Complex systems usually fail through a gradual deterioration of component subsystems until a critical subsystem failure renders the unit inoperative. Given exponential life distributions of chance failures for individual subsystems, absorbing Markov chains capture the statistics of this deterioration. Dependent subsystem failures are readily accomodated. Matrix formulations yield estimates of the system mean time to failure for all states of deterioration. Examples illustrate several distinct forms of subsystem interdependencies; and their implications for systems design, maintenance and repairs are examined.