ON THE SEMI-MARKOVIAN RANDOM WALK WITH DELAY AND WEIBULL DISTRIBUTED INTERFERENCE OF CHANCE

In this paper, a semi-Markovian random walk with delay and a discreteinterference of chance (X(t)) is considered. It is assumed that therandom variables {ζn} , n ≥ 1 which describe the discrete interferenceof chance have Weibull distribution with parameters (α, λ), α > 1, λ >0. Under this assumption, the ergodicity of this process is discussed andthe asymptotic expansions with three terms for the first four momentsof the ergodic distribution of the process X(t) are derived, when λ → 0.Moreover, the asymptotic expansions for the skewness and kurtosis ofthe ergodic distribution of the process X(t) are established.

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[1] Aliyev, R.T., Khaniyev, T.A and Kesemen, T. Asymptotic expansions for the moments of a semi-Markovian random walk with gamma distributed interference of chance, Communications in Statistics-Theory and Methods, 39 (1), 130–143, 2010.
[2] Aliyev, R., Kucuk, Z. and Khaniyev, T. Three-term asymptotic expansions for the moments of the random walk with triangular distributed interference of chance, Applied Mathematical Modelling, 34 (11), 3599–3607, 2010.
[3] Anisimov, V.V. and Artalejo, J.R. Analysis of Markov multiserver retrial queues with negative arrivals, Queueing Systems: Theory and Applic, 39 (2/3), 157–182, 2001.
[4] Borovkov, A.A. Stochastic Process in Queueing Theory, (Springer, New York, 1976) .
[5] Feller, W. Introduction to Probability Theory and Its Appl. II,( New York, 1971 ).
[6] Gihman, I.I. and Skorohod, A.V. Theory of Stochastic Processes II, (Berlin, 1975 ).
[7] Khaniyev, T.A. and Mammadova, Z. On the stationary characteristics of the extended model of type (s,S) with Gaussian distribution of summands, Journal of Statistical Computation and Simulation,76 (10), 861–874 , 2006.
[8] Khaniyev, T.A., Kesemen, T., Aliyev, R.T. and Kokangul, A. Asymptotic expansions for the moments of a semi-Markovian random walk with exponential distributed interference of chance, Statistics & Probability Letters, 78 (6), 785–793, 2008.
[9] Lotov, V.I. On some boundary crossing problems for Gaussian random walks, The Annals of Probability, 24 (4), 2154–2171, 1996.
[10] Rogozin, B.A. On the distribution of the first jump, Theory Probability and Its Applications, 9 (3), 498–545, 1964.