Properties of Hyperexponential and Hypererlangian Distributions

The article considers two laws of distribution: hyperexponential and hypererlangian as input distributions for queuing systems. These two laws of distribution belong to a general form with a wide range of variation of the coefficients of variation. The main unique feature of these laws of distributions consists that they can unambiguously be defined by both two first initial moments, and three moments. An approach to the approximation of arbitrary distribution laws by these laws is proposed both at the level of two initial moments and at the level of three moments. The considered laws of distributions, starting from some values of the coefficients of variation, have a heavy tail, and, consequently, in queuing systems they will give a greater delay than other laws of distributions. Practical calculations show that approximation using two moments gives somewhat underestimated results in terms of average waiting time. The use of these laws of distributions in the queueing theory extends and complements the well-known incomplete formula for the average waiting time for systems of mass service with arbitrary laws of the distribution of intervals of input requirements and service time. The results obtained are important for modern teletraffic theory.