Bivariate replacement policy for a system subject to shocks

A bivariate replacement policy for a two-unit system under the shock-effect is studied. Unit 1 is subject to one of the two types of shocks. A type 1 shock causes a minor failure of unit 1, while a type 2 shock makes a catastrophic failure of unit 1. The probabilities of the shock types depend on the number of shocks since the last replacement. Each minor failure of unit 1 brings some cumulative damage to unit 2. When the cumulative damage of unit 2 reaches L, unit 2 fails, which simultaneously causes the failure of unit 1, resulting in a catastrophic failure of system. Moreover, unit 2 with cumula-tive damage x has a minor failure with probability omega eth x THORN at unit 1 failure instant, and this failure is removed by a minimal repair. A more general replacement policy is considered, in which the system is replaced when the cumulative damage to unit 2 has reached l (<= L), or the nth type 1 shock occurs, or when type 2 shock occurs or the accumulative damage to unit 2 exceeds L, whichever comes first. The expression for average cost rate is developed and the corresponding optimal policy is determined analytically and numerically.