Application of piecewise-linear approximation of probabilistic-time characteristics of queuing systems

Nowadays many modern industries depend on satellite technologies to a greater or lesser extent. To build such satellite communication systems it is necessary to have estimated parameters of high service quality, one including the information delay time. To implement a mathematical model of traffic distribution in a satellite network, such analytical expressions for time delays are used, which have a discontinuity of the second kind at the moment when the arrival rate becomes equal to the service rate. Removal of this discontinuity can reduce the time required for calculating optimal routes. To achieve this goal, a piecewise linear approximation is used. As a way of specifying line segments, two approaches are considered, which are compared using the integral least squares method and the issue of the number of lines used in the conditions of the problem is also considered. As a result, approximation dependences were obtained, which allows plotting a piecewise-linear function of the mean waiting time in the buffer for the M/M/1 system. The procedure for finding the optimal parameters for this function is described and the analytical method was used to obtain approximate formulas for finding the tangency points of depending on the incoming traffic intensity.