A recursive algorithm for generating the transition matrices of multistation series production lines

This paper is concerned with reliable multistation series production lines, where an operation is performed on each job by the single machine at each station and jobs for the first station arrive according to a Poisson distribution. The processing times of all the stations are exponentially distributed and no buffers are allowed between successive stations. The structure of the transition matrices of these specific types of production lines is examined and a one-stage recursive algorithm is provided for generating them. The transition matrices are block-structured and sparse and by applying the proposed algorithm, one can create the transition matrix of a K-station line from the (K-1)-station transition matrix. This process avoids the necessity of writing down explicitly all the feasible states and transitions of the model, which is tedious and time-consuming especially for long production lines (i.e. for K=12 stations, the number of states is 46, 368.