Investigation of applied stochastic models by means of random variables indeces

G. Sh. Tsitsiashvili

Dalʹnevostočnyj matematičeskij žurnal, Tome 4 (2003) no. 1, pp. 27-35

Résumé

Multiserver queuing system $G|G|m|\infty$ and discrete time risk model under stochastic interest force are considered using indeces of random variables. These models have large theoretical and practical interest. Attempts to investigate them using traditional methods lead to large technical difficulties. So an application of random variables indeces which simplifies such an analysis significantly is reasonable. Obtained results give sufficiently exaustive classification of different regimes in considered models. Main result of this article is following: suppose that $(w_{n,1},\dots,w_{n,m})$, $n\ge 0$, is Kiefer-Volfowitz Markov chain describing the system $G|G|m|\infty$ with random serving time $\eta_0$ which has regurlarly varying distribution tail. If index $ind(X)$ of nonnegative random variable $X$ is defined by the formula $ind(X)=\sup\{r:EX^r\infty\}$ then $ind(w_{n,\;i})=(m-i+1)\;ind(\eta_0)$, $1\le i\le m\le n$.