The problem of inventory control for a system operating in steady state is considered. Order strategy is found, which results in minimal costs of the average expenses of the system. Stationary distribution of the basic Markov process is used while finding it. The parameters that determine the optimal strategy are found in explicit form for the ( , ) s S -strategy.We will analyze the inventory control problem for some system operating in steady state mode. For ( , ) s S -strategy, we will find, in explicit form, values of the parameters that determine the optimal strategy.Let replenishment process be the sequence X X 1 2 , , ... of independent equally distributed positive random variables. Assume that the general distribution function F( ) × has density j( ) × , { } X i is time between two serial moments of inventory replenishment, T n is the nth partial sum { } X i , i.e., time when the nth replenishment takes place. Find T 0 0 º and N t , which is the greatest value of n, for which T t n £ , for any positive number t. In terms of replenished inventory, N t is the number of replenishments on the interval [ , ] 0 t . Expected value N t is called the number of replenishments, is denoted by M t ( ), and satisfies the equations [1, 2]where F ( ) ( ) n t is n-fold convolution F( ) t from itself. Assume that F( ) t is absolutely continuous, then we can differentiate M t ( ) and obtain replenishment density m t ( ) satisfying the equationsm t t n n ( ) ( ) ( ) = = ¥ å j 1 . Let us find additional random variable g t N T t t º -+1
