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We introduce a new version of particle filter in which the number of "children" of a particle at a given time has a Poisson distribution. As a result, the number of particles is random and varies with time. An advantage of this scheme is that descendants of different particles can evolve independently. It makes easy to parallelize computations. Moreover, particle filter with Poisson resampling is readily adapted to the case when a hidden process is a continuous time, piecewise deterministic semi-Markov process. We show that the basic techniques of particle MCMC, namely particle independent Metropolis-Hastings, particle Gibbs Sampler and its version with ancestor sampling, work under our Poisson resampling scheme. Our version of particle Gibbs Sampler is uniformly ergodic under the same assumptions as its standard counterpart. We present simulation results which indicate that our algorithms can compete with the existing methods.