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In this paper a serial, three echelon, push-pull supply chain is investigated. The supply chain consists of a provider, a distribution centre (buffer) and a retailer. The material flow between upstream stages is push type, while between downstream stages it is driven by continuous review, reorder point/order quantity inventory control policy. Exponentially distributed lead times between stages are assumed. External demand occurs according to pure Poisson, while the demand that cannot be met is lost. The system is modelled using matrix analytic methods as a Markov birth-and-death process. An algorithm is developed to generate the transition matrix for different parameters of the system. Then, the corresponding system of stationary linear equations is generated and the solution of the stationary probabilities is provided. Key performance metrics such as average inventories and customer service levels at each echelon of the system can be computed. The algorithm is programmed in Matlab© and its validity is tested using simulation, with the two approaches giving practically identical results. The contribution of our work is an exact algorithm for a lost sales push-pull supply network. This algorithm can be used to evaluate different scenarios for supply chain design, to explore the dynamics of a push-pull system, or as an optimization tool.