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research supported by NSF Stochastic Hybrid Systems Applications to Communication Networks Jo o P Hespanha Center for Control Engineering and Computation University of California at Santa Barbara Deterministic Hybrid Systems continuous dynamics guard conditions reset maps q t Q 1 2 discrete state continuous state x t Rn right continuous by convention we assume here a deterministic system so the invariant sets would be the exact complements of the guards 1 Stochastic Hybrid Systems continuous dynamics transition intensities instantaneous rates at which transitions occur reset maps NA t N transition counter which is incremented by one each time the Ath reset map A x is activated number of transitions on t1 t2 right continuous by convention equal to integral of transition intensity A x on t1 t2 Stochastic Hybrid Systems continuous dynamics transition intensities instantaneous rates at which transitions occur reset maps Special case When all A are constant transitions are controlled by a continuous time Markov process q 2 q 1 specifies q independently of x q 3 2 Formal model Summary State space q t Q 1 2 discrete state continuous state x t Rn Continuous dynamics Transition intensities Reset maps one per transition intensity of transitions Results 1 existence Under appropriate regularity Lipschitz assumptions there exists a measure consistent with the desired SHS behavior 2 simulation The procedure used to construct the measure is constructive and allows for efficient generation of Monte Carlo sample paths 3 Markov The pair q t x t Q Rn is a Piecewise deterministic Markov Process in the sense of M Davis 1983 Example I TCP congestion control transmits data packets server receives data packets r network client packets dropped with probability pdrop congestion control selection of the rate r at which the server transmits packets feedback mechanism packets are dropped by the network to indicate congestion 3 Example I TCP congestion control transmits data packets server



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