 Applied Mathematics, 2011, 2, 908-911 doi:10.4236/am.2011.27122 Published Online February 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Definition of Laplace Transforms for Distribution of the First Passage of Z er o Lev el o f th e Semi-Mark o v Ra nd om Process with Positive Tendency and Negative Jump Tamilla I. Nasirova, Ulviyya Y. Kerimova Baku State University, Baku, Azerbaijan E-mail: shaxbazi_a@yahoo.com, ulviyye_kerimova@yahoo.com Received May 5, 2011; revised May 26, 2011; accepted May 29, 2011 Abstract One of the important problems of stochastic process theory is to define the Laplace transforms for the distri-bution of semi-markov random processes. With this purpose, we will investigate the semi-markov random processes with positive tendency and negative jump in this article. The first passage of the zero level of the process will be included as a random variable. The Laplace transforms for the distribution of this random variable is defined. The parameters of the distribution will be calculated on the basis of the final results. Keywords: Laplace Transforms, Semi-Markov Random Process, Random Variable, Process With Positive Tendency And Negative Jumps1. Introduction There are number of works devoted to definition of the Laplace transforms for the distribution of the first pas-sage of the zero level. (Borovkov 2004)  defined the explicit form of the distribution, while (Klimov 1996)  and (Lotov V. I.)  indicated implicit form of the dis-tribution of the first passage of zero level. The presented work explicitly defines the Laplace transforms for the unconditional and conditional distribution of the semi-markov random processes with positive tendency and negative jump. 2. Problem Let’s assume that k and k, random variables 1kof independent random variable sequence 1,kkkevenly distributed in  probability face are given. Using these random variables we will derive the following semi-markov random process: ,,()FP11kiiXtz t, if 111ξξ1,kkiiiitk  X(t) process is the (asymptotic) semi-markov random processes with positive tendency and negative jump. Let’s include the01 random variable defined as be-low: 01min :0tXt where 01, is the time of the first passage of X(t) process. We need to find Laplace transform for distribution of 01 random variable. 3. Definition of Laplace Transform for the Distribution of 01Random Variable Let us set Laplace transform for the distribution of 01 random variable as L: 01010,0, 0, LEeLzEe Xzz1 In this case we can express the equation as 1,1 101110,, 0, zTz Thus, T and 01 are evenly distributed random vari-ables. Our goal is to find Laplace transform of relative and non-relative distribution of 01 random variable. According to the formula of total probability, we can put it as T. I. NASIROVA ET AL.909  00 111 111 11:0 :0|0 dddTzzEeXze PePeP     If to consider the following substitution 11;;ss yT we derive   011101100 01101100110110|0d;dd;d; ddddddPddssyzszs ssyssyzszsssyssssEeXze PsyePsyTePs PyePyPsLzsyeP szseLPzsP       0zss or 1101100ξddξdsszsssyLzeP zsP seLzsyPyPs (1) Let’s assume that zsy . In this case we will receive the following integral equation: 110100ξdddξdsszsssLzePzsP seL PzsP 1s (2) We will solve this integral equation in special case. Let’s assume that 1 random variable has the Erlan-gian distribution of m construction, while 1ξ random variable has the single construction Erlangian distribu-tion:   1211ξω11, 0,2!1 !1(), 0.mttPttttetmPtet t   where 0, 0() 1, 0.ttt In this case Equation (2) will be as follows: 100dd(1)!mzmmzsszmsLz eee semL  s (3) We can derive differential equation from this integral equation. For this purpose, we will multiply both sides of Equation (3) by ze: 100dd(1)!mzmmzssmseL zeseLm   s If we increment both sides by z, we will get: 10d(1)!zzmz smsseLzeLzeeseLzsm  s In this case we will receive the differential equation with (m+1) construction:  10 11mkkkmkmk mkzmzmCL zLzeeLz   (4) The general solution of this differential equation will be t 1212zzmkkmzC LeCeCe zk (5) By finding ,,mCC1 from Equation (3) we will get the following systematic equations:  1000' 1011000 d1!00 d1!001 dmm ssmmssmsmsmmkk xkmmxkLesemLL esLssmCLLe Lxx        dLs (6) Copyright © 2011 SciRes. AM T. I. NASIROVA ET AL. 910 By exploitation of Equation (5), Equation (6) becomes   1000 11011 111001 01 1!d1!1diiimmm msskmi imsk immm mxkxmii iisii imm mmmxkxkm mmmi iixki iiCesemCkCex CexmCkCkCeCe            Cex (7) After simplification of the last system we will get    1111 1111 111! 1!1!1!1!1mmmmiimmiiiimmm mii iimii iimmm mmkm mmimi imii iiiCm mCmkkmCk CCmkCCkC kCk               m (8) or     11101()00mmmmimimmmiiimmiimiimmkmmi imkikCkkkCkCk kCk         (9) By exploitation of  (mmiikk  Equation (9) becomes      1110100mmmmiimmimiiiimkmmi iikkCkkCCkkk C         or Copyright © 2011 SciRes. AM T. I. NASIROVA ET AL. Copyright © 2011 SciRes. AM 911 1110000mmmiiimiimiikCCC     (10) Thus, (9) is a linear dependence equation system, as   23110mmmCC CCk   Then the general solution of integral Equation (3) will be as follows: 1111zzmkkmLzC eek  This expression is the Laplace transform for relative distribution of 01 random variable. Then, we will need to find L. In accordance with formula of total prob-ability, z0LzdPX0Lz and as the distribution of X(0) and 1 random variables are same,   1111z011z011z011dP 0d1!d1!zzkmkmzmkzmmmLCeXzCe zemCezzmCkz Therefore,   11mmLCk This expression is the Laplace transform for non-rela- tive distribution of 01 random variable. Respectively, we will get the following characteristics for ߣm > ߤ:     0132 3223322012232232201012303021 (300 21 1z1( )zmEL mmm mLmmmmmmmDL Lmmmmmm mmmmzEmmm mzmDmm       4. Conclusions In this article we have defined Laplace transforms for relative and non-relative distribution of the first passage of zero level of semi-markov random process with posi-tive tendency and negative jump. 5. References  A. A. Borovkov, “On the Asymptotic Behavior of the Distributions of First-Passage,” Mathematics and Stati- stics, Vol. 75, No. 1, 2004, pp. 24-39. doi:10.1023/B:MATN.0000015019.37128.cb  V. I. Lotov, “On the Asymptotics of the Distributions in Two-Sided Boundary Problems for Random Walks Defined on a Markov Chain,” Siberian Advances in Mathematics, Vol. 1, No. 3, 1991, pp. 26-51.  G. P. Klimov, “Stochastic Queuening Systems,” Nauka, Moscow, 1966.  T. I. Nasirova and R. I. Sadikova, “Laplace Trans- formation of the Distribution of the Time of System Sojourns within a Band,” Automatic Control and Computer Sciences, Vol. 43, No. 4, pp. 190-194. doi:10.3103/S014641160904004X