# Poisson Twister Generator by Cumulative Frequency Technology
In the theoretical Poisson distribution, this situation is not allowed. For some applications that are not limited to strict constraints, this could be neglected. However, if the generator is used for the comprehensive modeling of real stochastic situations, it is better to avoid this. The second disadvantage is apparent if Knuth generator is launched repeatedly. The monitoring of the values in columns \(pK\) and \(nuK\) registers their inconsistency. In probability theory by Kolmogorov axiomatics [2; 23] a change in the probabilistic measure in a given space under any circumstances is categorically prohibited. The disturbance of the axioms of a space can make it difficult to interpret the results. This may be crucial when repeated tests are required, for example, in emergency situations, i.e. when it is necessary to reiterate the special cases. The third limitation of Knuth algorithm is that counter \(gc\), which summarizes the number of generated uniform random variables, turned out to be 197025. The cycle of Knuth generator is arbitrary; hence the value of \(gc\) should also be arbitrary. In the current case, value \(gc=197025\) is almost three times greater than the number \(N=65536\) of uniform random variables, from which [PERSON] algorithm generates Poisson stochastic variables. This entails uncontrolled repetitions and skipping of basic random variables together with a loss of their uniformity, which ultimately leads to an insufficient quality for the results obtained. So, summing up all the aforementioned points, the aim of this article is to propose a generator of stochastic variables in strict accordance with the theory of Poisson distribution by having no excessive and intermediate generations of uniform variables. This is the next step in searching for better algorithms for Poisson stochastic generation and their optimization. ## 2 Theory

Poisson stochastic process operates with random variables that linearly depend on a continuous parameter. Usually, such a parameter is the observation time of a stochastic event, but other interpretations are possible as well. In this particular case, an interest is represented both by the time moment \(t\) and by the time interval \(\tau\) following it. The randomness of events in two successive continuous time intervals \([0,t]\) and \((t,t+\tau]\) implies [23] that random events in the quantity \(\eta\) could occur during the total time duration \([0,\;t+\tau]\). If events in the quantity \(k\) are observed in the interval \([0,t]\), then a diminution \(\eta-k\) for these events should occur in the half-open interval \((t,t+\tau]\). The first axiomatic restriction is due to the fact that both intervals are independent, and also that the events in them separately occur as the substantive cases. The consequence of this restriction is that the probability of observing the events \(\eta\) on the common interval \([0,t+\tau]\) is the joint probability of independent events:

\[P_{\eta}([0,t+\tau])=P_{k}\big{(}[0,t]\big{)}\cdot P_{\eta-k}((t,t+\tau]) \big{)}. \tag{2}\]

In Poisson model the probability \(P_{\eta-k}((t,t+\tau])\) has a number of serious limitations, which can be formulated as follows:

1. The probability of events in the time interval \((t,t+\tau]\) does not depend on its origin \(t\): \[P_{\eta-k}((t,t+\tau])=P_{\eta-k}((0,\tau]).\] (3)
2. The probability of one event in the time interval \((0,\tau]\) depends linearly on the length of the interval \(\tau\) with given intensity \(\lambda\); and the probability \(o(\tau)\) of observing other events is negligible: \[P_{1}((0,\tau])=P_{1}(\tau)=\lambda\cdot\tau+o(\tau).\] (4)

In Equation (4) the notation \(\alpha=\lambda t\) (1) is used, which in the theory of probabilities [2] and the theory of random processes is in common use. Equation (4) excludes an observation of two or more events simultaneously in the time interval \(\tau\).