Nested renewal processes with Special Erlangian densities

We consider the class of nested renewal processes in which all densities are Special Erlangian. We first derive explicit expressions for the transient and steady-state distributions of the accumulated number of shocks and for their mean and variance. We also consider situations with infrequent replacements. The transient and asymptotic distributions of accumulated damage are also studied. When the shape parameter of the second-order renewal distribution is large, so that replacement times are approximately normally distributed, our results show that the steady-state distribution of the accumulated number of shocks is approximately uniform. Some numerical examples are presented.