Ph.D. Tezi Görüntüleme

Student:	Ali Akbar FATTAHPOUR MARANDİ
Supervisor:	Prof. Dr. İhsan ÜNVER ve Prof. Dr. Tahir KHANİYEV
Department:	Matematik
Institution:	Graduate School of Natural and Applied Sciences
University:	Karadeniz Technical University Turkey

Title of the Thesis:	ASYMPTOTIC EXPANSIONS FOR THE ERGODIC MOMENTS OF SEMI-MARKOVIAN RANDOM WALK WITH A GENERALIZED DELAYING BARRIER
Level:	Ph.D.
Acceptance Date:	4/5/2012
Number of Pages:	80
Registration Number:	Di901

Summary:

ASYMPTOTIC EXPANSIONS FOR THE ERGODIC MOMENTS OF SEMI-MARKOVIAN RANDOM WALK WITH A GENERALIZED DELAYING BARRIER

Karadeniz Technical University

The Graduate School of Natural and Applied Sciences

Mathematics Graduate Program

Supervisor: Prof. İhsan ÜNVER and Prof. Dr. Tahir KHANİYEV

2012, 80 Pages, 9 Pages Appendix

In this study, a number of very interesting problems of stock control, queuing and reliability theories are expressed by means of random walk with various types of barriers. These barriers can be reflecting, delaying, absorbing, elastic, and etc., depending on concrete problems at hand. In this topic, there are many interesting studies in literature.

Here, a semi – Markovian random walk process ( ) with a generalized delaying barrier is considered and ergodic theorem for this process is proved under some weak conditions. Then, the exact expressions and asymptotic expansions for the first four ergodic moments of the process are obtained. However, the results of these studies are not readily applicable to real-world problems probable characteristics of considered process have very complex mathematical structure. To avoid this difficulty, the asymptotic approach method is used in this study to calculate the ergodic moments of the process . Moreover, the characteristic function of the ergodic distribution of the process is expressed by characteristic function of a boundary functional .

Mentioned in sections, the first four initial moments of the boundary functionals, the ergodic distribution function and the ergodic moments of the process is expressed by means of a renewal function, the exact and asymptotic results for these are obtained. Then, using this representation, it is shown that the ergodic distribution of the “standardized” process converges to a limit distribution, when . Finally, the explicit form of the limit distribution is obtained.

Key Words: Semi - Markovian random walk, Delaying barrier, Ergodic distribution, Ergodic moments, Asymptotic expansion, Boundary functional, Ladder epoch, Weak convergence, Ladder height.