Equation Which Arises in Some Traffic and Queueing System Analysis

摘要

The ERT-method (Equivalent Random Theory) by R I Wilkinson is commonly used in dimensioning and optimization of alternative routing telephone networks. During the process of calculation also very small values may occur. As is known the approximate formulas by Y Rapp cannot be used in such cases (Rahko Sanders). In the paper the authors present a method for numerical calculation of the values of the ERT-parameters, when the overflow traffic is small and peaked. To use this method one must solve an equation of the form 1n(X) - X + C = 0. This equation also appears in some queueing problems, which are considered in the paper. The conditions for the existence of the roots and their boundaries are given. The iterative method for the equation solving is presented. It can be used for calculation of the roots with given accuracy. The authors also give an approximative solution. Some examples showing the accuracy of the methods are presented. The proofs of the theorems are in the appendices.