Inventory replenishment practices play significant role in supply chain networks and they are strongly associated with overall supply chain performance. This paper presents a state-space model of a stochastic series multi-node supply chain, controlled via local proportional-integral inventory replenishment policies. The model is triggered by identically distributed and uncorrelated random variables which represent uncertain customer demand. The proposed model is analyzed under stationarity, provided that control parameters belong to a certain range which is clearly specified. Simple recursive techniques are developed to calculate explicitly the associated covariance matrix in closed parametric form. This allows to analyze the effect of control parameters in the inventory practices on the bullwhip effect. It is shown that proportional-integral replenishment practices can be applied to explain in qualitative terms the main performance and stability properties of supply chains.