Comparison of hyperexponential distribution and other models for positively defined random variables

Due to the necessity of creating complex system models with a large number of possible states (for example, network-centric control systems), there is a need for qualitative and quantitative selection of a model that takes into account essential characteristics of random phenomena from the practical standpoint and is characterized by moderate computational difficulties in application. In the article we compare various models of positively defined random variable with hyper exponential distribution of special Hs -type based on empirical numerical characteristics: expectation and variance. We study the log-normal, gamma, and Weibull-Gnedenko distributions. All these distributions are examined under the parameters providing a decreasing intensity of “failures” and the coefficient of variation being greater than one. The uniform and average metrics are considered in the space of distribution functions as quantitative estimates of the proximity of the Hs -distribution to the rest of the models. The possibility of replacing two-parameter distributions of a positively defined random variable by the hyperexponential Hs -distribution of a special type is shown. Estimates of the effectiveness of such an approximation for various sets of parameters and examples of its application are given. An example of calculating the stationary probability characteristics of a system with server failures is considered, where the basic Weibull-Gnedenko distribution is replaced by the hyperexponential distribution of a special type. According to the criteria considered, for small coefficients of variation Hs -distribution has the best approximation properties in relation to the Weibull-Gnedenko distribution, for large ones - to the log-normal distribution. In general, the larger the coefficient of variation is, the more precisely the possibility of describing the observed random variable by the hyperexponential distribution should be tested. But in case of such approximation admissibility, the use of the hyperexponential distribution greatly simplifies the analytical modeling of complex systems due to the splitting of states into phases, the sojourn time having exponential distributions.