Redundant Systems with Appreciable Exchange Time

In dealing with redundant systems, the time required to exchange the failed unit for a spare one is often assumed to be negligible and only the time required for repair is considered. If exchange time is indeed appreciable, numerical methods are applied to analyze the system because it is usually thought to be too complicated to be solved analytically. This paper proposes an analytic solution, in which states of the system are grouped into rows; states having the same number of up-units are in the same row. Steady state probability of each state is expressed in terms of the probabilities of the states in the same row and in the row immediately above. Through these recursive relationships and simple substitutions, the steady state probabilities are solved.