The Reversibility Property of Production Lines

A production line is treated as a series arrangement of k work stations. An unlimited supply of raw production items is available at the first station, and each item passes through all of die stations in sequence. The service time for a single item at station j is assumed to be a random variable with a probability distribution peculiar to that station. In this mode of operation any station will at any time be either busy, or idle, or blocked. A measure of the productivity of such a line is its mean production rate r. It has been conjectured that the production rate remains invariant under reversal of the production line. Line reversal means that every item passes through the stations in the reverse order, that is, beginning with station k and ending with station 1. A general proof of the reversibility property is given. First it is shown that with predetermined service times the total time required to process n dissimilar items through k dissimilar stations does not change when the order of the stations and the order of the items is reversed. Then it is shown for the stochastic case that the order of the items does not affect the production rate.