By Beno?te de Saporta, François Dufour, Huilong Zhang
Mark H.A. Davis brought the Piecewise-Deterministic Markov procedure (PDMP) classification of stochastic hybrid types in an editorial in 1984. at the present time it's used to version various complicated structures within the fields of engineering, economics, administration sciences, biology, net site visitors, networks and plenty of extra. but, regardless of this, there's little or no within the method of literature dedicated to the improvement of numerical tools for PDMDs to resolve difficulties of useful value, or the computational keep an eye on of PDMPs.
This publication accordingly provides a suite of mathematical instruments which were lately built to take on such difficulties. It starts off by means of doing so via examples in different software domain names equivalent to reliability. the second one half is dedicated to the examine and simulation of expectancies of functionals of PDMPs. ultimately, the 3rd half introduces the advance of numerical ideas for optimum regulate difficulties akin to preventing and impulse keep watch over problems.
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Additional info for Numerical methods for simulation and optimization of piecewise deterministic Markov processes: application to reliability
2. 3] reflects that the temperature converges to an equilibrium state as long as there is some incoming fluid. The temperature can increase to the threshold 100◦ C if there is no incoming fluid. 9261◦C, so that the system is at an equilibrium state, and nothing happens until the failure of one of the components. The system stops as soon as one of the top events is reached: dry out (h < 4 m), overflow (h > 10 m) or hot temperature (θ > 100◦ C). 2. PDMP model As in [ZHA 08], we model the tank by a PDMP.
Moreover, the direction of the factory has decided that, whenever the machine has worked for a whole year without requiring reparation, it is sent to the workshop for maintenance. We assume that after a reparation or a maintenance, that both last a fixed time s, the machine is totally repaired and is not worn down. We, therefore, consider three modes: the machine is working (m = 1), being repaired (m = 2) or undergoing maintenance (m = 3). The state of the process at time t will be denoted by Xt = (mt , ζt ) where ζt is the time since the last change of mode.
If Tn (ω) < +∞, set: Sn+1 (ω) = F ← (XTn , Un (ω)), Tn+1 (ω) = Tn (ω) + Sn+1 (ω). If Tn+1 (ω) < +∞, choose XTn+1 with distribution Q(·, Φ(XTn , Sn+1 )). The trajectory is finally constructed by induction. With this construction, we can bound the probability that the jump times are smaller than a given threshold. – Let H be a survival function such that for all t ∈ R+ and for all x ∈ E, H(t) ≤ F (x, t). Then, there exists a sequence of independent random variables (Sn )n∈N with distribution H and such that for all T > 0 and N ∈ N: P(TN < T ) ≤ P(TN < T ), where TN = N n=0 Sn .