# Poisson Twister Generator by Cumulative Frequency Technology
Initial sequence contains \(N=2^{w}\) non-repeating numbers distributed uniformly and randomly in interval \([0:2^{w}-1]\). For utilization, the constants \(a\) and \(c\) of the twister generation of the random variable \(x_{i}=(ax_{i-1}+c)mod\ 2^{w}\) obtained from the value of the previous variable \(x_{i-1}\) are used. When the generator completes the creation of one series consisting of \(N=2^{w}\) random variables, it automatically proceeds to the creation of the next twisting series, in which \(N=2^{w}\) elements are obtained as well. So, for each pair of twister constants \(a\) and \(c\), the twisting uniform sequences in quantity \(N_{T}\) could be created:

\[P_{\eta-k}((t,t+\tau])=P_{\eta-k}((0,\tau]).\]4

Consequently, one complete twisting cycle (16) with unchanged \(a\) and \(c\) realizes the following quantity \(N_{s}\) for uniform random variables that allows the creation of the same number of Poisson stochastic variables:

\[P_{\eta-k}((t,t+\tau])=P_{\eta-k}((0,\tau]).\]5

To obtain a complete cycle of all Poisson stochastic series with quantity \(N_{s}=w\cdot 2^{2w}\) of the random variables in each complete twister (17), it is necessary to take into account the varieties of coefficients \(a\) and \(c\) in the twister transformation \(x_{i}=(ax_{i-1}+c)mod\ 2^{w}\). The values of these coefficients must not exceed the interval limit \(a,c\in[0:2^{w}-1]\) of all uniform variables in the single series. Quantity \(N_{a}\) of the coefficient \(a\) is defined as:

\[P_{\eta-k}((t,t+\tau])=P_{\eta-k}((0,\tau]).\]6

During the generation of uniform random variables, the values of the coefficients \(c\) have to be odd, i.e., \(c\ mod\ 2\ \
eq 0\). Their quantity \(N_{c}\) is defined as the following:

\[P_{\eta-k}((t,t+\tau])=P_{\eta-k}((0,\tau]).\]7

Collecting Equations (16)-(19) together, the estimate for the total number \(N_{\text{sac}}\) of the random variables is defined in the following manner:

\[P_{\eta-k}((t,t+\tau])=P_{\eta-k}((0,\tau]).\]8

Finally, the information concerning non-repeatable cycle \(N_{l}\) has the following estimation:

\[P_{\eta-k}((t,t+\tau])=P_{\eta-k}((0,\tau]).\]9

As an example, let us get the real values of these estimations for the bit length \(w=32\). In this case, each series contains \(N=2^{32}=4294967296\) Poisson stochastic variables, and in each of these series Poisson distribution is observed absolutely. In this case, the total number of generated Poisson stochastic variables is \(N_{sac}(w=32)=32\cdot 2^{4\cdot 32-3}=2^{5}\cdot 2^{125}=2^{130}\), and the value of the non-repeatable cycle (21) is defined as \(N_{l}(w=32)=2^{2^{5}2^{130}}=2^{2^{135}}\). Therefore, the complete discrete simulation of the random variables with Poisson distribution using the aforementioned basic twister generator confirms the important properties in terms of mathematical expectation and dispersion. Moreover, an automatic extension of the series of initial uniform random variables extends significantly the periods of non-repeatability (Equations (20) and (21)) for the series of Poisson stochastic variables. This notably exceeds the features of the known generators, which bases the probabilistic convergence algorithm. ## 5 Conclusions

An analysis of the source material shows that algorithms for generation of Poisson stochastic variables according to the probabilistic convergence technology lead to different realizations of distributions. Moreover, the skipping of elements in such distributions may also occur, which is inappropriate in the theory of Poisson stochastic processes. Since the quality of the generators depends on the basic generations used for uniform random variables, it has been proposed here to use the twister generator _nsDeonYuliCPoissonTwist32D_ to ensure the completeness of technology of the cumulative array of frequencies in Poisson space. This approach allows for reducing the time of operations and improving the quality of generation.