Conversion of Classical Queuing Systems to Systems with Time Lag

Abstract

The results of the classical queuing theory are used to model data transmission systems for various purposes, and the average delay of requests in the queue is used as the main characteristic for this. It is known that the average delay is directly proportional to the variances of random arrival and service intervals, which means its quadratic dependence on the coefficients of variation. Classical QS are applicable only in the case of fixed values of these coefficients of variation, which is a serious limitation. Unlike the classical theory, the article presents the results of research on QS described by the transformed distribution laws with a shift to the right. Such a transformation of the distribution laws with the introduction of a time shift parameter turns Markov QSs into non-Markov systems with delay. Spectral expansions and calculated formulas obtained on their basis for the average delay of requests in queue for sixteen QS with time lag are presented. These systems are obtained using four distribution laws used in queuing theory: exponential, hyperexponential, Erlang and hyper-Erlang. It has been theoretically and practically proven that in systems with time lag, the average delay is less than in classical systems. The resulting calculation formulas for the average delay expand and supplement the capabilities of classical systems. The spectral solutions allows you to calculate the average delay and its higher-order moments for these systems in mathematical packages for a wide range of flow parameters.