Stochastic Multiple Item Inventory System of (S, s) type is studied with general lead time distribution and Poisson demand. The demands occur according to Poisson process for one unit of a specific item at a time. The demand types are characterised by a discrete distribution An order is placed when the inventory level drops to si. Shortages are assumed to be lost. An explicit expression is provided for the marginal distributions of the Inventory level process. Using this, the expected total cost of the inventory system is obtained explicitly. The results for the special care of instantaneous lead time are deduced from the earlier results.