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We consider a generalized joint optimization policy of block replacement & periodic review spare inventory with random lead time. According to the relationship between geometric area in the graph of inventory level over time, and holding or shortage costs, a model analyzing four mutually exclusive & exhaustive possibilities is developed for the expected average cost per unit time, and is based on the stochastic behavior of the assumed system. The model reflects the cost of inventory holding, spare shortage, replacement, and ordering. And for the first time known to the authors, we deliver the sufficient and necessary conditions of the existence and uniqueness of the minimum in the joint models of this type. Because the model and its analysis are general, one existing result is shown to be subsumed by this model with some modifications. Some numerical cases verify the deduction, and give a general searching solution procedure. Finally, we introduce some discussions related to the models. The models mentioned in the paper can be readily applied in many fields such as economical fields, financial engineering, armament administration, and even medical fields, with some modifications. And the mathematical deduction in the paper will be a guideline for analyzing related stochastic models.