Queuing systems with semi-Markov flow in average and diffusion approximation schemes
A semi-Markov \((SM/M/1/\infty)\) flow including the Markov and the renewal flows as particular cases are studied. By means of using an asymptotic average and a diffusion approximation of the fluctuations for these systems, some quantitative results are obtained. Among others, the fluctuation of the counting process for these systems are derived. The so-called diffusion approximation in these series scheme is given. The main result of this paper is given in the proof of the diffusion approximation of this queueing process. With a large number of places, say \(N\;(N \rightarrow \infty)\), for a fixed time \(t\), it can be approximated by the variable \(N q t + N^{1/2}t\sigma Z\), where \(Z\) is the standard normal random variable, i.e. with the mean zero and variance 1. The method presented in this paper is general and could be used to obtain numerical solutions considered as the Markov flow, and the semi-Markov renewal flow. These results are in my opinion very valuable and suggest the possibility of the numerical solutions for these problems.
applications of Markov renewal processes
scientific article; zbMATH DE number 5565142