ON THE INVENTORY MODEL WITH TWO DELAYING BARRIERS

: In this paper the process of semi-Markovian random walk with negative drift under angle ) 90 0( o o < α < α , and positive jumps with probability )1 0( < ρ < ρ having two delaying screens at level zero and )0 a( a > is constructed. The exact expressions for Laplace transforms of the distributions of the first moments in order to reach to these screens by the process and, in particularly, the expectations and the variances of indicated distributions are obtained.

Abstract: In this paper the process of semi-Markovian random walk with negative drift under angle , and positive jumps with probability ) 1 0 ( < ρ < ρ having two delaying screens at level zero and ) 0 a ( a > is constructed. The exact 1. INTRODUCTION It is known that a numerous interesting problems in the fields of reliability, queuing, inventory theories, biomedicine etc., are given in terms of the stochastic processes with discrete chance interference. Particularly, these problems can often be modeled by using random walk with one or two barriers. There is large literature on theory and application about random walk with one or two barriers. For example, Spitzer (1964), Feller (1966), Skorohod & Slobedenyuk (1970), Borovkov (1975), Korolyuk & Turbin (1976), Nasirova (1984Nasirova ( , 1999, Lotov (1991), Alsmeyer (1992), El-Shehaway (1992), Zhang (1992), Khaniev & Ozdemir (1995,1998, Khaniev & Kucuk (2004), Unver (1997) , and positive jumps with probability ) 1 0 ( < ρ < ρ having two delaying screens at level zero and ) 0 a ( a > is constructed. The exact expressions for Laplace transforms of the distributions of the first moments in order to reach to these screens by the process, in particularly, the expectations and the variances of indicated distributions are obtained.  1 ζ and probability ρ is stored at the warehouse and then the new demand is served. Since storage capacity in the warehouse is limited, amount of inventory in the warehouse can't be greater than a. In this case, in the beginning of the inventory cycle (right after quantity order 1 ζ of size is received), the expected stock level is

DESCRIPTION OF THE PROBABILISTIC MODEL
. After serving all the customers, inventory in the warehouse is new arriving customer service is started after an order of size ) , a min( 0 1 ζ is received. The level of the stock is a stochastic process and we denote it by X(t). For this inventory model, it is important to determine the distributions of the first moments of the exhaustion and to fill the warehouse.

STRUCTURE OF THE PROCESS AND MATHEMATICAL STATEMENT OF THE PROBLEM
is a sequence of independent and identically distributed random variables defined on probability space are independent random variables. Construct the following process: . Delaying the process ) t ( X 1 with screen "0", we have ))) t ( and delaying the process ) t ( X 2 with screen "a", we have The process ) t ( X is called the process of semi-Markovian random walk with negative drift under the angle ) , positive jumps with probability ) 1 0 ( ≤ ρ ≤ ρ and two delaying screens at the level of zero and a (a>0). The first moments for reaching screens at the level of zero and a (a>0) by this process can be obtained by writing (4.1) Using the Wald equality from (3.1), (4.1.), (4,2) and (4.3), we obtain We denote the generating function of the random variable 0 1 ν by After that, by using the total probability formula for expectations, we have Using total probability formula for 2 k ≥ , we have Multiplying both sides of this equality by ) and after a simple transformation we obtain Suppose that the distribution of the random variable 0 1 ξ has the density function 0 x ), x ( P 0 1 > ξ and the distribution of the random variable 1 ζ has the density function . Then (4.4) can be written as follows: (4.5) We can solve this equation by using the Erlang class of distributions.
Using this distribution we can write expression (4.5) as From this integral equation we have the following differential equation are the roots of the characteristic equation The functions ) u ( c 1 and ) u ( c 2 can be obtained from the following boundary conditions (4.12) Using the fact that the distribution of the random variable ) 0 ( X coincides with the distribution of the random variable ) , a min( 1 ζ and in addition to this using the distribution of the random variable 1 ζ , which has a jump with of size is given by (4.8) together with (4.11) and (4.12). Substituting (4.8) into (4.13) and using the expression u ) 0 , u ( 0 = Ψ that was obtained earlier, we finally have (4.14) Now we can find the Laplace transform of the random variable 0 1 τ . In fact, the random variable 1 ξ has the exponential distribution, and we have ) Then we can write (4.2) as ) . Finally substituting (4.14) into (4.15), we get Note that, these formulas can also be obtained from (4.16). As the random variable 1 ξ is exponential the distributed, we have The expressions for ) 1 ( 0 Ψ′ and ) 1 ( 0 Ψ ′ ′ can be found by using (4.14). But we will use (4.13) to determine these derivations easily. Firstly, we have to determine expressions for 2 , . These expressions can be obtained from (4.9), (4.11) and (4.12) at u=l (Table 5.1).
By differentiating (4.13) with u=1 and substituting obtained expressions for ) Using the total probability formula for 2 k ≥ and multiplying both sides of this equality by ) 1 u 0 ( u k ≤ < , summing up for ∞ = , 2 k , and transforming, we obtain (6.4) Analogous to the Sec. 5 by using the distributions of the random variables 0 1 ξ and 1 ζ , which are given by (4.8) we can write (6.5) From this integral equation we have the following differential equation  Inserting (6.7) together with (6.8) into (6.9), we have Finally substituting (6.10) into (6.2), we have By differentiation of (6.9) with 1 u = and substituting the obtained expressions for 8. NUMERICAL RESULTS This section presents numerical results which are obtained by using Matlab5.    Varτ are calculated using formulas (5.8) and (5.9). Results are given in Table 8 Eτ is much greater in this case than that of µρ > λ . and with two delaying screens at level zero and ) 0 a ( a > is constructed. The exact expressions for Laplace transforms of the distributions of the first moments of reaching these screens by this process and in particularly the expectations and the variances of indicated distributions are obtained. The results obtained in this study can be practiced in queering and reliability theory.