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The paper is devoted to investigation of numerical algorithms for modeling of non-stationary random processes on the basis of approximation of the smoothed real processes by piecewise linear functions. The special class of non-stationary processes on Poisson flows having piecewise constant and piecewise linear trajectories, in addition the trajectories of this process are constructed with use of random walk model. The relative length of an overrun of a Poisson flow of points for the given border was considered. The exact formulas expressing dependence on time of probability density for this random variable and its expectation are received. On the basis of these formulas the formulas for expectation of considered processes are deduced. Asymptotic linearity of expectations of these processes is proved.
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