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Asymptotic expansions for a renewal-reward process with Weibull distributed interference of chance

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Abstract (Original Language): 
In this study, a renewal-reward process (X(t)) with aWeibull distributed interference of chance is investigated. Under the assumption that the process X(t) is ergodic, two-term asymptotic expansion is obtained for the ergodic distiribution of the process X(t), as  → 0. Also, the weak convergence theorem is proved for the ergodic distribution of the process X(t), as  → 0. Moreover, two-term asymptotic expansions are derived for nth-order moments n = 1; 2; ::: of the process X(t), as  → 0. Based on these results, the asymptotic expansions are obtained for the skewness and kurtosis of the process X(t), as  → 0.
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REFERENCES

References: 

[1] A.A. Borovkov, Stochastic Processes in Queuing Theory, Spinger-Verlag, New York, 1976.
[2] A. Csenki, Asymptotics for renewal-reward processes with retrospective reward structure,
Operation Research and Letters 26 (2000) 201–209.
[3] D. Beyer, S.P. Sethi, M. Taksar, Inventory Models with Markovian Demands and Cost
Functions of Polynomial Growth, Journal of Optimization Theory and Application 98(2)
(1998) 281–323.
[4] F. Chen, Y. Zheng, Waiting time distribution in (T,S) inventory systems, Operation Research
and Letters 12 (1992) 145–151.
[5] F. Chen, Y. Zheng, Sensitivity analysis of an (s,S) inventory model, Operation Research
and Letters 21 (1997) 19–23.
[6] F. Janssen, R. Heuts, T. Kok, On the (R, s, Q) inventory model when demand is modeled
as a compound Bernoulli process, European journal of operational research 104 (1998)
423–436.
[7] G. Alsmeyer, Second-order approximations for certain stopped sums in extended renewal
theory, Advances in Applied Probability 20 (1988) 391–410.
[8] G. Aras, M. Woodroofe, Asymptotic expansions for the moments of a randomly stopped
average, Annals of Statistics 21 (1993) 503–519.
[9] H.C. Tijms, Stochastic Models: An Algorithmic Approach, Wiley, New York, 1994.
[10] I.I. Gihman, A.V. Skorohod, Theory of Stochastic Processes II, Springer, Berlin, 1975.
[11] J.B. Levy, M.S. Taqqu, Renewal reward processes with heavy-tailed inter-renewal times
and heavy-tailed rewards, Annals of Statistics 6(1) (2000) 23–44.
[12] M. Brown, H.A. Solomon, Second-order approximation for the variance of a renewal-reward
process, Stochastic Processes and Their Applications 3 (1975) 301–314.
210
Nurgul Okur Bekar, Rovshan Aliyev, Tahir Khaniyev
[13] N. Okur Bekar, R.T. Aliyev and T.A. Khaniyev, Three-term asymptotic expansions for the
moments of the ergodic distribution of a renewal-reward process with gamma distributed
interference of chance, in: A. Ashyralyev, A. Lukashov (Eds.), First international conference
on analysis and applied mathematics: ICAAM 2012, volume 1470 of AIP Conf. Proc.,
pp.207-210.
[14] N.U. Prabhu, Stochastic Storage Processes, Springer, New York, 1981.
[15] R. Aliyev, T. Khaniyev, N. Okur Bekar, Weak convergence theorem for the ergodic distribution
of the renewal-reward process with a gamma distributed interference of chance,
Theory of Stochastic Processes 15(2) (2009) 42–53.
[16] S.G.J. Johansen, A. Thorstenson, Renewal reward processes with heavy-tailed inter-renewal
times and heavy-tailed rewards, International Journal of Production Economics 30 (1993)
179–194.
[17] S.P. Sethi, F. Cheng, Optimality of (s, S) policies in inventory models with markovian
demand, Operations Research 45(6) (1997) 931–939.
[18] S.M. Ross, Stochastic Processes, 2nd Ed, John Wiley and Sons, New York, 1996.
[19] T. Khaniyev, Z. Kucuk, Asymptotic expansions for the moments of the Gaussian random
walk with two barriers, Statistics and Probability Letters 69(1) (2004) 91–103.
[20] T. Khaniyev, Z. Mammadova, 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) (2006) 861–874.
[21] V. I. Lotov, On some boundary crossing problems for Gaussian random walks, Annals of
Probability 24(4) (1996) 2154-2171.
[22] W. Feller, An Introduction to Probability and Its Applications. II, J. Willey, New York,
1971.
[23] W.L. Smith, Renewal theory and its ramification, Journal of the Royal Statistical Society.
Series B (Methodological) 20(2) (1958) 243-302.

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