Bounds of the quantile entropy in the past lifetime of some ageing classes are explored firstly. The quantile entropy in the past lifetime of a random variable is shown to be increasing if its expected inactivity time is increasing. Some closure properties of the less quantile entropy in the past lifetime order are obtained under the model of generalized order statistics. Moreover, sufficient conditions are given for a function of a random variable and for a weighted random variable to have more quantile entropy in the past lifetime than original random variable.

In the study of the reliability of technical systems in reliability engineering, coherent systems play a key role. In this paper we consider a coherent system consisting of n components with independent and identically distributed components and propose two time-dependent criteria. The first criterion is a measure of the residual lifetime of live components of a coherent system having some of the components alive when the system fails at time t. The second criterion is a time-dependent measure which enables us to investigate the inactivity times of the failed components of a coherent system still functioning though some of its components have failed. Several ageing and stochastic properties of the proposed measures are then established.