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We present a new version of the time-parallel simulation with fix-up computations for monotone systems. We use the concept of monotony of a model related to the initial state of the simulation to derive upper and lower bounds of the sample-paths. For a finite state space with some structural constraints, we prove that the algorithm provides bounds at the first step. These bounds are improved at every fix-up computation steps leading to a natural trade-off between accuracy of the simulation results and efficiency of the parallel computations. We also show that many queueing networks models satisfy these constraints and show the links with the monotone version of the Coupling From The Past technique.