Analysis of the joint probability density function of the inventory in a two machine two buffer transfer line with unreliable machines

The behavior of the inventory for a transfer line with more than one production stage and with failure prone machines is analyzed. The general results are applied to a transfer line with two unreliable machines and two buffers. In this situation, the first machine is never starved but the second machine can be blocked. First, the partial differential equations that describe the evolution of the joint probability density functions of the inventory in each buffer of the transfer line operating under a buffer hedging point policy are derived. Then the problem is classified according to different assumptions on the way the machines fail and their relative maximum production rates. Finally, the partial differential equations are solved using a spatial approximation of the state space that preserves the characteristic lines of the inventory evolution. The numerical solution is then used to derive an estimation for the size of the buffer of finished parts that minimizes an objective function that penalizes part surplus and backlogs