İki Tutan Bariyerli Yarı-Markovian Rastgele Yürüyüş Sürecinin Bir Sinir Fonksiyonalinin Dağılımı Hakkında

Yıl 2017, Cilt: 7 Sayı: 2, 319 - 329, 15.12.2017

Selahattin Maden

Öz

Bu çalışmada, rastgele yürüyüşün
(1 ;1 )
 
L
Laplace dağılımına sahip olması durumunda, sıfır ve
 ( 0 )  
seviyelerinde tutan bariyerlere sahip bir yarı-Markovian rastgele yürüyüş süreci ve
bu sürecin sıfır seviyesindeki tutan bariyere ilk kez düşme anı, ( ), 0

matematiksel olarak
kurulmuştur. Daha sonra
0

rastgele değişkeninin Laplace dönüşümünün açık bir ifadesi
verilmiştir. Ayrıca bu Laplace dönüşümünü kullanarak,
0

rastgele değişkeninin beklenen değer
ve varyansı için basit formüller elde edilmiştir. 

Anahtar Kelimeler

Yarı-Markovian rastgele yürüyüş süreci, Laplace dağılımı, Tutan bariyer, Beklenen değer, Varyans, Laplace dönüşümü

On the Distribution of a Boundary Functional of the Semi-Markovian Random Walk Process with Two Delaying Barriers

Öz

In this study, a process of semi-Markovian random walk with delaying barriers at
0  and
  levels (
  0
) and first falling moment of the process into the delaying barrier at zerolevel,
( )
0
 , are mathematically constructed, in this case when the random walk happens
according to the Laplace’s distribution
(1 ;1 )
 
L . Then it is given an explicit expression of the
Laplace transformation of the distribution of random variable
0
 . Also the simple formulas for
expectation and variance of random variable
0

are obtained by the means of this Laplace
transformation.

Anahtar Kelimeler

Semi-Markovian random walk process, Laplace distribution, delaying barrier, expected value, variance, Laplace transformation

Kaynakça

• 1. Borovkov, A.A., (1975). On the random walk in the strip with two delaying barriers, Math. Zametki 17, 4, 647-657
• 2. Borovkov, A.A., , (1976). Stochastic Processes in Queueing Theory, Springer-Verlag: New York
• 3. Feller, W., (1968). An Introduction to Probability Theory and Its Applications, Vol. I, Wiley, New York
• 4. Khaniev, T.A., (1984). Distribution of a semi-Markovian Walk with two delay screens, Some question of the theory of stochastic processes, Collect sci. Works, Kiev, 106-113
• 5. Khaniev, T.A., Ünver, I., (1997). The study of the level zero crossing time of a semiMarkovian random walk with delaying screen, Turkish J. of Mathematics 21, 257- 268
• 6. Lotov, V.I,. (1991). On the asymptotic of distributions in two-sided boundary problems for random walks defined on a Markovian chain, Sib. Adv. Math. 1, №o. 3,26-51
• 7. Maden, S., Shamilova, B.Q., (2016). The Laplace Transform of a Boundary Functional of The Semi-Markovian Random Walk Process with Two Delaying Barriers, Ordu Univ. J. Sci. Tech., 6(1), 43-53
• 8. Maden, S., (2016). The Laplace Transform for The Ergodic Distribution of A SemiMarkovian Random Walk Process with Reflecting And Delaying Barriers, Ordu Univ. J. Sci. Tech., 6(2), 243-256
• 9. Maden, S., (2017). On Distribution of A Semi-Markovian Random Walk Process with Two Delaying Barriers, Ordu Univ. J. Sci. Tech., 7(1) , 33-41
• 10. Nasirova, Т. H., (1984). Processes of Semi-Markovian Random Walk, ELM, Baku
• 11. Nasirova, T.I., and Omarova, K.K., (2007). Distribution of The Lower Boundary Functional of The Step Process of Semi-Markovian Random Walk with Delaying Barrier at Zero, Ukrainian Math. Journal,, Vol. 59, No. 7, 1010–1018
• 12. Nasirova, T.I., and Sadikova, R.I., (2009). Laplace Transformation of the Distribution of the Time of System Sojourns within a Band, Automatic Control and Computer Sciences, Vol. 43, No. 4, 190–194
• 13. Nasirova, T.I., and Shamilova, B.G., (2014). Investigation of Some Probabilistic Characteristics of One Class of Semi-Markovian Wandering with Delaying Screens, Automatic Control and Computer Sciences, Vol. 48, No. 2, 109–119
• 14. Nasirova, T.I., Sadikova, R.I., and Ibaev, E.A., (2015). Determination of the Mean and Mean Square Deviations of the System Level, Automatic Control and Computer Sciences, Vol. 49, No. 1, 37–45
• 15. Omarova, K.K., Bakhshiev, S.B., (2010). The Laplace Transform for The Distribution of The Lower Bound Functional in a Semi-Markovian Walk Process with a Delay Screen at Zero, Automatic Control and Computer Sciences, Vol. 44, No. 4, 246-252