A policy of periodic replacement with minimal repair at failure is considered for a complex system. Under such a policy the system is replaced at periodic times. iT(i=1,2, $ldots$), while minimal repair is performed at any intervening system failures. The cost of the j-th minimal repair to the component which fails at age t is g(C(t). $c_j$ (t)), where C(t) is the age-dependent random part, $c_j$(t) is the deterministic part which depends on the age and the number of the minimal repair to the component, and g is a positive nondecreasing continuous function. The cost of replacement is expensive when the number of failures occurring in (0. T) is greater than a threshold level. The problem of determining the optimal replacement period, $T^{ast}$, which minimizes the total expected cost per unit time over an infinite time horizon is considered. Various special cases are considered.