Equation for probability density of the response of a dynamic system to Erlang renewal random impulse processes

Non-linear oscillators under Erlang renewal-driven random impulses are considered. As the driving counting process is a renewal process, the original state vector of the dynamic system is a non-Markov process. A non-Markov problem is then converted into a Markov one at the expense of introducing an auxiliary variable, a zero-one stochastic jump process. If the underlying Erlang process is characterized by an integer parameter k then the jump process is expressed in terms of k − 1 other auxiliary variables, all governed by stochastic differential equations driven by a Poisson process ( Nielsen, Iwankiewicz and Skjaerbaek 1995 ; Iwankiewicz and Nielsen 1999 ; Iwankiewicz and Nielsen 2000 ). According to the “phase approach” of the queueing theory an Erlang distributed inter-arrival time consists of k “phases”, which are the negative-exponential distributed variables. The “phases” are the Markov states and the scheme of the chain of those Markov states is drawn. The joint probability density-distribution function of the original state variables and of the Markov states of the jump process is governed by the general integro-differential forward Chapman-Kolmogorov equation. Based on the chain of the Markov states of the jump process the jump probability intensity function is defined for a continuous-jump state vector. The equations governing the joint probability density-distribution of the response of a dynamic system are derived. The resulting equations are a coupled set of one partial integro-differential equation and k − 1 partial differential equations. The set of equations for the characteristic function of the response is also derived.