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Martingale approach for tail asymptotic problems in the generalized Jackson network

Miyazawa M.

Probability and Mathematical Statistics

2017

Vol. 37, Fasc. 2

395--430

We study the tail asymptotic of the stationary joint queue length distribution for a generalized Jackson network (GJN for short), assuming its stability. For the two-station case, this problem has recently been solved in the logarithmic sense for the marginal stationary distributions under the setting that arrival processes and service times are of phase-type. In this paper, we study similar tail asymptotic problems on the stationary distribution, but problems and assumptions are different. First, the asymptotics are studied not only for the marginal distribution but also the stationary probabilities of state sets of small volumes. Second, the interarrival and service times are generally distributed and light tailed, but of phase-type in some cases. Third, we also study the case that there are more than two stations, although the asymptotic results are less complete. For them, we develop a martingale method, which has been recently applied to a single queue with many servers by the author.

A note on impotance sampling simulation for germ-grain model

Palmowski Z.

Probability and Mathematical Statistics

2003

Vol. 23, Fasc. 2

293--304

In this paper we demonstrate how to use the importance sampling method to simulate rare events in a germ-grain model. We analyze conditions under which two germ-grain models are mutually absolutely continuous. We also find the likelihood set process. We apply these results in simulating the probability that the radius of the occupied component of the origin in continuous percolation is greater than some R. This method is based on the reduction of the variance of estimator.

Simulation of Pickands constants

Burnecki K., Michna Z.

Probability and Mathematical Statistics

2002

Vol. 22, Fasc. 1

193--199

Pickands constants appear in the asymptotic formulas for extremes of Gaussian processes. The explicit formula of Pickands constants does not exist. Moreover, in the literature there is no numerical approximation. In this paper we compute numerically Pickands constants by the use of change of measure technique. To this end we apply two different algorithms to simulate fractional Brownian motion. Finally, we compare the approximations with a theoretical hypothesis and a recently obtained lower bound on the constants. The results justify the hypothesis.