circuits of an associated graph but also on more complex structures.

Appendix

Let us express system inequalities (1) on a reduced horizon. Such a form will simplify the calculations. The objective is to establish an equivalent model such that each place of the graph initially contains only zero or one token.

Roughly speaking, the general idea is to split each place containing i tokens into i places, where each place contains only one token (a place can initially contain a maximum number of m tokens). A systematic procedure is as follows.

Let us introduce new variable X, that is, $X(k)=(X_{0}^{t}(k) \; X_{1}^{t}(k) \; \ldots \; X_{i}^{t}(k) \; \ldots X_{m-1}^{t}(k)) ^{t}$ with X _i (k) = x(k − m + i + 1). By construction, we have X _m − 1(k) = x(k) and X _i (k) = X _i + 1(k − 1) for i going from 0 to m − 2.

So, system (1) becomes

$$ \left( \begin{array}{ll} G_{1}^{^{\prime }-} & G_{0}^{^{\prime }-} \\[6pt] G_{1}^{^{\prime }+} & G_{0}^{^{\prime }+} \end{array} \right) \times \dbinom{X(k-1)}{X(k)}\leq \left( \begin{array}{r} -T^{-} \\[6pt] T^{+} \end{array} \right), $$

where $G_{1}^{^{\prime }-}=( G_{m}^{-} \; 0 \; \ldots \; \ldots \; 0 )$, $G_{0}^{^{\prime }-}=( G_{m-1}^{-} \; G_{m-2}^{-} \; \ldots \; G_{1}^{-} \; G_{0}^{-} )$, $G_{1}^{^{\prime }+}=( G_{m}^{+} \; 0 \; \ldots \; \ldots \; 0 t)$ and $G_{0}^{^{\prime }+}=( G_{m-1}^{+} \; G_{m-2}^{+} \; \ldots \; G_{1}^{+} \; G_{0}^{+} )$.

This system is completed by X _i (k) = X _i + 1(k − 1) for i going from 0 to m − 2 which is equivalent to the following inequalities

$$ \left\{ \begin{array}{l} X_{i+1}(k-1)-X_{i}(k)\leq 0 \\[6pt] -X_{i+1}(k-1)+X_{i}(k)\leq 0 \end{array} \right. $$

for i = 0 to m − 2. The relevant matrix form is as follows:

$$ \left( \begin{array}{cc} H_{01}^{-} & H_{00}^{-} \\[6pt] H_{01}^{+} & H_{00}^{+} \end{array} \right) \times \dbinom{X(k-1)}{X(k)}\leq \dbinom{0}{0}, $$

where the dimension of the matrices $H_{01}^{-}=-H_{01}^{+}$ and $H_{00}^{-}=-H_{00}^{+}$ is ((m − 1).|TR|×m.|TR|). The matrix $H_{01}^{-}$ is a subdiagonal of identity matrices immediately above the main diagonal and the matrix $H_{00}^{-}$ is a main diagonal of negative identity matrices.

Finally, the concatenation of the two systems gives the algebraic form

$$ \left( \begin{array}{r} G^{-} \\[6pt] G^{+} \end{array} \right) \times \binom{X(k-1)}{X(k)}\leq \left( \begin{array}{l} -T^{-} \\[6pt] 0 \\[6pt] T^{+} \\[6pt] 0 \end{array} \right), $$

where $G^{-}=\left( \begin{array}{rr} G_{1}^{^{\prime }-} & G_{0}^{^{\prime }-} \\ H_{01}^{-} & H_{00}^{-} \end{array} \right)$ and $G^{+}=\left( \begin{array}{rr} G_{1}^{^{\prime }+} & G_{0}^{^{\prime }+} \\ H_{01}^{+} & H_{00}^{+} \end{array} \right)$.