Far Eastern Mathematical Journal

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Investigation of applied stochastic models by means of random variables indeces

G. Sh. Tsitsiashvili

2003, issue 1, P. 27–35

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 $(w_{n,1}, \ldots , w_{n,m})$, $n \ge 0$, 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 $\ind(w_{n,i})=(m?i+1) \ind(\eta_0), 1 \le i \le m \le n$.


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[1] P. Embrechts, C. Kluppelberg, T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer, Berlin, 1997.
[2] S. Asmussen, Ruin Probabilities, World Scientific, Singapore, 2000.
[3] T. Rolski, H. Schmidli, V. Schmidt, J. Teugels, Stochastic Processes for Insurance and Finance, Wiley, New York, 1999.
[4] V. V. Kalashnikov, D. Konstantinides, Ruin under interest force and subexponential claims: a simple treatment, Insurance: Mathematics and Economics, 27 (2000), 145–149.
[5] V. V. Kalashnikov, Kachestvennyj analiz povedeniya slozhnyx sistem metodom probnyx funkcij, Nauka, M., 1978, 248 s.
[6] V. M. Zolotarev, Sovremennaya teoriya summirovaniya nezavisimyx sluchajnyx velichin, Nauka, M., 1986, 416 s.
[7] D. J. Daley, The moment index of minima. Probability, statistics and seismology, J. Appl. Probab., 38A (2001), 33–36.
[8] J. Kiefer, J. Wolfowitz, On the theory of queues with many servers, Trans. Amer. Math. Soc., 78 (1955), 147–161.
[9] A. Scheller-Wolf, Further delay moment results for FIFO multiserver queues, Queuing Systems, 34 (2000), 38.
[10] G. Sh. Ciciashvili, Kolichestvennaya ocenka sovokupnogo e'ffekta v prostejshix mnogolinejnyx sistemax massovogo obsluzhivaniya, Problemy ustojchivosti stoxasticheskix modelej, Trudy seminara, VNIISI, M., 1988, 140–142 s.
[11] D. V. Lindley, The theory of queues with a single server, Proc. Camb. Phil. Soc., 48 (1952), 277–289.
[12] H. Nyrhinen, Finite and infinite time ruin probabilities in a stochastic economic environment, Stochastic Process. Appl., 92:2 (2001), 265–285.
[13] V. K. Malinovskii, Probabilities of ruin when the safety loading tends to zero, Adv. in Appl. Probab., 32:3 (2000), 885–923.
[14] V. Kalashnikov, R. Norberg, Power tailed ruin probabilities in the presence of risky investments, Stochastic Process. Appl., 98:2 (2002), 211–228.

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