| Abstract [eng] |
The method of modelling nonmarkovian systems using phase –type distributions is proposed. If the duration of time between two events in a system has the exponential distribution then the performance of the system can be described bythe continuous time Markov chain. However, in practice this condition is not satisfied. The direct modelling of such systems by continuous time Markov chains is impossible. In such cases an arbitrary continuous distribution of a positive random variable can be approximated with a sequence and mixture of exponential distributions which result in phase-type distribution.Consequently, the originalnonmarkovian system can by approximated by markovian one.The process of constructing models of some queuing systems withproposed method ispresented. The following tasks and problems were solved in this work. First of all, an algorithm for generating continuous time Markov chains of a specific system was developed and tested. The problem of approximation of a general type distributionfunction by the phase-type distribution was investigated. Several optimization algorithms for estimating parameters of phase-type distribution were compared. Also, a research on how approximation accuracy depends on the form of density function was carried out. In order to reduce computation time,innovative technologies were applied. All calculations were performed by programs written in Java and C++ programming languages. The literature of this topic was reviewed. It was noticed that many researchers investigatethe approximation of ageneral distribution function by a phase-type distributionswith a specific structure. In this worka general structure of phase-type distribution was used and a new approach to estimate the optimalparameters of phase-type distribution with the optimization algorithm of local unimodal sampling was proposed. |
