Inventory model with hyperexponential distribution of demand's batch size

Consider a mathematical model of inventory control. Let the product flow be continuous with the fixed rate v = 1. Let s(t) be an inventory level at the moment t. The demand occurs according to a Poisson process with piecewise constant intensity X(s), | Xj, s < S, X( s) - X₂, s > S, where S is the threshold inventory level of s (t). The values of purchases are independent and identically distributed random variables from the n-th order hyperexponential distribution with the first moment equal to 1. For stationary distribution probability density function P(s) we obtained the equation да P'(s) + X(s)P(s) = J X(s + x)P(s + x)dB(x), 0 where P(s) satisfies the boundary conditions P(-да) - P(O)) - 0 . Then, the expression for the probability density function P(s) is derived: С x_v 1 У 2-+- v-1 v У. 2 xv e, s < S, P(s) - -у( s-S ) s > S, where z_v and у are positive roots of equations да M* X₂ - у - X₂ J edB( x). 0 z + xj - xj 2 b*-. m* Z Here x_v are components of the vector X, which is a solution of a system of linear algebraic equations AX - h, where A*v are elements of the matrix A, h* are elements of the vector h. The elements A*v and h* have the form X1 X₂ h* -· M* +у V 1 x_v -+2 - у V-1 vy *v " M* v and normalizing constant C is determined by the equation ( С =

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