SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATIONS OF THE MODELS OF QUEUEING THEORY BY THE METHOD OF GENERATIONG FUNCTION

Abstract

The object is mathematical models and their formalization by differential equation systems. The aim is to popularize stochastic models and differential equation systems which solution allows an analytical form. A model formulation and a process of finding a solution to equation systems are of interest. In the queueing theory many models are formalized by systems of linear differential equations with one or more parameters in which distribution of states of queueing systems are unknown functions. In such systems Markov processes are often grounding in the theory of differential equations construction; in a special case postulates of Poisson process are used. Analytical solution of equation systems exists but it is hard to find by traditional methods. In our study we offer a method which allows to find an analytical solution not only for probability distribution but also for moments of any order from one equation system. Description of procedure of differential equation generation for moments of random order varieties is presented. The method is based on the usage of generating (characteristic) functions. This method is effective because it allows to find solutions for moments (here it is expectation and variance) without complex probability calculations. It is especially important in empirical researches of systems that consist of many elements. For example, when we analyze function effectiveness of operating and designed scaling computing systems and supercomputers. Three models and their formalization by differential equation systems that correspond to stochastic processes and analytical solutions of diverse complexity are formulated. Connection between stochastic differential equations systems and their solutions with probability distributions that are classical in probability theory is shown.