In practice, production planning and inventory control, both managed in the assembly systems framework, are often subject to various sources of exogenous uncertainty. In this regard, the present paper focuses on a single-level multi-component inventory control problem for the assembly systems replenishment under stochastic component procurement lead times. In order to be closer to the common assumption of MRP software tools, the case of discrete distributions of component lead times is considered, these latter being thus expressed as a number of periods. Since the finished product is assembled by using several types of component at the same time, the assembly process is stopped even if only a type of component is delayed. The assembly stopping forced by a components delay or stock-out is penalised by backlogging costs. Hence, the problem objective aims to minimize the total cost composed of holding and backlog ones. To address this problem, a joint chance constrained model is proposed and solved via an equivalent linear reformulation, the special structure of which is also deeply discussed. Apart from the effectiveness of the provided equivalent linear reformulation, the practical advantage of the proposed approach resides in its release from backlogging costs, which are often difficult to be quantified in real-life industrial applications.