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We address the problem of controlling the production rate of a failure prone manufacturing system so as to minimize the discounted inventory cost, where certain cost rates are specified for both positive and negative inventories, and there is a constant demand rate for the commodity produced.The underlying theoretical problem is the optimal control of a continuous time system with jump Markov disturbances, with an infinite horizon discounted cost criterion.We use two complementary approaches. First, proceeding informally, and using a combination of stochastic coupling, linear system. arguments, stable and unstable eigenspaces, renewal theory, parametric optimization etc., we arrive at a conjecture for the optimal policy. Then we address the previously ignored mathematical difficulties associated with differential equations with discontinuous right hand sides, singularity of the optimal control problem, smoothness and validity of the dynamic programming equation etc., to give a rigorous proof of optimality of the conjectured policy.It is hoped that both approaches will find uses in other such problems also.We obtain the complete solution and show that the optimal solution is simply characterized by a certain critical number, which we call the optimal inventory levelo If the current inventory level exceeds the optimal, one should not produce at all, if less, one should produce at the maximum rate, while if exactly equal one should produce exactly enough to meet demand. We also give a simple explicit formula for the optimal inventory level.